Correlation functions dipole

That is, a(co) can be expressed in terms of the dipole correlation function form. 

Without the external field, the Stockmayer fluid near the wall exhibits symmetric density oscillations that die out as they reach the middle of the film. Near the surface, the fluid dipoles are oriented parallel to the walls. Upon turning on the electric field, the density profile of the Stockmayer fluid exhibits pronounced oscillations throughout the film. The amplitude of these oscillations increases with increasing field strength until a saturation point is reached at which all the fluid dipoles are oriented parallel to the field (perpendicular to the walls). The density profile remains symmetric. The dipole-dipole correlation function and its transverse [] and longitudinal [] com-... 

Fig. 4.1 a Typical time evolution of a given correlation function in a glass-forming system for different temperatures (T >T2>...>T ), b Molecular dynamics simulation results [105] for the time decay of different correlation functions in polyisoprene at 363 K normalized dynamic structure factor at the first static structure factor maximum solid thick line)y intermediate incoherent scattering function of the hydrogens solid thin line), dipole-dipole correlation function dashed line) and second order orientational correlation function of three different C-H bonds measurable by NMR dashed-dotted lines)... 

The authors finish by exploring the transferability of their force field parameters to a different zeolite, namely, silicalite. In this instance, a Fourier transform of the total dipole correlation function provides another model infrared (IR) spectrum for comparison to experiment, and again excellent agreement is obtained. Dominant computed bands appear at 1099, 806, 545, and464 cm while experimental bands are observed at 1100, 800,550, and 420 cm A Some errors in band intensity are observed in the lower energy region of tlie spectrum. 

In Fig. 2 we show the IR spectrum in the O-H/D stretching region of our 7D model. The data have been obtained by a Fourier-transform of the dipole-dipole correlation function, where the... 

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. 

To the extent that the dipole induced in a cluster of atoms may be represented by a sum of pair dipoles induced in dissimilar atoms, i and i, fiu, = ft(Rw), we may write for the total dipole ft = Xw Mu - The summation is over all atoms i and all atoms i. The dipole correlation function can then be written as a sum of three parts (Poll 1980),... 

Using the standard expression of the dipole-dipole correlation function the CC absorption cross section can be derived as [9,42] ... 

Interestingly enough, one sees differences between the various variants of Markovian and non-Markovian theories already in static linear absorption spectra. In the regime of second-order perturbation theory in the coupling to the electromagnetic field the linear absorption line-shape / (ui) can be calculated from the Fourier transform of the dipole-dipole correlation function as... 

When interpreting time-resolved mid-IR spectra, it is beneficial to consider the influence of rotational dynamics on the vibrational spectrum of a heteronuclear diatomic. It was shown more than 30 years ago that the vibrational absorption spectrum of a diatomic is related to its transition dipole correlation function (/z(0) /r(t)> through a Fourier transform (10) ... 

The FIR spectrum is proportional to the Fourier transform of the dipole correlation function, (M(t) -M(0). As discussed above, at short times the correlation function is dominated by pseudo-oscillatory modes. Insofar as the dipole moment of the liquid is relatively weakly dependent on these pseudo-oscillatory coordinates, it is generally safe to truncate Eq. (1) after the second term in describing this short-time behavior. Thus, an IR-active intermolecular mode is considered to be one for which M-1 0. 

An alternative approach to DS study is to examine the dynamic molecular properties of a substance directly in the time domain. In the linear response approximation, the fluctuations of polarization caused by thermal motion are the same as for the macroscopic rearrangements induced by the electric field [27,28], Thus, one can equate the relaxation function < )(t) and the macroscopic dipole correlation function (DCF) V(t) as follows ... 

Figure 9. The three-dimensional plot of experimental dipole correlation function versus time and temperature. The percolation threshold temperature Tp = 26.5°C. (Reproduced with permission from Ref. 47. Copyright 1996, The American Physical...

Figure 10. The dipole correlation function v /(f/T,-) demonstrates power-law behavior for the temperature region near the percolation onset (Ton — 12°C). (Reproduced with permission from Ref. 2. Copyright 2002, Elsevier Science B.V.)...

Furthermore, we will focus our attention only on the time-dependent behavior of the dipole correlation function b(t) defined by (55), which is given by... 

The dielectric relaxation at percolation was analyzed in the time domain since the theoretical relaxation model described above is formulated for the dipole correlation function T(f). For this purpose the complex dielectric permittivity data were expressed in terms of the DCF using (14) and (25). Figure 28 shows typical examples of the DCF, obtained from the frequency dependence of the complex permittivity at the percolation temperature, corresponding to several porous glasses studied recently [153-156]. 

In order to explain the non-Debye response (134) it is possible to use the memory function approach [22,23,31,266-268]. Thus, the normalized dipole correlation function k(f) (22) corresponding to a nonexponential dielectric relaxation process obeys the equation... 

In Section IV we derived a system of differential equations describing the rotational dynamics of the dipole correlation functions of the tagged molecule in different environments. In Eq. (4.4) the quantities depending on the environment are and the diffusion coeffidrats 2> (i j). 

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