Computational Chemistry

Note that this is really a sixfold integral, since there are three variables (x 1 ,y 1 ,z 1 )
for electron 1, and three (x 2 ,y 2 ,z 2 ) for electron 2, represented bydv 1 anddv 2
respectively. This equation can be written more compactly as

frð 1 ÞjJ^jð 1 Þfsð 1 Þ

¼

Xm

t¼ 1

Xm

u¼ 1

c$tjcujðrsjtuÞð 5 : 72 Þ

The notation

ðrsjtuÞ¼

ZZ

f$rð 1 Þfsð 1 Þf$tð 2 Þfuð 2 Þ

r 12

dv 1 dv 2 ð 5 : 73 Þ

is a common shorthand for this kind of integral, which is called atwo-electron
repulsion integral(or two-electron integral, or electron repulsion integral); the phy-
sical significance of these is outlined in Section 5.2.3.6.4). This parentheses notation
should not be confused with the Dirac bra-ket notation,hijða braÞandj ða ketÞ:
by definition

hifjg ¼

Z

f$ðqÞgðqÞdq ð 5 : 74 Þ

so

hirsjtu ¼

Z

ðfrð 1 Þfsð 1 ÞÞ$ftð 1 Þfuð 1 Þdv 1 ð 5 : 75 Þ

Actually, several notations have been used for the integrals of Eq.5.73and for
other integrals; make sure to ascertain which symbolism a particular author is using.
The second integral, from Eq. 5.66, is

K^jð 1 Þfsð 1 Þ¼cjð 1 Þ

Z

c$jð 2 Þfsð 2 Þ

r 12

dv 2

Substituting forcj(1) the basis function expansion∑cujfu(1) and forc*j(2) the
expansion

P

c$tjf$tð 2 Þ(cf. Eq.5.52):

K^jð 1 Þfsð 1 Þ¼fuð 1 Þ

Xm

t¼ 1

Xm

u¼ 1

c$tjcuj

Z

f$tð 2 Þfsð 2 Þ

r 12

dv 2

To get the desired expression for frð 1 ÞK^ð 1 Þfsð 1 Þ
we multiply this byf$r(1)
and integrate with respect to the coordinates of electron 1:

frð 1 ÞjK^jð 1 Þfsð 1 Þ

¼

Xm

t¼ 1

Xm

u¼ 1

c$tjcuj

ZZ

f$rð 1 Þfuð 1 Þf$tð 2 Þfsð 2 Þ

r 12

dv 1 dv 2

208 5 Ab initio Calculations