Adiabatic approximation, appropriate for vibrational excitation

The coupling matrix elements due to the kinetic energy operator, 

Equations (3.5) and (3.33) are completely equivalent. In the diabatic representation, it is the potential matrix (3.6) that couples the different vibrational channels whereas in the adiabatic representation the matrices (3.35) and (3.36) impart the coupling between translation and vibration. Computationally more advantageous is the diabatic representation because the calculation of the matrices U and Q is in practice laborious. 

In the adiabatic approximation one neglects the coupling elements on the right-hand side of (3.33) so that each wavefunction n is determined by a one-dimensional Schrodinger equation t 

Coupling to other vibrational channels is zero and transitions from one vibrational state to another are therefore prohibited. Within the adiabatic approximation the partial photodissociation wavefunctions separate into a translational and an internal part, the latter depending parametrically on R, i.e., 

Inserting (3.38) into the expression for the partial photodissociation amplitudes yields 

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