Physical Chemistry Third Edition

834 20 The Electronic States of Diatomic Molecules

this calculation, but the result is that the variation energy is minimized whencAcB.

An approximation to the first excited state wave functionψ 2 σ∗uis obtained when the

energy has its maximum value, and this corresponds tocA−cB.^3 Another procedure

is to choose values ofcAandcBso that the approximate orbital is an eigenfunction

of the same symmetry operators as the exact orbitals. The exact Born–Oppenheimer

orbitalψ 1 σgis an eigenfunction of the inversion operator with eigenvalue+1. In order

to obtain an LCAOMO with this eigenvalue, we must choose

cAcB (20.2-4)

In order to obtain a molecular orbital with the same symmetry properties as the exact

Born–Oppenheimer orbitalψ 2 σ∗u, we must choose

cA−cB (20.2-5)

We introduce the symbols for our two LCAOMOs:

ψσg 1 scg(ψ 1 sA+ψ 1 sB) (20.2-6)

ψσ∗u 1 scu(ψ 1 sA−ψ 1 sB) (20.2-7)

These LCAOMOs are eigenfunctions of all of the symmetry operators that belong to

the H+ 2 molecule.

Exercise 20.5

a.Argue thatψσg 1 shas an eigenvalue of+1 for thêσhoperator and for thêC 2 aoperator, where

̂C 2 ais a rotation operator whose symmetry element lies anywhere in thexyplane.

b.Argue thatψσ∗u 1 shas an eigenvalue of−1 for thêσhoperator and for thêC 2 aoperator.

Figure 20.7 schematically shows the orbital regions for theσg 1 sLCAOMO and

theσ∗u 1 sLCAOMO, as well as the orbital regions for the 1satomic orbitals. The

intersection of the two atomic orbital regions is called theoverlap region. For the

σg 1 sorbital the two atomic orbitals combine with the same sign in the overlap region,

producing a molecular orbital region with no nodal surface between the nuclei. For

theσ∗u 1 sorbital the atomic orbitals combine with opposite signs in the overlap region,

canceling to produce a nodal surface between the nuclei. The orbital regions of the

LCAOMOs have the same general features as the “exact” Born–Oppenheimer orbitals

whose orbital regions were depicted in Figure 20.4.

Figure 20.8 shows the electronic energy for each of these LCAOMOs for H+ 2 along

with the exact Born–Oppenheimer energies. As we expect from the variational theorem,

the approximate energies lie above the exact energies for all values ofr. The value ofDe

for theσg 1 sorbital is equal to 1.76 eV atre132 pm 1. 32 × 10 −^10 m 1 .32 Å.

LCAOMOs can be constructed that are linear combinations of more than two atomic

orbitals. For example, to representψ 1 σgandψ 2 σ∗uwe could have written a linear

combination of six atomic orbitals:

ψc 1 sAψ 1 sA+c 1 sBψ 1 sB+c 2 sAψ 2 sA+c 2 sBψ 2 sB+c 2 pzAψ 2 pzA+c 2 pzBψ 2 pzB

(20.2-8)

(^3) J. C. Davis, Jr.,Advanced Physical Chemistry, The Ronald Press, New York, 1965, p. 404.