Velocity autocorrelation function , mode

Figure 3. Hydrogen and oxygen velocity autocorrelation function from two-body MCY with vibrations allowed (MCYL), and computed infrared spectrum for intramolecular bending modes and bond stretching.

An alternative approach is to use the fact that an MD calculation samples the vibrational modes of the polymer for a period of time, f, from 0 to fmax and to calculate from the trajectory, the mass weighted velocity autocorrelation function. Transforming this function from the time domain into the frequency domain by a Fourier transform provides the vibrational density of states g(v). In practice this may be carried out in the following way ... 

In more complex molecular systems, increased coupling between the translational motion and both rotational and vibrational modes occurs. It is difficult to separate these effects completely. Nevertheless, the velocity autocorrelation functions of the Lennard—Jones spheres [519] (Fig. 52) and the numerical simulation of the carbon tetrachloride (Fig. 39) are quite similar [452a]. 

It is well known that the velocity autocorrelation function decays as f3/2 in the asymptotic limit due to the coupling between the tagged particle motion and the transverse current mode of the solvent [23, 56, 57]. The asymptotic limit of the Rn term can be calculated by assuming that Fs(q, t) and Ctt(q, t) have simple diffusive behavior. Thus the expression for Rn in this limit takes the following form ... 

There have been various approaches in the mode coupling theory [9, 37, 57, 176]. All these theories have exhibited the presence of t 3/2 of the velocity autocorrelation function in the asymptotic limit in three dimensions. Extending each of these theories for studies in two dimensions we can show that the velocity autocorrelation function has r1 tail in the asymptotic limit. Since the diffusion coefficient is related to Cv(t) through Eq. (337), it can be shown that D diverges in the long time due to the presence of this t l tail in the VACF. 

The barrier crossing dynamics is determined by the memory kernel K(c), and Russell and Sceats suggested that it be modeled such that it gave the same velocity autocorrelation function in the vicinity of the outer minimum Rq of K(R) as exjjected from a superposition of the damp>ed conformational normal modes of the polymer, j,J = 1, N — 1. The potential P(R) is expanded about Rq with a force constant ficoi, where coq is defined by an integral such as Eq. (7.20), but with the range of integration extended to infinity. The result is that... 

Consider Eq. (6.84). This result was obtained for a harmonic system of identical and equivalent atoms. We could however reverse our reasoning and define a vibrational spectrum for a dense atomic system from the velocity autocorrelation function according to Eq. (6.84). Since this function can be computed for all systems, including liquids and disordered solids, we may use (6.84) as a definition of a spectrum that may be interpreted as density of modes fimction for such media. We can then use it in expressions such as (4.33), and (6.92). Is this approach to dynamics in condensed phases any good ... 

The difference in the spectral density between the displacement and velocity autocorrelation functions can be understood from a normal-mode description (see Chapt. IV.F). Using Eq. 23 for the displacement autocorrelation function and differentiating it to obtain the velocity autocorrelation function, one finds that the terms in the latter are weighted by the square of the mode frequency relative to the former. Thus higher-frequency contributions are more important in the spectral density associated with the velocity autocorrelation function than the displacement autocorrelation function.153,332... 

Because relaxing solute normal mode frequencies are often much higher than typical solvent translational-rotational frequencies, the following short time picture of solute VER emerges [22,24]. In zeroth order the relaxing mode executes conservative harmonic notion in a hypothetical solvent that is nonresponsive to this motion. The velocity autocorrelation function /(f) for this zeroth order motion is... 

Berne et al. also showed that the form of the solute normal mode velocity autocorrelation function x(r) approached, at least qualitatively, the fast variable form of Figure 3.7 for co > 3(Omp (while having the usual thermal form for co cOmp see Figure 1 of reference 27a), suggesting the validity of Eq. (3.59) for high-frequency modes. 

Figure 3.7. The velocity autocorrelation function x t) of a high-frequency solute normal mode. The vibrational energy relaxation of a high-frequency solute normal mode occurs in a limit of extreme underdamping (3(cor) 2c0(, yielding for the mode s velocity autocorrelation...

These ring collision events are now a familiar part of the kinetic theory description of dynamic processes in simple dense fluids. A brief comparison of the theory for the velocity autocorrelation function with that for the chemically reacting fluid will help motivate our description. Recent developments in the theory of the velocity autocorrelation function have arisen out of an attempt to understand the slow t power law decay observed by Alder and Wainwright in a computer simulation of a dense hard-sphere fluid. This work also showed that the translational motion of a small hard sphere in a fluid of similar hard spheres has a significant collective (hydro-dynamic) component. On the theoretical side, this type of behavior was discussed from the kinetic theory point of view in terms of the ring collision events described above and provided a microscopic basis for the introduction of collective effects. In addition, it was shown that mode... 

In many respects, at a superficial level, the theory for the chemical reaction problem is much simpler than for the velocity autocorrelation function. The simplifications arise because we are now dealing with a scalar transport phenomenon, and it is the diffusive modes of the solute molecules that are coupled. In the case of the velocity autocorrelation function, the coupling of the test particle motion to the collective fluid fields (e.g., the viscous mode) must be taken into account. At a deeper level, of course, the same effects must enter into the description of the reaction problem, and one is faced with the problem of the microscopic treatment of the correlated motion of a pair of molecules that may react. In the following sections, we attempt to clarify and expand on these parallels. 

FIGURE 8.3 Time-resolved power spectra (TRPS) estimated by Fourier transforms (FTs) of the time-segmental velocity autocorrelation functions (TSVAFs) of the active center. The spectra indicate the intensities of the vibrational modes of the active center in the globin. The intensities over 3500 cm" are zero (not shown). The photodissociation is achieved at t = Ops, after the previous equilibrium MD stimulation is denoted by negative time duration. The intensities of power spectra are depicted as functions of the vibrational frequency. The ordinate axes indicate the absolute FT intensity in arbitrary units. The spectrum (-5 < t < Ops) is that of MbCO, while the other spectra are those of Mb -i- CO. (From Okazaki, I. et al., Chem. Phys. Lett., 337,151, 2001. With permission.)... 

The dynamics of different modes of molecular librations (hindered rotations) and intramolecular vibrations in supercritical water can now be analyzed in terms of velocity autocorrelation functions for the corresponding projections (Eqns. 22-27) (Kalinichev and Heinzinger 1992, 1995 Kalinichev 1993). The velocity autocorrelation functions calculated for the quantities Qi (Eqns. 25-27) are shown in Figure 19 for two extreme cases of high-density and low-density supercritical water. The Fourier transforms of these functions result in the spectral densities of the corresponding vibrational modes. They are shown in Figure 20 for the supercritical thermodynamic states listed in Table 5. 

At T 0 the sharp lines corresponding to the harmonic modes are broadened by anharmonic effects until, at high temperature, the simple relationship between vibrational density of states and dynamical matrix is lost. In this regime, and especially for large aggregates, MD is the most suitable tool to compute the vibrational spectrum. Standard linear response theory within classical statistical mechanics shows that the spectrum f(co) is given by the Fourier transform of the velocity-velocity autocorrelation function... 

The calculation of the vibrational spectrum from an (AI)MD trajectory involves Fourier-transforming the time-dependent velocity autocorrelation function [60] an alternative approach involves calculating the phonon frequencies by diagonalizing the Hessian matrix of a model obtained by structural optimization of the classical MD structure [53]. The AIMD-VACF approach naturally include finite-temperature anharmonic effects missing in the Hessian-harmonic approximation, but it does not produce accurate IR intensities (for which an autocorrelation function based on the exact dipole moments would be needed [61-63]). Despite these issues, it turns out that, in the case of 45S5 Bioglass , the two methods give similar frequencies of the individual modes [53]. 

Figure 11.5 Power spectra of velocity autocorrelation functions for AI, All, and Ain modes of trani-A-methylacetamide in aqueous solution obtained by GLOB/ADMP dynamics.

This function is a continuous analogue of the frequencies derived from the quasiharmonic approximation. Information about the intensities can be obtained by using the dipole moment autocorrelation function in place of the velocity. The advantage of using MD to build up information about the vibrational modes of the polymer is that the approach incorporates an averaging over many vibrational states of a complex molecule, which may be changing conformation... 

The rotational relaxation of water molecules is often discussed in terms of angular momentum autocorrelation functions (e g., Stillinger and Rahman 1972 Yoshii et al. 1998). For a flexible water model, a slightly different approach can also be used. In order to separate the various modes of molecular librations (hindered rotations) and intramolecular vibrations, the scheme proposed by Bopp (1986) and Spohr et al. (1988) can be employed. The instantaneous velocities of the two hydrogen atoms of every water molecule in the molecular center-of-mass system are projected onto the instantaneous unit vectors i) in the direction of the corresponding OH bond (ui and U2) ... 

The frequencies associated to each mode i can be obtained by Fourier transform of the autocorrelation function of normal-mode velocities Q. 

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