Quantum correction factor, harmonic

If one is interested in spectroscopy involving only the ground Born Oppenheimer surface of the liquid (which would correspond to IR and far-IR spectra), the simplest approximation involves replacing the quantum TCF by its classical counterpart. Thus pp becomes a classical variable, the trace becomes a phase-space integral, and the density operator becomes the phase-space distribution function. For light frequency co with ho > kT, this classical approximation will lead to substantial errors, and so it is important to multiply the result by a quantum correction factor the usual choice for this application is the harmonic quantum correction factor [79 84]. Thus we have... 

Figure C3.5.6 compares the result of this ansatz to the numerical result from the Wiener-Khintchine theorem. They agree well and the ansatz exhibits the expected exponential energy-gap law (VER rate decreases exponentially with Q). The ansatz was used to determine the VER rate with no quantum correction Q = 1), with the Bader-Beme harmonic correction [61] and with a correction based [M, M] on EgelstafPs method [62]. The Egelstaff corrected results were within a factor of five of experiment, whereas other eorrections were off by orders of magnitude. This ealeulation represents the present state of the art in computing VER rates in such difficult systems, inasmuch as the authors used only a model potential and no adjustable parameters. However the ansatz proeedure is clearly not extendible to polyatomic molecules or to diatomic molecules in polyatomic solvents.

A more interesting point was made by Bader and Berne, who noted that the vibrational relaxation rate of a classical oscillator in a classical harmonic bath is identical to that of a quantum oscillator in a quantum harmonic bath [71]. On the other hand, when the relaxation of the quantum system is calculated using the corrected correlation function of the classical bath [Eq. (31)], the predicted rate is slower by a factor of j/3h(i) coth(/3h(o/2), which can be quite substantial for high-frequency solutes. The conclusions of a number of recent studies were shown to be strongly affected by this inconsistency [42,43,72]. Quantizing the solvent by mapping the classical correlation functions onto a quantum harmonic bath corrects the discrepancy. 

The computational approaches up to 2006 were reviewed by Perry et al. °" Briefly, these methods are based on representing the SFG spectrum by the Fourier transform of a polarizability-dipole quantum time correlation function (QTCF). A fully classical approach to computing the SFG spectrum is then obtained by replacing the QTCF by a classical expression including a harmonic correction factor ... 

As a rule the quantum-mechanical force-fields and the corresponding normal frequencies are calculated in a harmonic approximation, while the experimentally accessible frequencies are influenced by anharmonic contributions. The Puley s scaling factors are also found to incorporate the relevant empirical corrections for the vibrational anharmonicity. 

However, there are fundamental problems in the derivation of a quantum transition state theory. TST requires the simultaneous knowledge of position and momentum the direction of the initial momentum at the dividing surface is a key ingredient to the theory. Thus, TST violates the uncertainty principle and a straightforward derivation of a quantum transition state theory is not possible. Ad hoc assumptions are required in the introduction of a QTST. Truhlar and coworkers, for example, introduce a specific one-dimensional path and add a tunneling correction, calculated along this path, to account for quantum effects in transition state theory calculations. Poliak and coworkers employ a harmonic approximation at the saddle point to obtain a quantum approximation for the dynamial factor. 

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