Chemistry Reference
In-Depth Information
It can easily be demonstrated that the Hamiltonian in this new basis is real:
=
ϑφ
χ)
=
ϑφ
ϑχ
=
|
H
|
|
H
|
H
|
|
H
|
φ
χ
ϑ(
ϑ
ϑ
φ
χ
(7.59)
Hence, in this case, it is always possible to rewrite the basis in such a way that the
Hamiltonian matrix is completely real, and the states behave in all respects as a
real twofold-degenerate irrep, which can be split by symmetry-lowering electro-
static fields. In particular, such states will be subject to Jahn-Teller distortions.
•
ϑ
2
1. In the case of a negative sign, it is impossible to obtain states that are
time invariant. This can be shown as follows. We start again with two states that
are each other's time inverse
(
|
Ψ
=−
and
ϑ
|
Ψ
)
and first show that these states must
be linearly independent:
=
ϑΨ
ϑ
2
Ψ
=−
Ψ
|
ϑΨ
|
ϑΨ
|
Ψ
=−
Ψ
|
ϑΨ
=
0
(7.60)
In contrast to the previous case, all attempts to construct a linear combination of
these basis states that is invariant under time reversal, fail. Indeed, suppose that
|
X
is a linear combination with coefficients
a
and
b
, such that
ϑ
|
X
=|
X
. Then
we have:
|
X
=
a
|
Ψ
+
bϑ
|
Ψ
(7.61)
−
b
ϑ
|
X
=¯
aϑ
|
Ψ
|
Ψ
≡
a
|
Ψ
+
bϑ
|
Ψ
Since the two kets are linearly independent, their respective coefficients must
coincide, and this is possible only for
a
0. Hence, it is not possible to
remove the degeneracy by time-even external fields. In particular, these states
will not be subject to the JT effect.
=
b
=
Time-Reversal Selection Rules
The argument used in Eq. (
7.59
) can be generalized to describe selection rules that
depend on time reversal [
12
]. We first introduce two parities,
τ
and
η
, which de-
scribe the time-dependence of the state and of the Hamiltonian:
ϑ
2
1
)
τ
E
=
(
−
(7.62)
ϑ
−
1
1
)
η
ϑ
H
=
(
−
H
The first label,
τ
, indicates the parity of the state functions, as we have just intro-
duced in this section. The second label,
η
, indicates whether the Hamiltonian is
time-even or time-odd. Time-even interactions are typically interactions associated
with the electrostatic potential, such as the Jahn-Teller and Stark effects. Time-odd
interactions are electrodynamic in nature, the most common one being the Zeeman
interaction. We shall now study a function space that is invariant under time reversal
