Vibrational dynamics autocorrelation function

The linear response theory [50,51] provides us with an adequate framework in order to study the dynamics of the hydrogen bond because it allows us to account for relaxational mechanisms. If one assumes that the time-dependent electrical field is weak, such that its interaction with the stretching vibration X-H Y may be treated perturbatively to first order, linearly with respect to the electrical field, then the IR spectral density may be obtained by the Fourier transform of the autocorrelation function G(t) of the dipole moment operator of the X-H bond ... 

There are many experiments which determine only specific frequency components of the power spectra. For example, a measurement of the diffusion coefficient yields the zero frequency component of the power spectrum of the velocity autocorrelation function. Likewise, all other static coefficients are related to autocorrelation functions through the zero frequency component of the corresponding power spectra. On the other hand, measurements or relaxation times of molecular internal degrees of freedom provide information about finite frequency components of power spectra. For example, vibrational and nuclear spin relaxation times yield finite frequency components of power spectra which in the former case is the vibrational resonance frequency,28,29 and in the latter case is the Larmour precessional frequency.8 Experiments which probe a range of frequencies contribute much more to our understanding of the dynamics and structure of the liquid state than those which probe single frequency components. 

The immediate question, then, is whether this scenario reflects what actually happens. Do the INM theories really work There is, in fact, some evidence on this score (45,52). If we compare the vibrational friction predicted by INM theory, Equation (20), with that revealed by an exact molecular-dynamics evaluation of the force autocorrelation function, Equations (4) and (13), we see some reasonably impressive agreement (Fig. 3) (52). Not only is the few hundred cm-1 spectral range of the friction predicted quite nicely, but the basic form of the response is as well. Each example shows that the friction diminishes as the frequency rises, beginning with a sharp drop from its maximum value at co = 0 and gradually going over to a much slower decay, behavior captured nicely by the INM formulas. 

Not all molecular properties are, however, of local nature, which lend themselves to efficient computational schemes. Molecular vibrations are typically non-local and delocalized. Nevertheless, it is possible even in such cases to design a tailored quantum chemical method, the Mode-Tracking protocol [164], for the selective calculation of only those vibrations relevant for a certain scientific context. For the selective calculation of various types of vibrational spectra we refer the interested reader to the reviews in Refs. [165, 166]. It should be mentioned that molecular dynamics simulations offer different routes to spectra through autocorrelation functions. 

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 ... 

All of this information about vibrational frequencies and transition intensities is observable directly in the frequency domain absorption spectrum, A"= o(w) The autocorrelation function picture is an alternative way of deriving a ball-and-spring physical picture (or dynamical mechanism) from the raw experimental data. Although there is a simple Fourier transform relationship between Ivn (to) and ( f (f)l F(O)), profoundly different intuitive pictures are used to make sense of experimental results and to guide the design of new experiments. 

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... 

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... 

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 spectrum. ° In particular, semiclassical methods based on the initial-value representation of the semiclassical propagator,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, the reviews Refs. 85, 86 and references therein). These methods cannot directly be applied to nonadiabatic dynamics, though, because the Hamilton operator for the vibronic coupling problem [Eq. (1)] involves discrete degrees of freedom (discrete electronic states) which do not possess an obvious classical counterpart. 

The approximate QD pump probe signal (Fig. 3.47) still yields a good agreement with the experimental and the exact QD theoretical results. The 320 fs oscillation structure dominates in all cases. Moreover, one of the slower pseudorotational vibrations with a period of about 1 ps can also be detected by our approximate QD method. A detailed analysis by means of the Fourier transform of the corresponding autocorrelation function [378] and of the induced wave packet dynamics reveals that this oscillation is caused by a slow pseudorotational vibration in the coordinate (f. As the Fourier spectrum shows the wave packet prepared in the B state is centered around 621 nm, i.e. between the vibrational states = 5 and 6 of the (f mode. The energy distance between these two eigenstates corresponds to a vibrational period of about 1 ps. It is one of the fastest pseudorotational vibrations which can be observed in the absorption spectrum and is the next slower vibration after the Qs mode. This interpretation of the 1 ps oscillation is confirmed by analyzing the induced wave packet dynamics. Here, an accumulation of the... 

The theoretical model for photodetachment is similar to that used to describe photodissociation outlined in the last section. As illustrated in Fig. 3.7, the initial wave packet on the neutral PES was chosen as the ground vibrational state of cis-HOCO, which has a lower energy than its tram counterpart. The anion vibrational eigenfunction was determined on a newly developed anion PES at the same CCSD(T)-F12/AVTZ level [130], as used to construct the neutral PES [100, 101]. The neutral wave packet was propagated to yield probabilities to both the HO-I-CO and H-I-CO2 asymptotes with a flux method [108] and the cosine Fourier transform of the Chebyshev autocorrelation function yielded the energy spectrum [44]. The discretization of the Hamiltonian and wavepacket, and the propagation were essentially the same as in our recent reaction dynamics study [107]. 

The distribution of vibration frequencies in condensed matter and molecules provides important structural information and can be calculated (a) from the Fourier transform of the velocity-velocity autocorrelation function, calculated from trajectories of 6000 time steps of 3.025 fs each, and (b) from the dynamical matrix calculated from the second-order energy derivatives of a well-equilibrated structure at 0 K. The results for a-GST (Fig. 17.8) show that the results for the two methods agree very well. The most pronounced features are peaks near 60 and 150 cm and a tail for frequencies above 180 cm . Projections onto the vibrations of different elements and structural units show that the tail is associated with the vibrations of the lightest element Ge, particularly atoms that are fourfold coordinated. 

If the PES is not purely repulsive, i,e if direct dissociation is (partly) prohibited by a potential barrier in the exit channel or some other dynamical effect, some part of the wave packet is trapped for a longer time and performs an oscillatory motion as observed for FNO(Si) in Figures. Whenever a part of the wave packet recurs to its starting position, the overlap with d>ex(0) increases and the autocorrelation function shows a series of recurrences as schematically illustrated in Figure 6(c), The recurrences reflect the vibrational motion of the molecule in the inner region of the PES, before it finally escapes through the transition state (point of no return) and dissociates. If T is the period of the oscillation in the time-domain, the Fourier transformation of S(t) leads to structures in the spectrum. 

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