Computational Physics

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)