Mathematical Methods for Physics and Engineering : A Comprehensive Guide

REPRESENTATION THEORY

(i) UnderIall eight basis functions are unchanged, andχ(I)=8.

(ii) The rotationsR,R′andQchange the value ofNin every case and so

all diagonal elements of the natural representation are zero andχ(R)=

χ(Q)=0.

(iii)mxtakesxinto−xandyintoyand, forN= 1 and 3, leavesNunchanged,

with the consequences (remember the forms of Ψx(N)andΨy(N)) that

Ψx(1)→−Ψx(1), Ψx(3)→−Ψx(3),

Ψy(1)→Ψy(1), Ψy(3)→Ψy(3).

Thusχ(mx) has four non-zero contributions,−1,−1, 1 and 1, together

with four zero contributions. The total is thus zero.

(iv)mdandmd′leave no atom unchanged and soχ(md)=0.

The character set of the natural representation is thus 8, 0, 0, 0, 0, which, either

by inspection or by applying formula (29.18), shows that

Dnat=A 1 ⊕A 2 ⊕B 1 ⊕B 2 ⊕2E,

i.e. that all possible irreps are present. We have constructed previously the

combinations of Ψx(N)andΨy(N) that transform according to A 1 and B 1.

The others can be found in the same way.

29.11.2 Matrix elements in quantum mechanics

In section 29.10 we outlined the procedure for determining whether a matrix

element that involves the product of three factors as an integrand is necessarily

zero. We now illustrate this with a specific worked example.

Determine whether a ‘dipole’ matrix element of the form

J=

∫

Ψd 1 xΨd 2 dτ,

whereΨd 1 andΨd 2 ared-state wavefunctions of the formsxy f(r)and(x^2 −y^2 )g(r)respec-

tively, can be non-zero(i)inamoleculewithsymmetryC 3 v(or 3 m), such as ammonia, and

(ii)in a molecule with symmetryC 4 v(or 4 mm), such as the MnI 4 molecule considered in

the previous example.

We will need to make reference to the character tables of the two groups. The table for
C 3 vis table 29.1 (section 29.6); that forC 4 vis reproduced as table 29.5 from table 29.4 but
with the addition of another column showing how some common functions transform.
We make use of (29.23), extended to the product of three functions. No attention need
be paid tof(r)andg(r) as they are unaffected by the group operations.
Case(i). From the character table 29.1 forC 3 v, we see that each ofxy,xandx^2 −y^2
forms part of a basis set transforming according to the two-dimensional irrep E. Thus we
may fill in the array of characters (using chemical notation for the classes, except that
we continue to useIrather thanE) as shown in table 29.6. The last line is obtained by