Computational Physics

4.7 The structure of a Hartree–Fock computer program 69

point for this calculation is the atomic orbitals of the ground state of the isolated
H-atom. In the latter, only the (spherically symmetric) 1s orbitals are filled, and in
the H 2 molecule, these orbitals will merge into a single molecular orbital which is
given by the sum of the two atomic orbitals plus a correction containing a substantial
contribution from the atomic pz-orbitals (the axis connecting the two nuclei is taken
to be thez-axis). This shows that it is sensible to include these pz-orbitals in the
basis, even though they are not occupied in the ground state of the isolated atom.
Such basis states, which are included in the basis set to make it possible for the
basis to represent the polarisation of the atom by its anisotropic environment, are
calledpolarisation orbitals. When calculating (dipole, quadrupole, ...) moments by
switching on an electric or magnetic field and studying the response of the orbitals
to this field, it is essential to include such states into the basis.

4.7 The structure of a Hartree–Fock computer program

In this section we describe the structure of a typical computer program for solv-
ing the Roothaan equations. The program for the helium atom as described in
Section 4.3.2 contained most of the features present in Hartree–Fock programs
already – the treatment given here is a generalisation for arbitrary molecules.
As the most time-consuming steps in this program involve the two-electron integ-
rals, we consider these in some detail in the next subsection. The general scheme
of the HF program is then given in Section 4.7.2.

4.7.1 The two-electron integrals

The two-electron integrals are the quantities

〈pr|g|qs〉=

∫

d^3 rd^3 r′χp(r)χr(r′)

1

|r−r′|

χq(r)χs(r′). (4.86)

(Do not confuse thelabel rwith the orbital coordinater!) The two-electron matrix
elements obey the following symmetry relations:

p↔qr↔sp,q↔r,s. (4.87)

This implies that, starting withK basis functions, there are roughlyK^4 /8two-
electron matrix elements, since each of the symmetries in (4.87) allows the reduction
of the number of different matrix elements to be stored by a factor of about 2.
The subset of two-electron matrix elements to be calculated can be selected in
the following way. Becausepandqcan be interchanged, we can takep≥q. As the
pairp,qcan be interchanged with the pairr,s, we may also takep≥r. The range
ofsdepends on whetherp=ror not. Ifp
=r,wehaves≤r, but forp=r, the