Physical Chemistry Third Edition

858 20 The Electronic States of Diatomic Molecules

The formula shown in Eq. (20.4-13) is the same as if theith electron were a

“smeared-out” charge with a density distribution equal to|ψi|^2. It is a theorem of

electrostatics that a spherically symmetrical distribution of charge has an effect out-

side of the charge distribution as though the charge were concentrated at the center

of symmetry. An electron moving in an undistorted and unhybridized atomic orbital

contributes to the charge density just as though it were at the nucleus. Electrons moving

in LCAOMOs that have unequal coefficients make a larger contribution to the negative

charge at the end of the molecule with the coefficient of larger magnitude, so that the

LiH molecule has a sizable dipole moment, with the hydrogen end negative.

The SI unit in which dipole moments are measured is the coulomb-meter (C m). The

experimental bond length of the LiH molecule is 1.596× 10 −^10 m, so that if the bond

were purely ionic with an undistorted Li+ion and an undistorted H−ion,

μionic(1. 6022 × 10 −^19 C)(1. 595 × 10 −^10 m) 2. 56 × 10 −^29 Cm

There is a common non-SI unit named theDebye, which is defined by

1 Debye 3. 335641 × 10 −^30 Cm (20.4-14)

The Debye unit was originally defined in terms of the c.g.s. unit of charge, the electro-

static unit (esu), such that 1 Debye equals 10−^18 esu cm (10−^10 esu Å). The charge on

a proton equals 4. 80 × 10 −^10 esu, so that molecular dipole moments generally range

from 0 to 10 Debye.

The Debye unit is named for Peter J. W.
Debye, 1884–1966, a Dutch-American
chemist who became famous for the
Debye–Hückel theory of ionic solutions,
but whose 1936 Nobel Prize in
chemistry was for his work on dipole
moments.

EXAMPLE20.11

Estimate the dipole moment of the LiH molecule from the orbitals of Eq. (20.4-8).

Solution

The lithium nucleus has charge 3e. The two nonbonding electrons in the 1σorbital contribute

as though they were at the lithium nucleus. The probability density of an electron in the 2σ

orbital is

|ψ 2 σ|^2 (0.47)^2 ψ^22 sp(1)Li+(0.47)(0.88)ψ 2 sp(1)Liψ 1 sH+(0.88)^2 ψ^21 sH

We neglect the second term in this expression because it is appreciably nonzero only in the

overlap region. The third term will make its contribution to the integral in Eq. (20.4-13) as

though the electron were centered at the hydrogen nucleus. The 2sp(1) hybrid orbital does not

have its center of charge exactly at the lithium nucleus, but we approximate the contribution

of the first term as though it did. The net charge at the lithium nucleus is

QLi 3 e− 2 e−2(0.47)^2 e 0. 56 e 8. 9 × 10 −^20 C

The net charge at the hydrogen nucleus is

QHe−2(0.88)^2 e− 0. 56 e− 8. 9 × 10 −^20 C

with a bond length of 1.595× 10 −^10 m,

μ≈(8. 9 × 10 −^20 C)(1. 595 × 10 −^10 m) 1. 42 × 10 −^29 Cm 4 .3 Debye

This estimate of the dipole moment is about 60% as large as the value for a purely ionic

bond, indicating a bond that is roughly 60% ionic in character. It agrees only roughly with

the experimental value, 5.88 Debye.