Dennis P. Slromrnenl
and Ellis R. Lippincon University of Maryland College Park, Maryland 20742
I
I
Comments on Infinite Point Groups
In a recent issue it was pointed out by Schaer and Cyvin2that the well known group theoretical equation
(6) Finally compare the basis vectors of the irreducible representations under GO with those obtained for the molecule assuming G.
We will now apply this method to the examples treated in the earlier paper.= fails to work when applied to linear groups. Equation (1) is used to determine the number of times, n(y), that a given symmetry species y will occur in the reducible representation of a particular molecule. .-.. r~ is the character of the reducible representation T;X , ( ~ ) is the character of the irreducible representation; and R is the index used to denote each of the symmetry operations of the group. Schafer and Cyvin2 present a method of circumventing eqn. (1) and obtaining the values of n(7) through use of the formula x. =
C n(r)x." R
(2)
which does hold for all of the characters of the infinite linear groups. Although their method is quite valid it depends upon the ability of the student to guess or know a priori which of the coefficients do not vanish. We have observed that the beginning student usually is unaware that the only irreducible representations occurring in vibrational systems with symmetry C,, are X + and II, and that Z,+, 2,+, II, and II, are the proper species to consider for D,&systems. We propose an alternate method of obtaining the same data, which among its other advantages, requires no previous knowledge of the symmetry species that are allowed. Our method is based upon the fact that the basis of a representation of a group Go will also be a representation (perhaps degenerate) of another group G which is a subgroup of Go. We will treat each of the examples of the earlier paper using our method.
Example 1. Linear zyz (1) Go = C-6 G = Cs,
(4) Using eqn. (1)
Obviously the 2A, symmetry species transform as 22+ species under the molecular symmetry C,. while the BIand BZ symmetry species transform together as the degenerate II species. Ezample 8. XYn (1) GO = D-A, G = Dxn
Outline of Method (1) Assume a. lower molecular symmetry which oorresponds to s subgroup G of the molecular group GO. (2) Place a set of cartesian coordinate vectors on each atom. (The a axis must he placed along the maximum symmetry axis of the parent g m ~ p . ) ~ (3) Using standard methods obtain the characters of rrrducibla: (4) Calculate the vdues of each n ( r ) by application of eqn. (1). (5) Subtract those symmetry species corresponding to translational and rotational motion, rt and Fa, respectively.
E CP(Z) Cdv) Cdx) i (3)rrrd 9 -3 -1 -1 -3
~ )Y Z )
3
(4) Using eqn. (1)
(5)
rr
=
m
= =
BL, B*.
+ B1" + Bsu
+ B,,
rUia rrrd- rl - T R = A,. Present address: Carthage College, Kenosha, Wisconsin 53140. SCH&ER,L., AND CYVIN,S. J., J. CAEM.EDUC.,48, 295 (1071) - - .- ,. Steps 2 through 5 constitute a standard treatment of vihrational problems. See for example the reference in footnote 4. COTTON, F. A,, "Chemical Applications of Group Theory" (1st Ed.) Wiley-Interscience, New York, 1963.
U&Y) 4 ~ 1 3
+ BL.,+ B P +~ Bsu
\
Then correlating the symmetry species throu& the basis vectors we obtain the answer
Volume 49, Number 5, May 197.7
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341
rse = +,+
+ +.+ + u.
Ezample 3. Linear symmetrical zryr ( 1 ) GO = D,h, a = D,,
Then clearly under D,n the solution to example 3 is
(4)Using eqn. (1)
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Journol of Chemical Education
Although we have taken the groups CzOand Dlh to be subgroups of C,. and Dm&,respectively, the method is quite general. That is, any subgroup of a given parent group may be used in treating the problem. I n conclusion we feel that there are three practical advantages to our method. First it requires no previous knowledge of or guesses as to which representations are allowed. Secondly, the significance of the basis of a representation is made clear. Finally, use of our method should sewe as a foundation for the future use of correlation tables.
