Classical time correlation functions

Such Lorentzian lineshapes will be also obtained from microscopic quantum (Section 9.3 see Eq. (9.49)) and classical (Section 8.2.4 Eq. (8.41)) models. 

For a stationary (e.g. equilibrium) system the time origin is irrelevant and 

In this case the correlation fiinction can also be calculated as the time average 

The equality between the averages computed by Eqs (6.27) and (6.29) is implied by the ergodic theorem of statistical mechanics (see Section 1.4.2). 

At t 0 the correlation function, CAid. t) becomes the static correlation function Cab(Q = AB. In the opposite limit t x -we may assume that correlations vanish so that 

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The classical time-correlation function, , does not obey the condition of detailed balance. Computer experiments provide detailed information about classical time-correlation functions. Is there any way to use the classical functions to predict quantum-mechanical time-correlation functions The answer to this question is affirmative. There exist approximations which enable the quantum-time-correlation functions to be predicted from the corresponding classical functions. Let us denote by v fn(t) the classical time-correlation function and by It(r) the one-sided quantum-mechanical correlation function ... 

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. 

In the oxygen VER experiments (3) the n = 1 vibrational state of a given oxygen molecule is prepared with a laser, and the population of that state, probed at some later time, decays exponentially. Since in this case tiojo kT, we are in the limit where the state space can be truncated to two levels, and 1/Ti k, 0. Thus the rate constant ki o is measured directly in these experiments. Our starting point for the theoretical discussion is then Equation (14). For reasons discussed in some detail elsewhere (6), for this problem we use the Egelstaff scheme in Equation (19) to relate the Fourier transform of the quantum force-force time-correlation function to the classical time-correlation function, which we then calculate from a classical molecular dynamics computer simulation. The details of the simulation are reported elsewhere (4) here we simply list the site-site potential parameters used therein e/k = 38.003 K, and a = 3.210 A, and the distance between sites is re = 0.7063 A. 

R. Ramrez, T. Lopez-Ciudad, P. Kumar, and D. Marx (2004) Quantum corrections to classical time-correlation functions Hydrogen bonding and anharmonic floppy modes. J. Chem. Phys. 121, p. 3973... 

We will often measure dynamical variables relative to their average values, that is, use A — A rather than A. Under such convention the limit in (6.30) vanishes. In what follows we list some other properties of classical time correlation functions that will be useful in our future discussions. 

Becuase of the formal similarity between C(f) and scalar products in quantum mechanics the mathematical techniques of quantum mechanics can be applied to a study of classical time-correlation functions. 

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)],... 

R. A. Rosenstein, Ber. Bunsenges. Phys. Chem., 77, 493 (1973). Classical Time Correlation Function Theory of Proton Transfer Reactions. 

Inspection of Fig. 5.18 shows that the autocorrelation functions for this particular model decay exponentially with time, and that the rate constant for this decay is the sum of the rate constants for forward and backward transitions between the two states (kon + The upper curve in Fig. 5.18B, for example, decays to He (0.368) of its initial value in 16.61 At, which is the reciprocal of (0.05 -t 0.01 )Mt. In classical kinetics, if a system with first-order reactions in the forward and backward directions is perturbed by an abrupt change in the concentration of one of the components, a change in temperature, or some other disturbance, it will relax to equilibrium with a rate constant given by the sum of the rate constants for the forward and backward reactions. The fact that the autocorrelation functions in Fig. 5.18 decay with the relaxation rate constant of the system is a general property of classical time-correlation functions [259-262]. One of the potential strengths of fluorescence correlation spectroscopy is that the relaxation dynamics can be obtained with the system at equilibrium no perturbation is required. 

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. 

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