## Methane Symmetry Operations

### 17. The Compound Operation Time-Reversal Hermitian
Conjugation

It is sometimes convenient, especially when investigating the legitimacy of
phenomenologically introduced interaction terms in the Hamiltonian operator, or
when determining their selection rules, to consider the transformation
properties of the term in question not only with respect to operations of the
molecular symmetry group, but also with respect to the operations of Hermitian
conjugation and time reversal [22]. (When
electron-spin and nuclear-spin functions are neglected, time reversal is simply
the operation of taking complex conjugates.)
Since the behavior of an interaction term under either of these two latter
operations can be reversed simply by multiplying the term by a pure imaginary
constant, Dorney and Watson [66,67] have suggested
considering the transformation properties of operators with respect to the
product of both operations. The transformation properties of terms with respect
to this compound operation are invariant to multiplication by a complex
constant. Since the Hamiltonian for a molecule in free space must be invariant
to both Hermitian conjugation and time reversal, only terms which are invariant
to the product of these two operations can occur in the Hamiltonian. Of course,
a term allowed in the Hamiltonian under
time-reversal ×
Hermitian-conjugation may still require multiplication by some complex constant
to be acceptable under either time reversal or Hermitian conjugation
individually.

Further discussion of the individual operations of time reversal and Hermitian
conjugation or of the compound operation of time-reversal followed by Hermitian
conjugation will not be given here, except to say that considerations of this
kind have raised questions [45,
68] concerning two of the higher-order
vibration-rotation interaction terms recently introduced by Susskind
[65] .