Dennis P. Strommen Carthaae CoNeae
I
Some Additional Commenk 0n hfhite Point & O U ~ S
Recently, J. M. Alvariiio reported to this Journal1 that the method of reduction of representations of infinite point groups developed by Strommen and Lippincott (S-L)2was in error. The purpose of this note is to correct what I believe to be an incomplete analysis of our earlier work and to demonstrate the flexibility of our method. I t was claimed that the S-L method compares characters rather than basis sets, since the representations and cP under C,, and 2;. asand a, under D,h "do not possess such basis functions." The basis functions the author is referring to are the Cartesian coordinate bdis functions. Let me first of all point out that the S-L method does not require Cartesian basis functions as I will demonstrate. Secondly, basis functions must and do exist for all representations althouah some of them are not listed in commonly available character tables. For example, the Cartesian basis for under C,, is xlyz - ~ 1 x although 2 most tables list only R,. Likewise, the representation for a under C,, transforms as (x3- 3xy2,3x2y - ~ 3 1 . 3 With respect to the derivation of the symmetries of electronic states, it was also claimed that a given electronic confieuration of a linear molecule "could ohviouslv not he solved" by the method uiS-I.. I will nmv dt:m~rnsrratethe flexibility o i the 3-1.ntethod b \ sol\.inr the wrv oroblem referred 10 in pap&.1 Let the electronic configurapart (b) of the tion of a D,h molecule he: (core) IIglII,'. Using CI, as a subgroup, the direct product II @ II corresponds to E @ E, which hv insoection vields: red = A, + A ? BT +Be. I chose C4" as su6group since two of the possihe re'presekations under D,h transform as (xz,yz) and (x2- y2,xy). Therefore, the appropriate suhgroup should not mix these binary products. I need not concern myself with the fact that (x,y) forms a basis for E under C4, as wellas (xz,yz), since all of theresults must be g-type (g @ g = g ) and under D,h, (XJ) transforms
Representation (C,,)
x-
x;,
x-
a
as nu.Using a standard character table, the following correlations can then he made between the representations Cq,, and Doih. Notice that this is the same answer as that obtained, by Correlations Between the Representations C,. and D,,
r
640 1 Journal of Chemical Education
Basis
A, A*
8, 82
2.9
:-
XY
y 2 ~
Representation (Dm,)
x; A,
what I believe to be the less efficient method proposed by Alvarifio. Notice also that the basis R, was used to correlate the representations and not the Cartesian equivalent xlyp YlX2. The final comment made by Alvariiio has some validity. Indeed, not all subgroups will give the desired information and one has to have some feel for thebasis of a representation in order to intelligently select the appropriate suhgroup as I did in the above example. But since it is these same bases that connect group theory to physical reality, it is clearly desirable that the student acquire some familiarity with them. In conclusion, I have shown that the S-L method is valid and as simple a procedure as any proposed up until now. Furthermore, it is less mechanical in nature and not likely to he forgotten once learned.
+
'Alvariiio, J. M., J. CHEM. EDUC., 55,307 (1978). ?3trornmen, D. P. and Lippincott, E. R., J. CHEM. EDUC., 49,341 . ~
(1972).
3Harris,D. C. and Bertolucci, M. D., "Symmetry and Spectroscopy," Oxford University Press, New York, 1978,p. 416.
