Calculation of heat of formation

The heat of formation is defined as:

DH_f = E_elect + E_nuc - E_isol + E_atom

where E_elect is the electronic energy, E_nuc is the nuclear-nuclear repulsion energy, -E_isol is the energy required to strip all the valence electrons off all the atoms in the system, and E_atom is the total heat of atomization of all the atoms in the system.

(For a single atom, there is no nuclear term, therefore:

DH_f = E_elect - E_isol + E_atom

But E_elect = E_isol for an isolated atom, therefore

DH_f = E_atom

where E_atom 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.)

E_elect is calculated from $frac{1}{2}{bf P(H + F)}$ , or

$begin{displaymath}E_{elect} = frac{1}{2}sum_{lambda=1}^6sum_{sigma=1}^6P_{lambdasigma}(H_{lambdasigma} + F_{lambdasigma}).end{displaymath}$

Using the data we have already derived, we can calculate E_elect as:

E_elect	=	3(+1.0000)(-51.7124 + -5.4823)
		+ 6(+0.6667)( -3.2457 + -6.4652)
		+ 3(-0.3333)( -0.6992 + -0.3611)

$begin{displaymath}E_{elect} = -210.0898 {rm eV}.end{displaymath}$

E_nuc is a relatively straightforward calculation, and is equal to 130.2902eV. The total energy of the system is thus -79.7996 eV.

We are now ready to calculate the DH_f. As the total energy and E_isol are in eV, we must first convert them into kcal/mol:

$begin{displaymath}Delta H_f = 23.061( -79.7996 + 71.4377) + 6(52.1020){rm kcal/mol}end{displaymath}$

DH_f = 119.780 kcal/mol.

It is convenient to combine E_isol and E_atom together, to simplify this calculation. In order to convert any total energy (E_elect + E_nuc)into a DH_f, the following operation must be performed:

$begin{displaymath}Delta H_f = 23.061(E_{elect} + E_{nuc} -sum_i E_{(isol-atom)} )end{displaymath}$

in which the index i is over all atoms in the system.

Users of MOPAC may wish to verify this calculation for a system of their own choice. To facilitate this, the data in the Table may prove useful.

Table: Values of E_(isol-atom)

Element	E_(isol-atom) (eV/atom)
	MINDO/3	MNDO	AM1	PM3
Hydrogen	-14.764312	-14.165588	-13.655739	-15.332633
Lithium		-6.793583
Beryllium		-27.541992
Boron	-67.584394	-70.200344	-69.601659
Carbon	-126.880346	-127.910952	-128.226140	-118.640263
Nitrogen	-192.410048	-207.466249	-207.307791	-162.513823
Oxygen	-309.652672	-320.451178	-318.682192	-291.924879
Fluorine	-475.817831	-477.502913	-483.109715	-438.336301
Aluminum		-47.931017		-50.311708
Silicon	-95.576505	-87.539565	-83.701885	-72.488357
Phosphorus	-154.270388	-156.236921	-124.436836	-121.236135
Sulfur	-231.996798	-228.891710		-186.333060
Chlorine	-347.185366	-354.374768	-373.455532	-316.452049
Zinc		-31.231065
Germanium		-80.129955
Bromine		-347.840783	-353.473742	-353.699430
Tin		-95.454929
Iodine		-341.704860	-347.970786	-289.422586
Mercury		-29.456154
Lead		-107.856099

These numbers may be used in conjunction with the semiempirical electronic and nuclear energies to calculate the heat of formation.