Essentials of Computational Chemistry

126 4 FOUNDATIONS OF MOLECULAR ORBITAL THEORY
and carry out the same evaluation of interelectronic repulsion we have
1
SD SDdr1dω1dr2dω2
r12
= 1
|ψa(1)|
2
2 2 1
|α(1)| |ψb(2)|
r12
2 |β(2)| 2 dr1dω1dr2dω2
− 2 ψa(1)ψb(1)α(1)β(1) 1
ψb(2)ψa(2)α(2)β(2)dr1dω1dr2dω2
r12
+ |ψa(2)| 2 2 1
|α(2)| |ψb(1)|
r12
2 |β(1)| 2
dr1dω1dr2dω2
= 1
2
2 1
|ψa(1)| |ψb(2)|
r12
2 dr1dr2
− 2 · 0
2 1
+ |ψa(2)| |ψb(1)|
r12
2
dr1dr2
= 1
2 (Jab + Jab)
= Jab
(4.51)
Note that the disappearance of the exchange correlation derives from the orthogonality of
the α and β spin functions, which causes the second integral in the second equality to be
zero when integrated over either spin coordinate.
4.5.5 The Hartree-Fock Self-consistent Field Method
Fock first proposed the extension of Hartree’s SCF procedure to Slater determinantal wave
functions. Just as with Hartree product orbitals, the HF MOs can be individually determined
as eigenfunctions of a set of one-electron operators, but now the interaction of each electron
with the static field of all of the other electrons (this being the basis of the SCF approximation)
includes exchange effects on the Coulomb repulsion. Some years later, in a paper
that was critical to the further development of practical computation, Roothaan described
matrix algebraic equations that permitted HF calculations to be carried out using a basis set
representation for the MOs (Roothaan 1951; for historical insights, see Zerner 2000). We
will forego a formal derivation of all aspects of the HF equations, and simply present them
in their typical form for closed-shell systems (i.e., all electrons spin-paired, two per occupied
orbital) with wave functions represented as a single Slater determinant. This formalism is
called ‘restricted Hartree-Fock’ (RHF); alternative formalisms are discussed in Chapter 6.
The one-electron Fock operator is defined for each electron i as
fi =− 1
2 ∇2 i −
nuclei Zk
+ V
rik
k
HF
i {j} (4.52)