Dy815

Introduction

Potential Energy Surface (PES)

The Potential energy surface (PES) is a central concept in computational chemistry, and PES can allow chemists to work out mathematical or graphical results of some specific chemical reactions.

The concept of PES is based on Born-Oppenheimer approximation - in a molecule the nuclei are essentially stationary compared with the electrons motion, which makes molecular shapes meaningful.

The potential energy surface can be plotted by potential energy against any combination of degrees of freedom or reaction coordinates; therefore, geometric coordinates, $q$, should be applied here to describe a reacting system. For example, as like Second Year molecular reaction dynamics lab (triatomic reacting system: HOF), geometric coordinate ($q_1$) can be set as O-H bond length, and another geometric coordinate ($q_2$) can be set as O-F bond length. However, as the complexity of molecules increases, more dimensions of geometric parameters such as bond angle or dihedral need to be included to describe these complex reacting systems. In this computational lab, $q$ is in the basis of the normal modes which are a linear combination of all bond rotations, stretches and bends. and looks a bit like a vibration.

Transition State

Transition state is normally considered as a point with the highest potential energy in a specific chemical reaction. More precisely, transition state is a stationary point (zero first derivative) with negative second derivative along the minimal-energy reaction pathway($\frac{dV}{dq}=0$ and $\frac{d^2V}{dq^2}<0$). $V$ indicates the potential energy, and $q$ here represents an geometric parameter.

In computational chemistry, along the geometric coordinates ($q$), one lowest-energy pathway can be found, as the most likely reaction pathway for the reaction, which connects the reactant and the product. The maximum point along the lowest-energy path is considered as the transition state of the chemical reaction.

Computational Method

As Schrödinger equation states, $\lang {\Psi} |\mathbf{H}|\Psi\rang = \lang {\Psi} |\mathbf{E}|\Psi\rang,$ a linear combination of atomic orbitals can sum to the molecular orbital: $\sum_{i}^N {c_i}| \Phi \rangle = |\psi\rangle.$ If the whole linear equation expands, a simple matrix representation can be written:

${E} = \begin{pmatrix} c_1 & c_2 & \cdots & \ c_i \end{pmatrix} \begin{pmatrix} \lang {\Psi_1} |\mathbf{H}|\Psi_1\rang & \lang {\Psi_1} |\mathbf{H}|\Psi_2\rang & \cdots & \lang {\Psi_1} |\mathbf{H}|\Psi_i\rang \\ \lang {\Psi_2} |\mathbf{H}|\Psi_1\rang & \lang {\Psi_2} |\mathbf{H}|\Psi_2\rang & \cdots & \lang {\Psi_2} |\mathbf{H}|\Psi_i\rang \\ \vdots & \vdots & \ddots & \vdots \\ \lang {\Psi_i} |\mathbf{H}|\Psi_1\rang & \lang {\Psi_i} |\mathbf{H}|\Psi_2\rang & \cdots & \lang {\Psi_i} |\mathbf{H}|\Psi_i\rang \\ \end{pmatrix} \begin{pmatrix} c_1 \\ c_2 \\ \vdots \\ c_i \end{pmatrix},$

where the middle part is called Hessian matrix.

Therefore, according to variation principle in quantum mechanics, this equation can be solve into the form of $H_c$ = $E_c$.

In this lab, GaussView is an useful tool to calculate molecular energy and optimize molecular structures, based on different methods of solving the Hessian matrix part ($H_c$ bit in the simple equation above). PM6 and B3LYP are used in this computational lab, and main difference between PM6 and B3LYP is that these two methods use different algorithms in calculating the Hessian matrix part. PM6 uses a Hartree–Fock formalism which plugs some empirically experimental-determined parameters into the Hessian matrix to simplify the molecular calculations. While, B3LYP is one of the most popular methods of density functional theory (DFT), which is also a so-called 'hybrid exchange-correlation functional' method. Therefore, in this computational lab, the calculation done by parameterized PM6 method is always faster and less-expensive than B3LYP method.

Exercise 1

In the exercise, very classic Diels-Alder reaction between butadiene and ethene has been investigated. In the reaction, an acceptable transition state has been calculated. The molecular orbitals, comparison of bond length for different C-C and the requirements for this reaction will be discussed below. (The classic Diels-Alder reaction is shown in figure 1.)

Figure 1. Reaction Scheme of Diels Alder reaction between butadiene and ethene

Molecular Orbital (MO) Analysis

Molecular Orbital (MO) diagram

The molecular orbitals of Diels-Alder reaction between butadiene and ethene is presented below (Fig 2):

Figure 2. MO diagram of Diels Alder reaction between butadiene and ethene

In figure 2, all the energy levels are not quantitatively presented. It is worth noticing that the energy level of ethene LUMO is the highest among all MOs, while the energy level of ethene HOMO is the lowest one. To confirm the correct order of energy levels of MO presented in the diagram, energy calculations at PM6 level have been done, and Jmol pictures of MOs have been shown in table 1 (from low energy level to high energy level):

table 1: Table for energy levels of MOs in the order of energy(from low to high)

ethene HOMO - MO 31	butadiene HOMO - MO 32	transition state HOMO-1 - MO 33	transition state HOMO - MO 34	butadiene LUMO - MO 35	transition state LUMO - MO 36	transition state LUMO+1 - MO 37	ethene LUMO - MO 38

In table 1, all the calculations are done by putting the reactants and the optimized transition state in the same PES. TO achieve this, all the components (each reactant and the optimized transition state) are set a far-enough distances to avoid any presence of interactions in this system (the distance usually set up greater than 6 Å, which is greater than 2 × Van der Waals radius of atoms). The pictures of calculation method and molecular buildup are presented below:

Figure 3. energy calculation method for MOs

Figure 4. molecular buildup for MO energy calculation

Frontier Molecular Orbital (FMO)

The table 2 below contains the Jmol pictures of frontier MOs - HOMO and LUMO of two reactants (ethene and butadiene) and HOMO-1, HOMO, LUMO and LUMO+1 of calculated transition state are tabulated.

table 2: Table of HOMO and LUMO of butadiene and ethene, and the four MOs produced for the TS

Molecular orbital of ethene (HOMO)	Molecular orbital of ethene (LUMO)
Molecular orbital of butadiene (HOMO)	Molecular orbital of butadiene (LUMO)
Molecular orbital of TS (HOMO-1)	Molecular orbital of TS (HOMO)
Molecular orbital of TS (LUMO)	Molecular orbital of TS (LUMO+1)

Requirements for successful Diels-Alder reactions

Combining with figure 2 and table 2, it can be concluded that the requirements for successful reaction are direct orbital overlap of the reactants and correct symmetry. Correct direct orbital overlap leads to a successful reaction of reactants with the same symmetry (only symmetric/ symmetric and antisymmetric/ antisymmetric are 'reaction-allowed', otherwise, the interactions are 'reaction-forbidden'). As a result, the orbital overlap integral for symmetric-symmetric and antisymmetric-antisymmetric interactions is non-zero, while the orbital overlap integral for the antisymmetric-symmetric interaction is zero.

Measurements of C-C bond lengths involved in Diels-Alder Reaction

Measurements of 4 C-C bond lengths of two reactants and bond lengths of both the transition state and the product gives the information of reaction. A summary of all bond lengths involving in this Diels-Alder reaction has been shown in the figure 5 and a table of dynamic pictures calculated by GaussView has also been shown below (figure 6):

Figure 5. Summary of bond lengths involving in Diels Alder reaction between butadiene and ethene

Figure 6: Corresponding (to figure 5) dynamic Jmol pictures for C-C bond lengths involving in Diels Alder reaction

The change of bond lengths can be clearly seen in figure 5, in first step, the double bonds in ethene (C₁-C₂) and butadiene (C₃-C₄ and C₅-C₆) increase from about 1.33Å to about 1.38Å, while the only single bond in butadiene (C₄-C₅) is seen a decrease from 1.47Å to 1.41Å. Compared to the literature values of C-C (sp³:1.54Å), C=C (sp²:1.33Å) and Van der Waals radius of carbon (1.70Å), the changes of bond lengths indicate that C=C(sp²) changes into C-C(sp³), and C-C(sp³) changes into C=C(sp²) at the meantime. As well, since the C₁-C₆ and C₂-C₃ is less than twice fold of Van der Waals radius of carbon (~2.11Å < 3.4Å), the orbital interaction occurs in this step as well. Because all the C-C bond lengths here are greater than the literature value of C=C(sp²) and less than the literature value of C-C(sp³), all the C-C bonds can possess the properties of partial double bonds do.

After the second step, all the bond lengths in the product (cyclohexene) go to approximately literature bond lengths which they should be.

Vibration analysis for the Diels-Alder reaction

The vibrational gif picture which shows the formation of cyclohexene picture is presented below (figure 7):

Figure 7. gif picture for the process of the Diels-Alder reaction

Although in this gif, it looks like the dissociation of bonds (which can be considered as an inverse process of Diels-Alder reaction), it can also be seen a synchronous process according to this vibrational analysis. So, GaussView tells us that Diels-Alder reaction between ethane and butadiene is a concerted pericyclic reaction as expected. Another explanation of a synchronous process is that in molecular distance analysis at transition state, C₁-C₆ = C₂-C₃ = 2.11 Å.

Calculation files list

optimized reactant: Media:EX1 SM1 JMOL.LOG Media:DY815 EX1 DIENE.LOG

optimized product: Media:PRO1-PM6(FREQUENCY).LOG

otimized/frequency calculated transition state:Media:DY815 EX1 TSJMOL.LOG

IRC to confirm the transition state: Media:DY815-EX1-IRC-PM6.LOG

Energy calculation to confirm the frontier MO: Media:DY815 EX1 SM VS TS ENERGYCAL.LOG

Exercise 2

Figure 8. Reaction scheme for Diels-Alder reaction between cyclohexadiene and 1,3-dioxole

The reaction scheme (figure 8) shows the exo and endo reaction happened between cyclohexadiene and 1,3-dioxole. In this section, MO and FMO are analyzed to characterize this Diels-Alder reaction and energy calculations will be presented as well.

Note:All the calculations in this lab were done by B3LYP method on GaussView.

Molecular orbital at transition states for the endo and exo reaction

MO diagram of the Endo Diels-Alder reaction

The energy calculation for the endo transition state combined with reactants has been done in the same PES. The distance between the transition state and the reactants are set far away to avoid interaction between each reactants and the transition state. The calculation follows the same energy calculation method mentioned in fig 3 and fig 4. Table 3 includes all the MOs needed to construct a MO diagram with the order of energy (from low to high).

table 3: Table for Endo energy levels of TS MO in the order of energy(from low to high)

1,3-dioxole HOMO - MO 79	ENDO TS HOMO-1 - MO 80	ENDO TS HOMO - MO 81	cyclohexadiene HOMO - MO 82	cyclohexadiene LUMO - MO 83	1,3-dioxole LUMO - MO 84	ENDO TS LUMO - MO 85	ENDO TS LUMO+1 - MO 86

Based on the table above, the MO diagram for endo can generated (figure 9).

Figure 9. MO for endo Diels-Alder reaction between cyclohexadiene and 1,3-dioxole

MOs of the Exo Diels-Alder reaction

The determination follows the exactly same procedures as mentioned in construction endo MO, except substituting the endo transition into exo transition state. Table 4 shows the information needed to construct exo MO.

table 4: Table for Exo energy levels of TS MO in the order of energy(from low to high)

cyclohexadiene HOMO - MO 79	EXO TS HOMO-1 - MO 80	1,3-dioxole HOMO - MO 81	EXO TS HOMO - MO 82	cyclohexadiene LUMO - MO 83	EXO TS LUMO - MO 84	EXO TS LUMO+1 - MO 85	1,3-dioxole LUMO - MO 86

Figure 10. MO for exo Diels-Alder reaction between cyclohexadiene and 1,3-dioxole

Therefore, the MO diagram can be constructed as above (figure 10). There are some differences between exo and endo Diels-Alder reaction.

Comparing Endo and Exo MO at transition states

Table 5 shows Jmol pictures for the calculated molecular orbitals at B3LYP level.

table 5: Table for molecular orbital of transition states (Exo and Endo)

ENDO HOMO-1 (MO 40) of transition states	ENDO HOMO (MO 41) of transition states	ENDO LUMO (MO 42) OF transition states	ENDO LUMO+1 (MO 43) OF transition states
EXO HOMO-1 (MO 40) of transition states	EXO HOMO (MO 41) of transition states	EXO LUMO (MO 42) OF transition states	EXO LUMO+1 (MO 43) OF transition states

The comparison done by energy calculations for these different transition states are tabulated below (table 6). The results are confirmed by energy calculation at B3LYP level, with putting these two transition states in the same PES surface but far enough (the distance > 2 * VDW radius of carbon) to avoid any interaction. The energy calculation method details can see fig 3 and fig 4 in exercise 1.

table 6: Table for Exo and Endo energy levels of TS MO in the order of energy(from low to high)

EXO TS HOMO-1 - MO 79	ENDO TS HOMO-1 - MO 80	ENDO TS HOMO - MO 81	EXO TS HOMO - MO 82	EXO TS LUMO - MO 83	ENDO TS LUMO - MO 84	EXO TS LUMO+1 - MO 85	ENDO TS LUMO+1 - MO 86

The energy levels of HOMO-1, LUMO and LUMO+1 for Exo TS are lower than the energy levels for Endo MO, while HOMO for Exo TS is higher comapared to HOMO for Endo TS.

The difference in energy levels at transition state can be explained by the different repulsions for different transition states feel. Because there is a difference in steric clash for different products, the transition states for different Diels-Alder reaction can have different interacting situation. In figure 11, the steric clash difference in prroducts is clearly shown.

Figure 11. different products with different steric clash

Characterize the type of DA reaction by comparing dienophiles in EX1 and EX2 (ethene and 1,3-dioxole)

This Diels Alder reaction can be characterized by analysing the difference between molecular orbitals for dienophiles, because the difference for two diene (butadiene vs cyclohexadiene) is not big and we cannot decide the type of DA reaction by comparing the dienes.

table 7: Table for Exo energy levels of TS MO in the order of energy(from low to high)

ethene HOMO - MO 26	1,3-dioxole HOMO - MO 27	ethene LUMO - MO 28	1,3-dioxole LUMO - MO 29

Because we can see that both HOMO and LUMO for 1,3-dioxole are higher in energy than HOMO and LUMOfor ehtene. It can be concluded that it is an inverse electron demand Diels-Alder reaction (in exercise 2), and this is caused by the oxygens donating into the double bond raising the HOMO and LUMO of the 1,3-dioxole. Due to the extra donation of electrons, the dienophile 1,3-dioxole, has increased the energy levels of its HOMO and LUMO.

Reaction energy analysis

calculation of reaction energies

table 7. thermochemistry data from Gaussian calculation

	1,3-cyclohexadiene	1,3-dioxole	endo-transition state	exo-transition state	endo-product	exo-product
Sum of electronic and thermal energies at 298k (Hartree/Particle)	-233.290938	-267.037365	-500.287655	-500.285409	-500.376556	-500.377578
Sum of electronic and thermal energies at 298k (KJ/mol)	-612505.404377	-701106.655215	-1313505.33826	-1313499.44139	-1313738.74785	-1313741.43111

The reaction barrier of a reaction is the energy difference between the transition state and the reactant. If the reaction barrier is smaller, the reaction goes faster, which also means it is kinetically favoured. And if the energy difference between the reactant and the final product is large, it means that this reaction is thermodynamically favoured.

Endo reaction:

reaction barrier = -1313505.33826 - (-701106.655215) - (-612505.404377) = 106.72 KJ/mol

energy difference between the reactant and the final product = (-1313738.74785) - (-612505.404377) - (-701106.655215) = -126.69 KJ/mol

Exo reaction:

reaction barrier = -1313499.44139 - (-701106.655215) - (-612505.404377) = 112.62 KJ/mol

energy difference between the reactant and the final product = (-1313741.43111) - (-612505.404377) - (-701106.655215) = -129.37 KJ/mol

We can see, from the calculating results, that exo pathway is preferred thermodynamically but endo reaction is favoured kinetically, which can be explained by secondary orbital effect.

Secondary molecular orbital overlap

Figure 12. Secondary orbital effect

As seen in figure 12, the secondary orbital effect occurs due to a stablised transition state where oxygen p orbitals interact with π ^* orbitals in the cyclohexadiene. And as for exo transition state, only first orbital interaction occurs and the exo product shows a less sterically crowded structure (seen in fig 11), resulting in the thermodynamically favoured pathway of reaction.

table 8. Frontier molecular orbital to show secondary orbital interactions

ENDO TS MO of HOMO	EXO TS MO of HOMO

Table 8 gives the information of the frontier molecular orbitals.

Exercise 3

In this section, Diels-Alder and its competitive reaction - cheletropic reaction will be investigated. Also, the side reaction - internal Diels-Alder reaction will be discussed. The figure below shows the reaction scheme (figure 13).

Figure 13. Secondary orbital effect

IRC calculation

IRC is the process of integrating the intrinsic reaction coordinate to calculate a pathway for a reaction process. Before doing the IRC calculation, optimised products and transition states have been done. The table below shows all the optimised structures (table 8):

table 9: Table for optimised products and transition states

	endo Diels-Alder reaction	exo Diels-Alder reaction	cheletropic reaction
optimised product
optimised transition state

After optimising the products, set a semi-empirical distances (C-C: 2.2 Å) between two reactant components and freeze the distances. IRC calculations in both directions from the transition state at PM6 level were obtained. The table below shows IRC calculations for endo, exo and cheletropic reaction.

Table 10. IRC Calculations for reactions between o-Xylylene and SO₂

	Total energy along IRC	IRC
IRC of Diels-Alder reaction via endo TS
IRC of Diels-Alder reaction via exo TS
IRC of cheletropic reaction

Reaction barrier and activation energy

The thermochemistry calculations have been done and presented in the tables below:

table 11. Summary of electronic and thermal energies of reactants, TS, and products by Calculation PM6 at 298K

Components	Energy/Hatress	Energy/kJmol-1
SO2	-0.118614	-311.421081
Xylylene	0.178813	469.473567
Reactants energy	0.060199	158.052487
ExoTS	0.092077	241.748182
EndoTS	0.090562	237.770549
Cheletropic TS	0.099062	260.087301
Exo product	0.027492	72.1802515
Endo Product	0.021686	56.9365973
Cheletropic product	0.000002	0.0052510004

The following figure and following table show the energy profile and the summary of activation energy and reaction energy.

Figure 14. Energy profile for ENDO, EXO and CHELETROPIC reaction

table 11. Activation energy and reaction energy(KJ/mol) of five reaction paths at 298K

	Exo	Endo	Cheletropic
Activation energy (KJ/mol)	83.69570	79.71806	102.03481
Reaction energy (KJ/mol)	-85.87224	-101.11589	-158.04724

Second Diels-Alder reaction (side reaction)

The second Diels-Alder reaction can occur alternatively. Also, for this second DA reaction, endo and exo pathways can happen as normal DA reaction do.

Figure 15. Second Diels-Alder reaction reaction scheme

The table 12 below gives the IRC information and optimised transition states and product involved in this reaction:

Table 12. IRC Calculations for THE SECOND DA reaction between o-Xylylene and SO₂

optimised endo product	IRC endo TS	optimised exo product	IRC exo TS	Total energy along endo IRC	Total energy along endo IRC