Dipole autocorrelation function

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. 

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In the case where x and y are the same, C (r) is called an autocorrelation function, if they are different, it is called a cross-correlation function. For an autocorrelation function, the initial value at t = to is 1, and it approaches 0 as t oo. How fast it approaches 0 is measured by the relaxation time. The Fourier transforms of such correlation functions are often related to experimentally observed spectra, the far infrared spectrum of a solvent, for example, is the Foiuier transform of the dipole autocorrelation function. ... 

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 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. 

Heisenberg formalism. The Heisenberg view leads to an expression equivalent to the Schrodinger formalism that stresses the time evolution of quantum systems it has a clear correspondence with classical mechanics it is most conveniently expressed in terms of the dipole autocorrelation function (Gordon 1968). 

Later studies showed the same phenomena in deuterium and deuterium-rare gas mixtures [335, 338, 305], and also in nitrogen and nitrogen-helium mixtures [336] in nitrogen-argon mixtures the feature is, however, not well developed. The intercollisional dip (as the feature is now commonly called) in the rototranslational spectra was identified many years later see Fig. 3.5 and related discussions. The phenomenon was explained by van Kranendonk [404] as a many-body process, in terms of the correlations of induced dipoles in consecutive collisions. In other words, at low densities, the dipole autocorrelation function has a significant negative tail of a characteristic decay time equal to the mean time between collisions see the theoretical developments in Chapter 5 for details. 

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. 

The dipole autocorrelation function is the expectation value over the equilibrium ensemble of an unperturbed system of atoms or molecules (i.e., without an applied field),... 

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

We note that classical dipole autocorrelation functions are real and symmetric in time, C(—t) = C(t). 

Density expansion. The method of cluster expansions has been used to obtain the time-dependent correlation functions for a mixture of atomic gases. The particle dynamics was treated quantum mechanically. Expressions up to third order in density were given explicitly [331]. We have discussed similar work in the previous Section and simply state that one may talk about binary, ternary, etc., dipole autocorrelation functions. 

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... 

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.

Fig. 5.3. The dipole autocorrelation function, long-time behavior (schematic). Xd and xc are times of the order of the mean duration of a collision and the mean time between collisions, respectively.

Like many other properties of the N-particle system, the electric dipole moment, p(p, q, t), is a function of the canonical variables and time. We define the classical dipole autocorrelation function, according to... 

The classical dipole correlation function is symmetric in time, C(—t) = C(f), as may be seen from Eq. 5.59 by replacing x by x — t the classical scalar product in Eq. 5.59 is, of course, commutative. Classical line shapes are, therefore, symmetric, J(—. Furthermore, classical dipole autocorrelation functions are real. 

Long-time behavior of correlation functions. The dipoles induced in successive collisions are correlated as Fig. 3.4 on p. 70 suggests. As a consequence, the dipole autocorrelation function has a negative tail of a duration comparable to the mean time between collisions, Fig. 5.3. Furthermore, the area under the negative tail is of similar order of magnitude as the area under the positive (or intracollisional) part of C(r). If the neg-... 

We will briefly consider several desymmetrization procedures that have been mentioned in the literature. These may simply employ various factors applied to the symmetric, classical profile, G(co), or alternatively attempt to correct the classical dipole autocorrelation functions in the time domain. 

The line shape is the Fourier transform of the dipole autocorrelation function,... 

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... 

Ternary and Other Induced Spectra. Three-particle induced dipoles and the associated ternary collision-induced absorption spectra and dipole autocorrelation functions have been studied for fluids composed of mixtures of rare gases, and for neat fluids of nonpolar molecules — that is for systems that are widely thought to interact with radiation only by virtue of interaction-induced properties. A convenient framework is thus obtained for understanding the variety of experimental observations. The computer simulation studies permit an insight into the involved basic processes, but were not intended for direct comparison with measurements [57]. Methods have been developed for computer... 

The dipole autocorrelation function, , defined previously. The full-time dependence of this function for liquid carbon monoxide has been successfully determined experimentally from Fourier inversion of infrared band shapes.2,15 In fact, this was one of the reasons this system was studied. This function has also been successfully evaluated in terms of models of the molecular reorientation process.58 s memory function, KD(t), is defined by... 

The functions X(Q) and Y(fl) are specified by the choice of the particular experiment. Prominent orientational correlation functions result when setting X(Q) = K(Q) = P/(cos0), where P/ is the Legendre polynomial of rank / and the angle 0 specifies the orientation of the molecule with respect to some fixed axis. For example, consider a molecule that possesses a vector property, say the molecular electric dipole p = pu, (u is a unit vector). Then, one defines the dipole autocorrelation function g (t) = (u,(t)u,(0)). Similarly, one defines a correlation function gilt) for second rank tensorial molecular properties. In general the normalized (g/(0) = 1) orientational correlation function of rank l is given by... 

The static susceptibility Xo is given by Eq. (84). Here, the quantity cj 0 (/co)/cl0 ° (0) coincides with the one-sided Fourier transform of the normalized dipole autocorrelation function C t) = (cosi (0) cosi (f))0, namely,... 

Our objective is to calculate from the system of Eqs. (1) (7) the dipole autocorrelation function of an assembly of encaged dipoles ... 

We have seen (Section 6.2,3) that a Lorentzian lineshape corresponds to an exponentially decaying dipole autocorrelation function. For the Hamiltonian of Eqs (9.36) and (9.39) this correlation function is C/x(f) = ik g = =... 

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