Harmonic-oscillator function, hydrogen bonds

Here, t/(f) is the reduced time evolution operator of the driven damped quantum harmonic oscillator. Recall that representation II was used in preceding treatments, taking into account the indirect damping of the hydrogen bond. After rearrangements, the autocorrelation function (45) takes the form [8]... 

Tbus the anharmonicity in the vibration is captured by the coefficient x, which adopts values typically less than 1 % of ft>e for bond stretches, but may be up to 5% for those involving hydrogen. It can be shown that any function capable of representing the variation of potential energy with displacement will lead to energy levels given by a power series in (v + 1/2), so the simple harmonic and Morse oscillators are particular cases of this general anharmonic oscillator, with the power series truncated after the first and second terms, respectively. 

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