Physical Chemistry Third Edition

840 20 The Electronic States of Diatomic Molecules

where the value ofcuis chosen to minimize the variational energy, and the value of

cCIis chosen to normalize the function. With the optimum value of the parametercu,

this function givesDe 4 .02 eV andre75 pm. The inclusion of a single additional

electron configuration in this relatively simple wave function has removed much of

the correlation error and has given a better result for the dissociation energy than a

sophisticated Hartree–Fock–Roothaan calculation with a single configuration.^5 We will

see later how this wave function introduces dynamical electron correlation. Table 20.1

summarizes some results of calculations on the ground state of the hydrogen molecule.

Excited states of the hydrogen molecule correspond to electron configurations other

than (σg 1 s)^2. In the electron configuration (σg 1 s)(σ∗u 1 s) there is one electron in a

bonding orbital and one in an antibonding orbital. The antibonding effect of one electron

approximately cancels the bonding effect of the other electron, and the molecule will

dissociate into two hydrogen atoms if placed in such a state.

The Valence-Bond Method for the Hydrogen Molecule

Orbital wave functions are not the only type of approximate molecular wave functions.

In 1927 Heitler and London^6 introduced thevalence-bond function:

ΨVBcVB[ψ 1 sA(1)ψ 1 sB(2)+ψ 1 sB(1)ψ 1 sA(2)][α(1)β(2)−β(1)α(2)] (20.3-7)

wherecVBis a normalizing constant. This wave function expresses the sharing of

electrons in a different way from an orbital wave function. It contains one term in

which electron 1 occupies an atomic orbital centered on nucleus A while electron

2 occupies an atomic orbital centered on nucleus B and another term in which the

locations are switched. This wave function includes dynamical electron correlation,

Table 20.1 Summary of Results of Calculations on the Hydrogen Molecule

De/eV re/pm

Experimental values 4.75 74. 1

Molecular Orbital Methods

Simple LCAOMO 2.65 84

Simple LCAOMO, variable orbital exponent1.197 3.49 73. 2

CI, two configurations, variable orbital exponent 4.02 75

CI, 33 configurations, elliptical coordinates 4.71

Valence-Bond Methods

Simple VB 3.20 80

Simple VB, variable orbital exponent1.16 3.78

VB with ionic terms,δ 0 .26, variable orbital exponent1.19 4.02 75

Other Variational Methods

13 terms, elliptical coordinates 4.72

100 terms, elliptical coordinate 4.75

Self-Consistent Field Methods

Best Hartree–Fock–Roothaan 3.64 74

Data from I. N. Levine,Quantum Chemistry, 5th ed., Prentice-Hall, Englewood Cliffs, NJ, 2000.

(^5) I. N. Levine,op. cit., p. 395ff (note 4).
(^6) W. Heitler and F. London,Z. Physik, 44 , 455 (1927).