Physical Chemistry Third Edition

1300 I Matrix Representations of Groups

Theb 2 basis function can combine with the 2pyfunction on the oxygen, and the 2px

function on the oxygen cannot combine with any of the other basis functions. Knowing

the symmetry species of the basis orbitals and including only those that have the same

symmetry species shortens the calculation, as did exclusion of the 2pxand 2pyorbitals

from the sigma molecular orbitals that we formed for the lithium hydride molecule in

Section 20.4.

A useful application of group theory is the prediction of whether an overlap integral

will vanish. In various places in earlier chapters, we have occasionally asserted that an

integral vanishes without actually calculating it by noting that the positive and negative

contributions cancel because of the symmetry of the integrand function. Group theory

provides a systematic means of doing this.∫^17 Consider an overlap integral of the type

ψ∗ 1 ψ 2 dqwhereψ 1 andψ 2 are two orbitals and whereqstands for the coordinates of

the electron. If the integrand changes sign under some symmetry operation belonging

to the molecule, this means that positive and negative contributions will cancel. This

generally means that the product of the two functions must have the symmetry species

A 1 in order not to vanish automatically. We determine whether this is the case by

forming thedirect product, which is a representation obtained as follows: Write the

characters for the irreducible representation for orbitalψ 1 and that for orbitalψ 2 , one

above the other with the symmetry operations in the same order. Multiply the two

characters for each operator together, to obtain a representation of the group that can

be reducible or irreducible. If the irreducible representation A 1 (the totally symmetric

representation) is obtained, the integral will not vanish. If a reducible representation is

obtained that contains the totally symmetric representation, the integral will not vanish.

The number of times that a given irreducible representation is contained in a reducible

representation is given by Eq. (I-19).

As an example, we show that the water molecule symmetry-adapted orbitalψb 2 

ψ 1 sHa−ψ 1 sHbhas a vanishing overlap with the oxygen 2sorbital. The symmetry

species of the oxygen 2sorbital isa 1 , and that of the symmetry-adapted orbital isb 2.

From the character table in Table A.26 of Appendix A,

C 2 v E C 2 σyz σxz

A 1 111 1

B 2 1 − 11 − 1

Direct product 1 − 11 − 1

The character of the direct product is that of B 2 , so the integral vanishes. If we had

not recognized the direct product as an irreducible representation, we could have used

Eq. (I-19) to determine whether the irreducible representation A 1 is contained in the

direct product:

nA 1 

1

4

(1− 1 + 1 −1) 0

so that it is not contained in it and we know that the overlap integral must vanish.

An orbital with a given symmetry species belongs to a level with the degeneracy

equal to the dimension of the irreducible representation. Using a character table, we

can quickly determine the dimension of a reducible representation by looking at the

trace of the matrix representing the identity E, which is equal to the dimension of the

(^17) A. W. Adamson,op. cit., pp. 762ff (note 1).