Heats of Formation versus Total Energies

Ab-initio and semiempirical methods represent the energy of a chemical system in two different ways. Ab-initio methods use a quantity called the "total energy", and semiempirical methods use the Heat of Formation. These quantities can be defined as follows:

Total Energy (TE) is the energy, in atomic units (Hartrees) released when the chemical system is formed from the constituent atomic nuclei and electrons. The zero of energy is the isolated atomic nuclei and isolated electrons. Ab-initio methods model the system at zero Kelvin, (0 K). No zero point energy, vibrational or translational terms are included in the TE. See Gaussian FAQ.

Heat of Formation (ΔH_f) is the change in enthalpy, in kcal mol^-1 or kj mol^-1, when one mole of a system is formed from its elements, with everything being in its standard state, i.e., at 298.15 K and one atmosphere of pressure. See HoF from SCF.

Semiempirical methods are parameterized using reference data for chemical systems in their standard state. Because the reference data represent systems at 298.15 K, any Zero Point Energy and all internal energy terms (vibrational, rotational, and translational -contributions) are automatically taken into account in the parameters, see Note on Thermochemistry. If the ΔH_f at a different temperature is needed, the internal energy at 298.15 K must be deleted, and the internal energy at the specified temperature added. The resulting quantity is the change in enthalpy, in kcal mol^-1, when one mole of a system at the specified temperature and one atmosphere of pressure is formed from its elements in their standard state, i.e., at 298.15 K and one atmosphere of pressure. This can be done using keyword THERMO.

Relationship of Total Energy to Heat of Formation

Steps in converting Total Energy to Heat of Formation

1. Add in Zero Point Energy.
2. Add in internal energy for 298.15 K.
3. Add in energy of removing valence electrons from atoms.
4. Add in energy of atomization of elements.
5. Convert from atomic units to kilocalories or Joules per mole.

     Steps in converting Heat of Formation to Total Energy

      1. Subtract off Zero Point Energy.
      2. Subtract off internal energy for 298.15 K.
    3. Subtract off energy of removing valence electrons from atoms.
      4. Subtract off energy of atomization of elements.
      5. Convert from kilocalories or Joules per mole to atomic units.

Semiempirical reference data includes all quantum and classical phenomena, therefore Zero Point Energies are automatically present, and do not need to be included in the calculation of ΔH_f. Calculation of the ZPE and internal energy can be done using the THERMO keyword. Energies of formation of atoms can be done one at a time, using individual atoms; look for "ELECTRONIC ENERGY". If this is done, take care to specify the correct ground state of the isolated atom.

Energy of atomization of elements is the experimental heat of formation of the isolated atom from the element in its standard state. For example, in nitrogen this is half the heat required to break a N₂ molecule into its separated atoms. Isolated nitrogen atoms have the configuration 1s² 2s² 2p³, and the state 4S_u, i.e. quartet S. Experimentally, this energy is 113 kcal/mol. This means that all semiempirical methods predict the heats of formation of isolated atoms with a zero error, except in those few instances when the predicted ground state is incorrect.

Which is better: Heats of Formation or Total Energies?

The choice of which quantity to use depends on what it is to be used for, thus ΔH_f are useful for practical chemistry work, and TE are useful for theoretical chemistry work. Why is this? All reactions are, by definition, balanced, so the difference in the choice of starting point for the energy calculation - whether nuclei plus electrons or elements - would cancel out. What would not cancel would be the zero point energy and internal energy needed to heat the system from 0 K to the specified temperature, typically 298.15 K. To the degree that these energies are important, if Total Energies are used, then post-SCF corrections must be made. These corrections do not need to be made in semiempirical methods, because they are already accounted for in the parameters.