Vibrational and Thermal Averaging

This then gives again two contributions a vibrationally averaged electronic polarizability 

In this section, we will describe in more detail how the vibrational averaging of the pure electronic polarizability and the calculation of the vibrational polarizability is carried out. We will hereby distinguish between diatomic and polyatomic molecules. 

For diatomic molecules the vibration-rotation wavefunctions can be obtained numerically as solutions of the one-dimensional Schrodinger equation 

For polyatomic molecules the electronic polarizability in the clamped-nucleus treatment is frequently expressed as the polarizability evaluated at an equilibrium geometry Rk,s plus a vibrational correction Aa  

When one is interested in the correction for the vibrational ground state u = 0 , i.e. the state where the vibrational quantum numbers of aU the vibrational modes of the molecule are equal to zero, one calls this the zero-point vibrational correction (ZPVC) = Aa =°. In order to calculate a vibrational correction 

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