that will be used to transform the Fock matrixFtoF^0 and to convert the transformed
coefficient matrixC^0 toC(Eqs.5.67–5.70). The integrals are those required for
Hcore, the one-electron part of the elementsFrsofF, and the two-electron repulsion
integrals (rs|tu), (ru|ts) (Eq.5.82), as well as the overlap integrals, which are needed
to calculate the overlap matrixS and thus the orthogonalizing matrix S#1/2
(Eq.5.67).
Efficient methods have been developed for calculating these integrals [ 32 ] and
their values will simply be given later. For our calculation the elementsFrsof the
Fock matrix (Eq.5.82) are conveniently written
Frs¼Hcorers ð 1 Þþ
Xm
t¼ 1
Xm
u¼ 1
Ptu ðrsjtuÞ#
1
2
ðrujtsÞ
¼TrsþVrsðHÞþVrsðHeÞþGrs
ð 5 : 100 Þ
HereHcore(1) has been dissected into a kinetic energy integral Tand two
potential energy integrals,V(H) andV(He). From the definition of the operator
H^core(Eq.5.64¼5.19) and the Roothaan–Hall expression for the integralHcore
(Eq.5.79) we see that (the (1) emphasizes that these integrals involve the coordi-
nates of only one electron):
Trsð 1 Þ¼
Z
fr #
1
2
r^21
fsdv
¼
Z
fr#
1
2
@^2
@x^2
þ
@^2
@y^2
þ
@^2
@z^2
fsdv
ð 5 : 101 Þ
0
0.5
1.0
2.0
3.0
0.370
0.333
0.244
0.070
0.009
0.588
0.485
0.271
0.027
0.0006
0
0.2
0.4
0.6
123
f(x, y, z) = f(|r–R|)
f(He)
f(H)
|r–R| Å
|r–R| f(H) = 0.3696exp(-0.4166|r–R 1 |) f(He) = 0.5881exp(– 0.7739|r–R 2 |)
Fig. 5.8 Electron density around the helium nucleus falls off more quickly than electron density
around the lower-charge hydrogen nucleus
216 5 Ab initio Calculations