Computational Chemistry

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