Spherical Transition Moment

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):
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> = -2^½T(1,-1)
<stateA|y|stateB> = i 2^½T(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.