H Solutions to Problems
257
sarily nonbonding and will be occupied by two electrons. The direct square of
this irrep yields symmetrized A ′
1 and E ′ states and an antisymmetrized A ′
2 state.
The expressions for these states are obtained from the coupling coefficients for
D 3 in Appendix F:
1 A
′
1 =
1
√
2
x(1)x(2) + y(1)y(2)
1
√
2
α(1)β(2) − β(1)α(2)
=
1
√
2
(xα)(xβ)
+
(yα)(yβ)
1 E
′
x =
1
√
2
(xα)(yβ)
+
(yα)(xβ)
1 E
′
y =
1
√
2
−
(xα)(xβ)
+
(yα)(yβ)
3 A 2 =
(xα)(yα)
Note that the distinction between zwitterionic and diradical states does not
hold in this case. Formally, TMM can be described as a valence isomer between three configurations in which one of the peripheral atoms has a double bond to the central atom and the other two sites carry an unpaired electron.
7.1 In a cube the d-shell also splits in e g + t 2g , but the ordering is reversed. Explicit
calculation of the potential shows that the splitting is reduced by a factor 8/9:
cube =−
8
9
octahhedron
7.2 Perform the matrix multiplication and verify that the product matrix is of
Cayley–Klein form. The multiplication is not commutative:
a 1 b 1
− ¯
b 1 ¯
a 1
×
a 2 b 2
− ¯
b 2 ¯
a 2
=
a 1 a 2 − b 1 ¯
b 2 a 1 b 2 +¯ a 2 b 1
−¯ a 1 ¯
b 2 − a 2 ¯
b 1 ¯
a 1 ¯
a 2 − ¯
b 1 b 2
(12)
7.3 The double group D ∗
3 contains 12 elements. In Table 7.5 we have listed the six
representation matrices for the elements on the positive hemisphere. The ˆ
C A
2
axis is along the x-direction, ˆ
C B
2 is at −60 ◦ and ˆ
C C
2 is at +60 ◦ . The derivation
of the multiplication table and the underlying class structure (see Table 7.6)is
based on a straightforward matrix multiplication.
257
sarily nonbonding and will be occupied by two electrons. The direct square of
this irrep yields symmetrized A ′
1 and E ′ states and an antisymmetrized A ′
2 state.
The expressions for these states are obtained from the coupling coefficients for
D 3 in Appendix F:
1 A
′
1 =
1
√
2
x(1)x(2) + y(1)y(2)
1
√
2
α(1)β(2) − β(1)α(2)
=
1
√
2
(xα)(xβ)
+
(yα)(yβ)
1 E
′
x =
1
√
2
(xα)(yβ)
+
(yα)(xβ)
1 E
′
y =
1
√
2
−
(xα)(xβ)
+
(yα)(yβ)
3 A 2 =
(xα)(yα)
Note that the distinction between zwitterionic and diradical states does not
hold in this case. Formally, TMM can be described as a valence isomer between three configurations in which one of the peripheral atoms has a double bond to the central atom and the other two sites carry an unpaired electron.
7.1 In a cube the d-shell also splits in e g + t 2g , but the ordering is reversed. Explicit
calculation of the potential shows that the splitting is reduced by a factor 8/9:
cube =−
8
9
octahhedron
7.2 Perform the matrix multiplication and verify that the product matrix is of
Cayley–Klein form. The multiplication is not commutative:
a 1 b 1
− ¯
b 1 ¯
a 1
×
a 2 b 2
− ¯
b 2 ¯
a 2
=
a 1 a 2 − b 1 ¯
b 2 a 1 b 2 +¯ a 2 b 1
−¯ a 1 ¯
b 2 − a 2 ¯
b 1 ¯
a 1 ¯
a 2 − ¯
b 1 b 2
(12)
7.3 The double group D ∗
3 contains 12 elements. In Table 7.5 we have listed the six
representation matrices for the elements on the positive hemisphere. The ˆ
C A
2
axis is along the x-direction, ˆ
C B
2 is at −60 ◦ and ˆ
C C
2 is at +60 ◦ . The derivation
of the multiplication table and the underlying class structure (see Table 7.6)is
based on a straightforward matrix multiplication.