Rep:Pth115TS

Introduction

Potential Energy Surface: Minima

Depiction of a Minimum and a Transition State on a PES^[1]

A potential energy surface (PES) is a representation of the energy of a molecule as a function of its geometrical configuration. Therefore, a molecule's potential energy is dependent on its internal coordinates^[2]. A minimum on a PES (which can be local or global) corresponds to a minimum-energy geometry of a molecule and is defined as a point on the PES where the first derivative of the potential energy (V) with respect to distance (x) is equal to zero $(\frac{\partial V}{\partial x} = 0)$ , whilst the second derivative is positive $(\frac{\partial^{2} V}{\partial x^{^{2}}} > 0)$ . In a chemical sense, a minimum on a PES corresponds to a stable or quasi-stable species (such as reactants, product or intermediates)^[2].

The latter mathematical condition also implies that the force constant (k) has to be larger than 0 at all times since the equation $k = \frac{\partial^{2} V}{\partial x^{2}}$ holds true. As the wavenumber (v) of a bond vibration depends on the strength of the bonds $(v \sim \sqrt{k})$ , one can anticipate to obtain exclusively positive vibration frequencies for a molecule in a minimum-energy configuration.

The computational development of PESs relies fundamentally on the Born-Oppenheimer approximation, which states that due to the considerably large difference between electronic and nuclear masses, nuclei move much slower than electrons and their respective motions can be considered separately. Consequently, the overall wavefunction of a molecule may be separated into a nuclear and an electronic term.

Transition States

The transition state of a reaction is defined as the point on a PES which behaves as a saddle point. This means that that specific point behaves like a potential energy minimum in all dimensions but one, for which it behaves like a maximum^[2]. Mathematically, for a system with 2 degrees of freedom (x and y), one can express these conditions as $\frac{\partial V}{\partial x} = 0, \frac{\partial V}{\partial y} = 0, \frac{\partial^{2} V}{\partial x^{2}} < 0$ and $\frac{\partial^{2} V}{\partial y^{2}} > 0$ . It is to be noted, however, that the inequalities of the second derivatives of potential energy may be switched (the former being greater than 0 whilst the latter is less than 0), depending on the nature of the saddle point.

The location of a transition state for a molecule with Gaussian can be verified by obtaining a single negative (imaginary) vibration frequency. This stems from the one negative second derivative, which corresponds to a negative force constant (as mentioned before, k is equivalent to the second derivative of the potential energy with respect to a degree of freedom).

As there are 3N-6 normal vibrational modes (and hence degrees of freedom and dimensions) for a non-linear molecule, there will be 3N-6 k values, each associated with a vibrational mode. Gaussian computes the values for k (and thus the vibration frequencies) by obtaining the eigenvalues of the Hessian matrix of 3N-6 dimensions.

Nf710 (talk) 22:08, 21 February 2018 (UTC) Good section. YOu can get all the force constants at the TS geom by diagonalising the hessian. The eigenvectors are the normal modes and the eigenvectors are the force constants. The normal modes are linear combinations of all the degrees of freedom and thats which its associated vibration looks like a bond breaking/forming.

Computations on Gaussian

Basis Sets

The computer programme Gaussian relies on the principle of the linear combination of atomic orbitals (LCAO) to estimate values for the energies for molecular orbitals (MO), using the Hamiltonian (H): $E = < Ψ | H | Ψ >$ . The LCAO to obtain MOs can be illustrated mathematically by the following expression: $| Ψ > = \sum_{i}^{N} c_{i} Φ_{i}$ , where φ_i represents an atomic orbital, c_i its weighting coefficient and |Ψ> the integral over an MO with which one is able to compute E with the Hamiltonian.

Nf710 (talk) 22:14, 21 February 2018 (UTC) When there is only 1 ket and not a bra-ket pair its not an integral its just a vector

PM6

PM6 optimisations utilise semi-empirical methods (which are based on the Hartree-Fock approach, amongst other approximations) to calculate the Hamiltonian. Semi-empirical approaches such as PM6 parametrise integrals based on experimental data^[3]. Due to this, optimisations at PM6 are performed fairly quickly and can be used on large molecules (with complex Hamiltonian expressions), however with significant amounts of unreliability and high chances of inaccuracy.

DFT: B3LYP/6-31(d)

B3LYP calculations produce better quality and more reliable results than PM6 computations. B3LYP makes use of Density Functional Theory (DFT) to actually calculate every term of the Hamiltonian expression (apart from one term, the exchange-correlation term, for which the DFT employs the Hatree-Fock approximation). The higher precision and accuracy of the B3LYP optimisation comes at the cost of taking significantly more time to process the calculations.

Nf710 (talk) 22:16, 21 February 2018 (UTC) Really good intro! nice uses of equations to explain the basis set. You could have added some into your description of the quantum chemical meethods.

Exercise 1

(Fv611 (talk) Excellent work and thorough discussion across the whole exercise. Good job!)

Reaction

This exercise looked at the Diels Alder reaction between 1,3-butadiene (acting as the diene) and ethene (the dienophile). The pericyclic reaction between the two molecules is a thermal [4+2] cycloaddition, indicating the number of π electrons on each of the reactants (4π e- on 1,3-butadiene and 2π e- on ethene). The reaction involves a σ-overlap between the π-orbitals of the two unsaturated compounds^[4].

The ethene and 1,3-Butadiene structures were both energetically minimised at PM6 level. The minimised structures of the unsaturated compounds were subsequently used to run an optimisation of the transition state structure of the Diels-Alder reaction, again at PM6 level. An IRC of the transition state was performed at PM6, and the structure of the product of the reaction (cyclohexene) was also optimised at PM6 level.

Curly Arrow Mechanism of [4+2] Cycloaddition between 1,3-Butadiene and Ethene	Illustration of example orbital interactions between diene and dienophile

MO Analysis

MO Diagram

MOs: 1,3-Butadiene

1,3-Butadiene MOs

E = -0.35898, HOMO

E = 0.01942, LUMO

MOs: Ethene

Ethene MOs

E = -0.39228, HOMO

E = 0.04256, LUMO

MOs: Transition State

Transition State MOs

E = -0.32755, HOMO-1

E = -0.32533, HOMO

E = 0.01732, LUMO

E = 0.03067, LUMO+1

Discussion

The way in which the molecular orbitals of the reacting species interact is restricted, so that only orbitals of the same symmetry (antisymmetric (a) or symmetric(s)) are ab to interact with each other. When this condition is fullfilled, the reaction may be classified as 'allowed'. In the case that there are no orbitals of the same symmetry present that are able to interact with each other to lower the overall energy of the system, the reaction is 'forbidden'.

When two antisymmetric orbitals interact, the overlap integral will be non-zero. The same is true when two symmetric orbitals interact. However, should a symmetric and an antisymmetric orbital interact, the overlap integral would be equal to zero. This may be rationalised by looking at the orbital phases that overlap in each of the three scenarios. Whenever two orbitals of like symmetry overlap there is favourable constructive interference (the orbital lobes that overlap are of the same phase) leading to a non-zero overlap integral. Although, if two orbitals of opposite symmetries overlap, there would be unfavourable destructive interference (the orbitals that overlap would be of opposite phases), resulting in an overlap integral that equates to zero. The discussion about overlap integrals is summarised in the table below.

Overlap Symmetries	Interference	Overlap Integral
Symmetric with Symmetric	Constructive	Non-Zero
Antisymmetric with Antisymmetric	Constructive	Non-Zero
Symmetric with Antisymmetric	Destructive	Zero

From the MO diagram shown above, one observes that the HOMO of the 1,3-butadiene molecule and the LUMO of ethene have matching symmetries and interact to give two transition state orbitals. The same is true for the LUMO of 1,3-butadiene and the HOMO of ethene.

As expected, the formation of the transition state raises the overall energy of the system (as the electrons are now located in frontier orbitals that are higher in energy than the HOMOs of the reactants).

C-C Bond Lengths

Tabulated Data

	C-C Bond Lengths/Å (Raw Information)	C=C Bond Lengths/Å (Raw Information)	C-C Bond Lengths/Å (2 decimal points)	C=C Bond Lengths/Å (2 decimal points)
Ethene	N/A	1.32754	N/A	1.33
1,3-Butadiene	1.47077 (C2-C3)	1.33349 (C1-C2) 1.33345 (C3-C4)	1.47 (C2-C3)	1.33 (C1-C2) 1.33 (C3-C4)
Transition State	1.41107 (C1-C2) 2.11492 (C3-C4) 2.11452 (C5-C6)	1.37976 (C1-C6) 1.37978 (C2-C3) 1.38176 (C4-C5)	1.41 (C1-C2) 2.11 (C3-C4) 2.11 (C5-C6)	1.38 (C1-C6) 1.38 (C2-C3) 1.38 (C4-C5)
Cyclohexene (Product)	1.50085 (C2-C3) 1.53718 (C3-C4) 1.53458 (C4-C5) 1.53716 (C5-C6) 1.50088 (C1-C6)	1.33697 (C1-C2)	1.50 (C2-C3) 1.54 (C3-C4) 1.53(C4-C5) 1.54 (C5-C6) 1.50 (C1-C6)	1.34 (C1-C2)

Cyclohexene (Product)	1,3-Butadiene	Transition State of Diels-Alder Reaction between Ethene and 1,3-Butadiene

Typical sp²C-C Bond Length/Å	1.34^[5]
Typical sp³ C-C Bond Length/Å	1.54^[5]

C-atom Van der Waals Radius/Å	1.7^[6]
2x C-atom Van der Waals Radius/Å	3.4

IRC

Animation	Graph
Ethene and 1,3-Butadiene Diels-Alder Reaction IRC Animation	Total Energy against IRC Graph for Diels-Alder Reaction between Ethene and 1,3-Butadiene

Discussion

Comparing the bond lengths of the ethene molecule before the reaction and in the transition state, an extension of the C-C bond is observable. This can be accounted for by the filling of ethene's π*-orbital during the reaction, which ultimately leads to the extension and cleavage of the C-C π-bond. The hybridisation of the ethene C-atoms changes from sp² to sp³.

As for the co-reactant, 1,3-butadiene, the collected C-C bond-length data shows a clear extension of the C1-C2 and C3-C4 bonds (originally double bonds, but single bonds after the reaction) and a contraction of the C2-C3 bond (originally a single bond, but a double bond after the reaction) as the diene reaches the transition state. This data is indicative of both C-C double bonds in 1,3-butadiene being broken during the Diels-Alder reaction, whilst a new π-bond is formed at C2-C3. Carbon atoms C1 and C4 change from sp² to sp³ hybridisation, whereas the hybridisations of C2 and C3 remain unchanged at sp².

In the cyclohexene product, the C-C bonds C3-C4 and C5-C6 are newly formed σ-bonds from the Diels-Alder process. In the transition state, the distances between carbon atoms C3 and C4, and C5 and C6 (2.11492 Å and 2.11452 Å respectively) are shorter than the sum two C-atom Van der Waals radii (3.4 Å). This demonstrates that there is favourable orbital overlap between the reactants that will lead to the formation of 2 new σ-bonds.

In both cases, ethene and 1,3-butadiene, the lengths of C-C double bonds that are broken at the transition state are closer in value to the bond lengths of the reactants than to the bond lengths of the products. For ethene: C-C double bond at transition state is 1.38 Å long, which is closer to 1.33 Å (reactant) than 1.53 Å (product). For 1,3-butadiene: C-C double bond at transition state is 1.38 Å long, which is closer to 1.33 Å (reactant) than 1.50 Å (product). By considering Hammond's postulate, one can deduce that we are dealing with an early transition state in this reaction (transition state resembles more the reactants than the products).

Vibrations

By inspection of the animation of the negative (and hence imaginary) bond vibration frequency at -948.86 cm^-1of the transition state, one can see that the bond formation between the diene and the dienophile is synchronous. This is as expected since the Diels-Alder reaction is concerted (i.e. all bond breaking and bond formation processes occur simultaneously).

Bond forming vibration

Log Files

File:PTH115-BUTADIENE-PM6.LOG

File:PTH115-CYCLOHEXENE-PM6.LOG

File:PTH115-ETHENE-PM6.LOG

File:PTH115-DA-PM6.LOG

File:PTH115-DA-IRC-PM6.LOG