On a free energy diagram, the transition state (TS) corresponds to the point with maximum free energy along the reaction coordinate, hence, it's the least stable state compared with the reactant and product. A chemical reaction can occur via different reaction pathways, where the mechanisms behind each pathway can be elucidated by study on TS. In a 2D free energy diagram, the TS accomodates at the local maximum with zero gradient and negative second derivative (Fig 1 [1]). However, in real situations, the reaction system can deviate away from the equilibrium position and displace in 3N-6 degrees of freedom, with each giving rise to a normal mode of vibration. A 3D potential energy surface can be plotted with energy against the total degrees of freedom along two reaction coordinates (Fig 2 [2]). The TS coincides with the local maximum and local minimum along two coordinates respectively, but only the local maximum gives a reasonable approximation for the real TS and the underlying mechanism.
Method
To yield an accurate determination of TS for a reaction system, a computer is essential to assist in doing a large amount of complex calculations. In this experiment, the TS of three different pericyclic reactions were located with Guassian. An optimisation process was carried out in each step to make sure the expected structure was obtained. The optimisation process consisted of following steps: firstly, the initial nuclear position was fixed while manually building up the molecular structure in Gaussian. A self consistent field calculation was then carried out by solving the Schrödinger equation for the fixed nuclear positions to obtain an energy E and a wavefunction Φ under Born-Oppenheimer approximation. The energy of the molecular structure varied with the changing nuclear position along the potential energy surface (EPS). As I mentioned before, the optimised TS structure should be found at the local maximum where the first derivative of PES is zero. Two different optimisation methods (i.e. the semi-empirical method PM6 and Density Functional Theory (DFT) method B3LYP with basis set 6-31G (d)) were adopted in this experiment. The particular type of Hamiltonian employed depends on the method used and the basis-set determines the number of functions included in the approximation of the electronic structure. In this experiment, possible geometries of transition states were firstly generated with PM6 method, then further optimized to B3LYP level. By checking the vibrational frequency for the optimised struture in Gaussian, the correct transition state can be confirmed. As the bond formation/ breaking process for a harmonic oscillator was investigated, a 'negative' stretching frequency for the first normal mode of the transition state should be expected. In principle, a TS should contain only one imaginary (negative) frequency. If there are more than one observed, that is due to higher order saddle point. On the other hand, no imaginary frequency should be observed for reactants and products as no bond formation/ breaking process was involved. However, one imaginary frequency at optimised TS structure is still insufficient to illustrate the reaction pathway and the corresponding PES, further Intrinsic reaction coordinate (IRC) calculation is required.
Exercise 1:Reaction of Butadiene with Ethylene
The [4+2] cycloaddition reaction between butadiene and ethylene is a typical type of Diels Alder reaction which normally involves a diene and a dienophile. Diels Alder reaction is commonly used as an efficient method for the synthesis of six membered ring with good control over regio and stereo selectivity. To gain a better understanding on the mechanism behind this reaction, it's important to investigate the TS formed during reaction. In the reaction scheme shown below, a normal demand (i.e. electron donation from HOMO of diene to LUMO of dienophile) Diels Alder reaction is studied for reaction between 1,3-butadiene (diene) with ethylene (dienophile) to form cyclohexene.
Optimisation of TS and results
The structure of product was built and optimised to PM6 level using Gaussian. The newly formed bond during Diels Alder reaction was broken manually and the C-C bond length was frozen at 2.20 Å, which was between the sum of Van der Waal's radii of two carbon atoms and a single C-C bond length (1.54 Å). This structure was then further optimised to the TS structure at PM6 level. The optimised structures for reactants, product and TS are shown below.
Butadiene
Ethlyene
Transition State
Cyclohexene
The transition state was determined and confirmed with a frequency calculation and IRC analysis. There is only one imaginary frequency observed in the vibrational normal modes which indicates the correct TS structure was found. The IRC graph displayed the energy change during bond forming process, where the minimums and maximum corresponded to reactant, product and transition states respectively.
Molecular Orbital (MO) Analysis
Butadiene
Ethylene
Transition state
Transition state
HOMO
HOMO of Butadiene
HOMO of Ethylene
HOMO of the TS
HOMO-1 of the TS
LUMO
LUMO of butadiene
LUMO of Ethylene
LUMO of the TS
LUMO+1 of the TS
In the bond formation process, the HOMO (highest occupied MO) and LUMO (lowest unoccupied MO) of the reactants have greatest contributions, hence, these MOs and the newly formed TS MOs are shown in the table above.
A MO diagram for the formation of TS can be constructed accordingly based on the energy and symmetry of the fragment orbitals of the reactants to investigate the principle behind MO mixing and interactions. In the diagram above, the symmetry of each orbital was indicated with 'S' and 'AS', which stand for symmetrical and anti-symmetrical respectively. Symmetrical orbital has a sigma mirror plane lies between the central sigma bond. It's commonly observed that the AS HOMO of butadiene is interacting with the AS LUMO of ethylene and S LUMO of butadiene is interacting with the S HOMO of ethylene to form new MOs, and no interactions are observed between S and AS orbitals, therefore, it's reasonable to conclude that only MO with same symmetry will mix and interact with each other to form new orbitals. From quantum mechanical view, the extent of orbital mixing depends on the overlap integral of wavefunctions between two fragment orbitals. In this case, orbitals with same symmetry (AS-AS or S-S) give rise to non-zero integral, otherwise (AS-S), zero integral is observed. A reaction can only occur when the integral is non-zero. Butadiene has a smaller HOMO-LUMO energy gap compared with ethylene because it's more electron rich. The calculated energy level for TS MO is slightly different from the expected value, this is due to the highly delocalised electron density in TS leads to the mixing between HOMO and LUMO to some extent. As a result, the HOMO is raised up and LUMO is lowered down.
Bond Length Analysis
Molecules
Bond lenghts
Molecules
Bond lengths
Butadiene
C3=C2 1.33 Å
C2-C1 1.47 Å
C1=C7 1.33 Å
Ethylene
C1=C4 1.33 Å
TS
C4=C5 1.35 Å
C1=C6 1.35 Å
C1=C2 1.44 Å
C2=C3 1.35 Å
C5 C6 2.2 Å
C3 C4 2.25 Å
Cyclohexene
C1=C2 1.33 Å
C1-C6 1.49 Å
C6-C5 1.54 Å
C4-C5 1.54 Å
C3-C4 1.54 Å
C2-C3 1.49 Å
The bond lengths of reactants, products and TS are shown in the above table. Typical sp3 and sp2 C-C bond lengths are shown below:
sp2-sp2 C=C bond length 1.34 Å
sp3-sp3 C-C bond length 1.54 Å
sp2-sp2 C-C bond length 1.47 Å
sp2-sp3 C-C bond length 1.50 Å
The C-C and C=C bond lengths observed in butadiene and ethylene are the same as the typical values.
During the Diels Alder reaction, the C=C bond length in ethylene elongates from 1.33 Å to 1.35 Å at TS. This suggests the weakening of C=C double bond and decrease in bond order towards C-C single bond. The C=C bond length in butadiene increases from 1.33 Å to 1.35 Å at TS and the C-C bond shortens from 1.47 Å to 1.44 Å at TS. The distances between the terminal carbons involved in bond formation at TS are 2.20 Å and 2.25 Å respectively. The Van der Waals radius of a C atom is 1.7 Å, but the measured separation between the carbon atoms are shorter than twice of the Van der Waals radii (3.4 Å) and longer than a C-C single bond, so the C-C single bond is partially formed. The bond lengths are then further decreased to 1.54 Å which is the same as sp3-sp3 C-C bond length. Based on the change in bond length, it's reasonable to conclude that the new C-C bonds are formed in the product. The other bond lengths in product are similar to typical values. To maintain these favourable bond lengths, the product adopts a twisted structure.
Vibrational Analysis
The animation of the first vibrational normal mode for the TS is shown below, the two terminal carbon atoms in ethylene and two carbon atoms in butadiene approach each other simultaneously. Hence, the bond forming step is synchronous.
Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole
Another example of Diels Alder reaction is shown below, which introduces further complexity on endo/exo selectivity. For this example, activation barrier and reaction energies are calculated to investigate which reaction pathway is more favoured.
Optimisation of substrates and frequency calculation
The structures of reactant, TS and product are optimised to PM6 level first, then to B3LYP/6-31G level. Frequency calculations were carried out to confirm each structure, the results were shown in the table below.
Cyclohexdiene
Frequency calculation of cyclohexadiene
1,3-dioxole
Frequency calculation of 1,3-dioxole
Endo TS
Frequency calculation of exo TS
Exo TS
Frequency calculation of endo TS
Endo product
Frequency calculation of endo product
Exo product
Frequency calculation of exo product
Only one imaginary frequency was observed for each TS, and no imaginary frequency for reactant and product. This is consistent with the previous conclusion.
MO analysis
EXO TS MO
HOMO-1 AS
HOMO S
LUMO S
LUMO+1 AS
ENDO TS MO
HOMO-1 AS
HOMO S
LUMO S
LUMO+1 AS
Based on the MO analysis for the TS using Gaussian, MOs from reactants are paired up and mixed accordingly, the resultant MOs are aligned with respect to energy.
This particular Diels-Alder reaction involves a cyclohexadiene(conjugated diene) and a 1,3-dioxole (dienophile). Reactivity is controlled by relative energies of frontier molecular orbitals. The key mechanism is electron donation from HOMO of one reactant to LUMO of the other. The closer the energy of the two interacting orbitals (e.g. HOMO and LUMO), the faster the reaction rate. Consequently, two types of Diels-Alder reaction are readily available to occur, which are classified as normal demand and inverse demand reactions. In normal demand DA reaction, the HOMO of a electron rich diene will interact with the LUMO of electron poor dienophile. In inverse demand DA reaction, the LUMO of electron poor diene will interact with the HOMO of electron rich dienophile.
Dioxole is more electron rich in this case because of the presence of two oxygen atoms in the ring, lone pairs on oxygen can form conjugation with the C=C double bond. Consequently, electrons are donated from HOMO of 1,3-dioxole to LUMO of cyclohexadiene which are closer in energy compared with the other pathway, so both exo and endo reactions are inverse demand DA reactions. This is proved quantitatively by doing a single point energy calculation for the reactants and product. The difference in MO energies between the exo and endo TS is due to the additional stabilisation effect on TS caused by p-orbital interactions between the oxygen atoms in 1,3-dioxole and the C=C double bond in cyclobutadiene while endo trajectory is adopted. Consequently, the endo reaction is more kinetically favourable.
Activation barrier and reaction energy
Substrates
Gibbs free energy at B3LYP (6-31G(d)) / kJmol-1
Cyclohexadiene
-612626
1,3-Dioxole
-701205
Sum of reactant energy
-1313831
Exo-TS
-1313662
Exo-Product
-1313895
Endo-TS
-1313670
Endo-Product
-1313900
B3LYP (6-31G(d)) / kJmol-1
Exo
Endo
Activation barrier (Eact) /kJmol-1
+ 169
+ 161
Reaction energy /kJmol-1
- 64
- 69
After extracting the free energies of the reactants, TS and products from Gaussian, the activation barrier can be calculated by subtracting the sum of free energy of reactants from the energy of transition state. And the reaction energy can be determined by calculating the energy difference between reactants and products. As is shown in the table above, the endo pathway has a lower activation barrier, therefore, higher reaction rate is expected and the endo product is kinetically favoured. More negative reaction energy for the endo reaction also suggests it's thermodynamically favourable. This can be interpreted using the MO diagram shown below.
EXO TS HOMO
Endo TS HOMO
In the endo transition state, the non-bonding p orbital on oxygen atoms in 1,3-dioxole has the correct orientation and symmetry to interact with the LUMO on cyclobutdiene and thereby stabilise the whole structure. This interaction is not available in the exo TS because of zero spatial overlap between the p orbitals on oxygen and cyclobutdiene. This stabilisation effect becomes less dominant in product, steric hindrance becomes significant instead. The endo product accommodates the bulky bridgehead and the dioxole ring on the opposite face, as a result, it experiences less repulsion than the exo product and thermodynamically more stable.
Exercise 3: Diels-Alder vs Cheletropic
In the o-xylylene-SO2 reaction shown below, an electron rich SO2 is used as dienophile. The abundant lone pairs on S atom induces a competitive side reaction (i.e. cheletropic reaction) to the normal Diels Alder reaction. To determine the most preferred reaction pathway, the TS for each possible reaction were investigated.
According to Huckle's theory, for a ring containing 4n π-electrons, it's anti-aromatic and very unstable; for a ring containing 4n+2 π-electrons, it's aromatic and energetically favourable. By inspection on the structure of each substrate, it's obvious that the cycloaddition reaction should be thermodynamically favourable in this case because xylylene is very unstable. It can have either 4 π-electrons in the ring (anti-aromatic) or 6 π-electrons in the ring (aromatic) but exist as a diradical which is extremely reactive. After undergoing cycloaddition reaction, a stable aromatic ring with an additional six-membered ring (five-membered-hetero-ring for cheletropic reaction) will form as the product. The tendency to form a stable structure makes this reaction energentically favourable. Besides that, the rigid cis-alkene in xylylene also makes it quite reactive towards Diels Alder reaction as no rearrangement is required to achieve the correct geometry.
IRC Analysis
Approaching trajectory of endo product
Approaching trajectory of exo product
Approaching trajectory of cheletropic product
IRC for endo product
IRC for exo product
IRC for cheletropic product
The approaching trajectory and IRC results further confirm the prediction I stated before. There is a good orbital overlap between the diene and dienophile, the formation of product from reactants does not involve a great extent of rearrangement in geometry, so it's kinetically favoured.
Reaction barrier and reaction energy
Same method as described in exercise 2 was adopted to calculate the activation barrier and reaction energy for exercise 3. The results are shown below.
Molecules
Gibbs free energy at PM6 / kJmol-1
SO2
-311.421
Xylyene
467.649
Sum of reactants energy
156.228
Exo DA TS
241.748
Endo DA TS
237.763
Cheletropic TS
260.087
Exo DA product
56.362
Endo DA product
56.971
Cheletropic product
-0.0105
PM6 / kJmol-1
Exo DA
Endo DA
Cheletropic
Activation energy /kJmol-1
+ 85.51
+ 81.54
+ 103.86
Reaction energy /kJmol-1
- 99.87
- 99.26
- 156.24
Reaction Profile
As is shown above, a reaction profile is constructed based on the activation barrier and reaction energy calculated in previous section. 0 energy was set for the reactants at infinite separation. The cheletropic reaction has the most negative reaction energy, so it's the most thermodynamically favourable pathway. The endo-DA reaction has the lowest activation barrier, hence, it's kinetically favourable. The endo and exo-DA reactions have similar activation barriers and reaction energies because neither the p-orbital conjugation nor the steric hindrance is significant in this case. The cheletropic product is the most stable as the large size of sulphur atom alleviates the 5-membered ring strain to some extent. All three reaction pathways are exothermic, according to Hammond's postulate, these are early barrier reactions, the structures of TS simulate the structure of reactants. The C-C and C=C bond lengths in TS are closer to those in the reactant rather than the product. This is proven by the optimisation results calculated using Gaussian. The results are shown below.
Optimisation results
ENDO TS
EXO TS
Cheletropic TS
possible side reactions
In xylylene, another cis-diene in the six-membered ring provides another possible reaction pathway for the Diels Alder reaction. The reaction scheme is shown below.
By inspection on the structure, both endo and exo reaction pathways are disfavourable. Both products have no aromaticity, and the bend bridge structure break down the correct geometry for conjugation between planar C=C double bonds observed in xylylene, thereby further destabilise the product and make the reaction pathway thermodynamically disfavourable. And the approaching of SO2 towards the cis-diene in the six-membered ring is subject to stronger steric hindrance in contrast with the terminal diene, hence, it's also kinetically disfavoured.
Approaching trajectory and IRC Analysis
Approaching trajectory of endo product
Approaching trajectory of exo product
IRC for endo product
IRC for exo product
Reaction barrier and reaction energy
Substrates
Gibbs free energy at B3LYP (6-31G(d)) / kJmol-1
SO2
-311.421
Xylyene
467.649
Sum of reactants energy
156.228
Exo-DA-TS
275.819
Endo-DA-TS
267.985
Exo-DA-product
176.704
Endo-DA-product
172.256
PM6 / kJmol-1
Exo DA
Endo DA
Activation energy /kJmol-1
+ 119.59
+ 111.76
Reaction energy /kJmol-1
+ 20.48
+ 16.03
Energy Profile
As is shown above, a reaction profile is constructed based on the activation barrier and reaction energy calculated in previous section. 0 energy was set for the reactants at infinite separation. The endo and exo-DA reactions with cis-diene in the ring have higher activation barriers than reactions with terminal dienes due to stronger steric hindrance in the former pathway, hence, it's not kinetically competitive. In addition, the reaction energies are positive in the former pathways, which means both reaction pathways are endothermic and thermodynamically less favourable than exothermic reaction. According to Hammond's postulate, this is a late barrier reaction, the structure of TS simulates the structure of product. The C-C and C=C bond lengths in TS are closer to those in product rather than reactant. This is proven by the optimisation results calculated using Gaussian. The results are shown below.
Optimisation Results
ENDO TS
EXO TS
Conclusion
Gaussian is a useful tool in the study of transition state. It can locate an accurate structure for TS via a series of optimisations. The reliability of the structure can be further justified using IRC analysis and frequency calculations. The correct TS can only have one imaginary frequency corresponding to the bond formation/breaking vibrational normal mode. The change in bond lengths provides valuable reference on the course of reaction. By running a MO analysis, the property of a Diels Alder reaction can be determined to be normal electron demand or inverse demand. A reasonable conclusion can also be drawn that only MO with same symmetry will interact and mix with each other to contribute for the reaction progress because the overlap integral is non-zero in this way. A full MO diagram can be built upon single point energy calculation. An energy calculation is effective in determining which reaction pathway is most kinetically and thermodynamically favourable among many competing side reactions. The pathway with lowest activation barrier is kinetically favoured. The pathway with most negative reaction energy is thermodynamically favoured. In conclusion, computational method can be used in cases where a large amount of complicated calculations and simulations are required to be carried out quickly, this will generate a more accurate answer compared with traditional experimental method more efficiently.
IRC for endo product
IRC for exo product
IRC for cheletropic product
