Physical Chemistry Third Edition

20.3 Homonuclear Diatomic Molecules 841

since there are no terms with the two electrons assigned to the same atom. When the

valence-bond function of Eq. (20.3-7) is used to calculate the variational energy the

valuesDe 3 .20 eV andre80 pm are obtained. These values are in better agreement

with experiment than the values obtained from the simple LCAOMO wave function of

Eq. (20.3-5), presumably because of the introduction of dynamical electron correlation.

We want to compare the simple LCAOMO wave function with the simple valence-

bond wave function. We write the space factor in the LCAOMO function of Eq. (20.3-5)

in the form

ψσg 1 s(1)ψσg 1 s(2)C^2 g[ψ 1 sA(1)+ψ 1 sB(1)][ψ 1 sA(2)+ψ 1 sB(2)]

C^2 g[ψ 1 sA(1)ψ 1 sA(2)+ψ 1 sB(1)ψ 1 sB(2)

+ψ 1 sA(1)ψ 1 sB(2)+ψ 1 sB(1)ψ 1 sA(2)] (20.3-8)

The last two terms on the right-hand side of Eq. (20.3-8) are the same as the space factor

of the simple valence-bond wave function. These terms are calledcovalent terms. The

other two terms are calledionic terms, because one term has both electrons on nucleus A

whereas the other has them on nucleus B.

The simple LCAOMO wave function gives the ionic terms equal weight with the

covalent terms while the simple valence-bond function omits them completely. A better

result is obtained by including the ionic terms with reduced weight. A modified valence-

bond wave function is:

ΨMVBcMVB[ΨVB+δΨI] (20.3-9)

wherecMVBis a normalizing constant,δis a variable parameter, andΨIcontains the

ionic terms:

ΨI[ψ 1 sA(1)ψ 1 sA(2)+ψ 1 sB(1)ψ 1 sB(2)][α(1)β(2)−β(1)α(2)] (20.3-10)

When the parameterδis optimized, this wave function is identical to the optimized

configuration-interaction function of Eq. (20.3-6). This fact enables us to understand

why configuration interaction can introduce dynamical electron correlation. The opti-

mized valence-bond function of Eq. (20.3-9) includes some electron correlation because

the optimum value ofδis smaller than unity, making the probability that the electrons

are on the same atom smaller than the probability that they are on different atoms.

Combining the two configurations in the optimized wave function of Eq. (20.3-6)

produces the same wave function as the optimized valence-bond function, reducing

the importance of parts of the wave function corresponding to electrons being close

together. Although it is not obvious from inspection of a configuration-interaction wave

function, addition of more configurations can produce additional dynamical electron

correlation through cancellation of parts of the wave function corresponding to small

electron–electron distances.

Exercise 20.8

By expressing the function of Eq. (20.3-6) in terms of atomic orbitals, show that it can be made

to be the same as the function of Eq. (20.3-9). Express the parameterscMVBandδin terms of

cCIandcu.

To obtain an additional comparison between the simple LCAOMO method and the

valence-bond method for the hydrogen molecule, we obtain the one-electron probability