Overtones of Degenerate Vibrations In an earlier paper in this Journal', D. I. Ford presented a general equation for generating the characters of a point group adapted to any desired symmetry species of the symmetric permutation group, S(n). This equation was then applied to constructing the totally antisymmetric electronic states for atoms in various crystal fields. This equation may also be applied to produce other types of permutational symmetry. Thus, for example, the totally symmetric state is required for the multiple exritatior of molecular vibrations.' In this case all of the W j ) characters will be unity in eqn. (2) of reference ( I 1, and the equation thus becomes:
1 k p l ~ ( G ' ) ~ l x ( G .~.).~x(G")~"I 2 x(G:lnl) = n! , , . S , ~ ,
(1)
We apply eqn. (1) t o a triple excitation of the Tzvibration of methane as follows. The tripleexcitation requires the use of the S ( 3 ) group; in general an n-fold excitation requires S(n). For S ( 3 ) the partitions are (1.'). (1, 2), and (3)leading to
--
(I3) (1,2)
(3)
x(GIP
x(C')x(G2) x(G8)
Then eqn. (1) becomes
The ( 2 representation in Td is
leading, when adapted to S ( 3 ) to
This reduces to lA1, IT1, and 2Tz vibrations. The procedure used here yields the same results as obtained by T i s m 3
R. L. Flurry, Jr. University of New Orleans Ncw Orleans, Louisiana 70122
Darl H. McDaniel University of Cincinnati Cincinnati, Ohio 45221
Volume 55. Number 1, January 1978 1 17
