Zero-point energy anharmonic oscillator

Attention has been called to the fact that there exists a usually neglected anharmonic term Go for the energy of the diatomic molecule-oscillator which depends on the reduced mass of the molecule. This term makes a contribution to the vibrational zero-point energy. It is shown that this contribution to the equilibrium constant for isotopic exchange reactions involving hydrogen isotopes may be non-negligible. 

There exist additional factors that can influence the v(OH) of isolated (non-H-bonded) hydroxyls. One of them is the anharmonicity of the O—H bond, which is related to the shape of the Morse potential that is overlaid with the harmonic potential. The harmonic frequencies are higher than the measured ones in general, an increase of the anharmonicity of a particular vibration lowers the O—H stretching frequency. For Morse potentials of equal shape, the anharmonicity decreases with decreasing wavenumber, because the zero-point energy of O—H bonds characterized by low frequencies hes closer to the bottom of the Morse potential curve, where the deviations from the harmonic oscillator model are smaller. However, the factors affecting the anharmonicity are far from being completely... 

Typical for the energy levels of a harmonic oscillator is that the distance between them is a constant = Wk/M. In reality, PES is anharmonic. The distance between the energy levels then decreases as we go to higher energies and approach the dissociation limit. Also note that the lowest energy, the zero-point energy, is larger than zero. 

The free energy of the system also includes entropic contributions arising from the internal fluctuations, which are expected to be different for the separate species and for the liganded complex. These can be estimated from normal-mode analyses by standard techniques,136,164 or by quasi-harmonic calculations that introduce approximate corrections for anharmonic effects 140,141 such approaches have been described in Chapt. IV.F. From the vibrational frequencies, the harmonic contribution to the thermodynamic properties can be calculated by using the multimode harmonic oscillator partition function and its derivatives. The expressions for the Helmholtz free energy, A, the energy, E, the heat capacity at constant volume, C , and the entropy are (without the zero-point correction)164... 

By force constants, we refer to derivatives of the electronic energy with respect to internal geometrical parameters of a molecule (Table 1). llie harmonic force constants are the second derivatives evaluated at an equilibrium structure, while higher derivatives may be put in the category of anharmonic force constants. From harmonic constants, harmonic frequencies are immediately obtained, and, typically, the harmonic frequencies are at least accurate enough to be used in making zero-point correaions to stabilities. Given harmonic constants and the lowest one or two orders of anharmonic force constants (third and fourth derivatives), transition frequencies of small polyatomics can often be extracted. Usually, this involves a perturbative treatment of the anharmonic parts of the potential in a produa basis of harmonic oscillator functions. 

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