Computational Physics

4.8 Integrals involving Gaussian functions 73

The convergence can finally be enhanced by extrapolating the values of the density
matrix. Various extrapolation schemes are used, one of the most popular being
Pulay’s DIIS scheme (see Section 9.4) [14].
Output:Output of the program are the Fock levels and corresponding eigen-
states. From these data, the total energy can be determined:

E=

1

2

[

∑

rs

hrsPrs+

∑

k

k

]

+Enucl. (4.96)

kare the Fock levels andEnucl represents the electrostatic nuclear repulsion
energy which is determined by the nuclear chargesZnand positionsRn. Invoking
Koopman’s theorem (see Section 4.5.3), the Fock levels may be interpreted as
electron removal or addition energies.

In Problem 4.12, the hydrogen molecule is treated again in Hartree–Fock rather
than in Hartree theory, as is done in Problem 4.9.

*4.8 Integrals involving Gaussian functions

In this section, we describe some simple calculations of integrals involving two or
four GTOs. We restrict ourselves to 1s-functions; for matrix elements involving
higherl-values,seeRefs.[ 15 , 16 ].Asnoticedalreadyin Section 4.6.2, the central
result which is used in these calculations is the Gaussian product theorem: denoting
the Gaussian function exp(−α|r−RA|^2 )byg1s,α(r−RA),wehave:

g1s,α(r−RA)g1s,β(r−RB)=Kg1s,γ(r−RP), (4.97)

with

K=exp

[

−

αβ

α+β

|RA−RB|^2

]

γ=α+β

RP=

αRA+βRB

α+β

. (4.98)

From now on, we shall use the Dirac notation:
g1s,α(r−RA)=|1s,α,A〉. (4.99)
The overlap integral:The overlap matrix for two 1s-functions can be calculated
directly using(4.97):

〈1s,α,A|1s,β,B〉= 4 π

∫

drr^2 Ke−γr

2

=

(

π

α+β

) 3 / 2

exp

[

−

αβ

α+β

|RA−RB|^2

]

. (4.100)