Calculation of ΔHf

The SCF calculation produces a density, P, and Fock matrix, F. These, together with the one-electron matrix, H, allow the total electronic energy to be calculated via

\begin{displaymath}E_{elect} = \frac{1}{2}\sum_{\mu}\sum_{\nu}P_{\mu\nu}(H_{\mu\nu}+F_{\mu\nu}). \end{displaymath}

The total core-core repulsion energy is given by:

\begin{displaymath}E_{nuc} = \sum_A\sum_{B<A}E_N(A,B). \end{displaymath}

The addition of these two terms represents the energy released when the ionized atoms and valence electrons combine to form a molecule.

A more useful quantity is the heat of formation of the compound from its elements in their standard state. This is obtained when the energy required to ionize the valence electrons of the atoms involved (calculated using semiempirical parameters), Eisol(A), and heat of formation for a gaseous atom from its standard state, Eatom(A), are added to the electronic plus nuclear energy. This yields:

ΔHf = (Eelect + Enuc) - ΣAEisol(A) + ΣAEatom(A),

or

ΔHf = Etot  - ΣAEisol(A) + ΣAEatom(A).

This is the quantity which MOPAC calls the "Heat of Formation". An alternative but equivalent definition of ΔHf , more suited for comparison with experimental ΔHf, is:

" ΔHf is the calculated gas-phase heat of formation at 298K of one mole of a compound from its elements in their standard state."

Things to note about this definition: unlike ab initio methods, which yield the energy at 0K, semiempirical methods give ΔHf  at 298K. This follows from the way in which semiempirical methods are parameterized: the reference ΔHf  are conventionally given at 298K. This means that semiempirical methods will reproduce ΔHf for 298K. Secondly, note that ΔHf  are for gas-phase systems. To calculate ΔHf  in the liquid or solid phases, additional terms are necessary.