Symmetric anharmonic oscillator potential

FIGURE 3 Potential energy as a function of displacement for a symmetric anharmonic oscillator potential. 

For a symmetric molecule and large perturbation, the electrons also experience anharmonic restoring forces. For a symmetric molecule, the potential energy as a function of electronic displacement (symmetric anharmonic oscillator) can be represented by the graph shown in Fig. 3. 

For the reflection symmetric two-level electron-phonon models with linear coupling to one phonon mode (exciton, dimer) Shore et al. [4] introduced variational wave function in a form of linear combination of the harmonic oscillator wave functions related with two levels. Two asymmetric minima of elfective polaron potential turn coupled by a variational parameter (VP) respecting its anharmonism by assuming two-center variational phonon wave function. This approach was shown to yield the lowest ground state energy for the two-level models [4,5]. 

Just as we corrected the expressions for the rigid rotor to allow for the centrifugal effect and an interaction with the vibration, we also must adjust the expression for the harmonic oscillator to account for the anharmonicity in the oscillation. The potential energy surface for the molecule is not symmetrical (Fig. 25.2). The parabola (dotted figure) represents the potential energy of the harmonic oscillator. The correct potential energy is shown by the full lines the vibration is anharmonic. The vibrational energy levels for such a system can be approximated by a series ... 

In the harmonic oscillator, approximation combination tones are forbidden as well as overtones. Darling and Dennison (33) in their classic paper gave the anharmonicity and the potential constants for the water molecule. vi is the symmetrical stretching... 

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