Spectral function dipole autocorrelator

The computation of spectra from the dipole autocorrelation function, Eq. 2.66, does not impose such stringent conditions on the integrand as our derivation based on Fourier transform suggests. Equation 2.66 is, therefore, a favored starting point for the computation of spectral moments and profiles the relationship is also valid in quantum mechanics as we will see below. 

The dipole autocorrelation function, C(t), is the Fourier transform of the spectral line profile, g(v). Knowledge of the correlation function is theoretically equivalent to knowledge of the spectral profile. Correlation functions offer some insight into the molecular dynamics of dense fluids. 

The theory of spectral moments and line shape is based on time-dependent perturbation theory, Eqs. 2.85 and 2.86, applied to ensembles of atoms, or equivalently on the Heisenberg formalism involving dipole autocorrelation functions, Eq. 2.90. 

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. 

Knowledge of the spectral density J co) is equivalent to knowledge of the dipole autocorrelation function C(t), and vice versa. 

Binary interactions. Dipole autocorrelation functions of binary systems are readily computed. For binary systems, it is convenient to obtain the dipole autocorrelation function, C(t), from the spectral profile, G(co). Figure 5.2 shows the complex correlation function of the quantum profile of He-Ar pairs (295 K) given in Figs. 5.5 and 5.6. The real part is an even function of time, 91 C(—t) = 91 C(t) (solid upper curve). The imaginary part, on the other hand, is negative for positive times it is also an odd function of time, 3 C(—t) — — 3 C(t) (solid lower curve, Fig. 5.2). For comparison, the classical autocorrelation function is also shown. It is real, positive and symmetric in time (dotted curve). In the case considered, the... 

Memory function. Spectral profiles may be computed from the memory function, K(t), which is related to the dipole autocorrelation function, C(t), according to... 

Spectral Function as a Dipole Autocorrelator We may simplify the formula (22b) for D(z), if we employ an equality10... 

I is the effective moment of inertia of a dipole (we consider here a linear molecule), determined by the relation (149). The spectral function L(z), calculated for thermal equilibrium, is linearly related to the spectrum C° of the dipolar autocorrelation function (ACF) C°(f) (VIG, p. 137 GT, p. 152) as... 

In our variant of the response theory the spectral function (SF) L(z) of the model is linearly related to the spectrum of the dipole autocorrelation function (ACF). 

Thus the nth vibrational spectral moment is equal to an equilibrium correlation function, the nth derivative of the dipole moment autocorrelation function evaluated at t=0. By using the repeated application of the Heisenberg equation of motion ... 

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 ... 

Spectral moments were obtained from the autocorrelation function of the A1A2AL component of the induced dipole component [294],... 

The time autocorrelation function can be written as a transition dipole correlation function, a form that is equally useful for an inhomogeneously broadened spectrum. This is the form that is extensively used to discuss the spectral effects of the environment (32-34). The dipole correlation function also provides for the novice an intuitively clear prescription as to how to compute a spectrum using classical dynamics. For the expert it points out limitations of this, otherwise very useful, approximation. The required transformation is to rewrite the spectrum so that the time evolution is carried by the dipole operator rather than by the bright state wave packet. The conceptual advantage is that it is easier to imagine what the classical limit will be because what is readily provided by classical mechanics trajectory computations is the time dependence of the coordinates and momenta and hence, of functions thereof. In other words, in our mind it is easier to... 

Another approach to the calculation of IR spectra of hydrogen-bonded complexes is based on linear response theory, in which the spectral density is the Fourier transform of the autocorrelation function of the dipole moment operator involved in the IR transition [62,63]. Recently Car-Parrinello molecular dynamics (CPMD) [73] has been used to simulate IR spectra of hydrogen-bonded systems [64-72]. 

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