Reducible Representations for Normal Vibrational Modes Clifford J. McGinn Le Moyne College, Syracuse, NY 13214 In determining the symmetry of normal vibrational modes the usual procedure is as follows: 1) Assign to each atom a set of unit vectors labeled xi, yi, and zi such that allxi a-e pointing in the x direction, all yj in they direction, and all zi in the z direction. The vectors xi, yj, and zj are, of course, orthogonal. 2) Using the matrices for the symmetry operations of the point group to which the system (molecularspecies) belongs determine the to which the entire set of vectors reducible representation, (., belongs. Thus, Icart.is reduced in the usual way to the sum of irreducible reoresentations. :I, Finally, remuw from the hst of irreducible reprewntstionr for rc:,?T. rhc three rr.milattonal modes. TiK, and t h thrrr ~ ttwo tor l ~ n r a r;vstemsl mtntional mcdr~,nii.. r o give ;,i6.
a
2
F
n
0 n
2coad n
(x,y),(Rx,Ry)
trans = 2++ a p€= a cart = (nZt)(2+ + a) ;.n 2 + + n n vib. = (n - l)Z+ (n 2)a e.g.,forn=3 m = 2 2 + + a
+ -
For D,h ( n even). Let n = 2 m and let
6= lT+E
A relatively simple method is offered here for determining w. Since the set x; transforms as E,yi a s p , and z, as p and the sum + ++ +equals itrans.,it follows that F5K
is equal to E. where E i s the representation for the number of atoms unmoved by each of the symmetry operations of the point group. Examples For the system Dsh (CO?,
A*"
+ E'
SOs, etc.) note that
Jtrans =
1= m&+,2=
mZ,+,n =mZ,++m2,+,trans= Z,++ a. rot = a# cart = m(Zpf ZU+)(Zu+ au) vih = n/2&+ + (n12 -l)(ZUf + rg, a,) e.g., n = 2 v i b = 2 ~ + , n = 4 1 ; ; 7 ; = 2 2 ~ + + Z , + + l i , + a u
+
Dsh
E
Az"
1
2
2C3 1 -1
3C2 -1 0
ah
-1 2
trans.
3
0
-1
1
2
cart.
12
0
-2
4 4
2S3 -1 -1 -2 1 -2
3cu
1
z
0
(xy)
+
+
1
2 2
For the tetrahedral system (CH4, Sod2-, etc.) the point group is Td and the x , y, and t axes transform as T1 and (trans. is
Tz 1 = (m + 1) Z,+ 2 = mZ,+ n = ( m + I)&* cart = ((m + l)Z,+ + rn2,+)(4++ a,) = (m + l)(Z.+ + a,) + m(&+ + a,) For the octahedral system (Fe(CN)fi3-, ete.) the point group is is Oh and the x , y, and z axes transform as TI, and TI,. Assume for simplicity that the molecule is AB6
m.
This method can he used to determine the symmetry of the normal vibrational modes for molecules belonging to infinite point group while avoiding the problems encountered in reducing Ei.'.* Consider C,,
+ mZUi
L
e.g., n = 3 n=S
vib = 2 Z,+
+ 2Zy++ 2au +
T~
' Schafer,L. and Cyvin, S. J., J. C ~ E MEouc., . 48, 295 (1971).
Strommen, D. P. and Lippincolt, E. R., J. CHEM., EOUC., 49, 341 (1972). Volume 59
Number 10
October 1982
813
