Computational Chemistry

special case of a molecule) spin orbitalscaandcb(cf. the minimization of energy
with respect to basis function coefficients in Section 4.3.4). The derivation of these
equations involves considerable algebraic manipulation, which is at times hard to
follow without actually writing out the intermediate expressions. The procedure has
been summarized by Pople and Beveridge [ 12 ], and a less condensed account is
given by Lowe [ 13 ].
It follows from the Schr€odinger equation that the energy of a system is given by

E¼

R

C$H^Cdt

R

C$Cdt

ð 5 : 13 Þ

This is similar to Eq. 4.40 inChapter 4, but here the total wavefunctionChas
been specified, and allowance has been made for the possibility ofCbeing a
complex function by utilizing its complex conjugateC; this ensures thatE, the
energy of the atom or molecule, will be real. IfCis complex thenC^2 dtwill not be
a real number, whileCCdt¼|C|^2 dtwill, as must be the case for a probability.
Integration is with respect to three spatial coordinates and one spin coordinate, for
each electron. This is symbolized bydt(t¼Greektau), which meansdxdydzdx, so
for a 2n-electron system these integrals are actually 4' 2 n-fold, each electron
having its set of four coordinates. We assume the use of orthonormal functions
(Section 4.3.4), since this makes several integrals disappear in the derivation of the
energy. Working with the usual normalized wavefunctions makes the denominator
unity, and Eq.5.13can then be written

E¼

Z

C$H^Cdt

or using the more compact Dirac notation for integrals (Section 4.4.1)

E¼ CjH^jC

ð 5 : 14 Þ

In Eq.5.14it is understood that the firstCis actuallyC, and that the integration
variables are the space and spin coordinates. The vertical bars are only to visually
separate the operator from the two functions, supposedly for clarity.
We next substitute into Eq.5.14the Slater determinant forC(andC) and the
explicit expression for the Hamiltonian. A simple extension of the helium Hamil-
tonian of Eq.5.5to a molecule with 2nelectrons andmatomic nuclei (themth
nucleus has chargeZm) gives

H^¼

X^2 n

i¼ 1

1

2

r^2 i#

X

allm;i

Zm

rmi

þ

X

alli;j

1

rij

ð 5 : 15 Þ

Just like the helium Hamiltonian, the molecular HamiltonianH^in Eq.5.15is
composed (from left to right) of electron kinetic energy terms, nucleus–electron

5.2 The Basic Principles of the ab initio Method 185