Time correlation function classical approach

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. 

A complete and detailed analysis of the formal properties of the QCL approach [5] has revealed that while this scheme is internally consistent, inconsistencies arise in the formulation of a quantum-classical statistical mechanics within such a framework. In particular, the fact that time translation invariance and the Kubo identity are only valid to O(h) have implications for the calculation of quantum-classical correlation functions. Such an analysis has not yet been conducted for the ILDM approach. In this chapter we adopt an alternative prescription [6,7]. This alternative approach supposes that we start with the full quantum statistical mechanical structure of time correlation functions, average values, or, in general, the time dependent density, and develop independent approximations to both the quantum evolution, and to the equilibrium density. Such an approach has proven particularly useful in many applications [8,9]. As was pointed out in the earlier publications [6,7], the consistency between the quantum equilibrium structure and the approximate... 

The focus of this chapter is exploration of the ability of mixed quantum classical approaches to capture the effects of interference and coherence in the approximate dynamics used in these different mixed quantum classical methods. As outlined below, the expectation values of computed observables are fundamentally non-equilibrium properties that are not expressible as equilibrium time correlation functions. Thus, the chapter explores the relationship between the approximations to the quantum dynamics made in these different approaches that attempt to capture quantum coherence. 

As discussed above, the conventional approach to VER in liquids involves a classical molecular dynamics simulation of the solute (with one or more vibrational modes constrained to be rigid) in the solvent. The required time-correlation functions are computed classically and then... 

The topic of this chapter is the description of a quantum-classical approach to compute transport coefficients. Transport coefficients are most often expressed in terms of time correlation functions whose evaluation involves two aspects sampling initial conditions from suitable equilibrium distributions and evolution of dynamical variables or operators representing observables of the system. The schemes we describe for the computation of transport properties pertain to quantum many-body systems that can usefully be partitioned into two subsystems, a quantum subsystem S and its environment . We shall be interested in the limiting situation where the dynamics of the environmental degrees of freedom, in isolation from the quantum subsystem [Pg.521]

The third alternative is to use the classical correlation functions to define an equivalent quantum mechanical harmonic bath. This approach was pioneered by Warshel as the dispersed polaron method [67, 68]. More recently, this idea has been used in studies of electron transfer systems in solution [64] and in the photosynthetic reaction center [65,69] (see also Ref. 70). This approach is based on the realization that the spectral density describing a linearly coupled harmonic bath [Eq. (29)] can be obtained by cosine transformation of the classical time-correlation function of the bath operator [Eq. (28)]. Comparing the classical correlation function for the linearly coupled harmonic bath [Eqs. (25) and (26)],... 

An alternative way to obtain the spectral density is by numerical simulation. It is possible, at least in principle, to include the intramolecular modes in this case, although it is rarely done [198]. A standard approach [33-36,41] utilizes molecular dynamics (MD) trajectories to compute the classical real time correlation function of the reaction coordinate from which the spectral density is calculated by the cosine transformation [classical limit of Eq. (9.3)]. The correspondence between the quantum and the classical densities of states via J(co) is a key for the evaluation of the quantum rate constant, that is, one can use the quantum expression for /Cj2 with the classically computed J(co). This is true only for a purely harmonic system [199]. Real solvent modes are anharmonic, although the response may well be linear. The spectral density of the harmonic system is temperature independent. For real nonlinear systems, J co) can strongly depend on temperature [200]. Thus, in a classical simulation one cannot assess equilibrium quantum populations correctly, which may result in serious errors in the computed high-frequency part of the spectrum. Song and Marcus [37] compared the results of several simulations for water available at that time in the literature [34,201] with experimental data [190]. The comparison was not in favor of those simulations. In particular, they failed to predict... 

Dual Lanczos transformation theory is a projection operator approach to nonequilibrium processes that was developed by the author to handle very general spectral and temporal problems. Unlike Mori s memory function formalism, dual Lanczos transformation theory does not impose symmetry restrictions on the Liouville operator and thus applies to both reversible and irreversible systems. Moreover, it can be used to determine the time evolution of equilibrium autocorrelation functions and crosscorrelation functions (time correlation functions not describing self-correlations) and their spectral transforms for both classical and quantum systems. In addition, dual Lanczos transformation theory provides a number of tools for determining the temporal evolution of the averages of dynamical variables. Several years ago, it was demonstrated that the projection operator theories of Mori and Zwanzig represent special limiting cases of dual Lanczos transformation theory. 

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 ... 

The most recent contribution to the field of nonlinear response theory comes from Keyes, Space, and collaborators [56,57]. Their approach results in an exact classical response function written in terms of classical time correlation functions (TCF). The response takes into account the nonlinear polarizability and is used in a fully anharmonic MD simulation to simulate the fifth-order response of CS2 (see Fig. 1.14). The results are strikingly similar to those obtained by Jansen et al. as well as to the simulations, both MD and adiabatic INM, of liquid Xe (see Figs. 1.8 and 1.10). The dominant features are the ridge along the probe delay and the distinct lack of signal along the pump delay. 

Within the semiclassical approach, each wavepacket is propagated by running a large number of classical trajectories. In order to calculate the correlation function, a double summation with respect to these large number of trajectories should be performed at each time step, which is, unfortunately, also very demanding computationally. 

Let us make a final comment, concerning the violation of the Green-Kubo relation. There is a close connection between the breakdown of this fundamental prescription of nonequilibrium statistical physics and the breakdown of the agreement between the density and trajectory approach. We have seen that the CTRW theory, which rests on trajectories undergoing abrupt and unpredictable jumps, establishes the pdf time evolution on the basis of v /(f), whereas the density approach to GME, resting on the Liouville equation, either classical or quantum, and on the convenient contraction over the irrelevant degrees of freedom, eventually establishes the pdf time evolution on the basis of a correlation function, the correlation function in the dynamical case... 

The spatial correlation approach is readily extended to the nonstationary case by defining time-dependent (classical and quantum) spatial correlation function and correlation length that is, one applies Eqs. (4.2) and (4.3) to time-dependent densities. An example, in which the time dependence of... 

A second recent development has been the application 46 of the initial value representation 47 to semiclassically calculate A3,8,13 (and/or the equivalent time integral of the flux-flux correlation function). While this approach has to date only been applied to problems with simplified harmonic baths, it shows considerable promise for applications to realistic systems, particularly those in which the real solvent bath may be adequately treated by a further classical or quasiclassical approximation. 

The linear-response approach has played an important role in the construction of most modern theories of transport processes (see, e.g. Ref. [3]). Moreover, it has had a profound impact on the development of classical molecular dynamics simulations - a field that was emerging at the same time. Linear-response theory showed how linear transport coefficients can be computed in a simulation, by studying the decay of fluctuations in equilibrium. More specifically, most transport coefficients can be expressed as time integrals of (au-to)correlation functions of microscopic fluxes (e.g. the... 

Basic requirements on feasible systems and approaches for computational modeling of fuel cell materials are (i) the computational approach must be consistent with fundamental physical principles, that is, it must obey the laws of thermodynamics, statistical mechanics, electrodynamics, classical mechanics, and quantum mechanics (ii) the structural model must provide a sufficiently detailed representation of the real system it must include the appropriate set of species and represent the composition of interest, specified in terms of mass or volume fractions of components (iii) asymptotic limits, corresponding to uniform and pure phases of system components, as well as basic thermodynamic and kinetic properties must be reproduced, for example, density, viscosity, dielectric properties, self-diffusion coefficients, and correlation functions (iv) the simulation must be able to treat systems of sufficient size and simulation time in order to provide meaningful results for properties of interest and (v) the main results of a simulation must be consistent with experimental findings on structure and transport properties. 

At the beginning of this section it is important to point out that the theoretical treatment we will briefly discuss now is sometimes called semiclassical just because the correlation functions are classical. A quantum mechanical treatment has been proposed (2, p. 284) which presents many formal similarities to the semiclassical treatment but which fundamentally differs from it by the way the correlation functions are defined (in terms of time-dependent operators and not of time functions). This distinction is rarely done in review articles about relaxation where the semiclassical approach is generally presented as the only way to handle the problem. To introduce the correlation functions and spectral densities, we will consider a spin system characterized by eigenstates a, b, c. and corresponding energies E, E. A perturbation, time-dependent Hamiltonian H(t) acts on this system. H(t) corresponds to the coupling of the spin system with the lattice. We shall make the assumption that H(t) can be written as a product A f(t). 

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