Fourier transform, velocity autocorrelation

BPTI spectral densities Cosine Fourier transforms of the velocity autocorrelation function... 

Fig. 8. Spectral densities for BPTI as computed by cosine Fourier transforms of the velocity autocorrelation function by Verlet (7 = 0) and LN (7 = 5 and 20 ps ). Data are from [88].

Another view of this theme was our analysis of spectral densities. A comparison of LN spectral densities, as computed for BPTI and lysozyme from cosine Fourier transforms of the velocity autocorrelation functions, revealed excellent agreement between LN and the explicit Langevin trajectories (see Fig, 5 in [88]). Here we only compare the spectral densities for different 7 Fig. 8 shows that the Langevin patterns become closer to the Verlet densities (7 = 0) as 7 in the Langevin integrator (be it BBK or LN) is decreased. 

In order to study the vibrational properties of a single Au adatom on Cu faces, one adatom was placed on each face of the slab. Simulations were performed in the range of 300-1000"K to deduce the temperature dependence of the various quantities. The value of the lattice constant was adjusted, at each temperature, so as to result in zero pressure for the bulk system, while the atomic MSB s were determined on a layer by layer basis from equilibrium averages of the atomic density profiles. Furthermore, the phonon DOS of Au adatom was obtained from the Fourier transform of the velocity autocorrelation function. ... 

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 ... 

As for the properties themselves, there are many chemically useful autocorrelation functions. For instance, particle position or velocity autocorrelation functions can be used to determine diffusion coefficients (Ernst, Hauge, and van Leeuwen 1971), stress autocorrelation functions can be used to determine shear viscosities (Haile 1992), and dipole autocorrelation functions are related to vibrational (infrared) spectra as their reverse Fourier transforms (Berens and Wilson 1981). There are also many useful correlation functions between two different variables (Zwanzig 1965). A more detailed discussion, however, is beyond the scope of this text. 

The velocity autocorrelation function (VAF) may be used to investigate the possibility of coupling between translational and rotational motions of the sorbed molecules. The VAF is obtained by taking the dot product of the initial velocity with that at time t. It thus contains information about periodic fluctuations in the sorbate s velocity. The Fourier transform of the VAF yields a frequency spectrum for sorbate motion. By decomposing the total velocity of a sorbate molecule into translational and rotational terms, the coupling of rotational and translational motion can be investigated. This procedure illustrates one of the main strengths of theoretical simulations, namely to predict what is difficult or impossible to determine experimentally. 

We mention this result here in order to assert that the spectral distribution of B(jf is the Fourier transform of the (force) autocorrelation function 0(t). In view of Eqn. (5.45), we can restate this result in terms of the velocity t>(/). The spectral distribution of the velocity autocorrelation function is directly related to the Fourier transform of 0 j), the force autocorrelation function. Thus, we see that the classical equation of motion when properly averaged over many particles provides insight into the relation between transport kinetics and particle dynamics [R. Becker (1966)]. 

The angular velocity and angular momentum acfs themselves are important to any dynamical theory of molecular liquids but are very difficult to extract directly from spectral data. The only reliable method available seems to be spin-rotation nuclear magnetic relaxation. (An approximate method is via Fourier transformation of far-infrared spectra.) The simulated torque-on acfs in this case become considerably more oscillatory, and, which is important, the envelope of its decay becomes longer-lived as the field strength increases. This is dealt with analytically in Section III. In this case, computer simulation is particularly useful because it may be used to complement the analytical theory in its search for the forest among the trees. Results such as these for autocorrelation functions therefore supplement our... 

By inserting the Fourier transform of the velocity autocorrelation function into Eq. (62)... 

The classical vibrational density of states can be calculated from an MD simulation by Fourier transform of the velocity autocorrelation function. Suppose we are in a frame of reference based on the center of mass of the unit cell, and let Vj be the velocity of theyth particle, hydrogen or oxygen in the present case. Then the full Boltzmann-weighted vibrational density of states is given by... 

The time scale of MD simulations is comparable to that probed by vibrational spectroscopies. The distribution of vibrational frequencies in the system /(co), known as the power spectrum, is calculated by Fourier transformation of the velocity autocorrelation function ... 

Simulations—isoergic and isothermal, by molecular dynamics and Monte Carlo—as well as analytic theory have been used to study this process. The diagnostics that have been used include study of mean nearest interparticle distances, kinetic energy distributions, pair distribution functions, angular distribution functions, mean square displacements and diffusion coefficients, velocity autocorrelation functions and their Fourier transforms, caloric curves, and snapshots. From the simulations it seems that some clusters, such as Ar, 3 and Ar, 9, exhibit the double-valued equation of state and bimodal kinetic energy distributions characteristic of the phase change just described, but others do not. Another kind of behavior seems to occur with Arss, which exhibits a heterogeneous equilibrium, with part of the cluster liquid and part solid. 

If we take the scalar product of an atom s velocity with its velocity a short time later, t, and take an average over all the atoms of the same t)q)e then we may represent it as a convolution v(0) v(i), also commonly written (v(O).v(t)). The Fourier transform of a convolution is equivalent to product of the Fourier transforms of the two functions taken separately. Self convolution, autocorrelation, yields the square of the functions imder Fourier transformation and only the real part of the transformation is available, the power spectrum, p (a). 

Figure 4.3 Velocity autocorrelation function Z(t), its time-integral (t) and its Fourier transform Z(/) for oxide ions in <5-Bi203 at 1094 K.

A number of other studies have subsequently examined the dynamics of individual ions in the liquid state. To cite just two examples, Urahata and Ribeiro [16] and Marguhs [61] have computed the vibrational density of states for a number of ionic liquids including [EMIM]C1, [OMIM]Cl, [BMIM][PF6] and [OMIM][PFe]. The density of states is obtained from a Fourier transform of the velocity autocorrelation functions, and can be compared, at least qualitatively, to experimental IR and Raman... 

S.2.2.2 Time-Segmental Velocity Autocorrelation Function and Its Fourier Transform... 

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.)... 

P((b) Fourier transform of the autocorrelation function of particle velocities... 

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 simplest approach to describe the ion dynamics in disordered materials is to assume completely uncorrelated, random ion movements [42]. In this case, the jump of an ion moving in a forward direction is only correlated to itself, thus the velocity autocorrelation function is proportional to a Dirac Delta function at = 0 (see Fig. la). The complex conductivity obtained by Fourier transform is then independent of frequency. This means that the real part of the conductivity shows no dispersion and at all frequencies the ac conductivity < (6 ) can be identified with the dc conductivity. By contrast, conductivity spectra of most ion-conducting materials show that o (o) varies with frequency. This is schematically illustrated... 

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]. 

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