76 The Hartree–Fock method
The functionF 0 (t)can be evaluated using the error function erf, which is avail-
able in most high-level programming languages as an intrinsic funtion. The error
function is defined by
erf(x)=
2
√
π
∫x
0
dx′e−x
′^2
. (4.116)
If your compiler does not have the error function as an intrinsic, you can calculate
it using a recursive procedure. The functionF 0 (t)is then considered as one in a
series of functions defined as
Fm(t)=
∫ 1
0
exp(−ts^2 )s^2 mds. (4.117)
The following recursion relation for Fm(t)can easily be derived via partial
integration:
Fm(x)=
e−x+ 2 xFm+ 1 (x)
2 m+ 1
. (4.118)
This recursion is stable only if performed downward (seeAppendix A2).
The two-electron integral:This has the form:
〈1s,α,A; 1s,β,B|g|1s,γ,C; 1s,δ,D〉
=
∫
d^3 r 1 d^3 r 2 e−α|r^1 −RA|
2
e−β|r^2 −RB|
2 1
r 12
e−γ|r^1 −RC|
2
e−δ|r^2 −RD|
2
. (4.119)
Using the Gaussian product theorem, we can write the Gaussian functions depend-
ing onr 1 andr 2 as new 1s-functions with exponential parametersρandσand
centresRP(ofRAandRC) andRQ(ofRBandRD):
〈1s,α,A; 1s,β,B|g|1s,γ,C; 1s,δ,D〉
=exp[−αγ /(α+γ)|RA−RC|^2 −βδ/(β+δ)|RB−RD|^2 ]
×
∫
d^3 r 1 d^3 r 2 e−ρ|r^1 −RP|
2 1
r 12
e−σ|r^2 −RQ|
2
. (4.120)
CallingMthe factor in front of the integral, and replacing the Gaussian exponentials
in the integrand by their Fourier transforms, and similarly for the 1/r 12 term, we
obtain
〈1s,α,A; 1s,β,B|g|1s,γ,C; 1s,δ,D〉
=M( 2 π)−^9
∫
d^3 r 1 d^3 r 2 d^3 k 1 d^3 k 2 d^3 k 3 (π/ρ)^3 /^2 e−k
(^21) / 4 ρ
eik^1 ·(r^1 −RP)
× 4 πk− 22 eik^2 ·(r^1 −r^2 )(π/σ )^3 /^2 e−k
32 /^4 σ
eik^3 ·(r^2 −RQ). (4.121)