Physical Chemistry Third Edition

20.3 Homonuclear Diatomic Molecules 839

The Zero-Order Approximation

In the zero-order approximation we ignore the electron–electron repulsion, represented

by the term proportional to 1/r 12. The zero-order electronic Hamiltonian operator is

now a sum of two hydrogen-molecule-ion Hamiltonians:

̂H(0)

el 

ĤHMI(1)+ĤHMI(2) (20.3-2)

This zero-order Hamiltonian leads to a separation of variables with a trial function that

is a product of two hydrogen-molecule-ion molecular orbitals

Ψ(0)ψ 1 (1)ψ 2 (2) (20.3-3)

The zero-order Born–Oppenheimer energy is a sum of two hydrogen-molecule-ion

electronic energies plusVnn:

E

(0)

BOEHMI,1+EHMI,2+Vnn (20.3-4)

Since the electrons can have different spins, the molecular space orbitals in the zero-

order ground-state function are theψ 1 σgspace orbitals that we discussed in the previous

section. The “exact” Born–Oppenheimer orbitals are complicated functions expressed

in an unfamiliar coordinate system. We replace them by theσg 1 sLCAOMO orbitals.

The zero-order orbital approximation is crude enough that this approximate replace-

ment produces no significant further damage. Including spin, the antisymmetrized and

normalized zero-order ground-state wave function is:

ψ(0)

1

√

2

ψσg 1 s(1)ψσg 1 s(2)[α(1)β(2)−β(1)α(2)] (20.3-5)

Electron configurations are assigned in much the same way as with atoms. This wave

function corresponds to the LCAOMO electron configuration (σg 1 s)^2. We say that

the hydrogen molecule has asingle covalent bondwith one pair of shared elec-

trons occupying a bonding space orbital that extends over both nuclei. This bond

is called aσ(sigma) bond because the two shared electrons occupy sigma orbitals.

This zero-order wave function gives dissociation energyDe 2 .65 eV and equilibrium

internuclear distance re84 pm 0. 84 × 10 −^10 m 0 .84 Å, compared with the

experimental values of 4.75 eV and 74.1 pm.

Improvements to the Zero-Order Approximation

A variation function with variable orbital exponents in the 1satomic orbitals making up

the LCAOMOs of Eq. (20.3-5) givesDe 3 .49 eV andre 73 .2 pm with an apparent

nuclear charge equal to 1.197 protons (a larger charge than the actual nuclear charge).

A careful Hartree–Fock–Roothaan calculation givesDe 3 .64 eV andre74 pm.^4

If this result approximates the best Hartree–Fock result, the correlation error is approx-

imately 1.11 eV.

Just as with atoms, configuration interaction (CI) can be used to introduce dynamical

electron correlation. A CI wave function using the two electron configurations (σg 1 s)^2

and (σ∗u 1 s)^2 is

ΨCICCI[ψσg 1 s(1)ψσg 1 s(2)+cuψσ∗u 1 s(1)ψσ∗u 1 s(2)][α(1)β(2)−β(1)α(2)] (20.3-6)

(^4) N. Levine,Quantum Chemistry, 5th ed., Prentice-Hall, Englewood Cliffs, NJ, 2000, p. 385ff.