Computational Chemistry

Jis called a coulomb integral; it represents the electrostatic (i.e. coulombic)
repulsion between an electron inciand one incj, i.e. between the charge clouds of
orbitalsciandcj. This may be clearer if one considers the integral as a sum of
potential energy terms involving repulsion between infinitesimal volume elements
dv(Fig.5.3). The 1 and 2 are just labels showing we are considering two electrons.
The integralsJandKallow each electron to experience the average electrostatic
repulsion of a charge cloud due to all the other electrons. This pretence that
electron–electron repulsion occurs between an electron and a charge cloud rather
than between all possible pairs of electrons as point particles isthe major deficiency
of the Hartree–Fock methodand transcending this approximation is the reason for
the development of the post-Hartree–Fock methods discussed later. SinceJrepre-
sents potential energy corresponding to a destabilizing electrostatic repulsion, it is
positive. As forHiiin Eq.5.18, the integration is with respect to spatial coordinates
because the spin coordinates have been integrated out. There are six integration
variables,x,y,zfor electron 1 (dv 1 ) andx,y,zfor electron 2 (dv 2 ), and so the integral
(5.20) is sixfold. Note that the ab initio coulomb integralJis not the same as what we
called a coulomb integral in simple H€uckel theory; that wasa¼

R

fiH^fidv
(Eq. 4.61a) and represents at least very crudely the energy of an electron in the
porbitalfi(Section 4.3.4). The ab initio coulomb integral can also be written

Jij¼

Z

c$ið 1 Þc$jð 2 Þ

1

r 12



cið 1 Þcjð 2 Þdv 1 dv 2 ð 5 : 21 Þ

but unlike Eq.5.20this does not notationally emphasize the repulsion (invoked
by the 1/r 12 operator) between electron 1 and electron 2, on the left and right,
respectively, of 1/r 12 in Eq.5.20.

yi

yj

contains electron 1 contains electron 2

dv 2

Potential energy between dv 1 and dv 2 is yi (1) yi (1) dv 1 yj (2) yj (2) dv 2

(product of the charges divided by their distance apart)

volume dv 1 contains

charge yi (1) yi (1) dv 1

volume dv 2 contains

charge yj (2) yj (2) dv 2

dv 1

r 12

1

Fig. 5.3 The coulomb integral (Jintegral) represents the electrostatic repulsion between two
charge clouds, due to electron 1 in orbitalciand electron 2 in orbitalcj.Jij¼

R

c$ið 1 Þcið 1 Þ

) 1

r 12

*

c$jð 2 Þcjð 2 Þdv 1 dv 2

5.2 The Basic Principles of the ab initio Method 187