Chemistry Reference
In-Depth Information
and transforms according to a degenerate irrep
Γ
with all characters real. As we
have seen, this can be either an orbital or a spinor irrep. If
|
φ
is an element of this
space, so is
ϑ
|
φ
,
we shall replace the bra-functions by their time-reversed partners. The interaction
element will then be of type
. Now, instead of considering matrix elements of type
φ
|
H
|
χ
. This may seem awkward, but in fact it does
not lead to inconsistencies. In the case of orbital irreps, basis functions will either
be real or may be arranged in complex-conjugate pairs, which are mutually time
inverses. For spinor irreps, we can always write the basis functions in time-reversal
pairs, such as the
α
and
β
spins.
The Hamiltonian for physical interactions must be Hermitian; hence, the bracket
will be equal to the complex-conjugate inverted bracket:
ϑφ
|
H
|
χ
ϑφ
|
H
|
χ
=
χ
|
H
|
ϑφ
(7.63)
Complex conjugation of the bracket can also be achieved by time reversal. But, as
an operator, time reversal can also enter into the bracket and operate on the compo-
nents:
=
ϑχ
ϑφ)
=
ϑχ
ϑ
2
φ
=
ϑ
−
1
1
)
τ
+
η
χ
|
H
|
ϑφ
|
ϑ(
H
|
ϑ
H
|
(
−
ϑχ
|
H
|
φ
(7.64)
By combining these results we may thus write
2
1
1
)
τ
+
η
ϑφ
|
H
|
χ
=
ϑφ
|
H
|
χ
+
(
−
ϑχ
|
H
|
φ
(7.65)
The transformations of a time-reversed bra,
, are de-
scribed by exactly the same matrices because complex conjugation is applied twice:
ϑf
i
|
, and its original ket,
|
f
i
R
D
ji
(R)
ϑf
i
|=
ϑf
j
|
j
(7.66)
R
D
ji
(R)f
j
|
f
i
=
j
This implies that the matrix elements in Eq. (
7.65
) are described by CG coupling
coefficients, which are symmetric with respect to exchange of the bra and ket parts.
This result leads to the following selection rules:
1. If the Hamiltonian and the system have the same parity under time reversal, in-
teraction can take place only with the symmetrized square of the irrep,
2
.
2. If the Hamiltonian and the system have opposite parity under time reversal, the
only allowed interaction elements must belong to the anti-symmetrized square of
the irrep,
[
Γ
]
2
.
As a case in point, the JT Hamiltonian for orbital systems is limited to the
symmetrized square, whereas the Zeeman Hamiltonian only arises if the anti-
symmetrized square contains the symmetries of an axial field. For systems with
an odd number of electrons, which therefore transform according to spinor repre-
sentations, the selection rules are exactly opposite.
{
}
Γ
