Quantum correction factor

Equation (C3.5.3) shows tire VER lifetime can be detennined if tire quantum mechanical force-correlation Emotion is computed. However, it is at present impossible to compute tliis Emotion accurately for complex systems. It is straightforward to compute tire classical force-correlation Emotion using classical molecular dynamics (MD) simulations. Witli tire classical force-correlation function, a quantum correction factor Q is needed 5,... 

For example, the rate constant of the collinear reaction H -f- H2 has been calculated in the temperature interval 200-1000 K. The quantum correction factor, i.e., the ratio of the actual rate constant to that given by CLTST, has been found to reach 50 at T = 200 K. However, in the reactions that we regard as low-temperature ones, this factor may be as large as ten orders of magnitude (see introduction). That is why the present state of affairs in QTST, which is well suited for flnding quantum contributions to gas-phase rate constants, does not presently allow one to use it as a numerical tool to study complex low-temperature conversions, at least without further approximations such as the WKB one. ... 

In the case of ohmic dissipation the product in (5.44) can be calculated explicitly and one obtains for the quantum correction factor... 

Table 4-1. Quantum correction factor k for the collinear reaction between H and H2...

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... 

The last term in (5.45) accounts for quantum corrections to the classical escape rate (5.32) [Dakhnovskii and Ovchinnikov 1985 Grabert and Weiss, 1984 Melnikov and Meshkov, 1983 Wolynes, 1981]. In the case of ohmic dissipation the product in (5.45) can be calculated explicitly and one obtains for the quantum correction factor... 

In this chapter we first review the general theoretical framework of VER, including a discussion of various different quantum correction factors. We then consider three specific systems, extending and developing the basic framework as needed, and in each case then make detailed comparison with experiment. 

The first system we consider is the solute iodine in liquid and supercritical xenon (1). In this case there is clearly no IVR, and presumably the predominant pathway involves transfer of energy from the excited iodine vibration to translations of both the solute and solvent. We introduce a breathing sphere model of the solute, and with this model calculate the required classical time-correlation function analytically (2). Information about solute-solvent structure is obtained from integral equation theories. In this case the issue of the quantum correction factor is not really important because the iodine vibrational frequency is comparable to thermal energies and so the system is nearly classical. 

D. Quantum Correction Factor Method and Other Methods... 

VER of the selected CD mode in the terminal methyl group of methionine (Met80) was previously addressed by Bu and Straub [9] They used equilibrium simulations for cyt c in water with the quantum-correction factor (QCF) method... 

Skinner and coworkers advocated to use the quantum correction factor (QCF) method [20], which is the replacement of the above formula Eq. (18) with... 

Equation (13,35) is the exact golden-rule rate expression for the bilinear coupling model. For more realistic interaction models such analytical results cannot be obtained and we often resort to numerical simulations (see Section 13.6). Because classical correlation functions are much easier to calculate than their quantum counterparts, it is of interest to compare the approximate rate ks sc, Eq. (13.27), with the exact result kg. To this end it is useful to define the quantum correction factor... 

At low temperature the quantum correction factor can be huge, as seen in Eq. (13.68). Since quantum correlation functions are not accessible by numerical simulations, one may evaluate numerically the corresponding classical correlation function and estimate theoretically the quantum correction factor. Some attempts to use this procedure appear to give reasonable results, however, it is not clear that the quantum correction factors applied in these works are of general applicability. 

Like Eq. (27.2), Eqs. (27.11) and (27.12) are also hybrid quantized expressions in which the bound modes are treated quantum mechanically but the reaction coordinate motion is treated classically. Whereas it is difficult to see how quantum mechanical effects on reaction coordinate motion can be included in VTST, the path forward is straightforward in the adiabatic theory, since the one-dimensional scattering problem can be treated quantum mechanically. Since Eq. (27.12) is equivalent to the expression for the rate constant obtained from microcanonical variational theory [7, 15], the quantum correction factor obtained for the adiabatic theory of reactions can also be used in VTST. 

As a direct test of the analytic theory for the centroid density, the quantum correction factor for the thermal rate constant of an Eckart barrier potential was calculated within the context of path-integral quantum transition-state theory [42-44,49]. The results are tabulated in... 

This approach allows for a fully quantum mechanical treatment of the dynamics, avoiding the nse of quantum correction factors used to denote classical dynamical approaches, with the concession that the potential energy surface must be expanded, ignoring higher order nonlinearity in the mode coupling. The potential energy surface is expanded with respect to the normal coordinates of the system, q, and bath, 01, and their freqnencies up to third and fourth order nonlinear conpling ... 

---