Spherical Transition Moment

This transition moment must be used for multiphoton or Raman transitions. It is also appropriate for single photon transitions classified as parallel or perpendicular, as noted below. For linear molecules the "Strength" number is the value of the vibronic only matrix element:

T(k,q) = <stateA, Λ+q| μ | stateB, Λ>

T(k,-q) = <stateA, -Λ-q| μ | stateB, -Λ>

is then computed by symmetry. The order of the states (which can be significant in cases involving multiple transition moments) is as displayed in the constants window. Strictly all the above matrix elements should have selection rules ΔS = 0 (and ΔΣ = 0 for linear molecules), but as an extension PGOPHER relaxes this requirement for ΔS ≠ 0 transitions and only enforces the |ΔΩ| = q rule. See the section on Forbidden Transitions in Linear Molecules for a more detailed discussion of this. See also Forbidden Transition Moment.

Relationship to other quantities.

For an electronic transition, this is the product of the square of the electronic transition moment and the Franck-Condon factor. While expressed in spherical tensor form above, the transformation to Cartesian components is straightforward for the normal electric dipole transitions, which have the rank 1. The q = 0 component is simply the z matrix element between the states. The q = 1 component is strictly the matrix element of the complex operator 2^-1/2(μ_x+iμ_y) between wavefunctions with values of Λ differing by one, but it can be shown to be the same as the matrix elements of the μ_x or μ_y operator between electronic states that transform as x or y. The latter will typically be produced by electronic structure calculations, so no conversion is necessary though the matrix elements calculated are not formally identical.