This transition moment type must
be used for multiphoton or
Raman
transitions. It is also appropriate for single photon transitions
classified as parallel or perpendicular. The notation used here is
slightly non-standard; the
Strength
numbers are actually taken as the sum and difference of the
two components such that the value of the vibronic matrix element
where
K increases is
(taking
q > 0):
<stateA
,
K+
q|
μ| stateB
,
K> =
T(
k,
q) +
T(
k,
-q)
and where it decreases is:
<stateA, K-q| μ| stateB, K> = T(k,q) - T(k,-q)
T(
k,
q) and
T(
k,
-q) are the
Strength parameters; they are not necessarily
symmetry related and two numbers are in general required for
q ≠ 0. With this
definition the operators multiplying
T(
k,
q) and
T(
k,
-q)
will have often have different symmetries in more symmetric point
groups (such as C
2v), so only one may be required. For
k = 1 the following identifications can be made:
<stateA|z|stateB> = T(1,0)
<stateA|x|stateB> = -2½T(1,-1)
<stateA|y|stateB> = i 2½T(1,1)
making the three possible components approximately equivalent to
the three Cartesian components, but with a different phase choice.
While the sign or phase of transition moment matrix elements is
often irrelevant, it will be important if more than one transition
moment can contribute to a given transition, either through
multiple components for a single vibronic transition, or if
different vibronic states are mixed, allowing interference between
different pathways to the same final state. In such circumstances
the relative signs are important and, given the definition above,
the order in which states are specified will make a difference to
the parameters required. Swapping the bra and ket over will change
the sign of the
T(
k,
-q) component.