66 The Hartree–Fock method
Figure 4.2. Positions in the Gaussian product theorem(4.79).These functions are calledprimitive basis functionsfor reasons which will be
explained below. These Gaussian type orbitals (GTO) have the nice property that
the product of two such functions centred on different nuclei again has a Gaussian
form as in(4.78):
PM(x,y,z)e−α(r−RA)2
QN(x,y,z)e−β(r−RB)2
=RN+M(x,y,z)e−(α+β)(r−RP)2
.
(4.79)
Here,RPis the ‘centre of mass’ of the two pointsRAandRBwith massesαandβ:
RP=
αRA+βRB
α+β(4.80)
(see also Figure 4.2), andRM+Nis a polynomial of degreeM+N. Equation (4.79) is
easy to prove; it is known as the ‘Gaussian product theorem’. This theorem makes
it possible for the integrals involved in the Hartree–Fock equations to be either
calculated analytically or reduced to an expression suitable for fast evaluation on a
computer. In Section 4.8 we shall derive some of these integrals.
The polynomialsPMinEq. (4.78)contain the angular-dependent part of the
orbitals, which is given by the spherical harmonicsYml(θ,φ).Forl=0, these
functions are spherically symmetric (no angular dependence) – hence a 1s-orbital
(having no nodes) is given as
χα(s)(r)=e−α(r−RA)
2. (4.81)
Note that we need not normalise our basis functions: the overlap matrix will ensure
proper normalisation of the final molecular orbitals. Forl =1, theLz-quantum
numbermcan take on the three different values 1, 0 and−1, so there are three
p-orbitals, and an explicit GTO representation is
χα(px)(r)=xe−α(r−RA)
2
, (4.82)and similarly for pyand pz. Proceeding in the same fashion forl=2, we find six
quadratic factorsx^2 ,y^2 ,z^2 ,xy,yzandxzbefore the Gaussian exponential, but there