Autocorrelation function classical calculation

The observed intensities also depend on the refractive index, vhich in general is frequency dependent [93], This dependence is unkno vn in most cases and has not been considered. We note, however, that for liquid water the refractive index is virtually constant between 300 and 3500 cm i [94]. The dipole autocorrelation function is calculated classically and quantum corrections [95, 96] are introduced through the factor 2/[l+exp(-ko/2 tkBT)]. Eq. (9.16) for the absorption spectrum has previously been applied in calculations of the far- and mid-IR spectra of liquid water [90, 97[ and crystals [85]. The quantum correction damps the intensities of the low frequency motions and more sophisticated schemes [98] may lead to more severe damping of the low frequencies - as found for liquid water [99]. 

A semiclassical description is well established when both the Hamilton operator of the system and the quantity to be calculated have a well-defined classical analog. For example, there exist several semiclassical methods for calculating the vibrational autocorrelation function on a single excited electronic surface, the Fourier transform of which yields the Franck-Condon spectmm [108, 109, 150, 244]. In particular, semiclassical methods based on the initial-value representation of the semiclassical propagator [104-111, 245-248], which circumvent the cumbersome root-search problem in boundary-value-based semiclassical methods, have been successfully applied to a variety of systems (see, for example, Refs. 110, 111, 161, and 249 and references therein). The mapping procedure introduced in Section VI results in a quantum-mechanical Hamiltonian with a well-defined classical limit, and therefore it... 

Induced dipole autocorrelation functions of three-body systems have not yet been computed from first principles. Such work involves the solution of Schrodinger s equation of three interacting atoms. However, classical and semi-classical methods, especially molecular dynamics calculations, exist which offer some insight into three-body dynamics and interactions. Very useful expressions exist for the three-body spectral moments, with the lowest-order Wigner-Kirkwood quantum corrections which were discussed above. 

Fig. 5.2. The dipole autocorrelation function of He-Ar at 295 K, according to a quantal (solid lines) and a classical calculation (dotted). The quantum correlation function is complex the real part is symmetric and positive (91) while the imaginary part (3) is anti-symmetric and negative at positive frequencies.

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 IR and Raman spectra are calculated by Fourier transformation of the dipole and polarizability tensor autocorrelation functions, respectively. T3s-A classical expression for the infrared absorption coefficient ct(co) at frequency CO reads ... 

Das and Bhattacharjee236 derive the frequency and shear dependent viscosity of a simple fluid at the critical point and find good agreement with recent experimental measurements of Berg et al.237 Ernst238 calculates universal power law tails for single and multi-particle time correlation functions and finds that the collisional transfer component of the stress autocorrelation function in a classical dense fluid has the same long-time behaviour as the velocity autocorrelation function for the Lorentz gas, i.e. 

Next we examine whether these vibrations are unique in the enzymatic environment or they are inherent in the substrates. In the left panel of Fig. 18 we compare the calculation in the enzyme with a simulation of the substrates in aqueous solution, in the absence of hPNP. The spectrum of the 0-5 —0-4 distance autocorrelation function of the classical MD of solvated substrates showed a peak at 330 cm-1, and of the unsolvated substrates at 285 cm-1, i.e. distinct from the peaks in the presence of the enzyme, revealing that hPNP is directly affecting the way in which these oxygens naturally vibrate. 

The IR spectrum is obtained as the Fourier transform of the autocorrelation function of the classical dipole moment M [90], calculated at every point of the MD trajectory... 

In Section II, four molecular mechanisms—a, b, c, and d—are briefly observed. These mechanisms allow us to model analytically the water/ice spectra in terms of classical theory based on calculation of autocorrelation function (ACF). Our description is restricted by the results of this theory. A more detailed consideration is given in the second part of the review. 

The treatment up to this point has been fully quantum mechanical Vjj is an operator in the bath degrees of freedom. For many calculations on liquids, however, one wants to treat these degrees of freedom (rotations and translations) classically the question then arises of what is the best way to replace a quantum correlation function with a classical one. A classical autocorrelation function is an even function of the time, a property shared by the anticommutator in (2.11) but not by the one sided correlation function of (2.10). It thus appears that the best place to make a classical approximation is in (2.11) in addition, doing so gives... 

The time-dependent theory of spectroscopy bridges this gap. This approach has received less attention than the traditional time-independent view of spectroscopy, but since 1980, it has been very successfully applied to the field of coordination chemistry.The intrinsic time dependence of external perturbations, for example oscillating laser fields used in electronic spectroscopy, is also expKdtly treated by modern computational methods such as time-dependent density functional theory, a promising approach to the efficient calculation of electronic spectra and exdted-state structures not based on adjustable parameters, as described in Chapter 2.40. In contrast, the time-dependent theory of spectroscopy outlined in the following often relies on parameters obtained by adjusting a calculated spectrum to the experimental data. It provides a unified approach for several spectroscopic techniques and leads to intuitive physical pictures often qualitatively related to classical dynamics. The concepts at its core, time-dependent wave functions (wave packets) and autocorrelation functions, can be measured with femtosecond (fs) techniques, which often illustrate concepts very similar to those presented in the following for the analysis of steady-state spectra. The time-dependent approach therefore unifies spectroscopic... 

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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