Quantum correction

Heller E J 1978 Quantum corrections to classical photodissociation models J. Chem. Rhys. 68 2066... 

The leading order quantum correction to the classical free energy is always positive, is proportional to the sum of mean square forces acting on the particles and decreases with either increasing particle mass or mcreasing temperature. The next tenn in this expansion is of order This feature enables one to independently calculate the leading correction due to quanmm statistics, which is 0(h ). The result calculated in section A2.2.5.5 is... 

If z = exp(pp) l, one can also consider the leading order quantum correction to the classical limit. For this consider tlie thennodynamic potential cOq given in equation (A2.2.144). Using equation (A2.2.149). one can convert the sum to an integral, integrate by parts the resulting integral and obtain the result ... 

The first temi is the classical ideal gas temi and the next temi is the first-order quantum correction due to Femii or Bose statistics, so that one can write... 

We have so far ignored quantum corrections to the virial coefficients by assuming classical statistical mechanics in our discussion of the confignrational PF. Quantum effects, when they are relatively small, can be treated as a perturbation (Friedman 1995) when the leading correction to the PF can be written as... 

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,... 

Figure C3.5.6 compares the result of this ansatz to the numerical result from the Wiener-Kliintchine theorem. They agree well and the ansatz exliibits the expected exponential energy-gap law (VER rate decreases exponentially with Q). The ansatz was used to detennine the VER rate with no quantum correction Q= 1), with the Bader-Beme hannonic correction [61] and with a correction based [83, M] on Egelstaff s method [62]. The Egelstaff corrected results were within a factor of five of experiment, whereas other corrections were off by orders of magnitude. This calculation 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 procedure is clearly not extendible to polyatomic molecules or to diatomic molecules in polyatomic solvents.

Behrens, P.H., Mackay, D.H.J., White, G.M., Wilson, K.R. Thermodynamics and quantum corrections from molecular dynamics for liquid water. J. Chem. Phys. 79 (1983) 2375-2389. 

Fig. 7.12 Experimental and calculated infrared spectra for liquid water. The black dots are the experimental values. The thick curve is the classical profile produced by the molecular dynamics simulation. The thin curve is obtained by applying quantum corrections. (Figure redrawn from Guilbt B 1991. A Molecular Dynamics Study of the Infrared Spectrum of Water. Journal of Chemical Physics 95 1543-1551.)...

For small p the contribution of paths with large x (n 0) to the partition function Z is suppressed because they are associated with large kinetic-energy terms proportional to v . That is why the partition function actually becomes the integral over the zeroth Fourier component Xq. It is therefore plausible to conjecture that the quantum corrections to the classical TST formula (3.49a) may be incorporated by replacing Z by... 

Although the correlation function formalism provides formally exact expressions for the rate constant, only the parabolic barrier has proven to be analytically tractable in this way. It is difficult to consistently follow up the relationship between the flux-flux correlation function expression and the semiclassical Im F formulae atoo. So far, the correlation function approach has mostly been used for fairly high temperatures in order to accurately study the quantum corrections to CLST, while the behavior of the functions Cf, Cf, and C, far below has not been studied. A number of papers have appeared (see, e.g., Tromp and Miller [1986], Makri [1991]) implementing the correlation function formalism for two-dimensional PES. 

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. ... 

The last term in (5.44) accounts for quantum corrections to the classical escape rate (5.31) [Wolynes 1981 Melnikov and Meshkov 1983 Grabert and Weiss 1984 Dakhnovskii and Ovchinnikov 1985]. 

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

As the temperature drops, (5.80) starts to incorporate quantum corrections. When friction increases, T u decreases and the prefactor in (5.80) increases. This means that the reaction becomes more adiabatic. However, the rise of the prefactor is suppressed by the strong decrease in the leading exponent itself The result (5.80) may be recast in a TST-like form. If the transition were classical, the rate constant could be calculated as the average flux towards the product valley... 

The values in brackets are those calculated from Eqs. 9 and 10 by using the parameters for the pure substances recommended by Hirschfelder, Curtiss, and Bird33 (with the quantum corrections for the hydrogen-hydrogen interaction). 

The parameters obtained here from measurements of B12 and Dn over wide ranges of temperature are probably as reliable as any that have been proposed for the interaction of molecules of different species. Unfortunately they do not provide an adequate test of Eqs. 9 and 10, since each of the systems has as one of its components either helium or hydrogen (for which there are significant quantum corrections) or carbon dioxide (which does... 

The approach to the evaluation of vibrational spectra described above is based on classical simulations for which quantum corrections are possible. The incorporation of quantum effects directly in simulations of large molecular systems is one of the most challenging areas in theoretical chemistry today. The development of quantum simulation methods is particularly important in the area of molecular spectroscopy for which quantum effects can be important and where the goal is to use simulations to help understand the structural and dynamical origins of changes in spectral lineshapes with environmental variables such as the temperature. The direct evaluation of quantum time- correlation functions for anharmonic systems is extremely difficult. Our initial approach to the evaluation of finite temperature anharmonic effects on vibrational lineshapes is derived from the fact that the moments of the vibrational lineshape spectrum can be expressed as functions of expectation values of positional and momentum operators. These expectation values can be evaluated using extremely efficient quantum Monte-Carlo techniques. The main points are summarized below. 

Eq. (2.16) is not an entirely new result. After this work had been concluded and we were looking around in search of bibliographical material, we came upon a paper by Englert and Schwinger [24] dealing with the introduction of quantum corrections to the Thomas-Fermi statistical atom. These authors attain the same result expressed by eq. (2.16) (for... 

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

By integrating out the coordinate dependence, they also obtain an approximate quantum-corrected formula for the momentum distribution which leads to the definition of an effective temperature. That is, the approximate distribution is still Gaussian in the momenta, but with an increased temperature for each particle... 

We now use a trick to partition this exact expression for the chemical potential into classical and quantum correction parts [29]. To do this we multiply and divide inside the logarithm of the excess term by the classical average... 

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