Great! That is all we need to start using the computational chemistry software.
Upon dismantling the reaction into a logical mechanism we can simply draw each structure in the software and calculate the needed values. We'll go through the results of this process to provide an overview of how we'll proceed, then go through a step-by-step example.
After drawing the reactant structure in the software we run a basic geometry optimization with a frequency calculation to obtain the thermodynamic quantities: entropy, enthalpy, zero point energy, and ground state energy. The data output at STP in tabular form for vinyl allyl ether is:
After drawing a plausible configuration of the transition state we run a transition state search and are able to find the true transition state. Which is characterized by an imaginary frequency in the direction of the reaction and also obtain thermodynamic quantities. The data output at STP in tabular form for the transition state is:
To obtain the thermodynamic quantities for the product structure we just repeat step 1 using the product structure instead of the reactant. The data output at STP in tabular form for 4-pentenal is:
With these values we can calculate the heat of reaction, the change in entropy, the change in Gibbs free energy, estimate the activation energy, and calculate the preexponential factor.
The heat of reaction for a general reaction such as:
(8)
is:
(9)
because we are dealing with a unimolecular reaction the general reaction equation reduces to:
(10)
and thus the heat of reaction would be:
(11)
We can use this equation as is if the program we are using gives us a heat of formation of the reactant, transition state, and product, but if the program does not, then we will have to resort to a slightly modified version of the above heat of reaction equation. This modification does not mean that we are calculating something different it just means that we are calculating the heat of reaction differently.
Some programs do not calculate the heat of formation but instead quantum mechanically calculate what we will call the ground state energy (GSE). This GSE is usually calculated in two different ways which only differ from each other in what the programs define as zero energy. Spartan, for example, defines zero energy of the system as the energy associated with all of the nuclei and electrons an infinite distance from each other. In other words, zero energy is when there are no interactions between any subatomic particles in the system. Thus Spartan calculates the amount of energy needed or released when all of the subatomic particles are brought together to form the molecule or molecules. These calculations usually are characterized by a large negative GSEs. As an example you could think of disassociating all of the subatomic particles of the vinyl allyl ether molecule to an infinite distance from each other and realize that one would need a lot of energy to do that, thus when you do just the opposite a lot of energy will be given off hence the large negative GSE. An alternate way of calculating the GSE is to reference molecular or atomic fragments as zero energy (just as Cerius2 does) and calculate the energy needed or released by bringing the fragments together. Atomic fragments are defined usually as an atom in its natural bonding state (e.g., hydrogen with one proton and one electron or carbon with four sp3 hybridized orbitals with one electron residing in each hybrid orbital).
Another little speed bump is that the GSE does not take into account the energy contained in the vibration, rotation, and translation of the molecule. This energy can be accounted for by using statistical mechanics (the program will do this). In statistical mechanics one can use the molecular frequencies calculated by the program and calculate thermodynamic quantities. We will be using the statistical mechanically calculated enthalpy and entropy, this enthalpy and entropy only takes into account the energy in the vibrations, rotations, and translations so it is not representative of the total energy of the molecule, because a large portion of the energy in a molecule is contained in its bonds and physical conformation. That is why we must take the sum of the GSE and the statistical mechanically calculated enthalpy to arrive at a closer approximation of the true energy of the molecule. With that said we will now define a slightly modified version of the heat of reaction equation:
The enthalpy of a species will be defined as:
(12)
where the superscript SM denotes the statistical mechanically calculated enthalpy. Substituting this definition into the usual definition of the heat of reaction we arrive at:
(13)
= -18.63 kcal/mol
The entropy can be calculated in the normal fashion by taking the difference of the products and reactants entropy:
(14)
From the tabularized data above:
Therefore:
= 0.34 cal/mol
Since the heat of reaction is not of the usual form the calculation of the Gibbs free energy will need to be calculated by the following method:
From the tabularized values above:
Therefore:
= -18.73 kcal/mol
The equilibrium constant can be calculated using the Gibbs free energy change of the reaction:
(15)
From the tabularized values above:
Therefore:
The activation energy of this reaction can be calculated by:
The preexponential factor (A) can be calculated using a bit of transition state theory that will be explained more in depth in the next example. For now the equation to calculate the preexponential factor is:
(16)
First, to calculate the change of entropy between the transition state and the reactants:
(17)
With values from the tables above:
Therefore:
= -6.67 cal/mol K
Now the preexponential is calculated as:
Which equals 2.165E11 reciprical seconds
