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Methane Symmetry Operations - Time-Reversal Hermitian Conjugation

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] .


Created September 29, 2016, Updated November 15, 2019