Physical Chemistry Third Edition

20.3 Homonuclear Diatomic Molecules 843

For the He 2 molecule we letZ2. Electron–electron repulsion terms are omitted

from this zero-order Hamiltonian and the constant nuclear repulsion termVnnhas also

been omitted. At the end of the calculation we must addVnnto the electronic energy

to obtain the zero-order Born–Oppenheimer energy.

The zero-order Hamiltonian operator of Eq. (20.3-13) leads to a zero-order electronic

wave function that is a product of four hydrogen-molecule-ion-like orbitals and to a

zero-order energy that is equal to the sum of four orbital energies. To choose the four

orbitals corresponding to the ground state, we apply the Aufbau principle: We choose

the set of orbitals with the lowest sum of orbital energies consistent with the Pauli

exclusion principle. The zero-order LCAOMO ground-state wave function including

spin but without antisymmetrization is

Ψ(0)ψσg 1 s(1)α(1)ψσg 1 s(2)β(2)ψσ∗u 1 s(3)α(3)ψσ∗u 1 s(4)β(4) (20.3-15)

Since no two electrons occupy the same spin orbital, this function can be antisym-

metrized.

Exercise 20.9

Antisymmetrize the function of Eq. (20.3-15) by writing it as a 4-by-4 Slater determinant. How

many terms would there be if you expanded this determinant?

The wave function of Eq. (20.3-15) corresponds to the electron configuration

(σg 1 s)^2 (σ∗u 1 s)^2 , with two electrons occupying bonding orbitals and two occupying

antibonding orbitals. In order to agree with the general chemistry definition, we define

thebond orderas:

Bond orderBO

1

2

(nbonding−nantibonding) (20.3-16)

wherenbondingis the number of electrons occupying bonding orbitals andnantibondingis

the number of electrons occupying antibonding orbitals. The bond order of the diatomic

helium molecule in its ground state is 0. The repulsive effect of the antibonding orbitals

roughly cancels the attractive effect of the bonding orbitals. The He 2 molecule does

not exist in its ground state, but has been observed spectroscopically in excited states.

Molecules with More Than Four Electrons

For homonuclear diatomic molecules with more than four electrons, we require addi-

tional molecular orbitals beyond theσg 1 sandσ∗u 1 sorbitals. For Li 2 and Be 2 we can

use theσg 2 sand theσ∗u 2 sorbitals in Eq. (20.2-9) and Eq. (20.2-10). The electron

configuration of Li 2 in the ground state is

(σg 1 s)^2 (σ∗u 1 s)^2 (σg 2 s)^2

and that of Be 2 is

(σg 1 s)^2 (σ∗u 1 s)^2 (σg 2 s)^2 (σ∗u 2 s)^2

The bond order for Li 2 equals unity, and the bond order for Be 2 is zero, explaining why

Be 2 does not exist in its ground state.