The Tabular Method for Reducing Representations Robert L. Carter University of Massachusetts-Boston, Boston. MA 02125 Most chemical applications of group theory require decomposing a reducible representation into its component irreducible representations. In simple cases, when the dimension of the representation is small, the reduction can be accomolished hv insoection or "trial-and-error" methods. In most other cases a more systematic approach is preferable. Virtually all texts on group theory for chemistry present the standard equation for systematic reduction (see eq 1below), followed by an illustration of its use in decomposing a sample reducible representation. Following this lead, students learn to accomplish systematic reduction by laboriously writing out in standard mathematical notation a series of specific equations for all the irreducihle representations. The work becomes particularly cumbersome ;hen the number of classes of operations, c, in the group is large, since c equations with c terms will he required. For example, a reducible representation in Oh requires 10 equations with a total of 100 terms. In such circumstances. all but the most careful students are likely to miswrite a term or two or to overlook the relationshios amone the results that serve as checks to accuracy. Most of the frustrations inherent in "brute force" application of the general reduction equation can be avoided by oreanizine the work with the tabular procedure described in The method, which significantly reduces the lathys bor and increases the accuracy of the process, has been used by anumber of practitioners for man;years."In light of this, it is odd that no general text on chemical applications of group theory gives-a presentation of the tab& a p p r o a ~ h . ~ This paper, then, is meant to bring the technique to a wider audience of nonspecialists, especially students: The general reduction equation can be written ~~~
~
.
~
~~~~~
&
For each irreducible representation of the group, eq 1 answers the question "how many times does this species contribute to the reducible representation, r,?"The answer, nj, might be zero or any integer less than or equal to d,ldj. In principle, eq 1must he solved to find ni for each irreducible representation of the group, starting with the totally symmetric representation and working through all the others one hy one. Actually, the process can he stopped a t any point when a sufficient number of irreducihle representations has been found to account for the dimension of r,, consistent with ea 2. For purposes of illustration, consider the following reducible representation of the point group L:'
~
A Tdcharacter table, such as the one shown below, is needed to apply eq 1to the reduction of this representation.
-
The order of the erouo. .. h.. is the number of o~erationscomposing the group or, equivalently, the sum of the numbers of operations in all the classes: i.e.. h = 1 8 3 6 6 = 24. since Tdconsists of five claises'of operations, there are five irreducible representations. In principle, then, eq 1must he solved five times. Rather than writing out five explicit equations of the form of eq 1,we use the following worksheet:
+ + + +
where nj = number of times the irreducible representation i occurs in the reducible representation; h = order of the e, = dimension of the class erouo: c = classes of the mouo: " (i.e., number of operations in the class); xi = character of the irreducible representation for the operations of the class; X, = character of the reducible representation for the operations of the class. The set of values found by eq 1 must be such that the sum of the products of the dimensions of the irreducible representations, dj, multiplied by the number of times they contribute to the reducible representation, nj, equals the dimension of the reducible representation, d,:
-
..
' I am indebted to David L. Powell of the College of Wooster, who
introduced me to a form of this technique over 25 years ago. The only exception of which I am aware is my previous brief emosition of the method in a soecialized text: Carter. R. L. in Infrared a n > ~-~ n m- a n~~schoscoov:~ra&. - - ~ ~ ? , . E. G.: Grasselli.J. G...Eds.:. Dekker: New York. 1976; Part A. Vol. 1, chapter 2,pp 95-97. This representation has been constructed solely for the purposes of illustration and has no particular physical meaning. "he format of the worksheet is compatible with electronic spreadsheets, which could be used to carry out the arithmetic automatically. However, in most cases the labor of setting up the spreadsheet probably exceeds that of doing the work manually. -
?
~
-
~
The first line below the header row shows the characters of r,. The first column lists, in the order of the character table, the Mulliken symbols for the irreducihle representations of Ta. In the laree central oortion. alone the row for each irredicible representation, we will fill in ~heproductsg,xjx,.For everv row. this involves takine the character for each class from the character table, mukplying it by the number of operationsin the class, and then multiplying that product by the character of r, listed on the worksheet. Then, we sum across the row to obtain 2,and fiuallv divide the sum hv the k across each row group order, h, which for ~d is 24. ~ h result is ni for the irreducible rep~esentation.~ Volume 68
Number 5 May 1991
373
The complete worksheet, generated in this manner, is shown below: T
r,
I
E
8C3
8
-1
3C2
6%
4
0
6m
I
2
1
2124
-2
+ +
This shows that r, = A2 2E TI. The proof is that this combination of irreducible representations adds to give F,:
sign changes or multiplications were made from one row to the next. (It is also possible, of course, that the original reducihle representation was faulty.) Thus, in the row for E , if we had inadvertently failed to change the sign of the product for 8C3 from the row for Ap, the sum would have been 16 - 8 24 0 0 = 32. which is not divisible by 24. The error would be caught by referring back to the character table and seeing that the +1character of 8Cs for Ap becomes -1 for E. ~ h i i ethe same error could be detected in the explicit expression for na in the tabular form there is no hunting for which term corresponds to which class of operations, since the worksheet is laid out in the same format as the character table. Finally, the tabular format, with its column of n,results, facilitates checkine to see that the sum of the dimensions of the component irreducible representations is the same as the dimension of the reducible representation, as required by eq eq 2 works out as 2. In our example where T, = Ap 2E + TI, d, = (1) (1) (2) (2) (1) (3) = 8, which is the dimension of r, asshowm by its character for the operationE in the header row of the worksheet. As one proceeds throueh the work, i t is a good idea to keep an eye oithe running sum of the dimensions. If a t any point the sum exceeds the dimension of the reducible representation, an error probably has been made in one or more of the ni computations. In this case, the error would be of a type that would give a sum fortuitously divisible by the order. Aside from error detection, noting the sum of the dimensions as successive ni results are obtained may prevent carrying out unnecessary work. For example, the reducible representation may consist exclusively of species that are listed in the u m e r part of the character table. If so. the dimension of ti; reducible representation will be sat&fied before reachine the bottom row of the worksheet, This was the case for our r,, where the dimension of the representation was satisfied at the ~ o i nthat t n, was determined to be 1 for TI.In other words; the last line in the worksheet is unnecessarv in this case. Therefore, as a general practice, one shouldcontinue row by row in the worksheet oniy until sufficient number of irreducible representations has been found to satisfy the dimension of the reducible representation. Of course, following the running total of the dimension is possible (and indeed recommended) even if the traditional method is employed, but with the results lined up in a column it becomes easier to make this checkine a routine art of the process. Regardless of the method usei, the decomposirionshould always be verified bv addine the f w n d irreducible representatio"ns to see thattheir sum gives back the reducible representation.
+ + +
-
+
Oreanizine the work of ea 1in this way has several advant a g e l ~ i r s tsince , successive irreducib1e;epresentations differ from one another by changes in sien or by multiplications of characters, it is e a s i t o generate the terms for su&essive ni computations simply by making the appropriate alterations from either the first or previous rows in the worksheet. For example, in Td, the difference between A, and Ap is a sign change for the characters of 6S4and 6ad. Thus, the row for Ap can be written quickly by copying the first three numbers and changing the signs on the last two from the A, row. I t shouldalso be noted that the first line itself can begenerated verv easilv. without referrine to the character table. hv writing'downihe products of thedimensions of the claskes times the characters of the reducible representation, g,~,, since all characters for any totally symmetric representation are +l. Furthermore, for succeeding lines, wherever zero characters appear in the reducible representation, whole columns can be skipped. In the case of our r, the character of 684 is 0, so for all calculations of ni we can ignore the penultimate column. These shortcuts speed the work significantly, compared to writing out every term of every eiplicit expression of the reduction formula. Second, the work is laid out in a form that aids checking for arithmetic errors. For example, we know that the sum must be divisible by the group order. If it is not, probably an error has been made in one or more of the terms in the row. Usually this can be detected by checking that the correct
+
+
-
a
Nominations Sought for Chromatography Award
- -
Nominations are now beine souebt for the 1992 Eastern Analvtieal Svmoosiurn for Ontstnndinp. , ~ , (EAS) Award ~ ~ , Achievementa~nChromato~aphy.Theawardronr~stsofnplaqur.a~hich will bepresenteaataspec~alawardsymposium arranged in recognllion of the rcclplent. The Award wdl be presented at rhs 1992 EAS ~n Somerset, SJ,ln November 1992. Estahhshed in 19H6, this award i s dcsigncd t o recognize an indlvdual who ha3 made major contributionr to the field of chromatok~aphy.A nomlnatmn letter drscribing the nominee, includma .;peufic accomplishmrnts, should be submittedalung.\r~thabwgraphwal sketch by August 1,1990. Send dl materials toChatrman, EAS AwardsComm~rrcc, Eastcrn Anal).tical Sympoe~umI n c , 1'0. Hox 633, Montchamn, DE 197111.0fi:33.
374
Journal of Chemical Education
~
~~
~
~
