Jose M. Alvariiio' and Alberto Charnorro Departmento de Quimica Fisica and Departmento de Fisica Universidad del Pais Vasco. Spain
I
continuous point G ~ O U ~ S
I
A simple derivation of the closed formula for the reduction of representations
1
Following the paper by one of us ( I ) three interesting communications on the reduction of representations of continuous point groups have appeared recently in this Journal (. 2 4 ).. We would also like to make some further comments on the topic. They are motivated by the impression that we have been a~oarentlvunaware of the fact that a closed formula does also edist for the reduction of representations of continuous point groups. This reads (5) a. =
5 1x(R)xi(R)dR' 5 S d~
(1)
where the integration is taken over the range of the continuously changing parameter R (in this case the azimuthal angle cp), and the summation goes over the classes of distinct, discrete o~erations.Other svmbols in ean. . (1) . . have the usual meaning. Eauation (1)is (not easilv) derived from the extension of the great orthogon&ty theorem to continuous groups (6). While just calling the attention to the general omission of eqn. (1) should be intrinsically interesting, we should also like to present a derivation of it starting from the well-known corresponding formula for finite groups, i.e.
..
which is the explicit form of eqn. (1) for C,,. For example (a) of (1) this yields ox+ = (lillr){J2-
an = (114r) {Jar
(2
(2
+ 2 eosd). 1. + J2-2.
+ 2 cos 4). 2 cos 4 . d4 + JZr
1. d*] = 2 2 . O . d#] = 1
as i t must be. For D-h one starts fromDnh. One first realizes that the elements of the Dh's axial g & ~ pare~ for any point group classifiable in four sets of distinct operations, namely proper rotations, improper rotations, binary axes and symmetry, vertical planes. The angle 6 serves to label and distinguish among different operations within the same set. For example, in Dsh (order of the group = 12 = 4 X 3) the sets are ICa(2r/3),Ca(4r/3).Ca(2r) =El IS~(2rl3),Sa(4r13),Sa(2*) = ah1 ICa(2rl3),C2(4rl3).C~(2r)J Iav(2r/3),av(4d3),av(2s)J
and from eqn. (2) where h is the order of the group. Let us first consider the finite, axial C, group. I t consists of the elements (symmetry operations) (C,, Cm2,C n 3 . . . ,C,, I El. I t is a one-parameter group, the parameter being the angle 4. The order of the group is n. The rotation angle is given in general by +A= k(2rln), k = 1.2,. . . , n and the minimal step in 4 is evidently
where only the terms from the first set have been explicitly written, and the summations go through bk = 2a/3,4a/3,2a. For D.h one should have instead symbolically
where p.r., i.r., C2 and uvabove the 2s mean summation over the proper rotations, over the improper rotations, over the binary axes, and over the symmetry planes, respectively, and ( )means the pertinent x(R)xi(R) products. Equation (6) can be written in a more convenient fashion, i.e.
Equation (2) takes in this case the form
ai = (118s) E
sets
- ."
-
In the limit, when n a, (&+I - &) combined with eqns. (3) and (4) yields for C,
and in the limit, i.e. for D,h dq4, and this
Let us now extend this result to C,,, by starting with C,,. Here one has a new class of element, the infinite reflection planes a,. The group order is 2n and the infinite reflection planes are also characterized by the angle 4. So, eqn. (2) takes the form
--
""
which upon multiplying and dividing by Za, becomes in the limit a s n
ai = (118~)1 Bets
1 ( ) (2rIn) m*
X2'
x(R)xi(R)d9
The four sets in D,h are (E,C(@)I,(i,S(4)),lCzl,and (av)so explicitly in example (b) of (1)
Here we have taken for the n,@II,representation: x(d(4)) = x(S4) = 4 cos2+, x(C2) = x(uV)= 0.In the same way one easily gets az,- = 1and a&, = 1.For other cases, the following relationships are useful
Lzr
cm k4.m~14-d4 = r 6 ~ 1
where 6kl is the Kronecker delta, and
SoZ"
cos k+ d$ = 0 fork integer.
Author to whom correspondence should be addressed.
I t should be noted that the proof above presented of eqn. Volume 57, Number 11. November 1980 1 785
(1) is rigorous enough for the usual representations of the point groups. For these the characters x(R) are continuous and hounded over the range of R and the limiting processes considered converge. In general one can he sure that the characters are bounded for unitary finite-dimensional representations. In fact let a;j(R) he the element belonging to the ith-row and jth-column of the matrix representing the element g(R) in a unitary finite-dimensional representation. The unitarity implies
x (lii(~)a*bjl~) =
6iir
n heing the dimension of the representation. Nonetheless, it would he wrong to infer from these considerations that the general theory of continuous groups can be drawn from that of the finite groups. Lie group theory has a structure and richness which goes far beyond the scope of the simpler discrete groups framework. Acknowledgment
We thankDr. Luis Lain, Departamento de Quimica Fisica, Universidad del Pais Vasco. for h e l ~ f u lsueeestions and comments.
and taking i = k Literature Cited laiJ2+
i*;
loij12= 1
wherefrom one gets la;; 1 5 1. This means in turn that
li I
lx(R)l = Xai; 5
786 1 Journal of Chemical Education
Xi la;;l 5 n
