Position time correlation functions

Figure 1 Position time correlation functions for the weakly anharmonic potential at two different temperatures o/P = 1 and P = 8. Shown are the exact (dots), CMD (solid line), and classical MD (dashed line) results.

The importance of computing the moments of experimental bands comes from their connection with molecular properties through statistical averages, as was initially remarked by Gordon (18). The fluctuation-dissipation theorem establishes the correspondence between absorption lineshapes I(o)) and the angular positions time-correlation function Ci(t) ... 

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. 

FIG. 21. The influence of potential on step fluctuations, x(t), may be described by means of a time correlation function F(t) = ((x(t) - x(0) ). At negative potentials, fluctuations are due solely to mass transport along the steps, while at more positive potentials the magnitude of the fluctuations increases rapidly. This is attributed to the onset of adatom exchange with terraces as well as the electrolyte, which occurs even at the potential well below the reversible value for Ag/Ag+. (From Refs. 207, 208.)... 

The important step of identifying the explicit dynamical motivation for employing centroid variables has thus been accomplished. It has proven possible to formally define their time evolution ( trajectories ) and to establish that the time correlations ofthese trajectories are exactly related to the Kubo-transformed time correlation function in the case that the operator 6 is a linear function of position and momentum. (Note that A may be a general operator.) The generalization of this concept to the case of nonlinear operators B has also recently been accomplished, but this topic is more complicated so the reader is left to study that work if so desired. Furthermore, by a generalization of linear response theory it is also possible to extract certain observables such as rate constants even if the operator 6 is linear. 

From Bochner s theorem it is seen that power spectra are everywhere positive and bounded, and furthermore, time-correlation functions have power spectra that can be regarded as probability distribution functions. 

This equality implies that p". >neq (y, t2 — t ) is not positive definite, a price that we have to pay to ensure the equivalence between the density and trajectory picture in the non-Poisson case. Thus, the two-time correlation function is evaluated using only density prescriptions, and the result turns out to be identical to Eq. (148), which is known to correspond to the prescription of renewal theory [see Eq. (147)]. In the Poisson case the equilibrium distribution is flat. Thus, the contributionp s " cq(y, t2 — t ) vanishes. 

Figure 9 shows the instantaneous conformation of a portion of a polymer chain with the unit vector m affixed to the central bond of the nine-bond sequence AB whose end-to-end vector is denoted by The instantaneous position of the chain-affixed coordinate system Axyz is denoted by R. The laboratory coordinate system is represented by Oxyz. The time correlation function involves the product of the external and internal correlation functions. These two are assumed to be independent. The internal autocorrelation function is given by... 

To obtain the diffusion constant, D, we consider two alternative equilibrium time correlation function approaches. First, D can be obtained from the long time limit of the slope of the time-dependent mean square displacement of the electron from its starting position. The quantum expression for this estimator is... 

The nonequilibrium solvation function iS (Z), which is directly observable (e.g. by monitoring dynamic line shifts as in Fig. 15.2), is seen to be equal in the linear response approximation to the time correlation function, C(Z), of equilibrium fluctuations in the solvent response potential at the position of the solute ion. This provides a route for generalizing the continuum dielectric response theory of Section 15.2 and also a convenient numerical tool that we discuss further in the next section. 

H. Kono, Y. Nomura, and Y. Fujimura, Adv. Chent. Phys. 80, 403 (1991). Here, the time-correlation function formalism is used to classify the RSR into the fluorescencelike, the Raman-like and the interference-like components. This classification corresponds to the nomenclature II in [6] and the interference-like component is not positive definite in general. 

In the first step the positions of all atoms in the cell are optimized. Cell parameters are usually borrowed from experiment. In some cases they are optimized [84] and in some cases not [85]. Harmonic frequency calculations verify that the computed structure corresponds to the global PES minimum. In the second step the anharmonic OH stretching [83, 84] frequency is estimated using ID potential curves calculated as a function of the displacement for the hydrogen atom. In the third step classical molecular dynamics (MD) simulations are performed. The IR [85] or vibrational spectrum [82, 83] of the crystal is computed from the Fourier transform of the corresponding time correlation function (see Section 9.3.1). 

We introduce the discrete notation in order to clarify the ensuing discussion. What we want to demonstrate is how the time-correlation function varies with time. In Fig. 2.2.1 we present the noise signal A(t). Note that many of the terms in the sum Eq. (2.2.3) are negative. For example, in Fig. 2.2.1 AjAj+n is negative. Consequently, this sum will involve some cancellation between positive and negative terms. Now consider the case . The sum contributing to this is J AjAj = J Aj2. Since... 

It is instructive to compute the time correlation function in the simple case that the ground and excited state potentials are harmonic but differ in their equilibrium position and frequency. This is particularly simple if the initial vibrational state is the ground state (or, in general, a coherent state (52)) so that its wave function is a Gaussian. We shall also use the Condon approximation where the transition dipole is taken to be a constant, independent of the nuclear separation, but explicit analytical results are possible even without this approximation. A quick derivation which uses the properties of coherent states (52) is as follows. The initial state on the upper approximation is, in the Condon approximation, a coherent state, i /,(0)) = a). The value of the parameter a is determined by the initial conditions which, if we start from a stationary state, are that there is no mean momentum and that the mean displacement (x) is the difference in the equilibrium position of the two potentials. In general, using m and o> to denote the mass and the vibrational frequency... 

Compared to crystalline materials, the production and handling of amorphous substances are subject to serious complexities. Whereas the formation of crystalline materials can be described in terms of the phase rule, and solid-solid transformations (polymorphism) are well characterised in terms of pressure and temperature, this is not the case for glassy preparations that, in terms of phase behaviour, are classified as unstable . Their apparent stability derives from their very slow relaxations towards equilibrium states. Furthermore, where crystal structures are described by atomic or ionic coordinates in space, that which is not possible for amorphous materials, by definition, lack long-range order. Structurally, therefore, positions and orientations of molecules in a glass can only be described in terms of atomic or molecular distribution functions, which change over time the rates of such changes are defined by time correlation functions (relaxation times). 

It should be noted that the imaginary time correlation function in Eq. (2.4) provides a measure of the localization of quantum particles in condensed media [17-19,52]. From this point on, the notation denotes an averaging by integrating some centroid-dependent function over the centroid position q weighted by the normalized centroid density p q ) Z. An alternative method for defining the correlation function... 

In Paper I, general imaginary-time correlation functions were expressed in terms of an averaging over the coordinate-space centroid density p (qj and the centroid-constrained imaginary-time-position correlation function Q(t, qj. This formalism was extended in Paper III to the phase-space centroid picture so that the momentum could be treated as an independent variable. The final result for a general imaginary-time correlation function is found to be given approximately by [5,59]... 

By definition, the centroid variable occupies a central role in the behavior of the centroid-constrained imaginary-time correlation function in Eq. (2.1). However, it is even more interesting to analyze the role of the centroid variable in the real-time quantum position correlation function [4, 8]. This information can in principle be extracted from the exact centroid-constrained correlation function C (t, q ) through the analytic continuation t— if. Such a procedure, however, is generally not tractable unless there is some prior simplification of the problem. One such simplification is achieved [4, 8] through use of the optimized reference quadratic action functional, given by [3, 21-23]... 

In the early papers [4,8], the development of the CMD method was guided in part by the effective harmonic analysis and, in part, by physical reasoning. In Paper III, however, a mathematical justification of CMD was provided. In the latter analysis, it was shown that (1) CMD always yields a mathematically well-defined approximation to the quantum Kubo-transformed position or velocity correlation function, and (2) the equilibrium path centroid variable occupies an important role in the time correlation function because of the nature of the preaveraging procedure in CMD. Critical to the analysis of CMD and its justification was the phase-space centroid density formulation of Paper III, so that the momentum could be treated as an independent dynamical variable. The relationship between the centroid correlation function and the Kubo-transformed position correlation function was found to be unique if the centroid is taken as a dynamical variable. The analysis of Paper III will now be reviewed. For notational simplicity, the equations are restricted to a two-dimensional phase space, but they can readily be generalized. 

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