Rep:Ac4515 TS

Introduction

Potential energy surfaces: Transition states and minima

The concept of potential energy surfaces (PES) is key in computational chemistry. A PES is a graphical or mathematical illustration of the relationship between the energy of a molecule, or group of molecules, and its respective geometric coordinates. As the PES is dependent on the geometry of the molecule, and non-linear molecules have 3N-6 degrees of freedom, the PES will have 3N-6 dimensions for non-linear systems. The PES will have many minima, each with corresponding energies and all are stationary points which satisfy: $\frac{\partial E}{\partial q} = 0$ for all geometric coordinates of q and $\frac{\partial^{2} E}{\partial q^{2}} > 0$ for all q (in all directions)^[1]. However, where the distinction comes in between minima and a transition state (or saddle point) is that along the reaction coordinate the transition state is a maximum and satisfies : $\frac{\partial^{2} E}{\partial q^{2}} < 0$ . Furthermore, a transition state would only have a single negative force constant on the PES, while contrastingly a minimum wouldn't have any. Therefore a transition state structure is confirmed by Gaussian when a particular structure is found to have a single negative vibrational mode. This is a mathematical explanation to distinguish between transition states and minima, below is a diagram to represent this as well^[1].

Nf710 (talk) 21:38, 21 February 2018 (UTC) You can diagonalise the hessian at the TS geom and the eigen values will be the curvature of the normal modes (dimensions)

Computational methods used in Gaussian: PM6 and B3LYP

Two different computational methods were employed during this task, the semi-empirical method PM6 and the density dunctional theory (DFT) method B3LYP. Mainly PM6 was used to due to time constraints, but on exercise 2 both methods were used in tandem to yield a higher accuracy.

Firstly, the PM6 is the quicker and less expensive of the two methods but does, however, predict structures with far less accuracy and reliability. A PM6 optimisation uses a Hartree-Fock calculation to obtain the Hamiltonian for the Schrödinger equation. Where the method makes approximations for the Hamiltonian is when terms are approximated using experimental data ^[2]. These empirical approximations is what makes the method sometimes unreliable and inaccurate.

The second method, B3LYP, is thought of as a hybrid function as it uses two mathematical theories to calculate the Hamiltonian. It uses DFT to calculate all terms in the Hamiltonian apart from the exchange-correlation where it brings in Hartree-Fock theorem ^[3]. This more thorough method for solving the Hamiltonian makes the method more expensive but does, however, gain results with increased accuracy when compared to PM6.

Nf710 (talk) 21:56, 21 February 2018 (UTC) Good understanding here. Some equations would have been nice to complement the discussion

Exercise 1

(Fv611 (talk) Very good job across the whole section. Well done!)

Reaction scheme

MO Diagram

From the visualised MOs in the table below and the MO diagram above, a conclusion can be made that only orbitals of the symmetry interact, i.e. symmetric-symmetric and anitsymmetirc-antisymmertic orbitals are able to interact. Interactions of like-symmetry are said to be allowed and interactions of opposite symmetry are symmetry-forbidden and disallowed. A further conclusion that can be drawn is that when MOs of like-symmtery combine they form a pair of MOs with the same symmetry. For example, the asymmetric combination of the butadiene HOMO and ethene LUMO produce an antisymmetric HOMO-1 and LUMO+1 pair.

The overlap integral produces a value from zero to one and quantitatively measures the overlap of two orbitals and how constructive or destructive an interaction is. A value closer to one, the more favourable and constructive the interaction. The overlap integral is zero when the orbitals (also wave functions) positive and negative aspects cancel out, for example, this occurs for a symmetric-antisymmetric interaction, hence why no such interaction is seen in the MO diagram. The integral is non-zero for like-symmetry interactions such as symmetric-symmetric and antisymmetric-antisymmetric. All interactions producing the transition state MOs visualised have come from a combination of reactant MOs with like-symmtery.

Observing how each of the electron pairs from the HOMOs of the reactants have changed to when occupying the HOMOs of the transition state, it's clear these electron pairs have gained energy. This is a result of the transition state having an energy barrier needed to be reached and overcome to form products.

Ethene MO's

Ethene HOMO

Ethene LUMO

Butadiene MO's

Butadiene HOMO

Butadiene LUMO

Transition state MO's

TS HOMO-1

TS HOMO

TS LUMO

TS LUMO+1

Variations in C-C bond lengths

Table 1: Bond lengths in the reactants, TS and products. (Look to reaction scheme for C atom numbering).

C-C Bond	Butadiene bond length/Å	Ethene bond length/Å	TS bond length/Å	Cyclohexene bond length/Å
C1-C2	1.34	n/a	1.38	1.50
C2-C3	1.47	n/a	1.41	1.34
C3-C4	1.34	n/a	1.38	1.50
C4-C5	n/a	n/a	2.11	1.54
C5-C6	n/a	1.33	1.38	1.53
C6-C1	n/a	n/a	2.11	1.54

C-atom Van der Waals radius: 1.70 Å ^[4]

Typical sp³ C-C bond length: 1.54 Å ^[5]

Typical sp² C-C bond length: 1.34 Å ^[5]

Firstly examining the C5-C6 bond variation, the bond length increases from ethene to the transition state and to finally the cyclohexadiene product. Both carbons in the bond start off with a sp2 hybridisation and end in the products as sp3 hybridised. The greater electron donation as the reaction progresses in to the ethene π* orbital increases the length of the bond until the π bond is broken and just a sigma bond remains. The initial C5-C6 bond length (1.33 Å) in ethene closely resembles the literature value (1.34 Å).

The C-C single bond in butadiene (C2-C3) is shorter than the literature value. This can be explained by the conjugation of the π system in butadiene, it increases the double bond character of the bond and therefore decreases the overall bond length. Observing this bond as the reaction progresses its bond length decreases and eventually a π bond is formed in the products. The C2-C3 double bond length in the product is consistent with literature. The C=C bonds in butadiene (C1-C2 and C3-C4) lengthen as expected as the they change from double to single bonds as the reaction progresses. The hybridisation of C2 and C3 stay unchanged (sp2) from reactants to product while C1 and C4 go from sp2 to sp3 hybridisation.

The C-C single bonds in the product have different bond lengths (1.54 Å and 1.50 Å), this is due some of the single bonds being between sp2 and sp3 centres. This shortens the C-C bond length.

For a bonding interaction to occur between two carbon atoms the distance between them has to be smaller than twice the Van der Waals radius of a C atom (3.40 Å). The partially formed bonds in the transition state (C4-C5 and C6-C1) meet this criterion, thus two new C-C sigma bonds are able to form.

Reaction path vibration at the transition state

Illustrated below is the reaction path vibration for the transition state. The vibration is imaginary which is consistent with being the vibration of the transition state. Furthermore, the vibration shows the formation of the two sigma bonds is synchronous. Both C-C bonds are formed simultaneously and in a concerted manner, as expected for a conventional Diels-Alder reaction.

Appropriate LOG Files

File:AC4515PM6-BUTADIENE.LOG

File:AC4515PM6-ETHENE-OPT.LOG

File:AC4515TS-PM6-2ND.LOG

File:Ac4515-IRC OF TS-PM6.LOG

File:Ac4515-ex-1-OPTIMISED PRODUCTS- PM^.LOG

Exercise 2

Reaction Scheme

Transition State MO Diagrams

(Fv611 (talk) Again, very nice MO diagrams and good discussion.)

Examining both the MO diagrams below (figure 4 and 5), the interactions that are symmetry allowed are the symmetric combinations of the dieneophile (1,3 dioxole) HOMO and the LUMO of the diene (cyclohexadiene) producing the symmetric HOMO and LUMO molecular orbitals of the transition state. The other symmetry allowed interaction is between the antisymmetric combination of the HOMO of the diene and the LUMO of the dienophile producing the antisymmetric HOMO-1 and LUMO+1 molecular orbitals of the transition state. Similar to exercise one, the TS bonding MOs (HOMO and HOMO-1) are higher in energy than the HOMOs of the reactants they were produced from. Again, this is a result of the energy barrier between the reactants and transition state.

Exo TS MO Diagram	Endo TS MO Diagram
Figure 4: Exo transition state MO diagram	Figure 5: Endo transition state MO diagram

Exo TS HOMO-1

Exo TS HOMO

Exo TS LUMO

Exo TS LUMO+1

Endo TS HOMO-1

Endo TS HOMO

Endo TS LUMO

Endo TS LUMO+1

Single point energy- Electron demand of the Diels-Alder reaction

A single point energy calculation was done to obtain an ordered list of the MO's of the diene (cyclohexadiene) and the dienophile (1,3 dioxole). In a conventional Diels-Alder pathway the diene and dienophile are electron rich and electron deficient respectively, this yields frontier orbitals with a HOMO supplied by the diene and the LUMO from the dienophile. This interaction is said to be a normal electron demand Diels-Alder.

For the Diels-Alder reaction of cyclohexadiene with 1,3 dioxole the opposite is the case: the diene is electron poor and the dienophile is electron rich. Therefore the frontier orbitals interacting strongly in this reaction are the HOMO of the dienophile and the LUMO of the diene (look to table 2) and the reaction is said to be an inverse electron demand Diels-alder. The electron demand of this reaction can be explained by the electron donation of the oxygen atoms on 1,3 dioxole (for a normal electron demand DA the dieneophile has an EWG substituent). The increased electron density given by the oxygen atoms raises the energy of the dienophile HOMO and LUMO resulting in a electron rich dienophile and an inverse electron demand.

Table 2: Relative energies of the HOMO and LUMO's of Cyclohexadiene and 1,3 Dioxole.

MO

Energy / Hartrees

Jmol

HOMO Diene

-0.32217

HOMO Dienophile

-0.32207

LUMO Diene

+0.02111

LUMO Dienophile

+0.02979

Thermochemistry

Table 3: Reaction barriers and reaction energies of DA pathways

Pathway	Reaction Barrier Hartrees	Reaction Barrier kJ/mol	Reaction Energy Hartrees	Reaction Energy kJ/mol
Exo Diels-Alder	+0.063852	+167.64	-0.024305	-63.81
Endo Diels-Alder	+0.060866	+159.80	-0.025674	-67.41

Data tabulated in table 3 shows the endo pathway to have the lower reaction barrier (energy required to reach the TS) and the pathway that forms the energetically preferred product (lower reaction energy). The endo-pathway therefore leads to the kinetic and thermodynamic products for this particular Diels-Alder reaction.

The kinetic product forms via the lowest energy transition state. Due to the secondary orbital interactions (table 4 below) seen in the endo pathway and an absence of this effect in the exo pathway, there is a greater stabilising effect on the transition state via the endo route. In the endo HOMO the secondary orbital interactions come about as a result of the p orbtials of the oxygens having the correct orientation to interact with pi system of the diene fragment, thus having a stabilising effect overall. This effect is not seen in the exo HOMO as the p orbitals are not oriented properly for favorable overlap to occur.

The most energetically stable product is the thermodynamic product, the exo product is disfavored energetically compared to the endo product due to a steric clash between the bridge of the cyclohexadiene ring and the di-oxygen ring (seen in figure 6) which causes the an increase in energy of the exo product. This type of steric clash is not seen in the endo product because the two rings are in different planes.

Table 4: Jmols showing secondary orbital interactions of the HOMO of the exo and endo transition states

Exo TS HOMO

Endo TS HOMO