Total moment correlation function

A remark applies to the experimental values of deduced from experimental bandshapes. Since we deal with total (multimolecular) correlation functions, their moments might also contain contributions arising from the intercorrelations of different molecules - or multimolecular effects - since cross terms in the statistical averages do not necessarily vanish. On this account it is necessary to estimate the extent of high-frequency collective effects if any, whenever torques are to be computed from the spectral moments. 

The dipole moment is the total dipole of the sample, p = Y.i Pi The correlation function describes the response of the system to the weakly coupled radiation field. The effects of the field are modeled by the response of the individual atoms or molecules unaffected by the weak coupling. The Hamiltonian describes the interaction of the field and matter (first-order perturbatiuon theory). The correlation function describes how the perturbed system approaches equilibrium. 

It is well known that in n-body complexes rotational transitions of the order n may occur [400]. However, we will assume here that the interaction forces are pairwise additive and isotropic so that rotations and translations are uncorrelated. In this case, at most double transitions occur [400] and the correlation function of the total dipole moment can be written as... 

Figs. 12, 13 and 14 show the correlation functions goo(f), goH(r) and gHH(r) at four different points on the phase diagram, as reported in Table 4 together with the values of the total potential energy and of the molecular dipole moment predicted by the simulations. 

With local information given by INM analysis in mind, we next see the character of rotational relaxation in liquid water. The most familiar way to see this, not only for numerical simulations [76-78] but for laboratory experiments, is to measure dielectric relaxation, by means of which total or individual dipole moments can be probed [79,80]. Figure 10 gives power spectra of the total dipole moment fluctuation of liquid water, together with the case of water cluster, (H20)io8- The spectral profile for liquid water is nearly fitted to the Lorentzian, which is consistent with a direct calculation of the correlation function of rotational motions. The exponential decaying behavior of dielectric relaxation was actually verified in laboratory experiments [79,80]. On the other hand, the profile for water cluster deviates from the Lorentzian function. As stated in Section III, the dynamics of finite systems may be more difficult to be understood. 

Time dependence of the normalized total moment-moment time correlation function [ J M(t)] for the water molecules, both in the CsPFO micellar solution (solid line) and in neat water. Circles are the simulation data and the continuous line is a multiexponential fit. 

The above equation is the correlation function of the total dipole moment. This result was derived in a slightly different manner by Schroer [73], Coffey et al. [74], and Risken and Vollmer [75] and is implicit in the work of Budo [76]. 

We now turn to the situation in which the photons emitted from the two atomic transitions are not distinguishable. This can happen when the atomic transition dipole moments are exactly parallel. Then the detector responds to the total field (30) for which the correlation functions are given by Eqs. (141) and (142). However, even for p 1, we can still distinguish between photons emitted from the s) —> 2) and a) —> 2) transitions as they can have different polarizations. This is easy to see from Eqs. (107) and (108), where the dipole moments pv and pa of the s) —> 2) and a) —> 2) transitions, respectively, are oriented in different directions, unless p, = p2 and then pH = 0. 

U typically relaxes through molecular reorientation (or dephasing) but % depends upon the relative positions and orientation of molecules in the fluid. The question which arises is - what aspects of the total moment correlation function (or its associated spectrum) reflect on the properties of M, or,to what extent can the spectrum be used to study reorientatTon ... 

In terms of the projected quantities the total moment correlation function becomes... 

The calculation of dielectric properties in computer simulations is complicated by the long-ranged nature of the coulomb interactions and by the fact that the total moment correlation function is a "collective" function which requires long runs to properly average. These issues have been discussed by Tildesley in these lectures and in several reviews. 

Fig 6 Two decompositions of the total moment correlation function for a model of methyl cyanide, from Ref. 55. 

We now can formulate the fluctuation-dissipation theorem. It relates the correlation function of the fluctuations of the component of the total dipole moment along the field direction, p-y, to the decay function of the polarization,... 

The spectral density is related to the time Fourier transform of the sample total dipole moment auto-correlation function (a.c.f.)... 

Dielectric relaxation is sensitive to the time correlation function of the collective variable P(t) that is the total dipole moment, just as quasielastic light scattering is sensitive to the time correlation function of the collective variable YTj= exp(iq r (t)) that is the spatial Fourier component of the concentration. 

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