Computational Chemistry

we get from Eq.5.86

EHF¼

Xn

i¼ 1

eiþ

Xm

r¼ 1

Xm

s¼ 1

Xn

i¼ 1

c$ricsiHcorers ð 5 : 89 Þ

Using Eq.5.81, Eq.5.89can be written in terms of the density matrix ele-
mentsP:

EHF¼

Xn

i¼ 1

eiþ

1

2

Xm

r¼ 1

Xm

s¼ 1

PrsHcorers ð 5 : 90 Þ

This is the key equation for calculating the HF electronic energy of a molecule. It
can be used when self-consistency has been reached, or after each SCF cycle
employing thee’s andc’s yielded by that particular iteration, andHcorers , which
latter does not change from iteration to iteration, since it is composed only of the
fixed basis functions and an operator which does not containe’s orc’s: from
Eqs.5.64¼5.19and5.79

Hcorers ¼ frj#

1

2

r^2 i#

X

allm

Zm

rmi

jfs

*+

ð 5 : 91 Þ

Hcorers does not change because the SCF procedure refines the electron-electron
repulsion (till the field each electron feels is “consistent” with the previous one),
butHrscorein contrast represents only the contribution to the kinetic energy plus
electron– nucleus attraction of the electron density associated with each pair of
basis functionsfrandfs.
Equation5.90gives the HF electronic energy of the molecule or atom – the
energy of the electrons due to their motion (their kinetic energy) plus their energy
due to electron–nucleus attraction and (within the HF approximation) to electron–
electron repulsion (their potential energy). Thetotalenergy of the molecule,
however, involves not just the electrons but also the nuclei, which contribute
potential energy due to internuclear repulsion and kinetic energy due to nuclear
motion. This motion persists even at 0 K, because the molecule vibrates even at this
temperature; this unavoidable vibrational energy is called thezero point vibrational
energyorzero point energy(ZPVE or ZPE; Section 2.5, Fig. 2.20 and associated
discussion). Calculation of the internuclear repulsion energy is trivial, as this is just
the sum of all coulombic repulsions between the nuclei:

VNN¼

X

allm;v

ZmZv

rmv

ð 5 : 92 ¼ 5 : 16 Þ

Calculation of the ZPE is more involved; it requires calculating the frequencies
(i.e. the normal-mode vibrations – Section 2.5) and summing the energies of each

212 5 Ab initio Calculations