Further Comment on Infinite Point Groups S. 0. Huang and P. G. Wang Northeastern University. ~oston,MA 021 15 It was pointed out that the well-known group theoretical equation
fails to work when applied to linear mole~ules.',~In eq 1n(y) gives the number of times the symmetry species (or irreducible representation), y, should occur in the reducible representation r; XR are the characters of r, for every symmetry operation R; & b e the characters of y; and g is the number of operations (i.e., group order). Schafer and Cyvin proposed a method to obtain n(y) through use of the formula1
xn = P ( x ) x j r '
(2)
which is true because the character XR of F is equal to the sum of the characters for R of all the x that combine to form r. Then it is possible to build a set of linear equations to evaluate n(x). Although this method is very valid, it is necessary to guess which of the coefficients, n(x), does not vanish. Stromman and Lippincott presented another approach based upon the fact that the basis of a representation of a group Go will also be the basis of a representation (perhaps degenerate) of another group G that is a subgroup of The second method does work, and it is useful in understanding of correlation tables. But, it is not a direct one too. In order to decompose the reducible representation of linear groups directly, it is necessary to apply the concept of continuous group^.^ Since this appears to he too advanced for most of the general and undergraduate courses, we can still present another elementary, direct approach to decompose the reducible representation of infinite point groups by using eq 1if we stick to the rules of finite point groups. The present method is based on the idea that for finite point groups, the g factor, order of group, is the number of symmetry operationa, which is also equal to the sum of characters of the totally symmetric A1 representation, since all the characters of A1 representation for every symmetry operation are equal to 1. Let us assume this is also true for infinite point groups:
Thoughg here is infinite by thin definition for C,, and D.,, it could lead to some meaningful results when we apply .. . eq- 1 to linear molecules, since we usually meet a ratio of two infinities, which in turn might be of finite value. Since all the irreducible iepresentations are orthogonal to each other, especially A , representation is orthogonal to all 34
Journal of Chemical Education
other representations, the sum of characters for irreducible representation (other than A,) should be 0, and the sum of the squared characters should he g for finite groups. By analogy, for Il representation in C,,:
This should be true for all other irreducible representations in C,, and D,h except A,. Now, we can use eq 1to decompose the reducible representations for linear molecules, considering the above mentioned relationships. Once again, we restrict our examples to those treated in the earlier papers.I.2 Example 1. Llnear xyz (C,.) (1) Choose Cartesian coordinates on atoms in the molecule as basis set E ZCi rmd9 3(1+ 2 cob$)
... .. .
mu"
3
(2) Use eq 1
rearrange into two series
n(Xf) = 3 3 * (1+ 2 * I + .
..+
rn r
1)
where the first series is g by definition, and the second series is zero in view of eq 4.
' Schafer,L; Cyvin, S.J. J. Chem. Educ. 1971,48,295.
. 341. Sirommen. D. P.: Lippincon, E. R. J. Chem. ~ d u c1972,49, Wigner. E. P. Gmup Theo'y, English ed.; Academic: New York.
1959.
Example 2. Linear xyz (Dm,,) (1)Choose Cartesian coordinates on atoms in the molecule as basis set E ZCi .. . c I ZSO .. . mC, r,, 9 3 ( 1 + 2 ~ 0 ~ +.). . 3 -3 ( - i + z ~ + ) ... -1 (2) Use eq 1
Similarly,
and
1 n ( ~ i ) = g [ 9 * l + 2 r ( 3 + 6 c o s + ) r l +. . . + o r * 3 r 1 - 3 *i+z*(-1+2~0~+)*1+...+-~(-1)*1]
rearrange into four series
where only the first series is g by definition, all other series vanish, which are irreducihle representations other than in D-h.
xi
In conclusion, we feel that the decomposition formula does work for linear molecules if we stick to the rules of finite point groups. Furthermore, the present method is useful in solvingprohlems related to C,, D,h (n = odd numbers), etc.
Volume 67 Number 1 January 1990
35
