Physical Chemistry Third Edition

894 21 The Electronic Structure of Polyatomic Molecules

21.8 Applications of Symmetry to Molecular

Orbitals

Our treatment of polyatomic molecules thus far has not exploited the symmetry proper-

ties of the molecules. We have largely restricted our descriptions of chemical bonding

to orbitals made from no more than two atomic orbitals, and have included hybrid

orbitals in our basis functions to achieve this goal. The approximate LCAO molecu-

lar orbitals that we have created are not necessarily eigenfunctions of any symmetry

operators belonging to the molecule.

Consider the H 2 O molecule. We now construct a basis set containing linear com-

binations of atomic orbitals that are eigenfunctions of the symmetry operators that

belong to the molecule. We orient the molecule with the center of mass at the origin,

the oxygen atom on thezaxis and the hydrogen atoms in theyzplane, and with thez

axis bisecting the bond angle. The symmetry operators that belong to the H 2 O molecule

are the identity operator, thêC 2 zoperator, aσˆvreflection with its symmetry element in

the plane of the molecule (theyzplane), and aσˆvoperator with its symmetry element

perpendicular to the plane of the molecule (thexzplane). The canonical orbitals result-

ing from a Hartree–Fock–Roothaan calculation can be eigenfunctions of all of these

operations.

We begin with a minimal basis set that contains the oxygen 1s,2s,2px,2py, and

2 pzorbitals and the 1sorbitals of the two hydrogen atoms. The oxygen orbitals are

eigenfunctions of these four operators, but the two hydrogen 1sorbitals are not. We

replace the hydrogen 1sorbitals by linear combinations that are eigenfunctions of the

symmetry operators:

ψa 1 ψ 1 sHa+ψ 1 sHb (21.8-1)

ψb 2 ψ 1 sHa−ψ 1 sHb (21.8-2)

We call these new basis functionssymmetry orbitalsorsymmetry-adapted basis func-

tions. The labels on these linear combinations are explained in Appendix I. The new

basis set consists of the oxygen orbitalsψ 1 s,ψ 2 s,ψ 2 px,ψ 2 py,ψ 2 pz, and the symmetry

orbitalsψa 1 andψb 2. The canonical orbitals will be simpler linear combinations of

these basis orbitals than if we used the original basis set.

Exercise 21.14

Find the eigenvalue of each basis orbital for each of the operators that belong to the H 2 O molecule.

The LCAO molecular orbitals are written in the form

ψc 1 sOψ 1 sO+c 2 sOψ 2 sO+c 2 pzOψ 2 pzO+c 2 pyOψ 2 pyO+c 2 pxOψ 2 pxO

+ca 1 ψa 1 +cb 2 ψb 2 (21.8-3)

Only basis orbitals of the same symmetry can be included in any one LCAO molecular

orbital if it is to be an eigenfunction of the symmetry operators. Thea 1 basis function

can combine with the 1s,2s, and 2pzfunctions on the oxygen. Theb 2 basis function

can combine with the 2pyfunction on the oxygen, and the 2pxfunction on the oxygen

cannot combine with any of the other basis functions. Table 21.2 contains the values

of the coefficients determined by the Hartree–Fock–Roothaan method for the seven

canonical molecular orbitals, using Slater-type orbitals (STOs) as basis functions, with