Physical Chemistry Third Edition

20.4 Heteronuclear Diatomic Molecules 857

Solution

Place a chargeQatr 2 and a charge of−Qatr 1. Thexcomponent ofμis

μxQx 2 +(−Q)x 1 Q(x 2 −x 1 )

Theyandzcomponents are similar.

|μ|μ

(

μ^2 x+μ^2 y+μ^2 z

) 1 / 2



(

Q^2 (x 2 −x 1 )^2 +Q^2 (y 2 −y 1 )^2 +Q^2 (z 2 −z 1 )^2

) 1 / 2

Q|r 2 −r 1 |

Exercise 20.17

A quadrupole is the next level beyond a dipole for describing a charge distribution. A particular

simple quadrupole consists of a charge+Qat the origin, a charge−Qat the point (1, 0), a

charge+Qat (1, 1), and a charge−Qat (0, 1). Show that if there are no other charges, the

dipole moment of this collection of charges vanishes.

The classical expression for the dipole moment in Eq. (20.4-9) contains no momen-

tum components, so its quantum mechanical operator is the operator for multiplication

by this expression. The expectation value of the electric dipole of a molecule in a state

Ψis

〈μ〉

∫

Ψ∗μΨdq

∫

μ|Ψ|^2 dq (20.4-11)

whereqis an abbreviation for the coordinates of all particles in the molecule. The

factor|Ψ|^2 is the probability density for finding the charged particles in the system.

The operator contains no spin dependence so spin functions and spin integrations can

be omitted.

In the Born–Oppenheimer approximation the nuclei are fixed and we can sum their

contributions without integrating. The nuclear contribution is given by

μnuc

∑nn

A 1

eZArA (20.4-12)

wherennis the number of nuclei andrAis the position vector of nucleus number A,

which containsZAprotons. With a one-term orbital wave function in which the orbitals

are orthogonal to each other, each orbital makes its contribution to the probability

density independently, as in Eq. (18.3-8), which also holds for molecular orbitals. In

an antisymmetrized orbital wave function the total electron probability density is the

same as in the one-term function, as shown in Eq. (18.4-5). The expectation value of

the dipole moment is given by

〈μ〉μnuc−e

∑ne

i 1

∫

ψi(i)∗riψi(i)d^3 riμnuc−e

∑ne

i 1

∫

ri|ψi(i)|^2 d^3 ri (20.4-13)

whereψiis theith occupied space orbital and whereneis the number of electrons.