Doubly harmonic model

To understand the complete role of vibration in determining electrical properties, it is useful to consider a diatomic molecule in the harmonic oscillator approximation, where the stretching potential is taken to be quadratic in the displacement coordinate. The doubly harmonic model takes the various electrical properties to be linear functions of the coordinate. This turns out to be most reasonable in the vicinity of an equilibrium structure, but it breaks down at long separations. Letting x be a coordinate giving the displacement from equilibrium of a one-dimensional harmonic oscillator, the dipole moment, dipole polarizability, and dipole hyperpolarizability, within the doubly harmonic (dh) model, may be written in the following way ... 

Using the definition of electrical properties as derivatives of the molecular energy with the doubly harmonic model gives... 

There are two steps to take in going from the doubly harmonic model to a highly realistic treatment. One is to allow for the potential to be anharmonic, and the other is to allow the properties to be more complicated than just... 

The doubly degenerate nontotally symmetric v, mode (411.3cm ) is modeled using two one-dimensional modes which have the same wavenumber for the ground electronic state and by two inverted harmonic oscillator surfaces for the excited electronic state. A value of 190 cm is used for the inverted harmonic oscillator in this calculation. 

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