The best way to make comparisons of stability, reaction energy, and so on, is to write an isodesmic (bond conserving) reaction that embodies the behavior of interest.

• An isodesmic reaction is one in which the numbers of bonds of each formal type are conserved, and only the relationships among the bonds are altered.

• Using isodesmic relationships is especially important if the calculations to be employed are of mixed type: that is, some RHF and some UHF.

For example, as part of a research project I want to know how easy it is for a free radical to remove the SH hydrogen from thioglycerol, HOCH₂CH(OH)CH₂SH.

• I do not attempt to calculate the bond energy by calculating the energy of thioglycerol, the thioglyceryl radical, and a hydrogen atom,
  • this would require me to compare an RHF energy (thioglycerol) and two UHF energies (the two radicals)

• I write an isodesmic reaction in which a model thiyl radical removes the SH hydrogen, and compute the energies of all of the species (3-21G*):

  

• For most accurate results, I also compute the vibrational spectra of all species at the same level.
  • This results in the zero point vibrational energies that also are listed.

• The DE for the reaction is calculated as: [-513.277265 + (-661.549729)] - [-512.679695 + (-662.149978)] = 0.001679 hartrees, or 1.05 kcal/mol.
  • The positive sign implies that the reaction as written is endothermic; that is, the thioglycerol SH hydrogen is harder to remove than the SH hydrogen of the model thiol.

  • However, the magnitude of the difference is quite small, making a correction for zero-point energy differences important.

  • Hence, I do another calculation: [69.79 + 69.69] - [63.16 + 76.48] = -0.16 kcal/mol. The corrected DE then is [1.05 - 0.16] = 0.89 kcal/mol.

• An alternative but equivalent way of using the zero point correction is simply to add the zero point correction to the calculated total energy for each species (making sure both are in the same units, of course).

• It has been suggested also that zero point energies should be scaled, just as are the frequencies from which they are calculated, to improve accuracy; a factor of 0.89 often is used. [Pople, J. A.; Scott, A. P.; Wong, M. W.; Radom, L. Israel J. Chem., 1993, 33, 345.]

A particularly useful kind of isodesmic reaction is the bond separation reaction:

• This is a formal transformation in which all bonds between heavy (non-hydrogen) atoms in the molecule of interest are separated into the simplest two-heavy-atom molecules containing these same bonds.

• The set of parents involving H, C, N, O, and F, for example consists of ethane, ethylene, acetylene, methylamine, methylene imine, hydrogen cyanide, methanol, formaldehyde, fluoromethane, hydrazine, diazene, hydroxylamine, nitroxyl, fluoramine, hydrogen peroxide, and hypofluorous acid.

• Stoichiometric balance is obtained by adding one-heavy-atom hydrides to the left side of the equation.

For example, to learn something about the stabilization afforded butadiene by conjugation, I might write the bond separation reaction:

The DE for this reaction is computed as [(2)(-77.600988) + (-78.793948)] - [(2)(-39.976878) + (-154.059457)] = 0.017289 hartrees, or 10.849 kcal/mol.

• Correction for zero point energy difference by manipulating these terms in the same way leads to a correction of 1.33 kcal/mol, for a total DE = 10.849 + 1.33 = 12.2 kcal/mol.

• The value from experimental enthalpies of formation is 11.3 kcal/mol.

Carried out as described here, these simple calculations generate energies at 0 K. Using Gaussian, one can calculate energies at other temperatures. By default, parameters are provided for 298 K. Here is a section of Gaussian output:

Note that the thermal energy correction includes the zero point correction! It is wrong to add them both.

Values for any other temperature may be obtained by using the ReadIsotopes keyword (see manual) to input another temperature.

For example, let's calculate DH at 298 K for the reaction:

The relevant numbers from B3LYP/6-31G* calculations are:

Add the row for each species, noting that the electronic energy of H⁺ is zero because it hasn't got any electrons. Further, it's thermal correction is translational only, 3/2 RT.