Computational Chemistry

to fluctuate, so that even a spherical atom can have temporary nonzero dipole
moments. Another point is that we usually consider the dipole moments ofneutral
molecules only, not of ions, because the dipole moment of a charged species is not
unique, but depends on the choice of the point in the coordinate system from which
the position vectors are measured.
Let us look at the calculation of the dipole moment within the Hartree–Fock
approximation. The quantum mechanical analogue of Eq. (5.205) for the electrons
in a molecule is

m¼ Cj

X^2 n

j¼ 1

erjjC

*+

ð 5 : 206 Þ

Here the summation of charges times position vectors is replaced by the integral
over the total wavefunctionC(the square of the wavefunction is a measure of
charge) of the dipole moment operator (the summation over all electrons of the
product of an electronic charge and the position vectors of the electrons). To
perform an ab initio calculation of the dipole moment of a molecule we want an
expression for the moment in terms of the basis functionsf, their coefficientsc, and
the geometry (for a molecule of specified charge and multiplicity these are the only
“variables” in an ab initio calculation). The Hartree–Fock total wavefunctionCis
composed of those component orbitalscwhich are occupied, assembled into a
Slater determinant (Section 5.2.3.1), and thec’s are composed of basis functions
and their coefficients (Sections 5.3). Equation (5.206), with the inclusion of the
contribution of the nuclei to the dipole moment, leads to the dipole moment in
Debyes as (ref. [ 1 g], p. 41)

m¼# 2 : 5416

XN

A

ZARA

Xm

r

Xm

s

Prshifrjrjfs

"

ð 5 : 207 Þ

Here the first term refers to the nuclei charges and position vectors and the
second term (the double summation) refers to the electrons.Prs¼the density matrix
elements (Sections 5.2.3.6.4 and 5.2.3.6.5), cf.:

Ptu¼ 2

Xn

j¼ 1

c$tjcuj ð 5 : 208 ¼ 5 : 81 Þ

ThePsummation is over the occupied orbitals (j¼1, 2,..., n; we are
considering closed-shell systems, so there are 2nelectrons) and the double summa-
tion in Eq. (5.207) is over thembasis functions. The operatorris the electronic
position vector.
How good are ab initio dipole moments? Hehre’s extensive survey of practical
ab initio methods [ 39 ] indicates that fairly good results are given by HF/6–31G//
HF/6–31G (dipole moment from a HF/6–31G calculation on a HF/6–31G
geometry) calculations, and that MP2/6–31G//MP2/6–31G calculations are

342 5 Ab initio Calculations