Time-correlation function potential

Zatsepin V. M. Time correlation functions of one-dimensional rotational Brownian motion in n-fold periodical potential. Theor. and Math. Phys. 

Starting with a crude model of a polymer melt, consisting of anharmonic springs connecting repulsive beads, with a bending potential to represent the polymer stiffness, the authors show how the time correlation function, C(f), should be expected to behave as a function of the polymer stiffness. 

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.)... 

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 relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

The frequency-time correlation function is dependent on the frequency and the force constants of the vibrational mode whose dephasing is being considered. They are determined by fitting the vibrational bond energies to a Morse potential of the following form ... 

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. 

The details of the pair potential used in the simulations are given in Table I. This consists of an -trans model of the sec-butyl chloride molecule with six moieties. The intermolecular pair potential is then built up with 36 site-site terms per molecular pair. Each site-site term is compost of two parts Lennard-Jones and charge-charge. In this way, chiral discrimination is built in to the potential in a natural way. The phase-space average R-R (or S-S) potential is different from the equivalent in R-S interactions. The algorithm transforms this into dynamical time-correlation functions. 

For Li, the time correlation function decays slowly compared to Na", due to the strong Li -water interaction. Thus, at low temperatures we could not detect differences in the dynamics given by the 12-6 or 10-3 functions. However, at high temperature, the 10-3 potential yields shorter times than the 12-6. For Mg", we observe that the 6-4 function yields a larger coordination... 

In this section we present some applications of the LAND-map approach for computing time correlation functions and time dependent quantum expectation values for realistic model condensed phase systems. These representative applications demonstrate how the methodology can be implemented in general and provide challenging tests of the approach. The first test application is the spin-boson model where exact results are known from numerical path integral calculations [59-62]. The second system we study is a fully atomistic model for excess electronic transport in metal - molten salt solutions. Here the potentials are sufficiently reliable that findings from our calculations can be compared with experimental results. 

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. 

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). 

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... 

Figure 18 Time cross-correlation functions for three Raman transitions in iodobenzene (from the ground state to the B excited electronic state with v = 1, 2, 3 quanta in the vu vibrational mode. (Left) Computed for a harmonic B state potential and convoluted with a 125-fs-wide window function. The spectrum is computed from this cross-correlation function. (Right) The time correlation function determined from the Raman frequency spectrum (the excitation profile) via the maximum entropy formalism, as discussed in the text, using nine Lagrange multipliers kr. (From Ref. (102).)...

Figure 19 The Raman spectrum and time cross correlation function when the motion on the excited electronic state potential is anharmonic, compare to Figs. 17 and 18, which are for a harmonic approximation. (Top, a) Computed time correlation function using a wide window function (b) The maximal entropy representation of this function, determined from the spectrum. Note the clear separation of time scales due to the anharmonicity (cf. Fig. 20). (Bottom) The Raman excitation spectrum obtained from the computed time correlation function (a). The arrows are the sequence of computations (a) is determined from the dynamics. The spectrum is determined from (a). The maximum entropy cross-correlation function (b) uses only the spectrum as input.

Figure 4. Plot of the imaginary-time correlation function q T)q 0)) for the nonquadratic potential described in Section II.E [Eq. (2.66)]. The correlation function is plotted as a function of the dimensionless variable u = T/h with /3 = 5. The solid circles are the numerically exact results, while the solid line is for the optimized LHO theory in Eqs. (2.24)-(2.28) used in the centroid-based formulation of the correlation function in Eq. (2.50).

In the past, comparisons between NMR relaxation and MD simulations have concentrated on internal motions, since these often involve sub-nanosecond time scales that could be examined with limited computer resources. In this approach, overall rotational motion is removed by an rms fitting procedure (for example, on backbone atoms in regular secondary structure), and computing time-correlation functions from the result. Typical results are shown in the upper panel of Figure 8.1 similar plots have been presented many times before [4,12,10,11]. Many backbone vectors are like Thr 49, and decay in less than 0.1 ns to a plateau value which can be identified as the order parameter for that vector. Most regions of regular secondary structure resemble this, although there can be exceptions, and there is potentially important information in the decay rates and plateau values that are obtained. 

C(t) is the time correlation function of equilibrixun fluctuations of the solvent response potential at the position of the solute ion. The electrostatic potential in C(t) will be replaced by the electric field or by higher gradients of the electrostatic potential when solvation of higher moments of the charge distribution is considered. 

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