Autocorrelation function dipole correlation

I quantities x and y are different, then the correlation function js sometimes referred to ross-correlation function. When x and y are the same then the function is usually called an orrelation function. An autocorrelation function indicates the extent to which the system IS a memory of its previous values (or, conversely, how long it takes the system to its memory). A simple example is the velocity autocorrelation coefficient whose indicates how closely the velocity at a time t is correlated with the velocity at time me correlation functions can be averaged over all the particles in the system (as can elocity autocorrelation function) whereas other functions are a property of the entire m (e.g. the dipole moment of the sample). The value of the velocity autocorrelation icient can be calculated by averaging over the N atoms in the simulation ... 

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. ... 

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 ... 

For the analysis of the dynamical properties of the water and ions, the simulation cell is divided into eight subshells of thickness 3.0A and of height equal to the height of one turn of DNA. The dynamical properties, such as diffusion coefficients and velocity autocorrelation functions, of the water molecules and the ions are computed in various shells. From the study of the dipole orientational correlation function... 

When l l, the above gives the so-called cross-correlation functions and the associated cross-correlation rates (longitudinal and transverse). Crosscorrelation functions arise from the interference between two relaxation mechanisms (e.g., between the dipole-dipole and the chemical shielding anisotropy interactions, or between the anisotropies of chemical shieldings of two nuclei, etc.).40 When l = 1=2, one has the autocorrelation functions G2m(r) or simply... 

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. 

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 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. 

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.

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-... 

In our approach [1, 2] termed the dynamic method the complex susceptibility x = x — ix" is determined by a law of undamped motion of a dipole in a given potential well and by dissipation mechanism often described as stosszahlansatz in the underlying kinetic or Boltzmann equation. In this review we shall refer to this (dynamic) method as the ACF method, since it is actually based on calculation of the spectrum of the dipolar autocorrelation function (ACF). Actually we use a one-particle approximation, in which the form of an employed potential well (being in many cases rectangular or close to it) is taken a priori. Correlation of the particles coordinates is characterized implicitly by the Kirkwood correlation factor g, its value being taken from the experimental data. The ACF method is simple and effective, because we do not employ the stochastic equations of motions. This feature distinguishes our method from other well-known approaches—for example, from those described in books [13, 14]. 

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... 

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 = =... 

There has been much controversy in the past several years concerning the relation of the dispersion of the dielectric constant to the molecular dipole-moment correlation function (see Titulaer and Duetch, 1974). Fatuzzo and Mason (1967) have shown that the autocorrelation function of the net dipole moment of a sphere imbedded in a medium of the same dielectric constant is related to the frequency-dependent dielectric constant by... 

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... 

The velocity autocorrelation function is an example of a single-particle correlation function, in which the average is calculated not only over time origins but also over all the atoms. Some properties are calculated for the entire system. One such property is the net dipole moment of the system, which is the vector sum of all the individual dipoles of the molecules in the system (clearly the dipole moment of the system can be non-zero only if each individual molecule has a dipole). The magnitude and orientation of the net dipole will change with time and is given by ... 

Examples of linear response functions (susceptibilities) include the frequency dependent electrical conductivity (the Fourier transform of an equilibrium current autocorrelation function), dielectric susceptibility, which is the transform of a dipole moment autocorrelation function, along with stress, heat flux, and an assortment of velocity correlation functions. 

Another common application of correlation functions is the calculation of IR absorption spectra. The Uneshape function, l (o), is given by the Fourier transform of the autocorrelation function of the electric dipole moment M,... 

Figure 1 The dipole autocorrelation function of the band of benzene. The top line is the dipole correlation function for benzene dissolved In cyclohexane and the bottom line is that for benzene vapour. Hill IR and Steele D, unpublished work.

One of the first applications of RQMC was to the rotational dynamics of carbonyl sulfide (OCS) molecules solvated in helium clusters, for cluster sizes (tV = 3,10) [42]. This and related work, described shortly, rest on the absorption spectrum given by the Fourier transform of the reptilian imaginary time electric dipole correlation function. Similarly, the optical activity is extracted from the autocorrelation of the molecular orientation vector. This work by Moroni and coworkers and/or Boroini and co-workers was closely followed by several other investigations of rotational dynamics in doped clusters, summarized as follows ... 

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