Classical autocorrelation function, vibrational

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... 

Spectral lineshapes were first expressed in terms of autocorrelation functions by Foley39 and Anderson.40 Van Kranendonk gave an extensive review of this and attempted to compute the dipolar correlation function for vibration-rotation spectra in the semi-classical approximation.2 The general formalism in its present form is due to Kubo.11 Van Hove related the cross section for thermal neutron scattering to a density autocorrelation function.18 Singwi et al.41 have applied this kind of formalism to the shape of Mossbauer lines, and recently Gordon15 has rederived the formula for the infrared bandshapes and has constructed a physical model for rotational diffusion. There also exists an extensive literature in magnetic resonance where time-correlation functions have been used for more than two decades.8... 

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... 

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. 

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]. 

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