Flux correlation functions

Thachuk M and Schatz G C 1992 Time dependent methods for calculating thermal rate coefficients using flux correlation functions J. Chem. Phys. 97 7297-313... 

Using these distribution functions, we can write the reactive flux correlation function in the compact form... 

According to (3.95), CLTST approximates Cf(t) by the -function, being, in a sense, a zerotime limit of the first of the equations (3.101). In the representation of eigenfunctions n>, n > the flux-flux correlation function equals... 

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. 

We start from the quantum mechanically exact flux-flux correlation function expression [53]... 

Figure 25. Electron-transfer rate the electronic coupling strength at T = 500 K for the asymmetric reaction (AG = —3ffl2, oh = 749 cm ). Solid line-present full dimensional results with use of the ZN formulas. Dotted line-full dimensional results obtained from the Bixon-Jortner formula. Filled dotts-effective ID results of the quantum mechanical flux-flux correlation function. Dashed line-effective ID results with use of the ZN formulas. Taken from Ref. [28].

Ryaboy, V. and Lefebvre, R. Flux-flux correlation function study of resonance effects in reactive collision, J. Chem. Phys., 99 (1993), 9547-9552... 

The Hamiltonian in Eq. (39) has bear used to calculate the adiahatic free energy as a function of the solvent coordinate using the umbrella sampling method, and reactive flux correlation function calculations have been used to determine the adiabatic rate constant. The results were qualitatively similar to the results based on the two-state model. 

Following FerrelK, the second term in Equation 2 can be expressed as a Green-Kubo integral over a flux-flux correlation function. The transport is due to a velocity perturbation caused by two driving forces, the Brownian force and frictional force. The transport coefficient due to the segment-segment interaction can be calculated from the Kubo formula(9 ... 

The instanton method takes into account only the dynamics of the lowest energy doublet. This is a valid description at low temperature or for high barriers. What happens when excitations to higher states in the double well are possible And more importantly, the equivalent of this question in the condensed phase case, what is the effect of a symmetrically coupled vibration on the quantum Kramers problem The new physical feature introduced in the quantum Kramers problem is that in addition to the two frequencies shown in Eq. (28) there is a new time scale the decay time of the flux-flux correlation function, as discussed in the previous Section after Eq. (14). We expect that this new time scale makes the distinction between the comer cutting and the adiabatic limit in Eq. (29) to be of less relevance to the dynamics of reactions in condensed phases compared to the gas phase case. 

For a quantum mechanical system in thermal equilibrium a transport coefficient Aab may be determined from the time integral of a flux-flux correlation function [64]. 

The rate coefficient of a reactive process is a transport coefficient of interest in chemical physics. It has been shown from linear response theory that this coefficient can be obtained from the reactive flux correlation function of the system of interest. This quantity has been computed extensively in the literature for systems such as proton and electron transfer in solvents as well as clusters [29,32,33,56,71-76], where the use of the QCL formalism has allowed one to consider quantum phenomena such as the kinetic isotope effect in proton transfer [31], Here, we will consider the problem of formulating an expression for a reactive rate coefficient in the framework of the QCL theory. Results from a model calculation will be presented including a comparison to the approximate methods described in Sec. 4. 

We see that the rate constant may be determined as the time integral of the canonical averaged flux autocorrelation function for the flux across the dividing surface between reactants and products. It is also clear that we only need to calculate the flux correlation function for trajectories starting on the dividing surface, for otherwise F(p(0), q(0)) = 0 and there will be no contributions to the product formation. 

Example 5.1 The flux correlation function of a free particle... 

To give an idea of the form of the flux autocorrelation function, we consider the dynamics of a free particle with a constant potential energy of Eo, H = P2 /(2m) + Eo, which to a first approximation can describe the dynamics along a relevant reaction coordinate in the barrier region of the potential surface. The flux correlation function (5.115) can, in the coordinate representation, be written in the form [2] (see Appendix F)... 

It is possible to show that when there is a linear relation between the force and the flux, the transport coefficient is equal to a time integral of the flux correlation function. 

For the H-rCH4 H2+CH3PO] and O+CH4 OH+CHspI] reactions, accurate Multi Conhgurational Time Dependent Hartree (MCTDH) calculations have been performed. These calculations include all vibrational modes and are for a total angular niomcntum J = 0. The flux correlation function formalism[68. 69, 70] was employed to calculate cumulative reaction probabilities for J — 0. 

Consider second the frequency-dependent flux Jt(co). This is an unusual concept, but can be readily defined. The zero-frequency flux is defined from the time integral of the transition state flux correlation function ... 

Transport properties are typically expressed as time integrals of flux-flux correlation functions. Letting B = A = he the flux corresponding to the operator A, the quantum expression for a transport coefficient takes the general form,... 

The quantum mechanical expression for a transport property was given in (23) and its generalization to a time-dependent transport coefficient, defined as the finite time integral of a general flux-flux correlation function involving the fluxes of operators A and B, is... 

Fig. 14.11 The reactive flux correlation function, Eq. (14.99) plotted against time. After the initial transient period this function becomes essentially constant on the time scale shown.

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