Physical Chemistry Third Edition

I Matrix Representations of Groups 1297

I.4 Bases for Representations

The last entry in each row of the character table for the three one-dimensional

representations specifies the coordinate that was acted on by the 1 by 1 matrix when

we generated the representation. To generate each one of these representations we could

have examined the effect of each of the symmetry operations on one of the Cartesian

coordinatesx,y, andz. TheC 2 operation changesxto−x, and multiplication by− 1

accomplishes the same thing, so that the 1 by 1 matrix representing theC 2 operation

has−1 as its only element.

We say that we usedzas thebasisof the A 1 representation,xas the basis of the B 1

representation, andyas the basis of the B 2 representation. We could have examined

the effect of the symmetry operators on any other functions ofx,y, andz, including

atomic orbitals or molecular orbitals, and could use these functions as the basis of a

representation. Since the coordinatezis unchanged by any of the symmetry operations

in theC 2 vgroup, we would get the same representations of this group by using the

functionsz^2 ,xz, andyzas the bases as we do by usingx,y, andz.

A theorem that we quote later implies that there must be four irreducible represen-

tations of theC 2 vgroup. The fourth representation can be obtained by usingxyas a

basis function. This function gives the characters

E C 2 σyz σxz function

A 2 11 − 1 − 1 xy

and this is included in a complete character table for theC 2 vgroup.^15

The hydrogen-like atomic orbitals have definite symmetry properties, and are eigen-

functions of specific symmetry operators. When we use one of the real 2porbitals on

the oxygen atom as a basis for a one-dimensional representation of theC 2 vgroup, we

can obtain the characters by determining what the eigenvalue is when each symmetry

operator in the group is applied to the function, since the eigenvalue is the single ele-

ment of the 1 by 1 matrix. The result is the same character table as obtained withx,y,

andzas the bases:

E C 2 σyz σxz Function

A 1 1 111ψ 2 pz

B 1 1 − 1 − 11 ψ 2 px

B 2 1 − 11 − 1 ψ 2 py

Character tables can be obtained for any group. Table A.26 of Appendix A lists character

tables for a few common point groups. The following notation is used: If there areC 2

axes perpendicular to the principal rotation axis (this occurs in the D groups), aC 2

operation is labeled as aC 2 ′axis if it passes through outer atoms of the molecule, and

as aC 2 ′′axis if it passes between outer atoms. A vertical mirror plane is labeled as a

σvplane if it passes through outer atoms and as aσdaxis if it passes between outer

atoms. The right column gives functions ofx,y, andzthat could be used as bases for

the representations, as well as some rotations that match the representation.Rzstands

for rotation about thezaxis, etc.

(^15) A. W. Adamson,op. cit., p. 748ff (note 1).