There are two isomers of 1,2-dichloroethylene, cis (or Z) and trans (or E). Suppose we would like to know which isomer is more stable, in the gas phase, at the two temperatures 298.15 K (warm room temperature) and 1200 K (incinerator temperature). Then the task is to compute the free energy change accompanying the following reaction.
cis-CHCl=CHCl → trans-CHCl=CHCl
There are many reasonable choices of quantum calculation to use. For this example, we'll begin with inexpensive density-functional calculations using the hybrid B3LYP functional and the 6-31G(d) basis sets. Geometry optimizations of the cis (C2v point group) and trans (C2h point group) give the electronic energies -997.778563 and -997.778378 hartree, respectively. This corresponds to an energy difference of 0.5 kJ/mol (1 hartree = 2625.5 kJ/mol), suggesting that the cis isomer is slightly more stable. However, these are "equilibrium" energies, which do not correspond to any physically realizable situation.
The vibrational zero-point energy (ZPE) must be added to each molecular energy to find the energies at absolute zero temperature (0 K). Running the Gaussian03 [1,2] output files through the script thermo.pl yields ZPEs of 88.4 kJ/mol and 87.5 kJ/mol for cis and trans, respectively. So the enthalpy (and free energy) change for the reaction is -0.4 kJ/mol at 0 K, slightly favoring the trans isomer. The vibrational frequencies were scaled by 0.9806 as recommended by Scott and Radom for ZPEs , but using unscaled frequencies has a negligible effect on the reaction energy.
At temperatures above absolute zero, thermal corrections are needed. This is simply because a warm molecule has more energy that a cold one. Using unscaled vibrational frequencies, the enthalpy corrections and entropy values are listed the table below. They were obtained, as before, using thermo.pl. Note that 1200 K is higher than the default temperature range for the script. The listed result was obtained by specifying "1200" as the "Special temperature to include in output table".
Combined with the results for 0 K, the following results are obtained for the reaction energetics.
Since these changes are close to zero, we might want to do further calculations in the hope of obtaining more confidence in the predictions. For example, we might compute equilibrium energies using the coupled-cluster method CCSD(T)/cc-pV(T+d)Z, accepting the B3LYP geometries. For the cis and trans isomers, the CCSD(T) energies are -996.710765 and -996.710265 hartree, respectively, corresponding to an equilibrium reaction energy of 1.3 kJ/mol. This is 0.8 kJ/mol higher than the B3LYP value of 0.5 kJ/mol, above. Thus, these CCSD(T) energies suggest that ΔrG = 0.8 kJ/mol at 298.15 K and -0.6 kJ/mol at 1200 K. For comparison, the experimentally derived values are 2.5 ± 0.4 kJ/mol at 298.15 K and 1.6 ± 0.8 kJ/mol at 1200 K .