Computational Physics

64 The Hartree–Fock method

It is convenient to introduce the density matrix which (for RHF) is defined as

Ppq= 2

∑

k

CpkC∗qk. (4.68)

This is the matrix representation of the operator

ρ= 2

∑

k

|φk〉〈φk| (4.69)

which is recognised as the usual definition of the density matrix in quantum theory
(the factor 2 is due to the spin). Using(4.68), the Fock matrix can be written as

Fpq=hpq+

1

2

∑

rs

Psr[ 2 〈pr|g|qs〉−〈pr|g|sq〉], (4.70)

and the energy is given by

E=

∑

pq

Ppqhpq+

1

2

∑

pqrs

PpqPsr

[

〈pr|g|qs〉−

1

2

〈pr|g|sq〉

]

. (4.71)

For the UHF case, it is convenient to define an orbital basisχp(r)and the spin-
up orbitals are now represented by the vectorC+and the spin-down ones byC−.
Using these vectors to reformulate the Hartree–Fock equations, the so-called Pople–
Nesbet equations are obtained:

F+C+=+SC+

F−C−=−SC−

(4.72)

with

F±pq=hpq+

∑

k±

∑

rs

Crk±∗±C±sk±[〈pr|g|qs〉−〈pr|g|sq〉]

+

∑

k∓

∑

rs

Crk∓∗∓Csk∓∓〈pr|g|qs〉.

(4.73)

In practice, real orbital basis functions are used, so that complex conjugates can be
removed from theCpkinEqs. (4.65),(4.68)and(4.73). In the following, we shall
restrict ourselves to RHF.

4.6.2 Basis functions: STO and GTO

In this subsection, we discuss the basis functions used in the atomic and molecular
Hartree–Fock programs. As already noted inChapter 3, the basis must be chosen
carefully: the matrix diagonalisation we must perform scales with the third power
of the number of basis functions, so a small basis set is desirable which is able to