Computational Chemistry

attraction potential energy terms, and electron–electron repulsion potential energy
terms (cf. Fig.5.1). This is actually theelectronicHamiltonian, since nucleus-nucleus
repulsion potential energy terms have been omitted; from the Born–Oppenheimer
approximation (Section 2.3) these can simply be added to the electronic energy after
this has been calculated, giving the total molecular energy for a molecule with
“frozen nuclei” (calculation of the vibrational energy, the zero-point energy, is
discussed later). Calculation of the internuclear potential energy is trivial:

VNN¼

X

allm;n

ZmZn

rmn

ð 5 : 16 Þ

Substituting into Eq.5.14the Slater determinant and the molecular Hamiltonian
gives, after much algebraic manipulation

E¼ 2

Xn

i¼ 1

Hiiþ

Xn

i¼ 1

Xn

j¼ 1

ð 2 Jij#KijÞð 5 : 17 Þ

for the electronic energy of a 2n-electron molecule (the sums are over then
occupied spatial orbitalsc). The terms in Eq.5.17have these meanings:

Hii¼

Z

c$ið 1 ÞH^

core

ð 1 Þcið 1 Þdv ð 5 : 18 Þ

where

H^coreð 1 Þ¼#^1

2

r^21 #

X

allm

Zm

rm 1

ð 5 : 19 Þ

The operatorH^coreis so called because it leads toHii, the electronic energy of a
single electron moving simply under the attraction of a nuclear “core”, with all the
other electrons stripped away;Hiiis the electronic energy of, for example, H, He+,

Hþ 2 , or CH^94 þ(of course, it is different for these various species). Note thatH^core(1)
represents the kinetic energy of electron 1 plus the potential energy of attraction of
that electron to each of the nucleim; the 1 in parentheses in these equations is just
a label showing that the same electron is being considered inc$i,ciandH^core
(we could have used, say, 2 instead). The integration in Eq.5.18is respect to spatial
coordinates only, (dv¼dxdydz, notdt) because spin coordinates have been “inte-
grated out”: on integration, i.e. on summation over the discrete spin variable, these
give 0 or 1 [ 12 , 14 ]. We are left with the three spatial coordinates as integration
variables (x,y,z) for the electron and so the integral (5.18) is threefold.

Jij¼

Z

c$ið 1 Þcið 1 Þ

1

r 12



c$jð 2 Þcjð 2 Þdv 1 dv 2 ð 5 : 20 Þ

186 5 Ab initio Calculations