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SOLUTIONSMANUAL TO ACCOMPANY ATKINS' PHYSICAL CHEMISTRY 343
sB into sC, sC into sD, and sD into sB: ( sC sD sB )← ( sB sC sD ). �is is
written using matrix multiplication as
( sC sD sB ) = ( sB sC sD )
D(C3)
³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ
⎛
⎜
⎝
0 0 1
1 0 0
0 1 0
⎞
⎟
⎠
Similarly
For σv: ( sB sD sC ) = ( sB sC sD )
D(σv)
³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ
⎛
⎜
⎝
1 0 0
0 0 1
0 1 0
⎞
⎟
⎠
For E: ( sB sC sD ) = ( sB sC sD )
D(E)
³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ
⎛
⎜
⎝
1 0 0
0 1 0
0 0 1
⎞
⎟
⎠
�e characters of the representatives are
χ(E) = 3 χ(C3) = 0 χ(σv) = 1
�is result can be arrived at much more quickly by noting that: (1) only
the diagonal elements of the representative matrix contribute to the trace;
(2) orbitals which are unmoved by an operation will result in a 1 on the
diagonal; (3) orbitals which are moved to other positions by an operation
will result in a 0 on the diagonal.�e character is found simply by count-
ing the number of orbitals which do not move. In the present case 3 are
unmoved by E, none are unmoved by C3, and 1 is unmoved by σv. �e
characters are therefore {3, 0, 1}.
�is representation is decomposed using the method described in Sec-
tion 10C.1(b) on page 408. �e number of times n(Γ) that a given irre-
ducible representation Γ occurs in a representation is given by [10C.3a–
408],
n(Γ) = 1
h
∑
C
N(C)χ(Γ)(C)χ(C)
where h is the order of the group, N(C) is the number of operations in
class C, χ(Γ) is the character of class C in the irreducible representation Γ,
and χ(C) is the character of class C in the representation being reduced.
In the case of C3v, h = 6. �e number of times that the irreducible
representation A1 occurs is
n(A1) = 1
6 (N(E)×χ(A1)(E)×χ(E) + N(C3)×χ(A1)(C3)×χ(C3)
+ N(σv)×χ(A1)(σv)×χ(σv))
= 1
6 (1×1×3 + 2×1×0 + 3×1×1) = 1