Representations with Imaginary Characters The Doubling Problem Robert L. Carter University of Massachusetts-Boston, Boston, MA02125
Most point groups have irreducible representations consisting of real-number characters. However, a few are represented using characters that are functions of the imaginary number.
These include • • •
•
the cyclic groups, C„ with n > 2 the C„h groups with n > 2 the improper-axis groups, two of the cubic groups, T and Th
bined forms of the two pairs of complex-conjugate representations are shown. The real-character representations are indicated by {Eg\ and {EJ This emphasizes that they are constructed, reducible representations of the group—rather than genuine, doubly degenerate, irreducible representations.
All these groups contain pairs of irreducible representations that are complex conjugates of one another. The paired representations appear on successive lines in the character tables. They are joined by braces (! I) and given the usual Mulliken symbol E of a doubly degenerate representation. The Origin of the Representations Using the symbol of a doubly degenerate representation tends to foster a misunderstanding about the true nature of the paired complex-conjugate representations. They appear to be a manufactured decomposition of a genuine doubly degenerate representation that were separated to avoid violating the well-known rule that the number of irreducible representations of a group must be equal to the number of classes in the group. Actually, as Cotton1 demonstrates for the cyclic groups, the imaginary-character representations arise quite naturally from the great orthogonality theorem, and thus are not an “after-the-fact" separation. Nonetheless, for many routine applications to physical problems (e.g., vibrational selection rules, symmetries of molecular orbitals), the complex-conjugate pairs may be added to obtain sets of real-number characters for two-dimensional representations.1 This procedure avoids the awkwardness of dealing with the imaginary terms, without limiting the ability to construct representations for any real physical property. The reducible representation must contain both complex-conjugate irreducible representations, if it contains either.2,3 However, it is easy to forget that a combined real-character representation—although a genuine representation—is not an irreducible representation of the group.
Caution in Using the Reducible Representation Failing to recognize the true nature of the combined realcharacter representations can lead to some surprising and erroneous results if they are used when attempting systematic decomposition of a reducible representation of one of these groups. To illustrate, let us construct a reducible representation of known irreducible representations, including a complex-conjugate pair. The group CUh, whose imaginary characters are simply ±i, will serve for this purpose.
The character table below shows the irreducible representations in their usual form. At the bottom, the com-
Let the test reducible representation be r
=
U?gl +
Au + 2 Bu
which by addition leads to the following set of characters.
Now, let us systematically decompose T by applying the usual reduction equation below. 'li
=
U
JlLScXiXr
(D
where n, is the number of times the irreducible representation i occurs in the reducible representation; h is the order of the group; c is the classes of the group; gc is the dimension of the class (i.e., number of operations in the class); %i is the character of the irreducible representation for the operations of the class; and y_r is the character of the reducible representation for the operations of the clas^.. In keeping with the usual practice with groups of this type, we will combine the paired imaginary representations to form the two-dimensional real-character representations \Eg\ and (£„}, as shown at the bottom of the preeed-
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ing character table. Then we use these two-dimensional representations with eq 1 and obtain the following reduction using the tabular method.4
all other (genuinely irreducible) representations of the group, the n, results will be correct as initially calculated by eq 1.
Appendix To understand
precisely why the doubling problem on the result of eq 1 of using a two-dimensional real-character representation as if it were an irreducible representation of the group. Let 11^ be the result of eq 1 obtained using characters Xab generated by adding the characters yfl and Xt> of the complex-conjugate irreducible representations Ta and Th, such occurs,
we
more
need to examine in general the effect
that
rab
ra
=
+
r*
As a perusal of the character tables shows, for all such pairs of representations,
The rii results, which appear in the “XJ8” column are clearly correct for all nondegenerate species and for the absent, combined representation {Eu]. However, the answer for \Eg) is twice what it should be. Even if we had not known the correct result beforehand, we could have recognized that the overall result for all nt is erroneous because the sum of the dimensions of the found species exceeds that of the original reducible representation, in violation of the following relationship. dr
—
^j rijdi
(2)
where dr is the dimension of the reducible representation; and di is the dimension of each irreducible representation. In general, if real-number combined representations are used with eq 1 in place of the corresponding individual complex-conjugate representations, the rii results for all such pairs will be twice the true values. (A more general explanation of this is presented as an appendix to this paper.) However, there is no problem if the individual imaginary-character representations are used to calculate rii. For example, for our test [the values of ri,- for Eg and Eu work out as follows. C4h
Eu
C4
O2
-i
i
-1 -t
-i i
-1 -1
i
G4^
/
-/ i -i
S4^
(Jh
S4
-1 -1
/
5
-i
5
1)
1
-/
-5 -5
-It
1
/
Z
£/8
8
1
8
1
0
0
0
0
nipulate. As a result, it is usually more practical to use the combined real-character representations, as long as one is aware of the doubling problem. If this practice is followed, the true number of times that any complex-conjugate pair of irreducible representations (or their combined realcharacter equivalent) contributes to the reducible representation can be found by dividing the result n, by 2. For Journal of Chemical Education
Xa
Xb or
=
Xa
=
Xb
where Xb is the complex conjugate of Xi>• When both Xa and Xb are real numbers, then Xa=Xb
and ~
Xab
2z„ =
However, for all pairs of complex-conjugate irreducible representations, the only real characters are either +1
which
means
or
-1
that Xab=±2
When Xa=Xb
the numerical characters
of the general form
are
where v = 1, 2, 3. For a system in which the principal axis (highest order) is Cn, =
As this shows, both irreducible representations comprising {Eg} contribute equally to T, as they must for any real T. From this it might seem that one should always use the individual imaginary-character representations when applying the general reduction equation (eq 1). However, in most point groups with imaginary characters (except groups like C4, C4h, and S4), the complex conjugate representations involve exponential functions of i or their trigonometric equivalents, which are very cumbersome to ma-
18
either
exp
2 Tit n
Then Xab
=
l
V
+ e
V*
=
2cos (2v;I o
T"
However,
is always a real-number constant (becoming 0 when n = 4). Therefore, all characters Xab of the combined representation rai, are 2 times some real-number value, which we shall call Xa(,. Xab
2X0£
~
For any complex-conjugate pair of Xa and
Xb,
Xah may be positive, negative, or 0.
For real-number pairs Xab
=
±1.
Thus, all characters of the combined representation rab contain the integer factor 2. As a result, when Xab characters are used in eq 1, the value of rij will always be divisible by 2 and will actually be twice the correct answer. ft'ab
'
l
X- Sc
b
Xr
XI Sc ^ab Xr
t
(3)
Thus, if
we
seek the true number of times ^a+b
that the following combination rab
=
+
^a+b
o
Ji
c
^c ^®b Xr
~
In this context, the half characters would act as realnumber substitutes for the characters of r„ or rb. By either approach, the value of na+b obtained implies that both T„ and Tb equally contribute na+b times to the reducible representation.
Literature Cited
rb
contributes to the reducible representation Tr, then the sult could be obtained as below. It-*
Simply put, the correct result for such complex-conjugate pairs is half the result obtained by eq 1. As eq 4 suggests, an equally valid but less efficient procedure would involve using the half characters
lx1 Xj Sc Xab Xr
ofo
c
Cotton, F. A. Chemical Applications of Group Theory, 3rd ed; Wi]ey-Interscience: New York, NY, 1990; pp 95-99. 2. Tinkliam, M. Group Theory and Quantum Mechanics', McGraw-Hill: New York, NY, 1.
re-
(4)
1964; p 147. 3. Hammermesh, M. Group Theory and Its Application to Physical Problems', Addison— Wesley: Reading, MA, 1962; p 118. 4. Carter, R. L. J, Ckem. Educ. 1991, 68, 373—374.
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