Vibrational averaging and centrifugal distortion corrections

Once we have removed the terms which couple different electronic states (at least to a certain level of accuracy), we can deal with the motion in the other degrees of freedom of the molecule for each electronic state separately. The next step in the process is to consider the vibrational degree of freedom which is usually responsible for the largest energy separations within each electronic state. If we perform a suitable transformation to uncouple the different vibrational states, we obtain an effective Hamiltonian for each vibronic state. Once again, we adopt a perturbation approach. 

In this case, the zeroth-order Hamiltonian is chosen to represent the vibrational energy of the anharmonic oscillator  

Here r], v) is the radial eigenvector with vibrational quantum number v in the electronic state r], A). It is assumed that the potential curve Vn(R) has a minimum value Vn(Rnr) equal to hcl, r at the equilibrium bond length R,r, i.e. this defines the energy origin of 

We use the operator on the left-hand side of equation (7.169) as the zeroth-order vibrational Hamiltonian. The remaining terms in the effective electronic Hamiltonian, given for example in equations (7.124) and (7.137), are treated as perturbations. In a similar vein to the electronic problem, we consider only first- and second-order corrections as given in equations (7.68) and (7.69) to produce an effective Hamiltonian 3Q, which is confined to act within a single vibronic state rj, v) only. Once again, the condition for the validity of this approximation is that the perturbation matrix elements should be small compared with the vibrational intervals. It will therefore tend to fail for loosely bound states with low vibrational frequencies. 

The first-order perturbation contribution is obtained by replacing each operator function X(R) by its anharmonic expectation value, 

We recall that the vibrational potential energy function is just the zeroth-order eigenvalue in the description of the electronic motion given in equation (7.76). Although this is not the most accurate choice we could make at this level (it is possible, for example, to add the adiabatic correction in equation (7.129)), it has the great advantage that it is isotopically independent. For heavier molecules, the spin correction term in equation (7.130) could be included without spoiling the isotopic inde- 

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