Computational Physics

4.6 Basis functions 65

model the exact solutions to the Fock equations accurately. A molecule consists
of atoms which, in isolation, have a number of atomic orbitals occupied by the
electrons. If we put the atoms together in a molecule, the orbitals with low energies
will be slightly perturbed by the new environment and the valence electrons will
now orbit around more than one nucleus, thus binding the molecule together. In
the molecule, the electrons now occupymolecular orbitals(MO). In constructing a
basis, it turns out to be efficient to start from the atomic orbitals. The one-electron
wave functions that can be constructed from these orbitals are linear combinations
of atomic orbitals (LCAO). The solutions to the HF equations, which have the form

φk(r)=

∑

p

Cpkχp(r), (4.74)

are the molecular orbitals, written in LCAO form.
Analytic forms of the atomic orbitals are only known for the hydrogen atom but
they can be used for more general systems. The orbitals of the hydrogen atom have
the following form:

χ(r)=fn− 1 (r)r−lPl(x,y,z)e−r/n, (4.75)

wherelis the angular momentum quantum number,Plis a polynomial inx,yand
zof degreelcontaining the angular dependence;fn− 1 (r)is a polynomial inrof
degreen−1;nis an integer (ris expressed in atomic unitsa 0 ). This leads to the
following general form of atomic orbital basis functions:

χζ(r)=rmPl(x,y,z)e−ζ|r−RA| (4.76)

which is centred around a nucleus located atRA. Functions of this form are called
Slater type orbitals (STO). The parameterζ, defining the range of the orbital,
remains to be determined;Plis taken the same as for the hydrogen atom. Foratomic
Hartree–Fock calculations, this basis yields accurate results with a restricted basis
set size. However, in molecular calculations, integrals involving products of two
and four basis functions centred at different nuclei are needed, and these are hard
to calculate since the product of the two exponentials,

e−ζ^1 |r−RA|e−ζ^2 |r−RB|, (4.77)

is a complicated expression inr. A solution might be to evaluate these integrals
numerically, but for a large basis set this is impractical.
Another basis set which avoids this problem, was proposed in 1950 by Boys [10],
who replaced the simple exponential in (4.76) by a Gaussian function:

χα(r)=PM(x,y,z)e−α(r−RA)

2

. (4.78)