Franck-Condon principle for polyatomic molecules: Axis-switching effects and transformation of normal coordinates
A rigorous derivation of the relationship between the vibrational coordinates belonging to two different electronic states of a polyatomic molecule is given. Contrary to common supposition, it is shown that this transformation is generally nonlinear and nonorthogonal. The nonlinearity arises because each electronic state has its own molecule-fixed axis system, and hence axis-switching effects, first discussed by Hougen and Watson ( Canad. J. Phys.43, 298-320 (1965)), must be taken into account. Explicit expressions giving the transformation to first order in the vibrational variables are presented, and it is shown that the transformation may be linearized for small-amplitude vibrations. As an example, the bent, symmetric XY2 molecule is treated in detail. Axis-switching effects are suggested as providing an alternative mechanism to the Herzberg-Teller theory of vibronic activity of nontotally symmetric modes.