Physical Chemistry Third Edition

864 20 The Electronic States of Diatomic Molecules

andΨIcontains a single ionic term:

ΨIψ 1 sLi(1)α(1)ψ 1 sLi(2)β(2)ψ 1 sH(3)ψ 1 sH(4)[α(3)β(4)−β(3)α(4)] (20.4-19)

The coefficientscVBandcIcould be optimized by the variation method. If|cVB|is

larger than|cI|this corresponds to a bond that is primarily covalent and if|cVB|is

smaller than|cI|this corresponds to a bond that is primarily ionic.

EXAMPLE20.15

Calculate the values ofcVBandcIthat make the wave functionΨMVBequivalent to the

LCAOMO wave function for LiH in Eq. (20.4-5) if the orbitals of Eq. (20.4-8) are used. Find

the percent ionic character, defined as [c^2 I/(cVB^2 +c^2 I)]×100%.

Solution

The molecular orbitals of Eq. (18.4-8) are

ψ 1 σψ 1 sLi

ψ 2 σc 2 sp(1)Liψ 2 sp(1)Li+c 1 sHψ 1 sH≈− 0. 47 ψ 2 sp(1)Li− 0. 88 ψ 1 sH

The modified valence-bond function for LiH in Eqs. (13.4-15) and (13.4-16) is

ΨMVBcVBΨVB+cIΨI

where

ΨVBψ 1 sLi(1)α(1)ψ 1 sLi(2)β(2)[ψ 2 sp(1)Li(3)ψ 1 sH(4)+ψ 1 sH(3)ψ 2 sp(1)Li(4)]

×[α(3)β(4)−β(3)α(4)]

ΨIψ 1 sLi(1)α(1)ψ 1 sLi(2)β(2)ψ 1 sH(3)ψ 1 sH(4)[α(3)β(4)−β(3)α(4)]

The molecular-orbital wave function can be written, omitting the spin factors

ΨMOψ 1 sLi(1)ψ 1 sLi(2)[− 0. 47 ψ 2 sp1Li(3)− 0. 88 ψ 1 sH(3)]

×

[

− 0. 47 ψ 2 sp(1)Li(4)− 0. 88 ψ 1 sH(4)

]

ψ 1 sLi(1)ψ 1 sLi(2)

[

0. 22 ψ 2 sp(1)Li(3)ψ 2 sp(1)Li(4)+ 0. 41

[

ψ 2 sp(1)Li(3)ψ 1 sH(4)

+ψ 1 sH(3)ψ 2 sp(1)Li(4)

]

+ 0. 77 ψ 1 sH(3)ψ 1 sH(4)

]

Exact equivalence cannot be achieved, because the termψ 2 sp(1)Li(3)ψ 2 sp(1)Li(4) does not

appear inΨMVB. This term corresponds to ionic bonding with both electrons on the lithium

atom, and is best omitted. By comparison, we have

cVB 0. 41

cI 0. 77

% ionic

(0.77)^2

(0.77)^2 +(0.41)^2

(100%)78%