J. M. Alvariho De~artamentode Quimica Fisica Facultad de Ciencias Unlversidad de Biloao. Spain
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On the Complete Reduction of Representations of Infinite Point Groups
It has been pointed out in this Journal ( I ) that the equation
.. ,. which is generally used for the reduction of redurihle representations (RR)of point groups, is of no utility for C - , or I)+ which have an infinite oiderlIn eqn. (1) ai gives the number of times the irreducible representation (IR) r' happens to occur in the RR; x(R) are the characters of the RR, r,for every symmetry element R; xt(R) are the characters for the given IR, r i , and h is the order of thegroup, i.e. the numher of elements. Equation (1) derives from: 1) the condition that a trace of a matrix is invariant under a similarity transformation, i.e.
for all R E G, where G stands for group, and 2) the great orthogonality theorem (2). The numher of classes of symmetry operations (i.e. the numher of different equations (2)) being, as a rule, finite and equal to the numher of IR's (i.e. to the numher of unknowns in the system (2)), all the ai's can be deduced from eqn. (2). In the case of infinite point groups, however, the system (2) consists of infinite equations with infinite unknowns, and Scbifer and Cyvin ( I ) say the prohlem is solvable if one is ahle to guess which of the ai do not vanish. They apply their method to determine the normal modes of vibration of linear molecules. The method works well in this case because it is a priori known that the only possihle solutions are Xi, X i , IIg and II, for D,h, and X+ and n for C,,. The necessity of this a priori knowledge prevents the application of the method to other problems (e.g. molecular electronic states) where such information is not available. l a t e r on Stnnnmen and 1.ippinrott ( 3 ) proposed an a l t ~ r native method which was intended to be freeofobjnctions. In particular no guess in respect to the IR's in which r decomposes was necessary. In short the method consists of working not with the infinite group GO = C,, or D-h, hut with a subgroup of it, G. r was consequently reduced in G and finally the direct sum in G was transformed into a direct sum in Go by means of a correlation between the IR's in both groups. The solution to this prohlem has a pedagogical interest. I t is for that reason unfortunate that the method proposed by Strommen and Li~oincott is not correct in several resuects. .. One of the keys of the method is the correlation between IR's of G and of GO. The correlations indicated in the paper (tables (6) of the 3 examples) are right, but two things m&i he noted. First, the correlation must he done by comparing characters of the symmetry operations which are common to G and Go, {R) = (G n GO) and not basis functions of the cartesian cwrdinates, as tahles (6) of Strommen and Lippincott's paper seem to imply. Otherwise IR's of GO which, like 2- or @ from C,, or Z,, X;, @g or h fromD,h, do not possess such basis functions (see character tahles in Hall's hook, for example) could never appear in the reduction of RR's. That would mean neither MO's nor molecular electronic states with that symmetry could exist. On the other side, while it is an easy task to find the set IRJ = (G n Go) in finite groups, it is a t least difficult, amhiguous, and therefore pedagogically unadvisable to do i t when Go is an infinite group. These groups possess
continuous symmetry operations (C*, 5'') which are not immediatelv comnarable to standard o~erationsof finite mouus. A final c;iticism to this method (3jconsists in the selection of the G suhgroup. I t is not true as Strommen and Lippiucott state in their paper that "any subgroup of a given parent group may be used in treating the prohlem." Referring to examples 2 and 3 of them if one takes G = Cz, (a suhgroup of D-h) ambiguities in the reduction arise. In example 2,2X: or 22; 2: could combine with II, or II, to give six possible or 2: reductions. The amhiguitv is still greater in exarn~le3. In short, the method r3;seems to work in the same casesas Schafer and Cpin's ( 1 ). Moreover it introduces in an unfurtunate way the idea of correlation. I propose an alternative method which is simple, valid for any problem, and free of the stated objections. In short, eqn. (2) is exhaustively exploited. In this case, i t has been said that (2) is a linear system of infinite equations with infinite unknowns. The magnitude of the problem can, however, be reduced to finite limits, as we shall see. The key of our method consists in realizing that (2) has a unique solution, as Schafer and Cyvin pointed out. One must just find one solution: this must he the solution. I t is then unnecessary to he ahle to guess which of the at's do not vanish. We will outline our method with the help of two examples. The first one is example 1of Schafer and Cyvin, i.e. to find the normal modes of vibration of a linear molecule. The second one consists of deriving the electronic states from a given electronic configuration of a linear molecule. This prohlem could obviously not he solved with the methods of SchaferCyvin or Strommen-Lippincott, Our procedure has four steps. Before going on to the examples we shall comment hriefly on step 1(see below). Infinite groups C,, and Dm,, have only one-dimensional ( 2 ) and two-dimensional (II, A, a,. . .) IR's. If one also considers that x(E) = 1, i.e. that the character of the identity operation equals the dimension of the representation, eqn. (2) takes the form x(E)=Xoi+2Xaj
(3)
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where the f i t sum refers to the X's and the second one to the n's, A's,. . . One must also hear in mind that there are four different 2's in D,h and two in C,.. With respect to Xi(C*) and xi(S') the following trigonometrical relations may he of interest: 2cos2@ = 4cos2* - 2; 2cos3@ = 8cos3@- 6cosm; 2cos4* = 16cos4@- 16cos2@ 2, etc. Let us now consider the two examples.
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a ) Normal Modes of Vlbratlon of Linear Molecules This is example 1of Schafer and Cyvin (I ). We take the character tahle for r,+(XYZ) from their paper (table 3)
Step 1. State the a priori possible reductions after eqn. 1.1) \..I
With x(E) = 4, the possibilities are
r=4x
r = 2X!II(A,. . .) r = 2n(2a,. . .) Step 2. Write the system (2) Volume 55, Number 5, May 1978 1 307
nmn=42 n e n = zz e n(a,. ..) n e n = zn(za,. . .) Step 3. Eliminate by logical deductions from 2some of the possibilities of I
Step 2. The system (2) is in this case
Equation (4) is impossible because the trigonometric coefficients (2eos@,. . .)appear only i n n , A,. . . ,and not in 2. Equation (6) is impossible, because at least ax+ must be f 0 (see last equation in step 2) Step 4. Analysis of the remaining possibilities In this case one has to consider just eqn. (5),i.e. These are really infinite possibilities,but we rememher that the reduction is unique, so if we find one solution this will be the solution. There are three at's to be found and they must satisfy the last equation as well as the corresponding form of the system derived in step 2. A systematic search is now undertaken, i.e. one proofs with r = 2 2 Bf II(az+, ax-, a, # 0, all other a; = O), then with r = 2Z (B A(az+, ax-, a a Z 0, all other a; = O), etc., till one solution is found. In this way one arrives easily to ax+ = 2, a n = 1,i.e.
0 = a=+- am-
Step 3. Out of the possibilities from step 1 i) Equation (I)is impossible (same reason as in eqn. (4) of example a) iii) Equation (9) is impossible hecause from the second equation of step 2. either az+ or om- must be z0. Step 4. The only possible solutions are of the type
n e n = 22 m n ( a , . ..) and one easily arrives to the reduction
r = 22+ e n All other examples cited by Schafer and Cyvin may be solvedin the same way.
so the states derived from the electronic configuration of fundamental 0 2 are
b) Electronic States of Llnear Molecules
In short, our method is simple to use, not unrigorous, and applicable to every problem of reduction of infinite groups representations. Its interest is mainly pedagogical, the possible direct products and their reductions being already tabulated ( 6 ) .With respect to example b) note, finally, that point group theory gives just the symmetry not the spin state of the terns. More elaborate studies would he needed for that (7).
The electronic states arising from a given electronic configuration of a linear molecule are to be found. The problem can be solved within C,., a subgroup of D-h, taking into account with respect to theg - u symmetry the following direct product relations (5):g @ g = u @ u = g a n d g @ u = u. Let the electronic configuration he
z,; z,; 4
Acknowledgment
corresponding, for example, to 0 2 fundamental. This function transforms according to the direct product
nen and the problem consists in reducing this representation taking into account that all states will have g @ g = g symmetry. Before all the characters of Il @ Il are C," IE ' C n e n I 4 4eos2@= o = 2 + 2 ~ 0 m as may be easily deduced from tables (2). Steps 1t o 4 are then followed. These are the results. Step I. With x(E)= 4 there are three kinds of possibilities
308 1 Journal of Chemical Education
This work was stimulated by discussions following a seminar on "Group Theory and Molecular Structure" held within the activities of the Spectroscopy and Energy Transfer Group of this Department. We thank all the members of the Group. Lnerature Cited (1) Schsefar,L., and Cyuin, S. J..J. CHEM. EDUC.,48,295 (1911). (2) At*ms, . - P. W.."Mole& QuanhlmMschani~q"(hfon3Univaaity Plea. O d d , 1370. Ch.
6.
(3) Strommon, D. P., and Lippineott, E.R., J.CHEM. EDUC., 49,341 (1972). (4) Hall, L. H., "Group Theory and Symmetry in Chemistry: McCraw-Hill, New York, ,om
Pm3, Oxford, 1970. (7) Ford, D. I.. J. CHEM. EDUC.,49,336 11972).
