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SOLUTIONSMANUAL TO ACCOMPANY ATKINS' PHYSICAL CHEMISTRY 347
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.
For the point group D2h, h = 8. Because the representation to be reduced
has characters of zero for all classes except E and σ x y , only these latter
two classes will make a non-zero contribution to the sum.�e number of
times that the irreducible representation Ag occurs is
n(Ag) = 1
8 (N(E)×χ(Ag)(E)×χ(E) + N(σ x y)×χ(Ag)(σ x y)×χ(σ x y))
= 1
8 (1×1×4 + 1×1×(−4)) = 0
Similarly
n(B1g) = 1
8 (1×1×4 + 1×1×(−4)) = 0
n(B2g) = 1
8 (1×1×4 + 1×(−1)×(−4)) = 1
n(B3g) = 1
8 (1×1×4 + 1×(−1)×(−4)) = 1
n(Au) = 1
8 (1×1×4 + 1×(−1)×(−4)) = 1
n(B1u) = 1
8 (1×1×4 + 1×(−1)×(−4)) = 1
n(B2u) = 1
8 (1×1×4 + 1×1×(−4)) = 0
n(B3u) = 1
8 (1×1×4 + 1×1×(−4)) = 0
�e four pz orbitals therefore span B2g + B3g +Au + B1u .
(b) �e SALCs of B2g, B3g, Au and B1u symmetries are generated by applying
the projection operator to the pA orbital using the method described in
Section 10C.2(b) on page 409.
Row E Cz2 C y2 Cx2 i σ x y σ yz σ zx
1 e�ect on pA pA pC −pB −pD −pC −pA pB pD
2 characters for B2g 1 −1 1 −1 1 −1 1 −1
3 row 1 × row 2 pA −pC −pB pD −pC pA −pB pD
4 characters for B3g 1 −1 −1 1 1 −1 1 −1
5 row 1 × row 4 pA −pC pB −pD −pC pA pB −pD
6 characters for Au 1 1 1 1 −1 −1 −1 −1
7 row 1 × row 6 pA pC −pB −pD pC pA −pB −pD
8 characters for B1u 1 1 −1 −1 −1 −1 1 1
9 row 1 × row 8 pA pC pB pD pC pA pB pD
�e SALCs are formed by summing rows 3, 5, 7, and 9 and dividing each