Physical Chemistry Third Edition

852 20 The Electronic States of Diatomic Molecules

We denote the distance from the hydrogen nucleus to electron numberibyriHand

the distance from the lithium nucleus to electron numberibyriLi.

The zero-order Hamiltonian operator of Eq. (20.4-1) gives a time-independent

Schrödinger equation in which the variables can be separated by assuming a prod-

uct wave function:

Ψ(0)ψ 1 (1)ψ 2 (2)ψ 3 (3)ψ 4 (4) (20.4-3)

where each factor is a molecular orbital. We construct LCAOMOs to represent these

molecular orbitals, but must now abandon our policy of including only two atomic

orbitals in our basis set. We take a basis set consisting of four atomic orbitals: the

lithium 1sorbital, the lithium 2sorbital, the lithium 2pzorbital, and the hydrogen 1s

orbital. This is called aminimalbasisset. All of these basis orbitals correspond tom0.

The symmetry operators belonging to the molecule are the identity operator, all of the

rotation operators about thezaxis, and all of thêσvoperators. The 2pxand 2pyorbitals

are not included since they have different eigenvalues for these symmetry operators.

Each of the basis functions is an eigenfunction of these operators with eigenvalues

equal to 1. Any linear combination of the basis functions will be a sigma orbital and

will be an eigenfunction of these symmetry operators with eigenvalue equal to 1.

Four independent LCAOMOs can be made from our basis set of four atomic orbitals.

We denote them by

ψjσc( 1 js)Liψ 1 sLi+c( 2 js)Liψ 2 sLi+c( 2 jp)zLiψ 2 pzLi+c( 1 js)Hψ 1 sH (20.4-4)

where the labeljcan equal 1, 2, 3, or 4. We have added aσsubscript to indicate

that all of these LCAOMOs are sigma orbitals. Values of the coefficients in the LCAO-

MOs can be obtained by the variation method or the Hartree–Fock–Roothaan method.^9

Table 20.3 shows the results of a Hartree–Fock–Roothaan calculation and Figure 20.14

is a correlation diagram showing schematically the atomic and molecular orbital ener-

gies. By the Aufbau principle, the ground-state wave function is (without antisym-

metrization)

Ψgσψ 1 σ(1)α(1)ψ 1 σ(2)β(2)ψ 2 σ(3)α(3)ψ 2 σ(4)β(4) (20.4-5)

where the orbitals are numbered from the lowest energy to the highest energy. The

1 σmolecular orbital is nearly the same as the lithium 1sorbital and is essentially

nonbonding. The 2σmolecular orbital is a bonding orbital. The molecule has two

Table 20.3 Results of a Hartree–Fock–Roothaan Calculation for

the LiH Ground State at an Internuclear Distance of 159 pm

MO c 1 sLi c 2 sLi c 2 pzLi c 1 sH

10. 9996 − 0. 0000 − 0. 0027 0. 0035

20. 0751 0. 3288 − 0. 2048 − 0. 7022

3 − 0. 0115 0. 7432 0. 6601 0. 1236

4 − 0. 1256 0. 8769 − 1. 0107 1. 2005

From A. M. Karo,J. Chem. Phys., 30 , 1241 (1959).

(^9) C. C. J. Roothaan,Rev. Modern Phys., 23 , 69 (1951).