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User:Yqw13

2016-02-12T10:38:41Z

Yqw13: /* Regioselectivity of Diels-Alder reaction */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13 gauche2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche5.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche6.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13_anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC plot above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interestingly found that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closest to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory '''B3LYP/6-31G(d)''' is suggested. Or more expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy (0 K), and electronic + thermal energy (298 K) are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to a higher resolution by larger basis set. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly more favoured. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is not significant hence relatively the small energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are approaching experimental values more, while the results from '''HF/3-21G''' are significantly overestimated.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4s + 2s] concerted, pericyclic pathway between a diene (4 carbons + 4 π electrons) and a conjugate dienophile (2 C + 2 πe-). There are overall 6 πe- s participating in the reaction and two σ-bonds being formed. Diene is usually more electron-rich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and so it is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, also π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number output from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Other factors, such as the geometry of diene, has to be considered, since s-cis conformation of diene is necessary for the synchronised breaking/formation of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, again the reaction path will be studied by '''IRC'''. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules that are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming yet gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7 accompanied with their symmetry. Symmetry of MO is determined respect to the vertical mirror plane. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely to occur owing to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled in the same manner as for Cope reaction. First, optimised butadiene and ethene molecules are modified to an estimated shape of cycloaddition TS using bicyclo-octane as a template, so that the optimal overlap of π system can be acquired. Product molecule (cyclohexene) is refined at '''HF/3-21G''' + '''B3LYP/6-31G(d)''' levels of theory. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects a complete minimisation of TS energy. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å) in TS. Typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. If considering the orientation of p-lobes, little reorganisation of central p-lobes on diene is required to form a new π-bond with neighboring p-lobe. However the terminal p-lobes of diene form σ-bonds with that of dienophile, and remarkable reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobes of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 + 0 = 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This vibration mode is favoured over that at lowest positive frequency (asynchronous), because [4+2] TS is able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2) hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 2; measure 2 3; measure 3 4; measure 4 5;
</script>
<uploadedFileContents>Yqw13 transition state 1.mol </uploadedFileContents>
</jmolApplet></jmol>
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the previous orbital symmetry of reagents (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory. HOMO-1 orbital might actually plays the role of HOMO during Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interactions. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition as secondary orbital overlap is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those on diene. Currently the stereoelectronic attraction on carbon outweighs the steric interaction with sp2 C-H bonds (planar) on diene, and the partly formed σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds directly clashes with sp3 C-H bonds (tetrahedral), i.e. greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 8; measure 2 3; measure 3 4; measure 7 8; measure 3 11
</script>
<uploadedFileContents>Yqw13 regioselec endo TS QST2 631G.mol</uploadedFileContents>
</jmolApplet></jmol>
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 7; measure 2 3; measure 3 4; measure 7 8; measure 5 9
</script>
<uploadedFileContents>Yqw13 regioselec exo TS QST2 631G.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but absent in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. It is clearly shown from MO that e- density around C=O overlaps with the bulk of π-system to some extent in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also small orbitals around sp3 C-H bonds but interaction hardly occurs. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) will be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|[[File:Yqw13 regioselec endo HOMO.png|200px]]Front[[File:Yqw13 regioselec endo HOMO 2.png|200px]]Side
|[[File:Yqw13 regioselec exo HOMO 1.png|200px]]Front[[File:Yqw13 regioselec exo HOMO 2.png|200px]]Side
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory in '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cycloaddition undergoes the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the small penalty of breaking weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product. This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually an exo (thermodynamic) product may be generated after a period of time. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised and electronic correlation is taken into account. <ref name=":0" /> It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene which leads to undesired product, such as [4s+2s] dimerisation. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is an unlikely situation unless strong driving force is applied, e.g. high concentration of butadiene, or heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involves secondary overlap that could be stabilised by interacting with polar solvent, resulting in more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose TS of the reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Diels-Alder cycloaddition includes a prototypical (butadiene and ethene) and a reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are able to behave as normal demand via [4s+2s] synchronised TS. Respect to regioselectivity, ''endo'' & ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore the lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T10:37:40Z

Yqw13: /* Regioselectivity of Diels-Alder reaction */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13 gauche2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche5.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche6.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13_anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC plot above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interestingly found that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closest to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory '''B3LYP/6-31G(d)''' is suggested. Or more expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy (0 K), and electronic + thermal energy (298 K) are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to a higher resolution by larger basis set. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly more favoured. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is not significant hence relatively the small energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are approaching experimental values more, while the results from '''HF/3-21G''' are significantly overestimated.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4s + 2s] concerted, pericyclic pathway between a diene (4 carbons + 4 π electrons) and a conjugate dienophile (2 C + 2 πe-). There are overall 6 πe- s participating in the reaction and two σ-bonds being formed. Diene is usually more electron-rich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and so it is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, also π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number output from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Other factors, such as the geometry of diene, has to be considered, since s-cis conformation of diene is necessary for the synchronised breaking/formation of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, again the reaction path will be studied by '''IRC'''. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules that are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming yet gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7 accompanied with their symmetry. Symmetry of MO is determined respect to the vertical mirror plane. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely to occur owing to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled in the same manner as for Cope reaction. First, optimised butadiene and ethene molecules are modified to an estimated shape of cycloaddition TS using bicyclo-octane as a template, so that the optimal overlap of π system can be acquired. Product molecule (cyclohexene) is refined at '''HF/3-21G''' + '''B3LYP/6-31G(d)''' levels of theory. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects a complete minimisation of TS energy. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å) in TS. Typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. If considering the orientation of p-lobes, little reorganisation of central p-lobes on diene is required to form a new π-bond with neighboring p-lobe. However the terminal p-lobes of diene form σ-bonds with that of dienophile, and remarkable reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobes of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 + 0 = 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This vibration mode is favoured over that at lowest positive frequency (asynchronous), because [4+2] TS is able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2) hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 2; measure 2 3; measure 3 4; measure 4 5;
</script>
<uploadedFileContents>Yqw13 transition state 1.mol </uploadedFileContents>
</jmolApplet></jmol>
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the previous orbital symmetry of reagents (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory. HOMO-1 orbital might actually plays the role of HOMO during Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interactions. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition as secondary orbital overlap is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those on diene. Currently the stereoelectronic attraction on carbon outweighs the steric interaction with sp2 C-H bonds (planar) on diene, and the partly formed σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds directly clashes with sp3 C-H bonds (tetrahedral), i.e. greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 8; measure 2 3; measure 3 4; measure 7 8; measure 3 11
</script>
<uploadedFileContents>Yqw13 regioselec endo TS QST2 631G.mol</uploadedFileContents>
</jmolApplet></jmol>
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 7; measure 2 3; measure 3 4; measure 7 8; measure 5 9
</script>
<uploadedFileContents>Yqw13 regioselec exo TS QST2 631G.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but absent in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. It is clearly shown from MO that e- density around C=O overlaps with the bulk of π-system to some extent in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction hardly occurs. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) will be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|[[File:Yqw13 regioselec endo HOMO.png|200px]]Front[[File:Yqw13 regioselec endo HOMO 2.png|200px]]Side
|[[File:Yqw13 regioselec exo HOMO 1.png|200px]][[File:Yqw13 regioselec exo HOMO 2.png|200px]]
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory in '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cycloaddition undergoes the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the small penalty of breaking weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product. This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually an exo (thermodynamic) product may be generated after a period of time. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised and electronic correlation is taken into account. <ref name=":0" /> It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene which leads to undesired product, such as [4s+2s] dimerisation. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is an unlikely situation unless strong driving force is applied, e.g. high concentration of butadiene, or heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involves secondary overlap that could be stabilised by interacting with polar solvent, resulting in more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose TS of the reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Diels-Alder cycloaddition includes a prototypical (butadiene and ethene) and a reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are able to behave as normal demand via [4s+2s] synchronised TS. Respect to regioselectivity, ''endo'' & ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore the lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T10:36:14Z

Yqw13: /* Regioselectivity of Diels-Alder reaction */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13 gauche2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche5.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche6.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13_anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC plot above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interestingly found that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closest to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory '''B3LYP/6-31G(d)''' is suggested. Or more expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy (0 K), and electronic + thermal energy (298 K) are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to a higher resolution by larger basis set. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly more favoured. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is not significant hence relatively the small energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are approaching experimental values more, while the results from '''HF/3-21G''' are significantly overestimated.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4s + 2s] concerted, pericyclic pathway between a diene (4 carbons + 4 π electrons) and a conjugate dienophile (2 C + 2 πe-). There are overall 6 πe- s participating in the reaction and two σ-bonds being formed. Diene is usually more electron-rich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and so it is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, also π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number output from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Other factors, such as the geometry of diene, has to be considered, since s-cis conformation of diene is necessary for the synchronised breaking/formation of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, again the reaction path will be studied by '''IRC'''. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules that are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming yet gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7 accompanied with their symmetry. Symmetry of MO is determined respect to the vertical mirror plane. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely to occur owing to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled in the same manner as for Cope reaction. First, optimised butadiene and ethene molecules are modified to an estimated shape of cycloaddition TS using bicyclo-octane as a template, so that the optimal overlap of π system can be acquired. Product molecule (cyclohexene) is refined at '''HF/3-21G''' + '''B3LYP/6-31G(d)''' levels of theory. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects a complete minimisation of TS energy. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å) in TS. Typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. If considering the orientation of p-lobes, little reorganisation of central p-lobes on diene is required to form a new π-bond with neighboring p-lobe. However the terminal p-lobes of diene form σ-bonds with that of dienophile, and remarkable reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobes of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 + 0 = 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This vibration mode is favoured over that at lowest positive frequency (asynchronous), because [4+2] TS is able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2) hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 2; measure 2 3; measure 3 4; measure 4 5;
</script>
<uploadedFileContents>Yqw13 transition state 1.mol </uploadedFileContents>
</jmolApplet></jmol>
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the previous orbital symmetry of reagents (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory. HOMO-1 orbital might actually plays the role of HOMO during Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interactions. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition as secondary orbital overlap is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those on diene. Currently the stereoelectronic attraction on carbon outweighs the steric interaction with sp2 C-H bonds (planar) on diene, and the partly formed σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds directly clashes with sp3 C-H bonds (tetrahedral), i.e. greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 8; measure 2 3; measure 3 4; measure 7 8; measure 3 11
</script>
<uploadedFileContents>Yqw13 regioselec endo TS QST2 631G.mol</uploadedFileContents>
</jmolApplet></jmol>
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 7; measure 2 3; measure 3 4; measure 7 8; measure 5 9
</script>
<uploadedFileContents>Yqw13 regioselec exo TS QST2 631G.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but absent in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. It is clearly shown from MO that e- density around C=O overlaps with the bulk of π-system to some extent in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction hardly occurs. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) will be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|[[File:Yqw13 regioselec endo HOMO.png|200px]][[File:Yqw13 regioselec endo HOMO 2.png|200px]]
|[[File:Yqw13 regioselec exo HOMO 1.png|200px]][[File:Yqw13 regioselec exo HOMO 2.png|200px]]
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory in '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cycloaddition undergoes the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the small penalty of breaking weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product. This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually an exo (thermodynamic) product may be generated after a period of time. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised and electronic correlation is taken into account. <ref name=":0" /> It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene which leads to undesired product, such as [4s+2s] dimerisation. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is an unlikely situation unless strong driving force is applied, e.g. high concentration of butadiene, or heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involves secondary overlap that could be stabilised by interacting with polar solvent, resulting in more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose TS of the reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Diels-Alder cycloaddition includes a prototypical (butadiene and ethene) and a reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are able to behave as normal demand via [4s+2s] synchronised TS. Respect to regioselectivity, ''endo'' & ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore the lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T10:34:36Z

Yqw13: /* Regioselectivity of Diels-Alder reaction */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13 gauche2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche5.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche6.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13_anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC plot above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interestingly found that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closest to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory '''B3LYP/6-31G(d)''' is suggested. Or more expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy (0 K), and electronic + thermal energy (298 K) are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to a higher resolution by larger basis set. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly more favoured. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is not significant hence relatively the small energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are approaching experimental values more, while the results from '''HF/3-21G''' are significantly overestimated.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4s + 2s] concerted, pericyclic pathway between a diene (4 carbons + 4 π electrons) and a conjugate dienophile (2 C + 2 πe-). There are overall 6 πe- s participating in the reaction and two σ-bonds being formed. Diene is usually more electron-rich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and so it is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, also π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number output from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Other factors, such as the geometry of diene, has to be considered, since s-cis conformation of diene is necessary for the synchronised breaking/formation of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, again the reaction path will be studied by '''IRC'''. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules that are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming yet gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7 accompanied with their symmetry. Symmetry of MO is determined respect to the vertical mirror plane. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely to occur owing to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled in the same manner as for Cope reaction. First, optimised butadiene and ethene molecules are modified to an estimated shape of cycloaddition TS using bicyclo-octane as a template, so that the optimal overlap of π system can be acquired. Product molecule (cyclohexene) is refined at '''HF/3-21G''' + '''B3LYP/6-31G(d)''' levels of theory. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects a complete minimisation of TS energy. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å) in TS. Typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. If considering the orientation of p-lobes, little reorganisation of central p-lobes on diene is required to form a new π-bond with neighboring p-lobe. However the terminal p-lobes of diene form σ-bonds with that of dienophile, and remarkable reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobes of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 + 0 = 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This vibration mode is favoured over that at lowest positive frequency (asynchronous), because [4+2] TS is able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2) hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 2; measure 2 3; measure 3 4; measure 4 5;
</script>
<uploadedFileContents>Yqw13 transition state 1.mol </uploadedFileContents>
</jmolApplet></jmol>
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the previous orbital symmetry of reagents (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory. HOMO-1 orbital might actually plays the role of HOMO during Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interactions. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition as secondary orbital overlap is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those on diene. Currently the stereoelectronic attraction on carbon outweighs the steric interaction with sp2 C-H bonds (planar) on diene, and the partly formed σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds directly clashes with sp3 C-H bonds (tetrahedral), i.e. greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 8; measure 2 3; measure 3 4; measure 7 8; measure 3 11
</script>
<uploadedFileContents>Yqw13 regioselec endo TS QST2 631G.mol</uploadedFileContents>
</jmolApplet></jmol>
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 7; measure 2 3; measure 3 4; measure 7 8; measure 5 9
</script>
<uploadedFileContents>Yqw13 regioselec exo TS QST2 631G.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but absent in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. It is clearly shown from MO that e- density around C=O overlaps with the bulk of π-system to some extent in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction hardly occurs. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) will be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|<jmol>
<jmolApplet>
<title></title><color>white</color><size>300</size>
<script>isosurface color red blue "images/5/58/Yqw13 regioselec endo TS QST2 mo47.cub.jvxl" translucent;</script>
<uploadedFileContents>Yqw13 regioselec endo TS QST2 mo47.cub.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title></title><color>white</color><size>300</size>
<script>isosurface color red blue "images/6/65/Yqw13 regioselec exo TS QST2 mo47.cub.jvxl" translucent;</script>
<uploadedFileContents>Yqw13 regioselec exo TS QST2 mo47.cub.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory in '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cycloaddition undergoes the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the small penalty of breaking weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product. This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually an exo (thermodynamic) product may be generated after a period of time. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised and electronic correlation is taken into account. <ref name=":0" /> It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene which leads to undesired product, such as [4s+2s] dimerisation. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is an unlikely situation unless strong driving force is applied, e.g. high concentration of butadiene, or heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involves secondary overlap that could be stabilised by interacting with polar solvent, resulting in more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose TS of the reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Diels-Alder cycloaddition includes a prototypical (butadiene and ethene) and a reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are able to behave as normal demand via [4s+2s] synchronised TS. Respect to regioselectivity, ''endo'' & ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore the lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T10:32:37Z

Yqw13: /* Regioselectivity of Diels-Alder reaction */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13 gauche2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche5.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche6.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13_anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC plot above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interestingly found that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closest to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory '''B3LYP/6-31G(d)''' is suggested. Or more expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy (0 K), and electronic + thermal energy (298 K) are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to a higher resolution by larger basis set. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly more favoured. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is not significant hence relatively the small energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are approaching experimental values more, while the results from '''HF/3-21G''' are significantly overestimated.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4s + 2s] concerted, pericyclic pathway between a diene (4 carbons + 4 π electrons) and a conjugate dienophile (2 C + 2 πe-). There are overall 6 πe- s participating in the reaction and two σ-bonds being formed. Diene is usually more electron-rich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and so it is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, also π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number output from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Other factors, such as the geometry of diene, has to be considered, since s-cis conformation of diene is necessary for the synchronised breaking/formation of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, again the reaction path will be studied by '''IRC'''. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules that are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming yet gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7 accompanied with their symmetry. Symmetry of MO is determined respect to the vertical mirror plane. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely to occur owing to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled in the same manner as for Cope reaction. First, optimised butadiene and ethene molecules are modified to an estimated shape of cycloaddition TS using bicyclo-octane as a template, so that the optimal overlap of π system can be acquired. Product molecule (cyclohexene) is refined at '''HF/3-21G''' + '''B3LYP/6-31G(d)''' levels of theory. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects a complete minimisation of TS energy. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å) in TS. Typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. If considering the orientation of p-lobes, little reorganisation of central p-lobes on diene is required to form a new π-bond with neighboring p-lobe. However the terminal p-lobes of diene form σ-bonds with that of dienophile, and remarkable reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobes of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 + 0 = 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This vibration mode is favoured over that at lowest positive frequency (asynchronous), because [4+2] TS is able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2) hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 2; measure 2 3; measure 3 4; measure 4 5;
</script>
<uploadedFileContents>Yqw13 transition state 1.mol </uploadedFileContents>
</jmolApplet></jmol>
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the previous orbital symmetry of reagents (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory. HOMO-1 orbital might actually plays the role of HOMO during Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interactions. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition as secondary orbital overlap is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those on diene. Currently the stereoelectronic attraction on carbon outweighs the steric interaction with sp2 C-H bonds (planar) on diene, and the partly formed σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds directly clashes with sp3 C-H bonds (tetrahedral), i.e. greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 8; measure 2 3; measure 3 4; measure 7 8; measure 2 9
</script>
<uploadedFileContents>Yqw13 regioselec endo TS QST2 631G.mol</uploadedFileContents>
</jmolApplet></jmol>
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 7; measure 2 3; measure 3 4; measure 7 8; measure 5 9
</script>
<uploadedFileContents>Yqw13 regioselec exo TS QST2 631G.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but absent in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. It is clearly shown from MO that e- density around C=O overlaps with the bulk of π-system to some extent in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction hardly occurs. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) will be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|<jmol>
<jmolApplet>
<title></title><color>white</color><size>300</size>
<script>isosurface color red blue "images/5/58/Yqw13 regioselec endo TS QST2 mo47.cub.jvxl" translucent;</script>
<uploadedFileContents>Yqw13 regioselec endo TS QST2 mo47.cub.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title></title><color>white</color><size>300</size>
<script>isosurface color red blue "images/6/65/Yqw13 regioselec exo TS QST2 mo47.cub.jvxl" translucent;</script>
<uploadedFileContents>Yqw13 regioselec exo TS QST2 mo47.cub.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory in '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cycloaddition undergoes the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the small penalty of breaking weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product. This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually an exo (thermodynamic) product may be generated after a period of time. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised and electronic correlation is taken into account. <ref name=":0" /> It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene which leads to undesired product, such as [4s+2s] dimerisation. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is an unlikely situation unless strong driving force is applied, e.g. high concentration of butadiene, or heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involves secondary overlap that could be stabilised by interacting with polar solvent, resulting in more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose TS of the reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Diels-Alder cycloaddition includes a prototypical (butadiene and ethene) and a reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are able to behave as normal demand via [4s+2s] synchronised TS. Respect to regioselectivity, ''endo'' & ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore the lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T10:30:01Z

Yqw13: /* Regioselectivity of Diels-Alder reaction */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13 gauche2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche5.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche6.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13_anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC plot above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interestingly found that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closest to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory '''B3LYP/6-31G(d)''' is suggested. Or more expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy (0 K), and electronic + thermal energy (298 K) are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to a higher resolution by larger basis set. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly more favoured. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is not significant hence relatively the small energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are approaching experimental values more, while the results from '''HF/3-21G''' are significantly overestimated.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4s + 2s] concerted, pericyclic pathway between a diene (4 carbons + 4 π electrons) and a conjugate dienophile (2 C + 2 πe-). There are overall 6 πe- s participating in the reaction and two σ-bonds being formed. Diene is usually more electron-rich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and so it is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, also π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number output from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Other factors, such as the geometry of diene, has to be considered, since s-cis conformation of diene is necessary for the synchronised breaking/formation of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, again the reaction path will be studied by '''IRC'''. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules that are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming yet gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7 accompanied with their symmetry. Symmetry of MO is determined respect to the vertical mirror plane. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely to occur owing to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled in the same manner as for Cope reaction. First, optimised butadiene and ethene molecules are modified to an estimated shape of cycloaddition TS using bicyclo-octane as a template, so that the optimal overlap of π system can be acquired. Product molecule (cyclohexene) is refined at '''HF/3-21G''' + '''B3LYP/6-31G(d)''' levels of theory. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects a complete minimisation of TS energy. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å) in TS. Typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. If considering the orientation of p-lobes, little reorganisation of central p-lobes on diene is required to form a new π-bond with neighboring p-lobe. However the terminal p-lobes of diene form σ-bonds with that of dienophile, and remarkable reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobes of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 + 0 = 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This vibration mode is favoured over that at lowest positive frequency (asynchronous), because [4+2] TS is able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2) hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 2; measure 2 3; measure 3 4; measure 4 5;
</script>
<uploadedFileContents>Yqw13 transition state 1.mol </uploadedFileContents>
</jmolApplet></jmol>
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the previous orbital symmetry of reagents (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory. HOMO-1 orbital might actually plays the role of HOMO during Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interactions. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition as secondary orbital overlap is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those on diene. Currently the stereoelectronic attraction on carbon outweighs the steric interaction with sp2 C-H bonds (planar) on diene, and the partly formed σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds directly clashes with sp3 C-H bonds (tetrahedral), i.e. greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
![[File:Yqw13 regioselec endo HOMO.png|150px]][[File:Yqw13 regioselec endo HOMO 2.png|150px]]
![[File:Yqw13 regioselec exo HOMO 1.png|150px]][[File:Yqw13 regioselec exo HOMO 2.png|150px]]
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but absent in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. It is clearly shown from MO that e- density around C=O overlaps with the bulk of π-system to some extent in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction hardly occurs. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) will be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|<jmol>
<jmolApplet>
<title></title><color>white</color><size>300</size>
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<uploadedFileContents>Yqw13 regioselec endo TS QST2 mo47.cub.xyz</uploadedFileContents>
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</jmol>
|<jmol>
<jmolApplet>
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</jmol>
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory in '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cycloaddition undergoes the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the small penalty of breaking weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product. This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually an exo (thermodynamic) product may be generated after a period of time. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised and electronic correlation is taken into account. <ref name=":0" /> It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene which leads to undesired product, such as [4s+2s] dimerisation. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is an unlikely situation unless strong driving force is applied, e.g. high concentration of butadiene, or heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involves secondary overlap that could be stabilised by interacting with polar solvent, resulting in more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose TS of the reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Diels-Alder cycloaddition includes a prototypical (butadiene and ethene) and a reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are able to behave as normal demand via [4s+2s] synchronised TS. Respect to regioselectivity, ''endo'' & ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore the lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

File:Yqw13 regioselec exo HOMO 2.png

2016-02-12T10:26:51Z

Yqw13:

File:Yqw13 regioselec exo HOMO 1.png

2016-02-12T10:25:56Z

Yqw13:

File:Yqw13 regioselec endo HOMO 2.png

2016-02-12T10:25:23Z

Yqw13:

File:Yqw13 regioselec endo HOMO.png

2016-02-12T10:24:25Z

Yqw13:

File:Regioselec endo TS QST2 mo47.cub.xyz

2016-02-12T09:28:14Z

Yqw13:

File:Regioselec endo TS QST2 mo47.cub.jvxl

2016-02-12T09:26:10Z

Yqw13:

User:Yqw13

2016-02-12T08:12:11Z

Yqw13: /* Regioselectivity of Diels-Alder reaction */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13 gauche2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche5.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche6.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13_anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC plot above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interestingly found that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closest to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory '''B3LYP/6-31G(d)''' is suggested. Or more expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy (0 K), and electronic + thermal energy (298 K) are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to a higher resolution by larger basis set. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly more favoured. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is not significant hence relatively the small energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are approaching experimental values more, while the results from '''HF/3-21G''' are significantly overestimated.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4s + 2s] concerted, pericyclic pathway between a diene (4 carbons + 4 π electrons) and a conjugate dienophile (2 C + 2 πe-). There are overall 6 πe- s participating in the reaction and two σ-bonds being formed. Diene is usually more electron-rich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and so it is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, also π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number output from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Other factors, such as the geometry of diene, has to be considered, since s-cis conformation of diene is necessary for the synchronised breaking/formation of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, again the reaction path will be studied by '''IRC'''. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules that are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming yet gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7 accompanied with their symmetry. Symmetry of MO is determined respect to the vertical mirror plane. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely to occur owing to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled in the same manner as for Cope reaction. First, optimised butadiene and ethene molecules are modified to an estimated shape of cycloaddition TS using bicyclo-octane as a template, so that the optimal overlap of π system can be acquired. Product molecule (cyclohexene) is refined at '''HF/3-21G''' + '''B3LYP/6-31G(d)''' levels of theory. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects a complete minimisation of TS energy. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å) in TS. Typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. If considering the orientation of p-lobes, little reorganisation of central p-lobes on diene is required to form a new π-bond with neighboring p-lobe. However the terminal p-lobes of diene form σ-bonds with that of dienophile, and remarkable reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobes of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 + 0 = 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This vibration mode is favoured over that at lowest positive frequency (asynchronous), because [4+2] TS is able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2) hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 2; measure 2 3; measure 3 4; measure 4 5;
</script>
<uploadedFileContents>Yqw13 transition state 1.mol </uploadedFileContents>
</jmolApplet></jmol>
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the previous orbital symmetry of reagents (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory. HOMO-1 orbital might actually plays the role of HOMO during Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interactions. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition as secondary orbital overlap is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those on diene. Currently the stereoelectronic attraction on carbon outweighs the steric interaction with sp2 C-H bonds (planar) on diene, and the partly formed σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds directly clashes with sp3 C-H bonds (tetrahedral), i.e. greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 8; measure 2 3; measure 3 4; measure 7 8; measure 2 9
</script>
<uploadedFileContents>Yqw13 regioselec endo TS QST2 631G.mol</uploadedFileContents>
</jmolApplet></jmol>
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 7; measure 2 3; measure 3 4; measure 7 8; measure 5 9
</script>
<uploadedFileContents>Yqw13 regioselec exo TS QST2 631G.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but absent in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. It is clearly shown from MO that e- density around C=O overlaps with the bulk of π-system to some extent in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction hardly occurs. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) will be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|<jmol>
<jmolApplet>
<title></title><color>white</color><size>300</size>
<script>isosurface color red blue "images/5/58/Yqw13 regioselec endo TS QST2 mo47.cub.jvxl" translucent;</script>
<uploadedFileContents>Yqw13 regioselec endo TS QST2 mo47.cub.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title></title><color>white</color><size>300</size>
<script>isosurface color red blue "images/6/65/Yqw13 regioselec exo TS QST2 mo47.cub.jvxl" translucent;</script>
<uploadedFileContents>Yqw13 regioselec exo TS QST2 mo47.cub.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory in '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cycloaddition undergoes the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the small penalty of breaking weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product. This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually an exo (thermodynamic) product may be generated after a period of time. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised and electronic correlation is taken into account. <ref name=":0" /> It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene which leads to undesired product, such as [4s+2s] dimerisation. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is an unlikely situation unless strong driving force is applied, e.g. high concentration of butadiene, or heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involves secondary overlap that could be stabilised by interacting with polar solvent, resulting in more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose TS of the reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Diels-Alder cycloaddition includes a prototypical (butadiene and ethene) and a reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are able to behave as normal demand via [4s+2s] synchronised TS. Respect to regioselectivity, ''endo'' & ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore the lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T08:09:12Z

Yqw13: /* Regioselectivity of Diels-Alder reaction */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13 gauche2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche5.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche6.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13_anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC plot above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interestingly found that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closest to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory '''B3LYP/6-31G(d)''' is suggested. Or more expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy (0 K), and electronic + thermal energy (298 K) are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to a higher resolution by larger basis set. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly more favoured. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is not significant hence relatively the small energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are approaching experimental values more, while the results from '''HF/3-21G''' are significantly overestimated.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4s + 2s] concerted, pericyclic pathway between a diene (4 carbons + 4 π electrons) and a conjugate dienophile (2 C + 2 πe-). There are overall 6 πe- s participating in the reaction and two σ-bonds being formed. Diene is usually more electron-rich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and so it is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, also π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number output from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Other factors, such as the geometry of diene, has to be considered, since s-cis conformation of diene is necessary for the synchronised breaking/formation of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, again the reaction path will be studied by '''IRC'''. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules that are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming yet gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7 accompanied with their symmetry. Symmetry of MO is determined respect to the vertical mirror plane. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely to occur owing to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled in the same manner as for Cope reaction. First, optimised butadiene and ethene molecules are modified to an estimated shape of cycloaddition TS using bicyclo-octane as a template, so that the optimal overlap of π system can be acquired. Product molecule (cyclohexene) is refined at '''HF/3-21G''' + '''B3LYP/6-31G(d)''' levels of theory. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects a complete minimisation of TS energy. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å) in TS. Typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. If considering the orientation of p-lobes, little reorganisation of central p-lobes on diene is required to form a new π-bond with neighboring p-lobe. However the terminal p-lobes of diene form σ-bonds with that of dienophile, and remarkable reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobes of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 + 0 = 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This vibration mode is favoured over that at lowest positive frequency (asynchronous), because [4+2] TS is able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2) hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 2; measure 2 3; measure 3 4; measure 4 5;
</script>
<uploadedFileContents>Yqw13 transition state 1.mol </uploadedFileContents>
</jmolApplet></jmol>
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the previous orbital symmetry of reagents (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory. HOMO-1 orbital might actually plays the role of HOMO during Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interactions. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition as secondary orbital overlap is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those on diene. Currently the stereoelectronic attraction on carbon outweighs the steric interaction with sp2 C-H bonds (planar) on diene, and the partly formed σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds directly clashes with sp3 C-H bonds (tetrahedral), i.e. greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 8; measure 2 3; measure 3 4; measure 7 8; measure 2 9
</script>
<uploadedFileContents>Yqw13 regioselec endo TS QST2 631G.mol</uploadedFileContents>
</jmolApplet></jmol>
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 7; measure 2 3; measure 3 4; measure 7 8; measure 5 9
</script>
<uploadedFileContents>Yqw13 regioselec exo TS QST2 631G.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but absent in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. It is clearly shown from MO that e- density around C=O overlaps with the bulk of π-system to some extent in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction hardly occurs. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) will be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|<jmol>
<jmolApplet>
<title></title><color>white</color><size>300</size>
<script>isosurface color red blue "images/5/58/Yqw13 regioselec endo TS QST2 mo47.cub.jvxl" translucent;</script>
<uploadedFileContents>Yqw13 regioselec endo TS QST2 mo47.cub.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory in '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cycloaddition undergoes the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the small penalty of breaking weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product. This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually an exo (thermodynamic) product may be generated after a period of time. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised and electronic correlation is taken into account. <ref name=":0" /> It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene which leads to undesired product, such as [4s+2s] dimerisation. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is an unlikely situation unless strong driving force is applied, e.g. high concentration of butadiene, or heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involves secondary overlap that could be stabilised by interacting with polar solvent, resulting in more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose TS of the reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Diels-Alder cycloaddition includes a prototypical (butadiene and ethene) and a reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are able to behave as normal demand via [4s+2s] synchronised TS. Respect to regioselectivity, ''endo'' & ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore the lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

File:Regioselec exo TS QST2 mo48.cub.jvxl

2016-02-12T07:59:28Z

Yqw13:

File:Yqw13 regioselec endo TS QST2 631G.mol

2016-02-12T07:50:41Z

Yqw13: Yqw13 uploaded a new version of File:Yqw13 regioselec endo TS QST2 631G.mol

File:Yqw13 regioselec exo TS QST2 631G.mol

2016-02-12T07:48:24Z

Yqw13: Yqw13 uploaded a new version of File:Yqw13 regioselec exo TS QST2 631G.mol

User:Yqw13

2016-02-12T07:45:18Z

Yqw13: /* Regioselectivity of Diels-Alder reaction */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13 gauche2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche5.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche6.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13_anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC plot above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interestingly found that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closest to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory '''B3LYP/6-31G(d)''' is suggested. Or more expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy (0 K), and electronic + thermal energy (298 K) are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to a higher resolution by larger basis set. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly more favoured. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is not significant hence relatively the small energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are approaching experimental values more, while the results from '''HF/3-21G''' are significantly overestimated.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4s + 2s] concerted, pericyclic pathway between a diene (4 carbons + 4 π electrons) and a conjugate dienophile (2 C + 2 πe-). There are overall 6 πe- s participating in the reaction and two σ-bonds being formed. Diene is usually more electron-rich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and so it is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, also π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number output from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Other factors, such as the geometry of diene, has to be considered, since s-cis conformation of diene is necessary for the synchronised breaking/formation of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, again the reaction path will be studied by '''IRC'''. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules that are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming yet gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7 accompanied with their symmetry. Symmetry of MO is determined respect to the vertical mirror plane. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely to occur owing to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled in the same manner as for Cope reaction. First, optimised butadiene and ethene molecules are modified to an estimated shape of cycloaddition TS using bicyclo-octane as a template, so that the optimal overlap of π system can be acquired. Product molecule (cyclohexene) is refined at '''HF/3-21G''' + '''B3LYP/6-31G(d)''' levels of theory. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects a complete minimisation of TS energy. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å) in TS. Typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. If considering the orientation of p-lobes, little reorganisation of central p-lobes on diene is required to form a new π-bond with neighboring p-lobe. However the terminal p-lobes of diene form σ-bonds with that of dienophile, and remarkable reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobes of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 + 0 = 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This vibration mode is favoured over that at lowest positive frequency (asynchronous), because [4+2] TS is able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2) hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 2; measure 2 3; measure 3 4; measure 4 5;
</script>
<uploadedFileContents>Yqw13 transition state 1.mol </uploadedFileContents>
</jmolApplet></jmol>
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the previous orbital symmetry of reagents (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory. HOMO-1 orbital might actually plays the role of HOMO during Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interactions. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition as secondary orbital overlap is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those on diene. Currently the stereoelectronic attraction on carbon outweighs the steric interaction with sp2 C-H bonds (planar) on diene, and the partly formed σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds directly clashes with sp3 C-H bonds (tetrahedral), i.e. greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 8; measure 2 3; measure 3 4; measure 7 8; measure 2 9
</script>
<uploadedFileContents>Yqw13 regioselec endo TS QST2 631G.mol</uploadedFileContents>
</jmolApplet></jmol>
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 7; measure 2 3; measure 3 4; measure 7 8; measure 5 9
</script>
<uploadedFileContents>Yqw13 regioselec exo TS QST2 631G.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but absent in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. It is clearly shown from MO that e- density around C=O overlaps with the bulk of π-system to some extent in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction hardly occurs. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) will be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|1
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory in '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cycloaddition undergoes the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the small penalty of breaking weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product. This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually an exo (thermodynamic) product may be generated after a period of time. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised and electronic correlation is taken into account. <ref name=":0" /> It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene which leads to undesired product, such as [4s+2s] dimerisation. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is an unlikely situation unless strong driving force is applied, e.g. high concentration of butadiene, or heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involves secondary overlap that could be stabilised by interacting with polar solvent, resulting in more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose TS of the reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Diels-Alder cycloaddition includes a prototypical (butadiene and ethene) and a reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are able to behave as normal demand via [4s+2s] synchronised TS. Respect to regioselectivity, ''endo'' & ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore the lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T07:43:03Z

Yqw13: /* Regioselectivity of Diels-Alder reaction */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13 gauche2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche5.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche6.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13_anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC plot above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interestingly found that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closest to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory '''B3LYP/6-31G(d)''' is suggested. Or more expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy (0 K), and electronic + thermal energy (298 K) are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to a higher resolution by larger basis set. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly more favoured. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is not significant hence relatively the small energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are approaching experimental values more, while the results from '''HF/3-21G''' are significantly overestimated.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4s + 2s] concerted, pericyclic pathway between a diene (4 carbons + 4 π electrons) and a conjugate dienophile (2 C + 2 πe-). There are overall 6 πe- s participating in the reaction and two σ-bonds being formed. Diene is usually more electron-rich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and so it is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, also π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number output from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Other factors, such as the geometry of diene, has to be considered, since s-cis conformation of diene is necessary for the synchronised breaking/formation of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, again the reaction path will be studied by '''IRC'''. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules that are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming yet gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7 accompanied with their symmetry. Symmetry of MO is determined respect to the vertical mirror plane. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely to occur owing to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled in the same manner as for Cope reaction. First, optimised butadiene and ethene molecules are modified to an estimated shape of cycloaddition TS using bicyclo-octane as a template, so that the optimal overlap of π system can be acquired. Product molecule (cyclohexene) is refined at '''HF/3-21G''' + '''B3LYP/6-31G(d)''' levels of theory. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects a complete minimisation of TS energy. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å) in TS. Typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. If considering the orientation of p-lobes, little reorganisation of central p-lobes on diene is required to form a new π-bond with neighboring p-lobe. However the terminal p-lobes of diene form σ-bonds with that of dienophile, and remarkable reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobes of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 + 0 = 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This vibration mode is favoured over that at lowest positive frequency (asynchronous), because [4+2] TS is able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2) hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 2; measure 2 3; measure 3 4; measure 4 5;
</script>
<uploadedFileContents>Yqw13 transition state 1.mol </uploadedFileContents>
</jmolApplet></jmol>
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the previous orbital symmetry of reagents (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory. HOMO-1 orbital might actually plays the role of HOMO during Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interactions. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition as secondary orbital overlap is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those on diene. Currently the stereoelectronic attraction on carbon outweighs the steric interaction with sp2 C-H bonds (planar) on diene, and the partly formed σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds directly clashes with sp3 C-H bonds (tetrahedral), i.e. greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 8; measure 2 3; measure 3 4; measure 7 8; measure 2 9
</script>
<uploadedFileContents>Yqw13 regioselec endo TS QST2 631G.mol</uploadedFileContents>
</jmolApplet></jmol>
!3
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but absent in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. It is clearly shown from MO that e- density around C=O overlaps with the bulk of π-system to some extent in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction hardly occurs. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) will be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|1
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory in '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cycloaddition undergoes the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the small penalty of breaking weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product. This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually an exo (thermodynamic) product may be generated after a period of time. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised and electronic correlation is taken into account. <ref name=":0" /> It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene which leads to undesired product, such as [4s+2s] dimerisation. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is an unlikely situation unless strong driving force is applied, e.g. high concentration of butadiene, or heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involves secondary overlap that could be stabilised by interacting with polar solvent, resulting in more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose TS of the reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Diels-Alder cycloaddition includes a prototypical (butadiene and ethene) and a reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are able to behave as normal demand via [4s+2s] synchronised TS. Respect to regioselectivity, ''endo'' & ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore the lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

File:Yqw13 regioselec exo TS QST2 631G.mol

2016-02-12T07:41:01Z

Yqw13:

File:Yqw13 regioselec endo TS QST2 631G.mol

2016-02-12T07:39:34Z

Yqw13:

User:Yqw13

2016-02-12T07:22:33Z

Yqw13: /* Cycloaddition of cis-butadiene and ethene */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13 gauche2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche5.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche6.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13_anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC plot above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interestingly found that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closest to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory '''B3LYP/6-31G(d)''' is suggested. Or more expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy (0 K), and electronic + thermal energy (298 K) are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to a higher resolution by larger basis set. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly more favoured. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is not significant hence relatively the small energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are approaching experimental values more, while the results from '''HF/3-21G''' are significantly overestimated.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4s + 2s] concerted, pericyclic pathway between a diene (4 carbons + 4 π electrons) and a conjugate dienophile (2 C + 2 πe-). There are overall 6 πe- s participating in the reaction and two σ-bonds being formed. Diene is usually more electron-rich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and so it is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, also π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number output from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Other factors, such as the geometry of diene, has to be considered, since s-cis conformation of diene is necessary for the synchronised breaking/formation of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, again the reaction path will be studied by '''IRC'''. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules that are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming yet gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7 accompanied with their symmetry. Symmetry of MO is determined respect to the vertical mirror plane. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely to occur owing to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled in the same manner as for Cope reaction. First, optimised butadiene and ethene molecules are modified to an estimated shape of cycloaddition TS using bicyclo-octane as a template, so that the optimal overlap of π system can be acquired. Product molecule (cyclohexene) is refined at '''HF/3-21G''' + '''B3LYP/6-31G(d)''' levels of theory. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects a complete minimisation of TS energy. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å) in TS. Typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. If considering the orientation of p-lobes, little reorganisation of central p-lobes on diene is required to form a new π-bond with neighboring p-lobe. However the terminal p-lobes of diene form σ-bonds with that of dienophile, and remarkable reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobes of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 + 0 = 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This vibration mode is favoured over that at lowest positive frequency (asynchronous), because [4+2] TS is able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2) hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!<jmol><jmolApplet>
<title></title>
<color>white</color>
<size>300</size>
<script>measure 1 2; measure 2 3; measure 3 4; measure 4 5;
</script>
<uploadedFileContents>Yqw13 transition state 1.mol </uploadedFileContents>
</jmolApplet></jmol>
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the previous orbital symmetry of reagents (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory. HOMO-1 orbital might actually plays the role of HOMO during Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interactions. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition as secondary orbital overlap is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those on diene. Currently the stereoelectronic attraction on carbon outweighs the steric interaction with sp2 C-H bonds (planar) on diene, and the partly formed σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds directly clashes with sp3 C-H bonds (tetrahedral), i.e. greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!2
!3
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but absent in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. It is clearly shown from MO that e- density around C=O overlaps with the bulk of π-system to some extent in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction hardly occurs. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) will be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|1
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory in '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cycloaddition undergoes the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the small penalty of breaking weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product. This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually an exo (thermodynamic) product may be generated after a period of time. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised and electronic correlation is taken into account. <ref name=":0" /> It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene which leads to undesired product, such as [4s+2s] dimerisation. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is an unlikely situation unless strong driving force is applied, e.g. high concentration of butadiene, or heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involves secondary overlap that could be stabilised by interacting with polar solvent, resulting in more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose TS of the reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Diels-Alder cycloaddition includes a prototypical (butadiene and ethene) and a reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are able to behave as normal demand via [4s+2s] synchronised TS. Respect to regioselectivity, ''endo'' & ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore the lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T07:21:17Z

Yqw13: /* Optimisation of Reactants and Products */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13 gauche2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche5.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche6.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13_anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti3.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 anti4.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC plot above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interestingly found that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closest to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory '''B3LYP/6-31G(d)''' is suggested. Or more expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy (0 K), and electronic + thermal energy (298 K) are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to a higher resolution by larger basis set. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly more favoured. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is not significant hence relatively the small energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are approaching experimental values more, while the results from '''HF/3-21G''' are significantly overestimated.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4s + 2s] concerted, pericyclic pathway between a diene (4 carbons + 4 π electrons) and a conjugate dienophile (2 C + 2 πe-). There are overall 6 πe- s participating in the reaction and two σ-bonds being formed. Diene is usually more electron-rich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and so it is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, also π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number output from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Other factors, such as the geometry of diene, has to be considered, since s-cis conformation of diene is necessary for the synchronised breaking/formation of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, again the reaction path will be studied by '''IRC'''. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules that are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming yet gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7 accompanied with their symmetry. Symmetry of MO is determined respect to the vertical mirror plane. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely to occur owing to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled in the same manner as for Cope reaction. First, optimised butadiene and ethene molecules are modified to an estimated shape of cycloaddition TS using bicyclo-octane as a template, so that the optimal overlap of π system can be acquired. Product molecule (cyclohexene) is refined at '''HF/3-21G''' + '''B3LYP/6-31G(d)''' levels of theory. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects a complete minimisation of TS energy. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å) in TS. Typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. If considering the orientation of p-lobes, little reorganisation of central p-lobes on diene is required to form a new π-bond with neighboring p-lobe. However the terminal p-lobes of diene form σ-bonds with that of dienophile, and remarkable reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobes of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 + 0 = 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This vibration mode is favoured over that at lowest positive frequency (asynchronous), because [4+2] TS is able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2) hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!1
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the previous orbital symmetry of reagents (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory. HOMO-1 orbital might actually plays the role of HOMO during Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interactions. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition as secondary orbital overlap is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those on diene. Currently the stereoelectronic attraction on carbon outweighs the steric interaction with sp2 C-H bonds (planar) on diene, and the partly formed σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds directly clashes with sp3 C-H bonds (tetrahedral), i.e. greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!2
!3
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but absent in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. It is clearly shown from MO that e- density around C=O overlaps with the bulk of π-system to some extent in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction hardly occurs. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) will be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|1
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory in '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cycloaddition undergoes the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the small penalty of breaking weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product. This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually an exo (thermodynamic) product may be generated after a period of time. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised and electronic correlation is taken into account. <ref name=":0" /> It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene which leads to undesired product, such as [4s+2s] dimerisation. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is an unlikely situation unless strong driving force is applied, e.g. high concentration of butadiene, or heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involves secondary overlap that could be stabilised by interacting with polar solvent, resulting in more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose TS of the reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Diels-Alder cycloaddition includes a prototypical (butadiene and ethene) and a reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are able to behave as normal demand via [4s+2s] synchronised TS. Respect to regioselectivity, ''endo'' & ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore the lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T07:19:05Z

Yqw13: /* Optimisation of Reactants and Products */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>Yqw13 gauche1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13_anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|G3
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|G4
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|G5
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|G6
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|A1
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|A2
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|A3
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|A4
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC plot above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interestingly found that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closest to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory '''B3LYP/6-31G(d)''' is suggested. Or more expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy (0 K), and electronic + thermal energy (298 K) are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to a higher resolution by larger basis set. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly more favoured. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is not significant hence relatively the small energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are approaching experimental values more, while the results from '''HF/3-21G''' are significantly overestimated.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4s + 2s] concerted, pericyclic pathway between a diene (4 carbons + 4 π electrons) and a conjugate dienophile (2 C + 2 πe-). There are overall 6 πe- s participating in the reaction and two σ-bonds being formed. Diene is usually more electron-rich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and so it is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, also π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number output from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Other factors, such as the geometry of diene, has to be considered, since s-cis conformation of diene is necessary for the synchronised breaking/formation of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, again the reaction path will be studied by '''IRC'''. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules that are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming yet gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7 accompanied with their symmetry. Symmetry of MO is determined respect to the vertical mirror plane. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely to occur owing to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled in the same manner as for Cope reaction. First, optimised butadiene and ethene molecules are modified to an estimated shape of cycloaddition TS using bicyclo-octane as a template, so that the optimal overlap of π system can be acquired. Product molecule (cyclohexene) is refined at '''HF/3-21G''' + '''B3LYP/6-31G(d)''' levels of theory. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects a complete minimisation of TS energy. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å) in TS. Typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. If considering the orientation of p-lobes, little reorganisation of central p-lobes on diene is required to form a new π-bond with neighboring p-lobe. However the terminal p-lobes of diene form σ-bonds with that of dienophile, and remarkable reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobes of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 + 0 = 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This vibration mode is favoured over that at lowest positive frequency (asynchronous), because [4+2] TS is able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2) hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!1
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the previous orbital symmetry of reagents (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory. HOMO-1 orbital might actually plays the role of HOMO during Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interactions. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition as secondary orbital overlap is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those on diene. Currently the stereoelectronic attraction on carbon outweighs the steric interaction with sp2 C-H bonds (planar) on diene, and the partly formed σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds directly clashes with sp3 C-H bonds (tetrahedral), i.e. greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!2
!3
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but absent in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. It is clearly shown from MO that e- density around C=O overlaps with the bulk of π-system to some extent in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction hardly occurs. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) will be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|1
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory in '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cycloaddition undergoes the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the small penalty of breaking weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product. This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually an exo (thermodynamic) product may be generated after a period of time. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised and electronic correlation is taken into account. <ref name=":0" /> It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene which leads to undesired product, such as [4s+2s] dimerisation. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is an unlikely situation unless strong driving force is applied, e.g. high concentration of butadiene, or heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involves secondary overlap that could be stabilised by interacting with polar solvent, resulting in more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose TS of the reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Diels-Alder cycloaddition includes a prototypical (butadiene and ethene) and a reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are able to behave as normal demand via [4s+2s] synchronised TS. Respect to regioselectivity, ''endo'' & ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore the lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

File:Yqw13 gauche6.mol

2016-02-12T07:18:25Z

Yqw13:

File:Yqw13 gauche5.mol

2016-02-12T07:17:11Z

Yqw13:

File:Yqw13 gauche3.mol

2016-02-12T07:16:53Z

Yqw13: Yqw13 uploaded a new version of File:Yqw13 gauche3.mol

File:Yqw13 gauche4.mol

2016-02-12T07:16:38Z

Yqw13:

File:Yqw13 gauche2.mol

2016-02-12T07:16:21Z

Yqw13:

File:Yqw13 gauche1.mol

2016-02-12T07:16:01Z

Yqw13:

File:Yqw13 anti4.mol

2016-02-12T07:15:46Z

Yqw13:

File:Yqw13 anti3.mol

2016-02-12T07:15:25Z

Yqw13:

File:Yqw13 anti1.mol

2016-02-12T07:14:59Z

Yqw13:

User:Yqw13

2016-02-12T07:13:08Z

Yqw13: /* Optimisation of Reactants and Products */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13_gauche1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13_anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|G3
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|G4
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|G5
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|G6
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|A1
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|A2
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|A3
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|A4
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC plot above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interestingly found that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closest to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory '''B3LYP/6-31G(d)''' is suggested. Or more expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy (0 K), and electronic + thermal energy (298 K) are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to a higher resolution by larger basis set. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly more favoured. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is not significant hence relatively the small energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are approaching experimental values more, while the results from '''HF/3-21G''' are significantly overestimated.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4s + 2s] concerted, pericyclic pathway between a diene (4 carbons + 4 π electrons) and a conjugate dienophile (2 C + 2 πe-). There are overall 6 πe- s participating in the reaction and two σ-bonds being formed. Diene is usually more electron-rich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and so it is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, also π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number output from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Other factors, such as the geometry of diene, has to be considered, since s-cis conformation of diene is necessary for the synchronised breaking/formation of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, again the reaction path will be studied by '''IRC'''. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules that are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming yet gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7 accompanied with their symmetry. Symmetry of MO is determined respect to the vertical mirror plane. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely to occur owing to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled in the same manner as for Cope reaction. First, optimised butadiene and ethene molecules are modified to an estimated shape of cycloaddition TS using bicyclo-octane as a template, so that the optimal overlap of π system can be acquired. Product molecule (cyclohexene) is refined at '''HF/3-21G''' + '''B3LYP/6-31G(d)''' levels of theory. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects a complete minimisation of TS energy. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å) in TS. Typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. If considering the orientation of p-lobes, little reorganisation of central p-lobes on diene is required to form a new π-bond with neighboring p-lobe. However the terminal p-lobes of diene form σ-bonds with that of dienophile, and remarkable reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobes of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 + 0 = 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This vibration mode is favoured over that at lowest positive frequency (asynchronous), because [4+2] TS is able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2) hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!1
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the previous orbital symmetry of reagents (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory. HOMO-1 orbital might actually plays the role of HOMO during Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interactions. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition as secondary orbital overlap is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those on diene. Currently the stereoelectronic attraction on carbon outweighs the steric interaction with sp2 C-H bonds (planar) on diene, and the partly formed σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds directly clashes with sp3 C-H bonds (tetrahedral), i.e. greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!2
!3
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but absent in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. It is clearly shown from MO that e- density around C=O overlaps with the bulk of π-system to some extent in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction hardly occurs. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) will be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|1
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory in '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cycloaddition undergoes the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the small penalty of breaking weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product. This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually an exo (thermodynamic) product may be generated after a period of time. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised and electronic correlation is taken into account. <ref name=":0" /> It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene which leads to undesired product, such as [4s+2s] dimerisation. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is an unlikely situation unless strong driving force is applied, e.g. high concentration of butadiene, or heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involves secondary overlap that could be stabilised by interacting with polar solvent, resulting in more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose TS of the reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Diels-Alder cycloaddition includes a prototypical (butadiene and ethene) and a reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are able to behave as normal demand via [4s+2s] synchronised TS. Respect to regioselectivity, ''endo'' & ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore the lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T07:12:24Z

Yqw13: /* Optimisation of Reactants and Products */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|G1
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>yqw13_anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|G3
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|G4
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|G5
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|G6
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|A1
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|A2
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|A3
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|A4
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC plot above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interestingly found that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closest to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory '''B3LYP/6-31G(d)''' is suggested. Or more expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy (0 K), and electronic + thermal energy (298 K) are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to a higher resolution by larger basis set. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly more favoured. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is not significant hence relatively the small energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are approaching experimental values more, while the results from '''HF/3-21G''' are significantly overestimated.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4s + 2s] concerted, pericyclic pathway between a diene (4 carbons + 4 π electrons) and a conjugate dienophile (2 C + 2 πe-). There are overall 6 πe- s participating in the reaction and two σ-bonds being formed. Diene is usually more electron-rich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and so it is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, also π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number output from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Other factors, such as the geometry of diene, has to be considered, since s-cis conformation of diene is necessary for the synchronised breaking/formation of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, again the reaction path will be studied by '''IRC'''. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules that are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming yet gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7 accompanied with their symmetry. Symmetry of MO is determined respect to the vertical mirror plane. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely to occur owing to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled in the same manner as for Cope reaction. First, optimised butadiene and ethene molecules are modified to an estimated shape of cycloaddition TS using bicyclo-octane as a template, so that the optimal overlap of π system can be acquired. Product molecule (cyclohexene) is refined at '''HF/3-21G''' + '''B3LYP/6-31G(d)''' levels of theory. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects a complete minimisation of TS energy. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å) in TS. Typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. If considering the orientation of p-lobes, little reorganisation of central p-lobes on diene is required to form a new π-bond with neighboring p-lobe. However the terminal p-lobes of diene form σ-bonds with that of dienophile, and remarkable reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobes of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 + 0 = 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This vibration mode is favoured over that at lowest positive frequency (asynchronous), because [4+2] TS is able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2) hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!1
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the previous orbital symmetry of reagents (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory. HOMO-1 orbital might actually plays the role of HOMO during Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interactions. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition as secondary orbital overlap is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those on diene. Currently the stereoelectronic attraction on carbon outweighs the steric interaction with sp2 C-H bonds (planar) on diene, and the partly formed σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds directly clashes with sp3 C-H bonds (tetrahedral), i.e. greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!2
!3
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but absent in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. It is clearly shown from MO that e- density around C=O overlaps with the bulk of π-system to some extent in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction hardly occurs. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) will be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|1
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory in '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cycloaddition undergoes the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the small penalty of breaking weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product. This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually an exo (thermodynamic) product may be generated after a period of time. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised and electronic correlation is taken into account. <ref name=":0" /> It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene which leads to undesired product, such as [4s+2s] dimerisation. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is an unlikely situation unless strong driving force is applied, e.g. high concentration of butadiene, or heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involves secondary overlap that could be stabilised by interacting with polar solvent, resulting in more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose TS of the reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Diels-Alder cycloaddition includes a prototypical (butadiene and ethene) and a reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are able to behave as normal demand via [4s+2s] synchronised TS. Respect to regioselectivity, ''endo'' & ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore the lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T07:11:35Z

Yqw13: /* Optimisation of Reactants and Products */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|G1
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|[[File:Yqw13 anti2 631.PNG |150px]]
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|G3
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|G4
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|G5
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|G6
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|A1
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|A2
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|A3
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|A4
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC plot above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interestingly found that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closest to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory '''B3LYP/6-31G(d)''' is suggested. Or more expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy (0 K), and electronic + thermal energy (298 K) are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to a higher resolution by larger basis set. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly more favoured. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is not significant hence relatively the small energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are approaching experimental values more, while the results from '''HF/3-21G''' are significantly overestimated.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4s + 2s] concerted, pericyclic pathway between a diene (4 carbons + 4 π electrons) and a conjugate dienophile (2 C + 2 πe-). There are overall 6 πe- s participating in the reaction and two σ-bonds being formed. Diene is usually more electron-rich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and so it is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, also π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number output from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Other factors, such as the geometry of diene, has to be considered, since s-cis conformation of diene is necessary for the synchronised breaking/formation of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, again the reaction path will be studied by '''IRC'''. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules that are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming yet gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7 accompanied with their symmetry. Symmetry of MO is determined respect to the vertical mirror plane. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely to occur owing to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled in the same manner as for Cope reaction. First, optimised butadiene and ethene molecules are modified to an estimated shape of cycloaddition TS using bicyclo-octane as a template, so that the optimal overlap of π system can be acquired. Product molecule (cyclohexene) is refined at '''HF/3-21G''' + '''B3LYP/6-31G(d)''' levels of theory. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects a complete minimisation of TS energy. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å) in TS. Typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. If considering the orientation of p-lobes, little reorganisation of central p-lobes on diene is required to form a new π-bond with neighboring p-lobe. However the terminal p-lobes of diene form σ-bonds with that of dienophile, and remarkable reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobes of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 + 0 = 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This vibration mode is favoured over that at lowest positive frequency (asynchronous), because [4+2] TS is able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2) hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!1
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the previous orbital symmetry of reagents (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory. HOMO-1 orbital might actually plays the role of HOMO during Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interactions. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition as secondary orbital overlap is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those on diene. Currently the stereoelectronic attraction on carbon outweighs the steric interaction with sp2 C-H bonds (planar) on diene, and the partly formed σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds directly clashes with sp3 C-H bonds (tetrahedral), i.e. greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!2
!3
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but absent in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. It is clearly shown from MO that e- density around C=O overlaps with the bulk of π-system to some extent in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction hardly occurs. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) will be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|1
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory in '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cycloaddition undergoes the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the small penalty of breaking weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product. This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually an exo (thermodynamic) product may be generated after a period of time. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised and electronic correlation is taken into account. <ref name=":0" /> It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene which leads to undesired product, such as [4s+2s] dimerisation. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is an unlikely situation unless strong driving force is applied, e.g. high concentration of butadiene, or heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involves secondary overlap that could be stabilised by interacting with polar solvent, resulting in more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose TS of the reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Diels-Alder cycloaddition includes a prototypical (butadiene and ethene) and a reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are able to behave as normal demand via [4s+2s] synchronised TS. Respect to regioselectivity, ''endo'' & ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore the lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T07:09:03Z

Yqw13: /* Regioselectivity of Diels-Alder reaction */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|G1
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|G2
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|G3
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|G4
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|G5
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|G6
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|A1
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|A2
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|A3
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|A4
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC plot above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interestingly found that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closest to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory '''B3LYP/6-31G(d)''' is suggested. Or more expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy (0 K), and electronic + thermal energy (298 K) are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to a higher resolution by larger basis set. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly more favoured. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is not significant hence relatively the small energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are approaching experimental values more, while the results from '''HF/3-21G''' are significantly overestimated.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4s + 2s] concerted, pericyclic pathway between a diene (4 carbons + 4 π electrons) and a conjugate dienophile (2 C + 2 πe-). There are overall 6 πe- s participating in the reaction and two σ-bonds being formed. Diene is usually more electron-rich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and so it is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, also π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number output from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Other factors, such as the geometry of diene, has to be considered, since s-cis conformation of diene is necessary for the synchronised breaking/formation of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, again the reaction path will be studied by '''IRC'''. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules that are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming yet gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7 accompanied with their symmetry. Symmetry of MO is determined respect to the vertical mirror plane. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely to occur owing to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled in the same manner as for Cope reaction. First, optimised butadiene and ethene molecules are modified to an estimated shape of cycloaddition TS using bicyclo-octane as a template, so that the optimal overlap of π system can be acquired. Product molecule (cyclohexene) is refined at '''HF/3-21G''' + '''B3LYP/6-31G(d)''' levels of theory. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects a complete minimisation of TS energy. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å) in TS. Typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. If considering the orientation of p-lobes, little reorganisation of central p-lobes on diene is required to form a new π-bond with neighboring p-lobe. However the terminal p-lobes of diene form σ-bonds with that of dienophile, and remarkable reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobes of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 + 0 = 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This vibration mode is favoured over that at lowest positive frequency (asynchronous), because [4+2] TS is able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2) hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!1
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the previous orbital symmetry of reagents (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory. HOMO-1 orbital might actually plays the role of HOMO during Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interactions. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition as secondary orbital overlap is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those on diene. Currently the stereoelectronic attraction on carbon outweighs the steric interaction with sp2 C-H bonds (planar) on diene, and the partly formed σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds directly clashes with sp3 C-H bonds (tetrahedral), i.e. greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!2
!3
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but absent in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. It is clearly shown from MO that e- density around C=O overlaps with the bulk of π-system to some extent in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction hardly occurs. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) will be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|1
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory in '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cycloaddition undergoes the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the small penalty of breaking weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product. This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually an exo (thermodynamic) product may be generated after a period of time. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised and electronic correlation is taken into account. <ref name=":0" /> It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene which leads to undesired product, such as [4s+2s] dimerisation. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is an unlikely situation unless strong driving force is applied, e.g. high concentration of butadiene, or heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involves secondary overlap that could be stabilised by interacting with polar solvent, resulting in more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose TS of the reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Diels-Alder cycloaddition includes a prototypical (butadiene and ethene) and a reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are able to behave as normal demand via [4s+2s] synchronised TS. Respect to regioselectivity, ''endo'' & ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore the lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T06:46:56Z

Yqw13: /* Diels-Alder Cycloaddition */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|G1
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|G2
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|G3
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|G4
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|G5
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|G6
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|A1
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|A2
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|A3
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|A4
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC plot above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interestingly found that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closest to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory '''B3LYP/6-31G(d)''' is suggested. Or more expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy (0 K), and electronic + thermal energy (298 K) are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to a higher resolution by larger basis set. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly more favoured. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is not significant hence relatively the small energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are approaching experimental values more, while the results from '''HF/3-21G''' are significantly overestimated.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4s + 2s] concerted, pericyclic pathway between a diene (4 carbons + 4 π electrons) and a conjugate dienophile (2 C + 2 πe-). There are overall 6 πe- s participating in the reaction and two σ-bonds being formed. Diene is usually more electron-rich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and so it is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, also π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number output from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Other factors, such as the geometry of diene, has to be considered, since s-cis conformation of diene is necessary for the synchronised breaking/formation of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, again the reaction path will be studied by '''IRC'''. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules that are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming yet gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7 accompanied with their symmetry. Symmetry of MO is determined respect to the vertical mirror plane. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely to occur owing to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled in the same manner as for Cope reaction. First, optimised butadiene and ethene molecules are modified to an estimated shape of cycloaddition TS using bicyclo-octane as a template, so that the optimal overlap of π system can be acquired. Product molecule (cyclohexene) is refined at '''HF/3-21G''' + '''B3LYP/6-31G(d)''' levels of theory. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects a complete minimisation of TS energy. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å) in TS. Typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. If considering the orientation of p-lobes, little reorganisation of central p-lobes on diene is required to form a new π-bond with neighboring p-lobe. However the terminal p-lobes of diene form σ-bonds with that of dienophile, and remarkable reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobes of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 + 0 = 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This vibration mode is favoured over that at lowest positive frequency (asynchronous), because [4+2] TS is able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2) hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!1
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the previous orbital symmetry of reagents (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory. HOMO-1 orbital might actually plays the role of HOMO during Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interactions. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition as secondary orbital overlap is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table. 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those in diene. Currently the stereoelectronic attraction on carbon outweighs the steric hindrance with sp2 C-H bonds (planar) on diene, and the forming σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds are directly opposite to sp3 C-H bonds (tetrahedral) with greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!2
!3
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but not in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. By rotating the MO, it is clearly shown that e- density around C=O overlaps to some extent with the bulk of π-system in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction is very unlikely. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) should be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|1
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory during '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cyycloaddition goes through the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the penalty of two weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product . This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually a exo (thermodynamic) product may be generated. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised. It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene, such as [4s+2s] dimerisation, which will lead to undesired product. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is the unlikely situation. Unless strong driving force is applied, such as high concentration of butadiene, and heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involveds secondary overlap that could be stabilised by interacting with polar solvent, leading to more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose the TS of reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Two Diels-Alder reactions --- prototypical (butadiene and ethene) and reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are thermally allowed via [4s+2s] synchronised TS, also they behaves as normal demand. From regioselectivity, ''endo'' and ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T06:29:15Z

Yqw13: /* Diels-Alder Cycloaddition */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|G1
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|G2
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|G3
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|G4
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|G5
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|G6
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|A1
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|A2
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|A3
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|A4
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC plot above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interestingly found that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closest to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory '''B3LYP/6-31G(d)''' is suggested. Or more expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy (0 K), and electronic + thermal energy (298 K) are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to a higher resolution by larger basis set. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly more favoured. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is not significant hence relatively the small energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are approaching experimental values more, while the results from '''HF/3-21G''' are significantly overestimated.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4s + 2s] concerted, pericyclic pathway between a diene (4 carbons + 4 π electrons) and a conjugate dienophile (2 C + 2 πe-). There are overall 6 πe- s participating in the reaction and two σ-bonds being formed. Diene is usually more electron-rich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and so it is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, also π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number output from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Other factors, such as the geometry of diene, has to be considered, since s-cis conformation of diene is necessary for the synchronised breaking/formation of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, again the reaction path will be studied by '''IRC'''. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules that are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming yet gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7 accompanied with their symmetry. Symmetry of MO is determined respect to the vertical mirror plane. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely to occur owing to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled. First, optimised butadiene and ethene molecules are modified to the proposed shape of cycloaddition TS using bicyclo-octane as a template. The reason is that optimal overlap of π system can be acquired in this manner. Product molecule (cyclohexene) is refined by '''HF/3-21G''' + '''B3LYP/6-31G(d)''' jobs. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects that TS energy has been minimised. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å). The typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. As predicted, little reorganisation of central p-lobe on diene is required in order to form new π-bond with neighboring p-lobe. However the terminal p-lobe forms σ-bond with that of dienophile, and reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobs of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This mode is favoured over that at lowest positive frequency (asynchronous), because the [4+2] TS ia able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2), hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!1
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the orbital symmetry of reagent before (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory, and HOMO-1 orbital actually plays the role of HOMO in Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interaction. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition, as secondary orbital interaction is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table. 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those in diene. Currently the stereoelectronic attraction on carbon outweighs the steric hindrance with sp2 C-H bonds (planar) on diene, and the forming σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds are directly opposite to sp3 C-H bonds (tetrahedral) with greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!2
!3
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but not in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. By rotating the MO, it is clearly shown that e- density around C=O overlaps to some extent with the bulk of π-system in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction is very unlikely. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) should be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|1
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory during '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cyycloaddition goes through the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the penalty of two weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product . This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually a exo (thermodynamic) product may be generated. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised. It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene, such as [4s+2s] dimerisation, which will lead to undesired product. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is the unlikely situation. Unless strong driving force is applied, such as high concentration of butadiene, and heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involveds secondary overlap that could be stabilised by interacting with polar solvent, leading to more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose the TS of reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Two Diels-Alder reactions --- prototypical (butadiene and ethene) and reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are [4s+2s] which is thermally allowed, also they behaves as normal demand. From regioselectivity, ''endo'' and ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T06:19:55Z

Yqw13: /* Study of Chair and Boat TS */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|G1
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|G2
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|G3
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|G4
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|G5
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|G6
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|A1
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|A2
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|A3
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|A4
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC plot above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interestingly found that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closest to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory '''B3LYP/6-31G(d)''' is suggested. Or more expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy (0 K), and electronic + thermal energy (298 K) are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to a higher resolution by larger basis set. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly more favoured. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is not significant hence relatively the small energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are approaching experimental values more, while the results from '''HF/3-21G''' are significantly overestimated.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4+2] concerted, pericyclic pathway between a diene (4 carbons with 4 π electrons) and the conjugate dienophile (2 C + 2 πe-). There are overall 6 π-e- s participating in the reaction and two σ-bonds being formed. Diene is usually more electronrich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and this is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, and π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Cycloaddition is referred as [4s + 2s] with the equation output of 1. Other factors, such as the geometry of diene, has to be considered, since scis conformation of diene is necessary for the synchronised breaking/forming of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, hence reaction path will be studied. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules which are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming, although it gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7. Symmetry of MO is determined respect to the vertical mirror plane, which are also indicated in Table 7. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely due to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled. First, optimised butadiene and ethene molecules are modified to the proposed shape of cycloaddition TS using bicyclo-octane as a template. The reason is that optimal overlap of π system can be acquired in this manner. Product molecule (cyclohexene) is refined by '''HF/3-21G''' + '''B3LYP/6-31G(d)''' jobs. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects that TS energy has been minimised. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å). The typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. As predicted, little reorganisation of central p-lobe on diene is required in order to form new π-bond with neighboring p-lobe. However the terminal p-lobe forms σ-bond with that of dienophile, and reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobs of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This mode is favoured over that at lowest positive frequency (asynchronous), because the [4+2] TS ia able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2), hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!1
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the orbital symmetry of reagent before (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory, and HOMO-1 orbital actually plays the role of HOMO in Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interaction. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition, as secondary orbital interaction is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table. 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those in diene. Currently the stereoelectronic attraction on carbon outweighs the steric hindrance with sp2 C-H bonds (planar) on diene, and the forming σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds are directly opposite to sp3 C-H bonds (tetrahedral) with greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!2
!3
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but not in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. By rotating the MO, it is clearly shown that e- density around C=O overlaps to some extent with the bulk of π-system in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction is very unlikely. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) should be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|1
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory during '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cyycloaddition goes through the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the penalty of two weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product . This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually a exo (thermodynamic) product may be generated. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised. It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene, such as [4s+2s] dimerisation, which will lead to undesired product. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is the unlikely situation. Unless strong driving force is applied, such as high concentration of butadiene, and heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involveds secondary overlap that could be stabilised by interacting with polar solvent, leading to more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose the TS of reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Two Diels-Alder reactions --- prototypical (butadiene and ethene) and reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are [4s+2s] which is thermally allowed, also they behaves as normal demand. From regioselectivity, ''endo'' and ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T06:08:24Z

Yqw13: /* Study of Chair and Boat TS */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|G1
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|G2
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|G3
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|G4
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|G5
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|G6
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|A1
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|A2
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|A3
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|A4
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
.[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of one σ-bond via a concerted transition structure. Reaction process is thoroughly reversible unless a stabilising group is applied to form an energy trap. There are two proposed conformations of TS --- "''chair''" and "''boat''" like, as shown in Figure 2. They will be analysed separately by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with a rough separation of 2.2 Å. Subsequently the model built is undergone two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible estimation of TS. Using combination job of '''Opt+Freq''' at '''HF/3-21G''' level, the structure is optimised to '''TS (Berny)'''. The success of calculation is highly dependent on predicted geometry of TS. In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to '''TS(Berny)''', which freezes and energetically relaxes the geometry at the moment of bond breaking/formation at '''HF/3-21G''' level. Unfreezing process is required afterwards.

Results from both methods should be able to converge to a similar TS. The parameters given in Table 4 shows slight inconsistency due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than the input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' TS conformation has been optimised. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy relative to the displacement. Negative k value indicates a stationary point in nuclear configuration space which is also a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. The physical significance is associated with the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vibrational frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by '''QST2''' method. Optimised Anti 2 (1,5-hexadiene) structure is carried down as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since the rearrangement process is not taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. TS is predicted by the maximal potential energy during linear interpolation between two molecules. The modified molecule is optimised using '''Opt+Freq''' at '''HF/3-21G''' level, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation is successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conformation such that MEP is traced from the saddle point to local minima, hence desired TS can be found. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended in order to enhanced the precision of IRC before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC figure above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interesting to see that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closer to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory such as '''B3LYP/6-31G(d)''' is suggested. Or most expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems only.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies of reactions via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy, and electronic + thermal energy are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to more powerful optimisation. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly favoured over ''boat''. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is relatively small, hence insignificant energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are more approaching experimental value, while the results from '''HF/3-21G''' are significantly higher.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4+2] concerted, pericyclic pathway between a diene (4 carbons with 4 π electrons) and the conjugate dienophile (2 C + 2 πe-). There are overall 6 π-e- s participating in the reaction and two σ-bonds being formed. Diene is usually more electronrich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and this is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, and π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Cycloaddition is referred as [4s + 2s] with the equation output of 1. Other factors, such as the geometry of diene, has to be considered, since scis conformation of diene is necessary for the synchronised breaking/forming of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, hence reaction path will be studied. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules which are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming, although it gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7. Symmetry of MO is determined respect to the vertical mirror plane, which are also indicated in Table 7. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely due to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled. First, optimised butadiene and ethene molecules are modified to the proposed shape of cycloaddition TS using bicyclo-octane as a template. The reason is that optimal overlap of π system can be acquired in this manner. Product molecule (cyclohexene) is refined by '''HF/3-21G''' + '''B3LYP/6-31G(d)''' jobs. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects that TS energy has been minimised. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å). The typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. As predicted, little reorganisation of central p-lobe on diene is required in order to form new π-bond with neighboring p-lobe. However the terminal p-lobe forms σ-bond with that of dienophile, and reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobs of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This mode is favoured over that at lowest positive frequency (asynchronous), because the [4+2] TS ia able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2), hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!1
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the orbital symmetry of reagent before (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory, and HOMO-1 orbital actually plays the role of HOMO in Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interaction. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition, as secondary orbital interaction is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table. 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those in diene. Currently the stereoelectronic attraction on carbon outweighs the steric hindrance with sp2 C-H bonds (planar) on diene, and the forming σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds are directly opposite to sp3 C-H bonds (tetrahedral) with greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!2
!3
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but not in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. By rotating the MO, it is clearly shown that e- density around C=O overlaps to some extent with the bulk of π-system in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction is very unlikely. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) should be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|1
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory during '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cyycloaddition goes through the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the penalty of two weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product . This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually a exo (thermodynamic) product may be generated. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised. It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene, such as [4s+2s] dimerisation, which will lead to undesired product. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is the unlikely situation. Unless strong driving force is applied, such as high concentration of butadiene, and heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involveds secondary overlap that could be stabilised by interacting with polar solvent, leading to more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose the TS of reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Two Diels-Alder reactions --- prototypical (butadiene and ethene) and reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are [4s+2s] which is thermally allowed, also they behaves as normal demand. From regioselectivity, ''endo'' and ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T05:43:29Z

Yqw13: /* Cope Rearrangement */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs sterics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|G1
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|G2
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|G3
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|G4
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|G5
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|G6
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|A1
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|A2
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|A3
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|A4
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures is tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. With the base of previous optimisation, Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of structure needs to be taken into account. As observed in '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction and the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to ensure the successful optimisation of geometry. For example, Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised.

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is listed in '''Table 3'''. All the calculations are set to 298.15 K by default, although the alternation can be achieved by the '''Temperature''' option. It is observed that all types of energies share the same value at 0 K. This can be the consequence of suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
·[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of a sigma bond via a concerted transition state. Reaction process is thoroughly reversible unless a stabilising group is applied to form the lower-energy trap. There are two proposed conformations of TS --- "''chair''" or "''boat''" like, which are shown in Figure 2. Each of them will be analysed by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with the rough separation of 2.2 Å. Subsequently the model built up is experienced two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible propose of TS. With the combination job of '''Opt+Freq''' at '''HF/3-21G''', the structure is optimised to TS (Berny). In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. The success of calculation is highly dependent on the predicted geometry of TS. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to TS(Berny), which freezes geometry at the moment of bond break/formation and energetically relaxes at '''HF/3-21G'''. Unfreezing process is required afterwards.

Results from both methods should be able to converge to similar TS. The parameters are given below, which shows very light difference due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' conformation is optimised as a TS. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy and displacement. The negative k indicates a stationary point in nuclear configuration space and also it is a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. Its physical significance corresponds to the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vib. frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by QST2 method. The optimised Anti 2 (1,5-hexadiene) structure is used as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. TS is predicted by the maxima potential energy during linear interpolation between two molecules. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since it would not be taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. The modified molecule is optimised using '''Opt+Freq''' at HF/3-21G, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation has been successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conforamtion so as to tracing the mep from saddle point to local minima and to find the desired TS. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended since the precision of IRC is enhanced before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC figure above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interesting to see that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closer to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory such as '''B3LYP/6-31G(d)''' is suggested. Or most expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems only.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies of reactions via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy, and electronic + thermal energy are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to more powerful optimisation. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly favoured over ''boat''. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is relatively small, hence insignificant energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are more approaching experimental value, while the results from '''HF/3-21G''' are significantly higher.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4+2] concerted, pericyclic pathway between a diene (4 carbons with 4 π electrons) and the conjugate dienophile (2 C + 2 πe-). There are overall 6 π-e- s participating in the reaction and two σ-bonds being formed. Diene is usually more electronrich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and this is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, and π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Cycloaddition is referred as [4s + 2s] with the equation output of 1. Other factors, such as the geometry of diene, has to be considered, since scis conformation of diene is necessary for the synchronised breaking/forming of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, hence reaction path will be studied. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules which are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming, although it gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7. Symmetry of MO is determined respect to the vertical mirror plane, which are also indicated in Table 7. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely due to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled. First, optimised butadiene and ethene molecules are modified to the proposed shape of cycloaddition TS using bicyclo-octane as a template. The reason is that optimal overlap of π system can be acquired in this manner. Product molecule (cyclohexene) is refined by '''HF/3-21G''' + '''B3LYP/6-31G(d)''' jobs. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects that TS energy has been minimised. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å). The typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. As predicted, little reorganisation of central p-lobe on diene is required in order to form new π-bond with neighboring p-lobe. However the terminal p-lobe forms σ-bond with that of dienophile, and reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobs of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This mode is favoured over that at lowest positive frequency (asynchronous), because the [4+2] TS ia able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2), hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!1
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the orbital symmetry of reagent before (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory, and HOMO-1 orbital actually plays the role of HOMO in Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interaction. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition, as secondary orbital interaction is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table. 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those in diene. Currently the stereoelectronic attraction on carbon outweighs the steric hindrance with sp2 C-H bonds (planar) on diene, and the forming σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds are directly opposite to sp3 C-H bonds (tetrahedral) with greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!2
!3
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but not in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. By rotating the MO, it is clearly shown that e- density around C=O overlaps to some extent with the bulk of π-system in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction is very unlikely. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) should be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|1
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory during '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cyycloaddition goes through the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the penalty of two weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product . This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually a exo (thermodynamic) product may be generated. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised. It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene, such as [4s+2s] dimerisation, which will lead to undesired product. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is the unlikely situation. Unless strong driving force is applied, such as high concentration of butadiene, and heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involveds secondary overlap that could be stabilised by interacting with polar solvent, leading to more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose the TS of reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Two Diels-Alder reactions --- prototypical (butadiene and ethene) and reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are [4s+2s] which is thermally allowed, also they behaves as normal demand. From regioselectivity, ''endo'' and ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T05:26:22Z

Yqw13: /* Cope Rearrangement */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with their corresponding electronic energies and point groups. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) shapes. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation at '''HF/3-21G''' level suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci (inversion operation). This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, whose stereoelectronic attraction outweighs steics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (Hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|G1
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|G2
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|G3
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|G4
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|G5
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|G6
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|A1
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|A2
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|A3
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|A4
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures are tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. On the basis of previous optimisation. Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of the structure need to be considered. Concluded from '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction hence the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to prove the successful geometry optimisation. Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised. (1 hartree = 627.509 kcal/mol)

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is computed, and the is listed in '''Table 3'''. All the calculations are set to 298.15 K by default. Geometry at 0 K can be achieved by the '''Temperature''' option, and vibrational temperature is tested. Obviously all types of energies share the same value at 0 K. This is the consequence of the suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
·[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of a sigma bond via a concerted transition state. Reaction process is thoroughly reversible unless a stabilising group is applied to form the lower-energy trap. There are two proposed conformations of TS --- "''chair''" or "''boat''" like, which are shown in Figure 2. Each of them will be analysed by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with the rough separation of 2.2 Å. Subsequently the model built up is experienced two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible propose of TS. With the combination job of '''Opt+Freq''' at '''HF/3-21G''', the structure is optimised to TS (Berny). In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. The success of calculation is highly dependent on the predicted geometry of TS. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to TS(Berny), which freezes geometry at the moment of bond break/formation and energetically relaxes at '''HF/3-21G'''. Unfreezing process is required afterwards.

Results from both methods should be able to converge to similar TS. The parameters are given below, which shows very light difference due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' conformation is optimised as a TS. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy and displacement. The negative k indicates a stationary point in nuclear configuration space and also it is a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. Its physical significance corresponds to the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vib. frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by QST2 method. The optimised Anti 2 (1,5-hexadiene) structure is used as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. TS is predicted by the maxima potential energy during linear interpolation between two molecules. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since it would not be taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. The modified molecule is optimised using '''Opt+Freq''' at HF/3-21G, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation has been successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conforamtion so as to tracing the mep from saddle point to local minima and to find the desired TS. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended since the precision of IRC is enhanced before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC figure above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interesting to see that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closer to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory such as '''B3LYP/6-31G(d)''' is suggested. Or most expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems only.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies of reactions via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy, and electronic + thermal energy are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to more powerful optimisation. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly favoured over ''boat''. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is relatively small, hence insignificant energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are more approaching experimental value, while the results from '''HF/3-21G''' are significantly higher.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4+2] concerted, pericyclic pathway between a diene (4 carbons with 4 π electrons) and the conjugate dienophile (2 C + 2 πe-). There are overall 6 π-e- s participating in the reaction and two σ-bonds being formed. Diene is usually more electronrich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and this is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, and π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Cycloaddition is referred as [4s + 2s] with the equation output of 1. Other factors, such as the geometry of diene, has to be considered, since scis conformation of diene is necessary for the synchronised breaking/forming of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, hence reaction path will be studied. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules which are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming, although it gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7. Symmetry of MO is determined respect to the vertical mirror plane, which are also indicated in Table 7. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely due to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled. First, optimised butadiene and ethene molecules are modified to the proposed shape of cycloaddition TS using bicyclo-octane as a template. The reason is that optimal overlap of π system can be acquired in this manner. Product molecule (cyclohexene) is refined by '''HF/3-21G''' + '''B3LYP/6-31G(d)''' jobs. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects that TS energy has been minimised. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å). The typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. As predicted, little reorganisation of central p-lobe on diene is required in order to form new π-bond with neighboring p-lobe. However the terminal p-lobe forms σ-bond with that of dienophile, and reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobs of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This mode is favoured over that at lowest positive frequency (asynchronous), because the [4+2] TS ia able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2), hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!1
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the orbital symmetry of reagent before (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory, and HOMO-1 orbital actually plays the role of HOMO in Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interaction. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition, as secondary orbital interaction is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table. 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those in diene. Currently the stereoelectronic attraction on carbon outweighs the steric hindrance with sp2 C-H bonds (planar) on diene, and the forming σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds are directly opposite to sp3 C-H bonds (tetrahedral) with greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!2
!3
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but not in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. By rotating the MO, it is clearly shown that e- density around C=O overlaps to some extent with the bulk of π-system in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction is very unlikely. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) should be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|1
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory during '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cyycloaddition goes through the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the penalty of two weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product . This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually a exo (thermodynamic) product may be generated. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised. It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene, such as [4s+2s] dimerisation, which will lead to undesired product. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is the unlikely situation. Unless strong driving force is applied, such as high concentration of butadiene, and heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involveds secondary overlap that could be stabilised by interacting with polar solvent, leading to more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose the TS of reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Two Diels-Alder reactions --- prototypical (butadiene and ethene) and reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are [4s+2s] which is thermally allowed, also they behaves as normal demand. From regioselectivity, ''endo'' and ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T05:22:08Z

Yqw13: /* Computation Introduction of Transition Structures of Pericyclic Reaction */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 05'' software in order for further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives of energy are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformers of reagents and products are connected with TS. It is worth mentioning that there may exist more than one saddle points in a complicated reaction, i.e. global and local peaks. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However PES method is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing the energies of different conformers hence their stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type exclusively for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of energy in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' can be conducted. Frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of molecular energy is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every point along the trajectory is optimised such that the gradients of the curve overlap with tangent of the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with the corresponding electronic energy and point group. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) types. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation of '''HF/3-21G''' method suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci and the inversion operation. This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, and its stereoelectronic effect outweighs steics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|G1
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|G2
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|G3
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|G4
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|G5
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|G6
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|A1
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|A2
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|A3
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|A4
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures are tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. On the basis of previous optimisation. Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of the structure need to be considered. Concluded from '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction hence the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to prove the successful geometry optimisation. Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised. (1 hartree = 627.509 kcal/mol)

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is computed, and the is listed in '''Table 3'''. All the calculations are set to 298.15 K by default. Geometry at 0 K can be achieved by the '''Temperature''' option, and vibrational temperature is tested. Obviously all types of energies share the same value at 0 K. This is the consequence of the suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
·[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of a sigma bond via a concerted transition state. Reaction process is thoroughly reversible unless a stabilising group is applied to form the lower-energy trap. There are two proposed conformations of TS --- "''chair''" or "''boat''" like, which are shown in Figure 2. Each of them will be analysed by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with the rough separation of 2.2 Å. Subsequently the model built up is experienced two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible propose of TS. With the combination job of '''Opt+Freq''' at '''HF/3-21G''', the structure is optimised to TS (Berny). In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. The success of calculation is highly dependent on the predicted geometry of TS. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to TS(Berny), which freezes geometry at the moment of bond break/formation and energetically relaxes at '''HF/3-21G'''. Unfreezing process is required afterwards.

Results from both methods should be able to converge to similar TS. The parameters are given below, which shows very light difference due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' conformation is optimised as a TS. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy and displacement. The negative k indicates a stationary point in nuclear configuration space and also it is a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. Its physical significance corresponds to the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vib. frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by QST2 method. The optimised Anti 2 (1,5-hexadiene) structure is used as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. TS is predicted by the maxima potential energy during linear interpolation between two molecules. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since it would not be taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. The modified molecule is optimised using '''Opt+Freq''' at HF/3-21G, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation has been successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conforamtion so as to tracing the mep from saddle point to local minima and to find the desired TS. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended since the precision of IRC is enhanced before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC figure above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interesting to see that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closer to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory such as '''B3LYP/6-31G(d)''' is suggested. Or most expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems only.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies of reactions via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy, and electronic + thermal energy are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to more powerful optimisation. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly favoured over ''boat''. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is relatively small, hence insignificant energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are more approaching experimental value, while the results from '''HF/3-21G''' are significantly higher.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4+2] concerted, pericyclic pathway between a diene (4 carbons with 4 π electrons) and the conjugate dienophile (2 C + 2 πe-). There are overall 6 π-e- s participating in the reaction and two σ-bonds being formed. Diene is usually more electronrich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and this is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, and π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Cycloaddition is referred as [4s + 2s] with the equation output of 1. Other factors, such as the geometry of diene, has to be considered, since scis conformation of diene is necessary for the synchronised breaking/forming of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, hence reaction path will be studied. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules which are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming, although it gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7. Symmetry of MO is determined respect to the vertical mirror plane, which are also indicated in Table 7. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely due to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled. First, optimised butadiene and ethene molecules are modified to the proposed shape of cycloaddition TS using bicyclo-octane as a template. The reason is that optimal overlap of π system can be acquired in this manner. Product molecule (cyclohexene) is refined by '''HF/3-21G''' + '''B3LYP/6-31G(d)''' jobs. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects that TS energy has been minimised. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å). The typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. As predicted, little reorganisation of central p-lobe on diene is required in order to form new π-bond with neighboring p-lobe. However the terminal p-lobe forms σ-bond with that of dienophile, and reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobs of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This mode is favoured over that at lowest positive frequency (asynchronous), because the [4+2] TS ia able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2), hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!1
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the orbital symmetry of reagent before (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory, and HOMO-1 orbital actually plays the role of HOMO in Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interaction. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition, as secondary orbital interaction is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table. 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those in diene. Currently the stereoelectronic attraction on carbon outweighs the steric hindrance with sp2 C-H bonds (planar) on diene, and the forming σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds are directly opposite to sp3 C-H bonds (tetrahedral) with greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!2
!3
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but not in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. By rotating the MO, it is clearly shown that e- density around C=O overlaps to some extent with the bulk of π-system in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction is very unlikely. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) should be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|1
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory during '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cyycloaddition goes through the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the penalty of two weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product . This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually a exo (thermodynamic) product may be generated. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised. It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene, such as [4s+2s] dimerisation, which will lead to undesired product. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is the unlikely situation. Unless strong driving force is applied, such as high concentration of butadiene, and heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involveds secondary overlap that could be stabilised by interacting with polar solvent, leading to more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose the TS of reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Two Diels-Alder reactions --- prototypical (butadiene and ethene) and reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are [4s+2s] which is thermally allowed, also they behaves as normal demand. From regioselectivity, ''endo'' and ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T05:01:17Z

Yqw13:

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematical methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 09'' software in order for the further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformer of reagents and products, are connected with TS. It is worth mentioning that there may be more than one saddle points found in a complicated reaction, i.e. either global or local peak. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However method of PES is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing energies of different conformers hence the stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type `exclusive for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of E in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' is conducted. Therefore frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of energy of a molecule is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every points along the trajectory are optimised such that the gradients of the curve overlap with tangent to the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with the corresponding electronic energy and point group. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) types. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation of '''HF/3-21G''' method suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci and the inversion operation. This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, and its stereoelectronic effect outweighs steics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|G1
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|G2
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|G3
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|G4
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|G5
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|G6
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|A1
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|A2
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|A3
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|A4
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures are tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. On the basis of previous optimisation. Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of the structure need to be considered. Concluded from '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction hence the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to prove the successful geometry optimisation. Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised. (1 hartree = 627.509 kcal/mol)

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is computed, and the is listed in '''Table 3'''. All the calculations are set to 298.15 K by default. Geometry at 0 K can be achieved by the '''Temperature''' option, and vibrational temperature is tested. Obviously all types of energies share the same value at 0 K. This is the consequence of the suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
·[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of a sigma bond via a concerted transition state. Reaction process is thoroughly reversible unless a stabilising group is applied to form the lower-energy trap. There are two proposed conformations of TS --- "''chair''" or "''boat''" like, which are shown in Figure 2. Each of them will be analysed by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with the rough separation of 2.2 Å. Subsequently the model built up is experienced two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible propose of TS. With the combination job of '''Opt+Freq''' at '''HF/3-21G''', the structure is optimised to TS (Berny). In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. The success of calculation is highly dependent on the predicted geometry of TS. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to TS(Berny), which freezes geometry at the moment of bond break/formation and energetically relaxes at '''HF/3-21G'''. Unfreezing process is required afterwards.

Results from both methods should be able to converge to similar TS. The parameters are given below, which shows very light difference due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' conformation is optimised as a TS. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy and displacement. The negative k indicates a stationary point in nuclear configuration space and also it is a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. Its physical significance corresponds to the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vib. frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by QST2 method. The optimised Anti 2 (1,5-hexadiene) structure is used as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. TS is predicted by the maxima potential energy during linear interpolation between two molecules. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since it would not be taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. The modified molecule is optimised using '''Opt+Freq''' at HF/3-21G, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation has been successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conforamtion so as to tracing the mep from saddle point to local minima and to find the desired TS. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended since the precision of IRC is enhanced before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC figure above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interesting to see that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closer to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory such as '''B3LYP/6-31G(d)''' is suggested. Or most expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems only.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies of reactions via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy, and electronic + thermal energy are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to more powerful optimisation. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly favoured over ''boat''. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is relatively small, hence insignificant energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are more approaching experimental value, while the results from '''HF/3-21G''' are significantly higher.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4+2] concerted, pericyclic pathway between a diene (4 carbons with 4 π electrons) and the conjugate dienophile (2 C + 2 πe-). There are overall 6 π-e- s participating in the reaction and two σ-bonds being formed. Diene is usually more electronrich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and this is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, and π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Cycloaddition is referred as [4s + 2s] with the equation output of 1. Other factors, such as the geometry of diene, has to be considered, since scis conformation of diene is necessary for the synchronised breaking/forming of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, hence reaction path will be studied. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules which are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming, although it gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7. Symmetry of MO is determined respect to the vertical mirror plane, which are also indicated in Table 7. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely due to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled. First, optimised butadiene and ethene molecules are modified to the proposed shape of cycloaddition TS using bicyclo-octane as a template. The reason is that optimal overlap of π system can be acquired in this manner. Product molecule (cyclohexene) is refined by '''HF/3-21G''' + '''B3LYP/6-31G(d)''' jobs. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects that TS energy has been minimised. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å). The typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. As predicted, little reorganisation of central p-lobe on diene is required in order to form new π-bond with neighboring p-lobe. However the terminal p-lobe forms σ-bond with that of dienophile, and reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobs of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This mode is favoured over that at lowest positive frequency (asynchronous), because the [4+2] TS ia able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2), hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!1
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the orbital symmetry of reagent before (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory, and HOMO-1 orbital actually plays the role of HOMO in Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interaction. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition, as secondary orbital interaction is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table. 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those in diene. Currently the stereoelectronic attraction on carbon outweighs the steric hindrance with sp2 C-H bonds (planar) on diene, and the forming σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds are directly opposite to sp3 C-H bonds (tetrahedral) with greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!2
!3
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but not in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. By rotating the MO, it is clearly shown that e- density around C=O overlaps to some extent with the bulk of π-system in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction is very unlikely. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) should be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|1
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory during '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cyycloaddition goes through the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the penalty of two weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product . This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually a exo (thermodynamic) product may be generated. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised. It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene, such as [4s+2s] dimerisation, which will lead to undesired product. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is the unlikely situation. Unless strong driving force is applied, such as high concentration of butadiene, and heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involveds secondary overlap that could be stabilised by interacting with polar solvent, leading to more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose the TS of reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Two Diels-Alder reactions --- prototypical (butadiene and ethene) and reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are [4s+2s] which is thermally allowed, also they behaves as normal demand. From regioselectivity, ''endo'' and ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

File:Yqw13 regioselec exo TS QST2 mo47.cub.jvxl

2016-02-12T04:52:01Z

Yqw13:

File:Yqw13 regioselec exo TS QST2 mo47.cub.xyz

2016-02-12T04:51:28Z

Yqw13:

User:Yqw13

2016-02-12T04:49:09Z

Yqw13: /* Regioselectivity of Diels-Alder reaction */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematics methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 09'' software in order for the further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformer of reagents and products, are connected with TS. It is worth mentioning that there may be more than one saddle points found in a complicated reaction, i.e. either global or local peak. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However method of PES is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing energies of different conformers hence the stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type `exclusive for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of E in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' is conducted. Therefore frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of energy of a molecule is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every points along the trajectory are optimised such that the gradients of the curve overlap with tangent to the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with the corresponding electronic energy and point group. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) types. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation of '''HF/3-21G''' method suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci and the inversion operation. This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, and its stereoelectronic effect outweighs steics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|G1
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|G2
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|G3
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|G4
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|G5
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|G6
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|A1
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|A2
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|A3
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|A4
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures are tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. On the basis of previous optimisation. Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of the structure need to be considered. Concluded from '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction hence the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to prove the successful geometry optimisation. Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised. (1 hartree = 627.509 kcal/mol)

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is computed, and the is listed in '''Table 3'''. All the calculations are set to 298.15 K by default. Geometry at 0 K can be achieved by the '''Temperature''' option, and vibrational temperature is tested. Obviously all types of energies share the same value at 0 K. This is the consequence of the suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
·[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of a sigma bond via a concerted transition state. Reaction process is thoroughly reversible unless a stabilising group is applied to form the lower-energy trap. There are two proposed conformations of TS --- "''chair''" or "''boat''" like, which are shown in Figure 2. Each of them will be analysed by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with the rough separation of 2.2 Å. Subsequently the model built up is experienced two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible propose of TS. With the combination job of '''Opt+Freq''' at '''HF/3-21G''', the structure is optimised to TS (Berny). In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. The success of calculation is highly dependent on the predicted geometry of TS. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to TS(Berny), which freezes geometry at the moment of bond break/formation and energetically relaxes at '''HF/3-21G'''. Unfreezing process is required afterwards.

Results from both methods should be able to converge to similar TS. The parameters are given below, which shows very light difference due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' conformation is optimised as a TS. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy and displacement. The negative k indicates a stationary point in nuclear configuration space and also it is a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. Its physical significance corresponds to the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vib. frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by QST2 method. The optimised Anti 2 (1,5-hexadiene) structure is used as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. TS is predicted by the maxima potential energy during linear interpolation between two molecules. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since it would not be taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. The modified molecule is optimised using '''Opt+Freq''' at HF/3-21G, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation has been successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conforamtion so as to tracing the mep from saddle point to local minima and to find the desired TS. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended since the precision of IRC is enhanced before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC figure above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interesting to see that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closer to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory such as '''B3LYP/6-31G(d)''' is suggested. Or most expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems only.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies of reactions via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy, and electronic + thermal energy are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to more powerful optimisation. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly favoured over ''boat''. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is relatively small, hence insignificant energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are more approaching experimental value, while the results from '''HF/3-21G''' are significantly higher.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4+2] concerted, pericyclic pathway between a diene (4 carbons with 4 π electrons) and the conjugate dienophile (2 C + 2 πe-). There are overall 6 π-e- s participating in the reaction and two σ-bonds being formed. Diene is usually more electronrich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and this is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, and π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Cycloaddition is referred as [4s + 2s] with the equation output of 1. Other factors, such as the geometry of diene, has to be considered, since scis conformation of diene is necessary for the synchronised breaking/forming of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, hence reaction path will be studied. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules which are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming, although it gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7. Symmetry of MO is determined respect to the vertical mirror plane, which are also indicated in Table 7. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely due to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled. First, optimised butadiene and ethene molecules are modified to the proposed shape of cycloaddition TS using bicyclo-octane as a template. The reason is that optimal overlap of π system can be acquired in this manner. Product molecule (cyclohexene) is refined by '''HF/3-21G''' + '''B3LYP/6-31G(d)''' jobs. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects that TS energy has been minimised. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å). The typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. As predicted, little reorganisation of central p-lobe on diene is required in order to form new π-bond with neighboring p-lobe. However the terminal p-lobe forms σ-bond with that of dienophile, and reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobs of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This mode is favoured over that at lowest positive frequency (asynchronous), because the [4+2] TS ia able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2), hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!1
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the orbital symmetry of reagent before (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory, and HOMO-1 orbital actually plays the role of HOMO in Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interaction. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition, as secondary orbital interaction is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table. 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those in diene. Currently the stereoelectronic attraction on carbon outweighs the steric hindrance with sp2 C-H bonds (planar) on diene, and the forming σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds are directly opposite to sp3 C-H bonds (tetrahedral) with greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="2" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
!2
!3
|-
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but not in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. By rotating the MO, it is clearly shown that e- density around C=O overlaps to some extent with the bulk of π-system in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction is very unlikely. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) should be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|1
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory during '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cyycloaddition goes through the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the penalty of two weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product . This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually a exo (thermodynamic) product may be generated. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised. It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene, such as [4s+2s] dimerisation, which will lead to undesired product. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is the unlikely situation. Unless strong driving force is applied, such as high concentration of butadiene, and heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involveds secondary overlap that could be stabilised by interacting with polar solvent, leading to more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose the TS of reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Two Diels-Alder reactions --- prototypical (butadiene and ethene) and reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are [4s+2s] which is thermally allowed, also they behaves as normal demand. From regioselectivity, ''endo'' and ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T04:46:02Z

Yqw13: /* Conclusion */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematics methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 09'' software in order for the further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformer of reagents and products, are connected with TS. It is worth mentioning that there may be more than one saddle points found in a complicated reaction, i.e. either global or local peak. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However method of PES is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing energies of different conformers hence the stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type `exclusive for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of E in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' is conducted. Therefore frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of energy of a molecule is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every points along the trajectory are optimised such that the gradients of the curve overlap with tangent to the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with the corresponding electronic energy and point group. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) types. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation of '''HF/3-21G''' method suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci and the inversion operation. This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, and its stereoelectronic effect outweighs steics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|G1
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|G2
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|G3
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|G4
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|G5
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|G6
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|A1
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|A2
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|A3
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|A4
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures are tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. On the basis of previous optimisation. Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of the structure need to be considered. Concluded from '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction hence the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to prove the successful geometry optimisation. Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised. (1 hartree = 627.509 kcal/mol)

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is computed, and the is listed in '''Table 3'''. All the calculations are set to 298.15 K by default. Geometry at 0 K can be achieved by the '''Temperature''' option, and vibrational temperature is tested. Obviously all types of energies share the same value at 0 K. This is the consequence of the suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
·[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of a sigma bond via a concerted transition state. Reaction process is thoroughly reversible unless a stabilising group is applied to form the lower-energy trap. There are two proposed conformations of TS --- "''chair''" or "''boat''" like, which are shown in Figure 2. Each of them will be analysed by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with the rough separation of 2.2 Å. Subsequently the model built up is experienced two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible propose of TS. With the combination job of '''Opt+Freq''' at '''HF/3-21G''', the structure is optimised to TS (Berny). In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. The success of calculation is highly dependent on the predicted geometry of TS. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to TS(Berny), which freezes geometry at the moment of bond break/formation and energetically relaxes at '''HF/3-21G'''. Unfreezing process is required afterwards.

Results from both methods should be able to converge to similar TS. The parameters are given below, which shows very light difference due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' conformation is optimised as a TS. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy and displacement. The negative k indicates a stationary point in nuclear configuration space and also it is a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. Its physical significance corresponds to the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vib. frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by QST2 method. The optimised Anti 2 (1,5-hexadiene) structure is used as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. TS is predicted by the maxima potential energy during linear interpolation between two molecules. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since it would not be taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. The modified molecule is optimised using '''Opt+Freq''' at HF/3-21G, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation has been successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conforamtion so as to tracing the mep from saddle point to local minima and to find the desired TS. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended since the precision of IRC is enhanced before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC figure above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interesting to see that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closer to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory such as '''B3LYP/6-31G(d)''' is suggested. Or most expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems only.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies of reactions via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy, and electronic + thermal energy are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to more powerful optimisation. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure, hence ''chair'' TS is slightly favoured over ''boat''. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is relatively small, hence insignificant energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are more approaching experimental value, while the results from '''HF/3-21G''' are significantly higher.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4+2] concerted, pericyclic pathway between a diene (4 carbons with 4 π electrons) and the conjugate dienophile (2 C + 2 πe-). There are overall 6 π-e- s participating in the reaction and two σ-bonds being formed. Diene is usually more electronrich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and this is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, and π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Cycloaddition is referred as [4s + 2s] with the equation output of 1. Other factors, such as the geometry of diene, has to be considered, since scis conformation of diene is necessary for the synchronised breaking/forming of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, hence reaction path will be studied. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules which are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming, although it gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7. Symmetry of MO is determined respect to the vertical mirror plane, which are also indicated in Table 7. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely due to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled. First, optimised butadiene and ethene molecules are modified to the proposed shape of cycloaddition TS using bicyclo-octane as a template. The reason is that optimal overlap of π system can be acquired in this manner. Product molecule (cyclohexene) is refined by '''HF/3-21G''' + '''B3LYP/6-31G(d)''' jobs. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects that TS energy has been minimised. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å). The typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. As predicted, little reorganisation of central p-lobe on diene is required in order to form new π-bond with neighboring p-lobe. However the terminal p-lobe forms σ-bond with that of dienophile, and reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobs of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This mode is favoured over that at lowest positive frequency (asynchronous), because the [4+2] TS ia able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2), hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!1
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the orbital symmetry of reagent before (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory, and HOMO-1 orbital actually plays the role of HOMO in Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interaction. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition, as secondary orbital interaction is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table. 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those in diene. Currently the stereoelectronic attraction on carbon outweighs the steric hindrance with sp2 C-H bonds (planar) on diene, and the forming σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds are directly opposite to sp3 C-H bonds (tetrahedral) with greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="4" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
!2
!3
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but not in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. By rotating the MO, it is clearly shown that e- density around C=O overlaps to some extent with the bulk of π-system in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction is very unlikely. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) should be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|1
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory during '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cyycloaddition goes through the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the penalty of two weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product . This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually a exo (thermodynamic) product may be generated. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised. It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene, such as [4s+2s] dimerisation, which will lead to undesired product. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is the unlikely situation. Unless strong driving force is applied, such as high concentration of butadiene, and heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involveds secondary overlap that could be stabilised by interacting with polar solvent, leading to more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
This study is base on the operation of ''Gaussian'' software. Molecules are mainly computed at '''HF/3-21G''' and '''B3LYP/6-31G(d) '''levels of theory, which provide slightly better optimisation than '''semi-empirical AM1'''. Besides '''opt'''imisation and '''freq'''uency jobs, '''TS(Berny)''' and '''QST2''' options are used to propose the TS of reaction. '''IRC''' is also introduced for simulating chemical path. Reactions of Cope rearrangement and Diels-Alder cycloaddition are investigated by energy optimisation, MO analysis and computation of transition structure. Specifically, ''chair ''TS of Cope reaction is preferred over ''boat'' TS. Two Diels-Alder reactions --- prototypical (butadiene and ethene) and reaction between maleic anhydrase and cyclobutadiene. The latter is for the purpose of regioselectivity study. Both reactions are [4s+2s] which is thermally allowed, also they behaves as normal demand. From regioselectivity, ''endo'' and ''exo'' TS geometries and relative energies are studied. ''Endo'' TS possess secondary orbital interaction therefore lower electronic energy, but ex''o'' TS does not. Methods of higher resolution should be tried if time is allowed.
{|
|-
|-
|-
|-
|-
|-
|-
|-
|-
|}

== Reference ==

User:Yqw13

2016-02-12T04:01:34Z

Yqw13: /* Regioselectivity of Diels-Alder reaction */

== Computation Introduction of Transition Structures of Pericyclic Reaction ==
Computation becomes a popular technique that makes effort to concentrate geometric energies and to solve problems of quantum chemical dynamics. With basis of theoretical chemistry and mathematics methods, computation contributes to many aspects of calculations including molecular geometry, molecular orbitals, transition state, etc. <ref name=":0">''Gaussian 03'' Revision, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., etc, Gaussian, Inc., Wallingford CT, 2004.
</ref> As a purpose of the study, transition structures (TS) of pericyclic reactions, specifically Cope rearrangement and Diels-Alder cycloaddition, will be simulated by ''Gaussian 09'' software in order for the further investigation of their properties such as total energy and selectivity. Essentially, the change in potential energy during a reaction is constructed as a function of the nuclear positions, named as potential energy surface (PES). In order to determine TS (saddle point) geometry of a reaction, 1st and 2nd derivatives are required to locate a stationary point where energy decreases in all direction (gradient = 0). Furthermore, minimum energy pathway (MEP) can be targeted among all possible trajectories on PES, where the most stable conformer of reagents and products, are connected with TS. It is worth mentioning that there may be more than one saddle points found in a complicated reaction, i.e. either global or local peak. <ref>P.Atkins, J.D.Paula, ''Atkins' Physical Chemistry , ''2010, '''22''', 852-856, Oxford Press</ref> In this specific case, it is assumed that only one TS is present during pericyclic pathway. However method of PES is no longer sufficient for computing complex structures, e.g. Diels-Alder TS. Additional applications are demanded including MO-based method, etc.

=== Molecular geometry '''opt'''imisation ===
This job is abbreviated as ''''opt'''<nowiki/>' with the purpose of geometry optimization. The input structure is adjusted in order to find the local minimal energy on the surface whose differentiation is zero. For example, Berny algorithm majorly acts on the recognition and optimizations of transition state via the '''Redundant''' option. This method is often applied when comparing energies of different conformers hence the stability. <ref name=":0" />

=== Vibrational '''Freq'''uency ===
This is another job type `exclusive for stationary points, naming ''''freq'''<nowiki/>'. Basically it approximates force constant by 2nd differentiation of E in terms of Cartesian-nuclear and mass-weighted coordinates, then vibrational frequencies and infrared intensities are further computed.The geometry is recommended to be optimised prior to frequencies calculation so that results will be more converged, in this case, a combination job '''Opt+Freq''' is conducted. Therefore frequency calculated also examines the performance of structure optimisation. It is noticeable that a successful minimisation of energy of a molecule is reflected by all positive vibrational frequencies, whereas for that of TS, one imaginary frequency will be present. <ref name=":0" />

=== Intrinsic Reaction Coordination (IRC) ===

This method is responsible for tracing in both directions from saddle point to minima in order to find the desired TS. Every points along the trajectory are optimised such that the gradients of the curve overlap with tangent to the path. TS geometry is adjusted to acquire a lower energy gradient hence lower pathway. Initial force constants is required to proceed IRC calculations, also isotopic input should be specified by labeling. <ref name=":0" />

== Cope Rearrangement ==

=== Optimisation of Reactants and Products ===
Different conformations of 1,5-hexadiene molecule were optimised at '''HF/3-21G''' level of theory which are listed in the table along with the corresponding electronic energy and point group. Proposed structures can be classified as 6 gauche and 4 anti-periplanar (anti) types. Commonly the energies of gauche conformers are predicted to be higher than those of anti, as they experience more steric effect. However the calculation of '''HF/3-21G''' method suggests that Gauche 3 possesses lower energy than Anti 2. Anti 2 is centro-symmetric with the point group of Ci and the inversion operation. This contradiction may be resulted from the overlap between π* orbital of C=C bonds and sigma bonds of vinyl protons in Gauche 3, and its stereoelectronic effect outweighs steics hindrance. <ref>Z. Zhou, B.W. Gung, R.A. Fouch, ''J.Am.Chem.Soc.'', 1995, '''117''', 1783-1788</ref>

{| class="wikitable"
! colspan="5" |Table 1. Conformations of 1,5-hexadiene after opt of HF/3-21G
|-
!Confomer
!Structure
!Energy HF/3-21G (hartree)
!Relative Energy (kcal mol-1)
!Point Group
|-
|''Gauche'' 1
|G1
|<nowiki>-231.687716</nowiki>
|3.1029
|C2
|-
|''Gauche'' 2
|G2
|<nowiki>-231.691667</nowiki>
|0.6237
|C2
|-
|''Gauche'' 3
|G3
|<nowiki>-231.692660</nowiki>
|0.0000
|C1
|-
|''Gauche'' 4
|G4
|<nowiki>-231.691532</nowiki>
|0.7094
|C2
|-
|''Gauche'' 5
|G5
|<nowiki>-231.689617</nowiki>
|1.9109
|C1
|-
|''Gauche'' 6
|G6
|<nowiki>-231.689160</nowiki>
|2.1967
|C1
|-
|''Anti'' 1
|A1
|<nowiki>-231.692603</nowiki>
|0.0368
|C2
|-
|''Anti'' 2
|A2
|<nowiki>-231.692535</nowiki>
|0.0789
|Ci
|-
|''Anti'' 3
|A3
|<nowiki>-231.689071</nowiki>
|2.2529
|C2h
|-
|''Anti'' 4
|A4
|<nowiki>-231.690971</nowiki>
|1.0607
|C1
|}
(1 hartree = 627.509 kcal/mol)

However the energy difference between two structures are tiny therefore further comparison is carried out after optimisation at '''B3LYP/6-31G(d)''' level of theory. On the basis of previous optimisation. Anti 2 geometry is readjusted by infinitesimal amount resulting in a slight decrease in electronic energy of ground state. Hence the molecule is further stabilised by a different method of calculation. Specific changes in bond lengths and dihedral angles between carbons are listed below. Since the molecule is centro-symmetric, only half of the structure need to be considered. Concluded from '''Table 2''', relief of dihedral angle (C1=C2-C3-C4) is the dominant factor that lowers steric interaction hence the energy, compared with minuscule changes of other parameter. As mentioned above, frequency are often computed to prove the successful geometry optimisation. Anti 2 is not TS therefore all frequencies present are positive and real, i.e. energy has been minimised. (1 hartree = 627.509 kcal/mol)

{| class="wikitable"
|-
! colspan="3" |[[File:Yqw13 anti2 631.PNG]]
|-
! colspan="3" |Figure 1. Labelled Anti 2 structure at 6-31G level of theory
|-
! colspan="3" |
|-
! colspan="3" |'''Table 2. The comparison between Anti 2 geometries opyimised at HF/3-21G and B3LYP/6-31(d) levels of theory'''
|-
!
!HF/3-21G
!B3LYP/6-31(d)
|-
|Electronic Energy (Hartrees)
|<nowiki>-231.69253</nowiki>
|<nowiki>-234.61171</nowiki>
|-
|Dihedral angle of C1=C2-C3-C4 (o)
|114.679
|118.586
|-
|Dihedral angle of C2-C3-C4-C5 (o)
|179.996
|180.000
|-
|C1=C2 bond length (Å)
|1.31616
|1.33350
|-
|C2-C3 bond length (Å)
|1.50887
|1.50420
|-
|C3-C4 bond length (Å)
|1.55292
|1.54816
|}
In addition, the relationship between thermal energy and temperature is computed, and the is listed in '''Table 3'''. All the calculations are set to 298.15 K by default. Geometry at 0 K can be achieved by the '''Temperature''' option, and vibrational temperature is tested. Obviously all types of energies share the same value at 0 K. This is the consequence of the suppressed molecular excitation such that only ground electronic state and zero-point vibrational energy are accessible at 0 K.
{| class="wikitable"
! colspan="3" |Table 3. Energy/Enthalpy terms of Anti 2 at temperatures of 0 K and 298 K
|-
!Energy/Enthalpy terms (Hartrees)
!0 K
!298 K
|-
|Electronic energy + Zero-point energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.469215</nowiki>
|-
|Electronic energy + Thermal energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.461866</nowiki>
|-
|Electronic enthalpy + Thermal enthalpy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.460922</nowiki>
|-
|Electronic free energy + Thermal free energy
|<nowiki>-234.484751</nowiki>
|<nowiki>-234.500800</nowiki>
|}

=== Study of ''Chair'' and ''Boat'' TS ===
·[[File:Yqw13 Cope.PNG|thumb|center|Figure 2. Cope rearrangement via ''chair'' or ''boat'' TS|400px]]

Cope rearrangement, also regarded as [3,3]-sigmatropic rearrangement, involves the migration of a sigma bond via a concerted transition state. Reaction process is thoroughly reversible unless a stabilising group is applied to form the lower-energy trap. There are two proposed conformations of TS --- "''chair''" or "''boat''" like, which are shown in Figure 2. Each of them will be analysed by different methods.

''Chair'' conformation (C2h) is set up by oppositely orienting two allylic fragments in parallel with the rough separation of 2.2 Å. Subsequently the model built up is experienced two distinct optimization processes. First method includes the computation of the force constant matrix with a sensible propose of TS. With the combination job of '''Opt+Freq''' at '''HF/3-21G''', the structure is optimised to TS (Berny). In case of the failure of TS optimisation, i.e. more than one negative frequency, calculation could be halted by the keyword '''Opt=NoEigen'''. The success of calculation is highly dependent on the predicted geometry of TS. Sometimes, optimisation could be reinforced by another methods yet at lower price, because it is not necessary to compute the entire force constant matrix. The second method involves the application of the keyword '''Opt=ModRedundant''' before optimising to TS(Berny), which freezes geometry at the moment of bond break/formation and energetically relaxes at '''HF/3-21G'''. Unfreezing process is required afterwards.

Results from both methods should be able to converge to similar TS. The parameters are given below, which shows very light difference due to different methods of calculation. Separation between terminals is consistently 2.02 Å which is smaller than input (2.20 Å). There is only one imaginary frequency present in vibration list so that ''chair'' conformation is optimised as a TS. The imaginary (negative) frequency is acquired from the negative force constant (k) due to the equation ω=√(k/μ). Force constant is calculated by the second derivative of potential energy and displacement. The negative k indicates a stationary point in nuclear configuration space and also it is a maxima. Comprehensively, TS locates at the maxima where geometry is going towards either reactant or product. Its physical significance corresponds to the stretching of bonds which are being formed or broken. Therefore a structure should be re-optimised if more than one negative frequencies are present. <ref>Robert N. Barnett, Uzi Landman, ''Phys. Rev. B'', 1993, '''48''', 2081–2097
</ref>
{| class="wikitable"
! colspan="3" |[[File:Yqw13 chair TS(2).gif]]
|-
! colspan="3" |Figure 3. Vibration of optimised ''chair'' TS
|-
! colspan="3" |
|-
! colspan="3" |Table 4. Properties of ''chair'' conformation obtained by different methods
|-
!
!1st Method
!2nd Method
|-
|Vib. frequency (cm-1)
|<nowiki>-817.87</nowiki>
|<nowiki>-817.90</nowiki>
|-
|Separation at terminals (Å)
|2.020
|2.020
|-
|Bond angle (o)
|120.51
|120.51
|}

.

''Boat'' TS conformation (C2v) is optimised by QST2 method. The optimised Anti 2 (1,5-hexadiene) structure is used as the shape of reactant and product, whose atom numbers are specified and matched as shown in Figure 3. TS is predicted by the maxima potential energy during linear interpolation between two molecules. Before processing the optimisation, Anti 2 structure needs to be shaped resembling the ''boat'' TS since it would not be taken into account by ''Gaussian'' software. More specifically, Anti 2 is manually bent by setting dihedral angle (C2-C3-C4-C5) to 0o, and internal angles (e.g. C2-C3-C4) to 100o. The modified molecule is optimised using '''Opt+Freq''' at HF/3-21G, then the interpolation is carried out by''' TS(QST2)''' option. Single negative frequency (-839.83 cm-1) suggests that ''boat'' TS optimisation has been successful.
{| class="wikitable"
! colspan="3" |Figure 4. Labelled reagent & product, and vibration of optimised ''boat'' TS
|-
|Reagent (Anti 2)
|Product (Anti 2)
|''Boat'' TS
|-
|[[File:Yqw13 anti2 reagent.png |250px]]
|[[File:Yq13 anti2 product.png |250px]]
|[[File:Yqw13 boat TS(2).gif]]
|}

This section applies Intrinsic Reaction Coordination (IRC) to the ''chair'' conforamtion so as to tracing the mep from saddle point to local minima and to find the desired TS. Total energy of 50 geometry steps are computed along the trajectory to determine the steepest path on PES. A sufficiently large number of steps is recommended since the precision of IRC is enhanced before the convergence of total energy. Symmetrical reaction coordinate allows IRC computing in a single direction (forward in this case).

.
{| class="wikitable"
! colspan="2" |Figure 5. IRC trace and Energy-coordination graph of chair TS
|-
|[[File:Yqw13 chair IRC(2).gif]]
|[[File:Yqw13 chair IRC energygraph.png |550px]]
|}

From the IRC figure above, the geometry at TS resembles Gauche 2 more than Gauche 3 (Table 1) although the latter possesses lowest energy. It is interesting to see that the trajectory does not necessarily pass through the conformation with lowest energy. According to Hammond's Postulate, smaller activation energy is achieved by resembling structures between reagent and TS, therefore the geometry whose energy is closer to TS is most likely to react. However '''HF/3-21G '''level of theory is insufficient for minimisation that gives the energy of -231.6152 Hartree. Performance of IRC at higher level of theory such as '''B3LYP/6-31G(d)''' is suggested. Or most expensively, IRC can be computed with force constant matrix at each single step, which provides the most reliable outcomes for small and medium systems only.
{| class="wikitable"
! colspan="7" |Table 5. Energy summary of ''chair'' and ''boat'' TS
|-
!Compound
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G(d)'''
|-
!
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
!Electronic energy
(Hartree)
!Electronic energy
+ Zero-point energy

at 0 K (Hartree)
!Electronic energy
+ Thermal energy

at 298 K (Hartree)
|-
|Chair TS
|<nowiki>-231.619322</nowiki>
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.461341</nowiki>
|<nowiki>-234.556980</nowiki>
|<nowiki>-234.414927</nowiki>
|<nowiki>-234.409007</nowiki>
|-
|Boat TS
|<nowiki>-231.602802</nowiki>
|<nowiki>-231.450930</nowiki>
|<nowiki>-231.445301</nowiki>
|<nowiki>-234.402340</nowiki>
|<nowiki>-234.402341</nowiki>
|<nowiki>-234.396006</nowiki>
|-
|Reactant (Anti 2)
|<nowiki>-231.692536</nowiki>
|<nowiki>-231.539542</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-234.611711</nowiki>
|<nowiki>-234.469209</nowiki>
|<nowiki>-234.461869</nowiki>
|}

Activation energies of reactions via both ''chair'' and ''boat'' conformations are present below. Electronic energy, electronic + zero-point energy, and electronic + thermal energy are compared at different levels of theory. There are two apparent facts. Firstly, energies minimised by '''B3LYP/6-31G(d)''' is significantly lower than that by '''HF/3-21G''' due to more powerful optimisation. Secondly, energies of ''chair'' conformation are marginally lower than that of ''boat'' structure. Comparing two geometries, it is evident that hydrogens at terminals of ''chair'' is staggered, but eclipsed at terminals of ''boat''. Since hydrogens are not on neighboring carbons, steric contribution is relatively small, hence insignificant energy difference between two geometries. Contribution of temperature is also studied by computing TS at 0 K and 298 K, respectively. Activation energies computed at '''B3LYP/6-31G(d)''' level of theory are more approaching experimental value, while the results from '''HF/3-21G''' are significantly higher.
{| class="wikitable"
! colspan="6" |Table 6. Activation energy of TS at different temperatures (kcal mol-1)
|-
!
! colspan="2" |'''HF/3-21G'''
! colspan="2" |'''B3LYP/6-31G(d)'''
!Expt.
|-
!
!0 K
!298 K
!0K
!298 K
!0K
|-
|Eact (''chair'')
|45.7083
|44.6963
|34.0687
|33.1845
|33.5 ± 0.5
|-
|Eact (''boat'')
|55.6040
|54.7606
|41.9549
|41.3238
|44.7 ± 2.0
|}
.

== Diels-Alder Cycloaddition ==
Among all the cycloaddition reactions, Diels-Alder is characterised by [4+2] concerted, pericyclic pathway between a diene (4 carbons with 4 π electrons) and the conjugate dienophile (2 C + 2 πe-). There are overall 6 π-e- s participating in the reaction and two σ-bonds being formed. Diene is usually more electronrich than dienophile hence electrons are transferred from HOMO of diene to LUMO of dienophile. This is regarded as the normal demand. But electron density can be reverted by attaching substituents, and this is called inverse demand. Fragmental orbitals (FO) from two reagents overlap 'end on' to generate σ bonds. Same symmetry and small energy difference of FO ensure significant overlapping effect. For example, Ψ2 FO (diene) ↔ π* (dienophile) is present at normal demand, and π (dienophile) ↔ Ψ3 FO (diene) at inverse demand. Moreover Woodward–Hoffmann rules are closely associated with the accessibility of Diels-Alder reaction. Based to the rule, a thermally allowed pericyclic reaction must possess an odd number from equation (4r+2)s + (4q)a, where <nowiki>''s''</nowiki> stands for superfacial orbital interaction and <nowiki>''a''</nowiki> for antarafacial. Cycloaddition is referred as [4s + 2s] with the equation output of 1. Other factors, such as the geometry of diene, has to be considered, since scis conformation of diene is necessary for the synchronised breaking/forming of the bonds.<ref>Carey, Francis A., Richard J., ''Adv. Org. Chem. 5th Ed, ''New York'', ''2007, 836–850 ISBN 0387448993</ref>

In this section, TS of two Diels-Alder cycloadditions will be computed and characterised using '''TS(QST2)'' '''''method at different levels of theory, hence reaction path will be studied. Geometries are optimised via '''HF/3-21G''', '''B3LYP/6-31G(d)''', and '''semi-empirical AM1''' methods.

=== Cycloaddition of ''cis''-butadiene and ethene ===
[[File:Diels alder mech.PNG|thumb|center|Figure 6. Diels-Alder reaction between cis-butadiene and ethene|400px]]Reagents are set up by building cis-butadiene and ethene molecules which are optimised via '''semi-empirical AM1'''. Alternatively one can minimise energy using '''HF/3-21G''' + '''B3LYP/6-31G(d)''' which is highly time-consuming, although it gives slightly better results. The frontier orbitals of each reagent, i.e. HOMO and LUMO, are illustrated in Table 7. Symmetry of MO is determined respect to the vertical mirror plane, which are also indicated in Table 7. As mentioned above, orbital overlap is magnified by the same symmetry. Therefore HOMO of butadiene (AS) with LUMO of ethene (AS), LUMO of butadiene (S) with HOMO of ethene (S) are two plausible interactions, although the latter is less likely due to more significant energy difference. Eventually only the interaction of two AS orbitals is accessible that obeys the mode of normal demand.
{| class="wikitable"
! colspan="3" |Table 7. HOMO and LUMO of reagents in prototypical cycloaddition
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Yqw13 butadiene HOMO.png|250px]]
|[[File:Yqw13 butadiene LUMO.png|250px]]
|-
|
|'''Anti-symmetric (AS)'''
|Symmetric (S)
|-
|Ethene
|[[File:Yqw13 ethene HOMO.png|250px]]
|[[File:Yqw13 ethene LUMO.png|250px]]
|-
|
|Symmetric (S)
|'''Anti-symmetric (AS)'''
|}
Subsequently, two optimised reagents are utilised to compute cycloaddition TS using '''TS(QST2)'' '''''method, where reagents and product should be built and labelled. First, optimised butadiene and ethene molecules are modified to the proposed shape of cycloaddition TS using bicyclo-octane as a template. The reason is that optimal overlap of π system can be acquired in this manner. Product molecule (cyclohexene) is refined by '''HF/3-21G''' + '''B3LYP/6-31G(d)''' jobs. The following label process is very essential for the success of TS computing.

Table 8 below illustrates the structure, reaction path, and two vibration modes of prototypic TS. Single imaginary frequency reflects that TS energy has been minimised. All carbon-carbon bonds in TS (except two σ-bonds forming) are in the state between single (sp3) and double (sp2) bond, which is also suggested by Figure 6 and Table 8. The TS static plot shows the tiny difference between bond lengths (~1.38 Å). The typical bond length for sp3 carbon is 1.54 Å, and that of sp2 carbon is 1.33 Å.<ref>Marye A., James K. ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen ''Springer, 1995

ISBN 978-3-86025-249-9.
</ref> Concluded from the comparison, sp2 C=C bond extends to a small amount, whereas the contraction of sp3 C-C bond is significant. As predicted, little reorganisation of central p-lobe on diene is required in order to form new π-bond with neighboring p-lobe. However the terminal p-lobe forms σ-bond with that of dienophile, and reorientation is required in this case. As a result, the proposed stage of TS might resemble sp2 more than sp3 due to the ease of C=C bond formation. Two partly formed σ-bonds (2.22 Å) are much longer than C-C single bond, but shorter than twice of Van der Waal's radius of carbon (1.70 Å),<ref>Bondi A., ''J. Phys. Chem.'', 1964, '''68''',''' '''441-451

doi:10.1021/j100785a001
</ref>. It proves the existence of orbital overlap between 2 carbons.

Superfacial (s) interaction exists between p-lobs of two reagents which gives rise to (6 πe-)s and (0 πe-)a. According to Woodward–Hoffmann rules, the output of equation (4r+2)s + (4q)a is 1 (odd), hence this reaction is thermally allowed. TS vibration at imaginary frequency demonstrates the synchronous formation of σ-bonds that is supported by IRC plot. This mode is favoured over that at lowest positive frequency (asynchronous), because the [4+2] TS ia able to rearrange towards a flat 6-membered ring. 6πe- obeys Hückle's Law (4n+2), hence the ring acquires aromatic character which stabilises TS and lower the energy. Furthermore, vibration at imaginary frequency involved the motion that benefits bond formation, whereas the orthogonal motion at real frequency does not. Thus IRC reaction path only includes the mode at former frequency rather than the latter.

{| class="wikitable"
! colspan="4" |Table 8. TS of prototypical cycloaddition via QST2 6-31G(d)
|-
!1
![[File:Prototype TS IRC.gif]]
![[File:Yqw13 prototype TS negfreq.gif]]
![[File:Yqw13 prototype TS realfreq.gif]]
|-
!TS static plot with bond length displayed
!IRC of TS
!TS vibration at imaginary frequency (-818.91 cm-1)
!TS vibration at lowest positive frequency (166.92 cm-1)
|}
Interestingly, HOMO of TS is symmetric respect to vertical plane, which contradicts to the orbital symmetry of reagent before (both AS). However the reason could be an issue of relative orbital energies. Noticed that energy difference between HOMO and HOMO-1 (~0.002) are five-time smaller than that between other HOMOs or LUMOs (~0.01). The HOMO plotted is likely to be a flaw at '''B3LYP/6-31G(d)''' level of theory, and HOMO-1 orbital actually plays the role of HOMO in Diels-Alder cycloaddition.
{| class="wikitable"
! colspan="4" |Table 9. Frontier orbitals of cycloaddition TS 6-31G(d)
|-
!
![[File:Yqw13 prototype TS LUMO.png|250px]]
![[File:Yqw13 prototype TS HOMO.png|250px]]
![[File:Yqw13 prototype TS HOMO-1.png|250px]]
|-
!Orbital
!'''LUMO'''
!'''HOMO'''
!'''HOMO-1'''
|-
!Symmetry
!S
!S
!AS
|-
!Relative energy
!0.14242
!-0.30086
!-0.30293
|}
.

=== Regioselectivity of Diels-Alder reaction ===
[[File: Yqw13 endo&exo.PNG|thumb|center|Figure 7. Diels-Alder endo and exo products|550px]]
Regioselectivity should be taken into account when the substituents attached on diene and dienophile exert additional orbital interaction. For example, 1,3-cyclohexadiene and maleic anhydrase are typical reagents for studying ''exo'' and ''endo'' selectivity of cycloaddition, as secondary orbital interaction is accessible. Resembling the method of protoypical reaction, the energies of 1,3-cyclohexadiene and maleic anhydrase are minimised using '''HF/3-21G''' and '''B3LYP/6-31G(d)''', respectively. Frontier orbitals of each reagent are illustrated in Table 10. Same symmetry and small energy difference are necessary for optimal orbitals interaction, however two possible outcomes arise from attached substituents --- ''endo'' & ''exo'' TS (Figure 7). Furthermore endo regioselectivity of a reaction displays the inverse demand, i.e. interaction between LUMO of the diene and HOMO of the dienophile.<ref name=":1">Ian Fleming, ''Molecular Orbitals and Organic Chemical Reactions,'' 2010, '''6''', 295-316</ref> Both TS will be computed by '''QST2''' at two levels of theory, and their relative electronic energies are analysed.

{| class="wikitable"
! colspan="3" |Table 10. HOMO and LUMO of reagents in regioselective cycloaddition
|-
!
!HOMO
!LUMO
|-
|1,3-Cyclohexadiene
|[[File:Yqw13 cyclohexdiene HOMO.png|250px]]
|[[File:Yqw13 cyclohexdiene LUMO.png|250px]]
|-
|
|'''AS'''
|S (diene part)
|-
|Maleic anhydrase
|[[File:Yqw13 maleic anhydr LUMO.png|250px]]
|[[File:Yqw13 maleic anhydr HOMO.png|250px]]
|-
|
|S
|'''AS'''
|}
The situation of bond stretch and contraction is similar to those of prototypical reaction. Formation of 2 σ-bonds are synchronised in both ''endo'' and ''exo'' reaciton via a 6-membered ring, so the the bond lengths should be symmetric during the formation. Specific distances between atoms are displayed in Table. 11. In this case fragment -(O=C)-O-(C=O)- is focused on as it carries p-lobes orthogonal to the plane of anhydrase. Secondary orbital overlap exists in ''endo'' TS, as illustrated in Figure 7, where the p-lobes on C=O is aligned with those in diene. Currently the stereoelectronic attraction on carbon outweighs the steric hindrance with sp2 C-H bonds (planar) on diene, and the forming σ-bonds are shorter compared with the value in prototypical reaction. In contrast, ''exo'' TS interaction is dominated by steric repulsion. Not only the absence of secondary interaction, C=O bonds are directly opposite to sp3 C-H bonds (tetrahedral) with greater steric effect and strain.<ref name=":1" /> As a consequence, both carbon separation and the partly formed σ-bonds are averagely larger than those in ''endo'' TS. Furthermore, total electronic energy of ''exo'' TS is higher than that of ''endo'' which will be mentioned below (Table 13).
{| class="wikitable"
! colspan="4" |Table 11. Structures and vibrations of ''endo'' and ''exo'' TS
|-
![[File:Yqw13 regioselec endo TS 631G.gif]]
!2
!3
![[File:Yqw13 regioselec exo TS 631G.gif]]
|-
!''Endo'' vibration at -449.14 cm-1
!''Endo'' TS and bond lengths
!''Exo'' TS and bond lengths
!''Exo'' vibration at -450.82 cm-1
|}
HOMO of two TS are illustrated here to explain the lower energy of ''endo'' TS caused by secondary orbital interaction, but not in ''exo''. Because total electronic energy of a molecule is solely related to occupied MOs. Both HOMOs have great electronic density between two reagents and small density around each C=O bond. By rotating the MO, it is clearly shown that e- density around C=O overlaps to some extent with the bulk of π-system in ''endo'' TS, while these two regions become staggered in ''exo'' TS. There are also tiny orbitals around sp3 C-H bonds but interaction is very unlikely. In conclusion, secondary overlap is referred to the non-direct interaction between orbitals with the same symmetry. It is not necessary to form a bond, but a net decrease of energy (stabilisation) should be observed. <ref name=":1" />
{| class="wikitable"
! colspan="3" |Table 12. Frontier orbitals of two TS
|-
!
!''endo'' TS
!''exo'' TS
|-
|HOMO
|1
|2
|}
Energies of TS are computed at '''HF/3-21G''' and '''B3LYP/6-31G(d)''' levels of theory during '''QST2''' jobs, which are summarised in Table 13. Compared with energies of Cope rearrangement, cyycloaddition goes through the exothermic pathway with deeper energy, since two strong σ-bonds are formed with the penalty of two weak π bonds. The data output is relatively consistent with the energy profile in Figure 8 --- Eendo TS < Eexo TS ; Eexo product < Eendo product . This is resulted from the compromise between stereoelectronic and steric effect by TS and product. So far ''endo'' is defined as the kinetic product which is rapidly formed due to lower energy barrier. However small Eact also facilitates the reversibility of the reaction and eventually a exo (thermodynamic) product may be generated. In term of calculation methods, '''B3LYP/6-31G(d)''' usually provides the further-minimised energy than '''HF/3-21G''' because a larger basis set is utilised. It is worth mentioning that the output of Eendo product is lower than Eexo product at '''HF/3-21G''' level of theory, yet reverse is obtained by '''B3LYP/6-31G(d)'''. Therefore higher resolution is recommended for optimising complex molecules with small energy difference.
[[File:Thermodynamic kinetic product diag.PNG|thumb|right|Figure 8. Energy profile of Thermodynamic & Kinetic product|500px]]
{| class="wikitable"
! colspan="3" |Table 13. Summary of reaction energy via two TS
|-
!Energy term (Hartree)
!HF/3-21G
!B3LYP/6-31G(d)
|-
|E1,3-cyclohexadiene + Emaleic anhydride
|<nowiki>-605.646744</nowiki>
|<nowiki>-612.723684</nowiki>
|-
|Eendo TS
|<nowiki>-605.610368</nowiki>
|<nowiki>-612.698361</nowiki>
|-
|Eendo product
|<nowiki>-605.721321</nowiki>
|<nowiki>-612.772769</nowiki>
|-
|Eactivation of ''endo'' reaction
|0.036376
(22.826267 kcal mol-1)
|0.025323
(15.890410 kcal mol-1)
|-
|Eexo TS
|<nowiki>-605.603591</nowiki>
|<nowiki>-612.694415</nowiki>
|-
|Eexo product
|<nowiki>-605.718735</nowiki>
|<nowiki>-612.845228</nowiki>
|-
|Eactivation of ''exo'' reaction
|0.043153
(27.078896 kcal mol-1)
|0.0292695
(18.366875 kcal mol-1)
|}

=== Further Discussion ===
One major effect is the side reactions of diene, such as [4s+2s] dimerisation, which will lead to undesired product. Figure 9 shows the possible dimerisation TS of 1,4-butadiene. Corresponding Eact are calculated in the same manner as above. The result indicates that Eact of dimerisation is much higher than that of prototypical cycloaddition, hence the side reaction is the unlikely situation. Unless strong driving force is applied, such as high concentration of butadiene, and heating. Solvent is another factor to consider that contributes to ''endo'':''exo'' ratios. For example, ''endo'' product will be significantly favoured under polar environment. According to the kinetics, ''endo'' TS involveds secondary overlap that could be stabilised by interacting with polar solvent, leading to more achievable kinetic product. <ref name=":1" />
{| class="wikitable"
|[[File:Dimerisation TS QST2.gif]]
|-
|Figure 9. TS vibration of undesired dimerisation reaction
|}
.

== Conclusion ==
{|
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|-
|-
|-
|-
|-
|-
|}

== Reference ==