Dielectric relaxation in dipolar fluids. Generalized mode approach

The concept of generalized collective modes, recently proposed for the investigation of simple fluids, is now applied to describe processes of di-electric relaxation in dipolar systems. The approach presented here is an extension of the dipole-density formalism to arbitrary numbers of dynami-cal variables and values of wavelengths. Generalized dipolar mode spectra of a Stockmayer fluid are evaluated over a wide scale of wavelengths up to the five-variable approximation. The wavevector-and frequency-dependent dielectric permittivity and dipole-moment time autocorrelation functions are calculated on the basis of analytical expressions using the dipolar modes. The obtained results are compared with those achieved in lower-order approximations and with molecular dynamics data. It is shown that the five-variable description quantitatively reproduces the entire frequency dependence of the dielectric constant at arbitrary wavenumbers.


Introduction
Processes of dielectric relaxation in polar uids have been intensively studied in theory 1{7], computer simulation 8{15] and pure experiment 16{18].Nevertheless, there is still lack of agreement between the theoretically predicted and experimental results.Moreover, some main problems remain unsolved even in the simplest case when the uid is treated as a system of point dipoles.Among various theoretical schemes, enabling us to describe the processes of dielectric relaxation in dipolar uids, it is necessary to point out two consequent approaches: the dipoledensity formalism of Madden and Kivelson 2] and the extended hydrodynamic description popularized by Bagchi and Chandra 4].
The usual Navier-Stokes hydrodynamics 19], being valid in the in nite-wavelength limit and large times, cannot be used at molecular length scales.In the extended hydrodynamic description, the microscopic operators of the number density n(r; ) = P N i=1 (r?r i ) ( ?i ), spatial P (r; ) = P N i=1 mv i (r?r i ) ( ?i ) and angular L(r; ) = P N i=1 Jw i (r?r i ) ( ?i ) momenta densities are considered as basic dynamical quantities 6].These quantities satisfy modi ed equations which take into account processes with short and intermediate time scales as well.
In their theory, uctuations of the total energy density ê(r; ) = P N i=1 e i (r ?r i ) ( ?i ), where e i = 1   2 mv 2 i + Jw 2 i + P N j(j6 =i) ' ij , are not taken into account because of the complexities of hydrodynamic equations.Vector i characterizes orientations of molecule i and, in the case of rigid nonpolarizable molecules, it can be associated with the unit vector directed along the particle's dipole moment, i.e., i = i = .Then, the dipole-moment uctuations can be reproduced on the basis of number-density correlations using the relation M(r; t) = h n(r; ; t)i where the averaging is performed over orientations.In such a way the dynamical polarization in a dense dipolar uid can be investigated, provided reasonable forms of the dissipative kernels are available.Despite the fact that such an approach allows us, in principle, to describe the processes of dielectric relaxation in a dipolar uid and connect them directly with thermodynamic and hydrodynamic properties, it is rather sophisticated and impractical in application.The main results are yet to be obtained here.
A somewhat di erent approach to describing the generalized hydrodynamics of dipolar systems has been recently proposed in 20].The main idea of this approach consists in the following.When vector M(k) of the dipole density is already included into the consideration as an orientational variable, it is no longer necessary to deal with the orientational dependence Q(r; ) fn; P ; ê; Lg of hydrodynamic variables.Then the basic set can be cast in the usual molecular form Q(k) = Z V h Q(r; )i e ?ik r dr fn(k); P (k); e(k); L(k)g, where n(k) = P N i=1 e ?ik r i , P (k) = P N i=1 mv i e ?ik r i , e(k) = P N i=1 he i i i e ?ik r i and L(k) = P N i=1 Jw i e ?ik r i .The advantage of such a representation lies in the evident simpli cation of the corresponding hydrodynamic equations.Moreover, owing to conservation laws of the total number of particles, momenta and energy, the basic set Q(k) can be considered as a set of indeed slow variables.For instance, time derivatives of n(k), P (k) and e(k) vanish when k !0. Finally, for isotropic and spatially homogeneous systems, the dipole density M(k) does not correlate at equilibrium with the basic hydrodynamic quantities in the static limit, i.e., hM(k) Q(k)i = 0.
For this reason, the dipole-density and hydrodynamic uctuations can be considered separately from each other as this has been assumed implicitly in the dipole-density formalism of Madden and Kivelson 2].The generalized collective modes related to dipole-density uctuations will be called dipolar modes in order to distinguish them from the hydrodynamic modes, concerning uctuations of hydrodynamic quantities.At the same time, the generalized hydrodynamic modes of a dipolar system can be studied within the same scheme as in the case of simple uids 21{24].It is also worth remarking that the correlations between the dipolemoment density and hydrodynamic quantities are not completely independent.They, being absent at the basic level, appear, however, at higher-orders of the description.This leads to additional non-Markovian e ects, when working within the dipole-density formalism.Nevertheless, these e ects are taken into account in an e ective way, including, besides the basic quantity M(k), its higher-order components as well.The hydrodynamic correlations can be included in the same way, too.Time constants appearing in the dissipative kernel are usually considered, as adjustable parameters.It is worth stressing that the previous applications of this theory were restricted to the long-wavelength regime only 12].
Recently, the concept of generalized collective modes, used earlier for the investigation of nonequilibrium properties of simple uids 21{25], has been applied to dipolar systems and actual computations have been performed in the whole wavevector range within the three-mode description 26].In particular, it has been concluded that the three-variable prescription is su cient to predict quantitatively the frequency dependence of dielectric quantities for a Stockmayer uid.But such a conclusion was based on the calculations carried out with the help of a tting procedure, because a higher-order static correlation function was not known.For this reason, it is not obvious that the described above pattern takes place indeed.
In the present paper, the generalized dipolar mode spectra of a Stockmayer uid are evaluated over a wide scale of wavelengths up to the ve-variable description without involving any adjustable parameters.Within the framework of the proposed approach, the frequency dependence of dielectric quantities is determined by extended continued fractions to which the Markovian approximation is applied.As a result, by using the dipolar modes it is shown that the threevariable theory reproduces qualitatively the wavevector-and frequency-dependent dielectric constant and only beginning from the ve-order description one can talk about a quantitative reproduction over the whole range of varying wavenumbers and frequencies.Moreover, we demonstrate that within the same approximation all times constants of memory kernels can be expressed in terms of static correlation functions, so that dynamic properties of the system are obtained using static uctuations exclusively.
is the Liouville operator of the system, ' ij denotes an intermolecular potential, v i and w i are the translational and rotational velocities, respectively, of molecule i with mass m and moment of inertia J. F(k; t) (5) which is diagonal in the static limit t !0, i.e., f (k) = f (k) where h i denotes the equilibrium average, designates the permanent magnitude of the molecule's dipole moment and the multiplier f2g is included in the case of transverse uctuations only.
The orthogonalized procedure can be simpli ed signi cantly taking into account that the basic function g(k; t) f 11 (k; t) is even with respect to time.Then one obtains that g (k) (as well as (k)) are equal to zero if + is an odd number, whereas nonzero elements can be expressed via their diagonal ones as g (k) = (?1) j ?j 2 g (k), where = ( + )=2.The processes of dynamical polarization in the system can be described by TCFs (5).In particular, the longitudinal " L (k; !) and transverse " T (k; !) components of the wavevector-and frequency-dependent dielectric permittivity are expressed via the rst element g(k; t) of the S S square matrix F(k; t) as 15]: " L (k; !) ? 1 9y" L (k; !) = g L (k) ?i!g L (k; !) ; " T (k; !) ? 1 9y = g T (k) ?i!g T (k; !) ; (7) where y = 4 N 2 .
9V k B T and k B , T are the Boltzmann constant and temperature of the system, respectively, g(k; !) = Z 1 0 g(k; t)e ?i!t dt L i! (g(k; t)) and L i! designates the Laplace transform.
It can be easily seen from ( 13) that the same function g(k; !) is obtained within the (S + 1)-th order continued fraction, too, if the memory functions obey the recurrent relation ?S (k; !) = S+1 (k) i! + ?S+1 (k; !) : (16) The solution (13) leads to exact results at arbitrary order S, provided the corresponding memory functions ?S are precisely determined.However, such a prescription is rather a formal one, because the exact calculation (9) of the dissipative kernel constitutes, in general, an unresolvable problem.We shall now consider a question of how to perform this calculation approximately.
be characteristic intervals of decaying in time of the autocorrelation functions from the S-order set and all the rest functions from higher-order sets, and S (k) = S+1 (k)= S (k) be their ratio.We assume in advance that there exist dynamical processes in the system, corresponding to essentially di erent scales of time and, therefore, beginning from some number S, the ratio S must become su ciently small, i.e., S (k) 1 at arbitrary wavevectors.This assumption is justi ed provided that the S-order set (4) forms an almost complete slow set of dynamical variables.It is obvious that in this case the memory kernel (9), which is built on projected higher-order variables, decays in time faster than TCFs (5), i.e., Thus, on characteristic time scales of varying TCFs, the dissipative kernel can be considered as a -function in time space, ?S (k; ) ? S (k) ( ), with the weight ?S (k) ?S (k; != 0).In the frequency representation the relation ( 18) takes the form: ?S (k; !) ?S (k) (19) that represents the well-known Markovian approximation 2, 4].
The memory kernel ?S (k) can be found in terms of elements (11) of the frequency matrix and the basic wavevector-depended correlation time putting !!0 in the S-order continued fraction (13).Then we obtain at different orders of the description the following result: ? 1 (k) = 1= cor (k), ? 2 (k) = cor (k) 2 (k) and (n = 2; 3; : : :) It is worth mentioning that the frequency independence of the memory kernel in the S-order description does not concern the memory functions of lower orders which depend on frequency according to the recurrent relation (16).
And now we consider a more general formulation of the Markovian approximation in higher-order descriptions.Namely, as far as the ratio S (k) is small enough at a given value of S, it will remain small at higher orders, too, or will even decrease with increasing S. If this statement indeed takes place, then not only the frequency dependence of ?S (k; !) can be neglected, but also the values of ?S (k) will begin to be almost independent from the order S of the description.Then we can write that ?S+1 (k) ?S (k) at su ciently great values of S and the basic correlation time (20) can be excluded from our consideration.Taking into account the explicit relations (21) and letting ? 1 (k) = ? 2 (k) and ? 2 (k) = ?3 (k), . . ., ?2n?1 (k) = ?2n (k) and ?2n (k) = ?2n+1 (k), where n = 2; 3; : : :, we obtain for the correlation time (s) cor (k) in the s-th approximation: (1)   cor (k) = 1= p 2 (k), (2)   cor (k) = p 3 (k)= 2 (k), . . ., and Finally, substituting values (22) into expressions (21) for memory functions within the same order of the approximation, we obtain In such a way, in view of ( 13), ( 14) and ( 23) the frequency dependence ( 7) of the dielectric constant can be reproduced using static correlation functions (15) exclusively.

Generalized collective modes
In the preceding subsection it was shown how to obtain analytic results for dynamical quantities in the frequency representation.However, in the Markovian approximation, the equation ( 8) for TCFs can be solved analytically in time space as well.This equation can now be written as @ @t F M (k; t) = ?T(k)F M (k; t) ; (25) where the generalized evolution operator T(k) = ?(k) + ?(k) is determined by explicit expressions (10), ( 11) and ( 21) (or ( 23)) for (k) and ?(k), respectively, and F M indicates the matrix F of TCFs calculated in the Markovian approximation.Let X (k) be an eigenvector associated with the eigenvalue Z (k) of the T(k)matrix, i.e., S X =1 T (k)X (k) = Z (k)X (k) ; (26) where ; = 1; : : : ; S and T designate the elements of T. Then the solution to di erential equation ( 24) is of the form i.e., each element of the F M -matrix can be expressed as a sum of S Lorentzians which are connected with the generalized collective modes Z (k).The amplitudes Q are de ned in terms of eigenvectors uniquely, using the initial condition lim t!0 F M (k; t) = F(k).The result is where the matrix X ?1 is the inverse of X fX g.The component Q (k) describes a partial contribution of the mode Z (k) to the time correlation function f (k; t).Applying the Laplace transform to equation ( 26) yields the following result in the frequency space which in the particular case of = = 1 can be considered as an alternative representation of extended continued fractions (13) for g(k; !) f 11 (k; !) in the Markovian approximation (18).The result (26) obtained in the S-mode description allows one to evaluate the S S matrix of longitudinal and transverse TCFs (5).
From Eqs. (8) and (24) it can be easily shown that R 1 0 dtF M (k; t) = R 1 0 dtF(k; t) or, in other words, lim !!0 F M (k; !) = lim !!0 F(k; !) and the Markovian approximation leads directly to exact results in the low-frequency limit.Moreover, from the initial condition F M (k) = F(k) (i.e., f M (k) = f (k), = 0; 1; : : : ; S ?1) it follows that 34], if the S-mode approximation is used, time derivatives of the genuine g(k; t) and approximated g M (k; t) functions coincide at t = 0 up to the 2(S ?1)-th order.For this reason, in a limit of S ! 1 the Markovian approximation exactly reproduces analytical in time functions.Note, however, that due to the presence of "long-time tails" and other anomalies in dipolar systems, the time correlation functions g(k; t) may be nonanalytic over a speci c wavenumber range.In such a case, they are calculated only approximately, even if the hypothetical limit S ! 1 is applied.
According to the basic results ( 13), ( 14), ( 21) and ( 22), the knowledge of static correlation functions g 2s (k) ( 15) is necessary at s = 0 4 to investigate the frequency dependence of the dielectric permittivity and evaluate the generalized dipolar mode spectra up to the ve-variable description.There are several possibilities to de ne the static correlation functions.The lowest-order function (s = 0) presents the well-known Kirkwood factor g(k) which is connected with the pair distribution function 29] and, therefore, can be calculated using one or another approach of the equilibrium statistical mechanics.The Kirkwood factor of the second order (s = 1) has an analytical representation 26], g L;T 2 (k) = k B T 3 ( 2 J + k 2 m ).The higher-order functions (s = 2 4) are related to the four-particle distribution function, and it is not a simple task to predict them theoretically.Usually, they are considered as adjustable parameters.It is obvious, however, that in such a way these functions cannot be determined uniquely.Because of this, to avoid any additional uncertainties in the calculation of collective modes and to observe the convergence of continued fractions in a pure form, we shall evaluate g 2s (k) using the molecular dynamics (MD) method.Details of our computer experiment are similar to those reported earlier 15].
We note that it is necessary to distinguish the correlation functions obtained directly in simulations for nite samples, G s (k), from in nite-system functions, g s (k).As it was shown previously, additional transformations to obtain g(k) from G(k) are necessary, namely, g(k) = (1=G(k) + D(k)) ?1 where D(k) takes into account the details of simulations 15].Moreover, the lowest-order time correlation functions related to in nite and nite systems obey the equality 1 Performing the Taylor expansion of (29) over inverse frequencies at ! ! 1, it can be easily shown that higher-order static correlation functions of the in nite system can be de ned as follows: The nite-system functions G(k), G 2 (k) and G 4 (k) were calculated directly in the simulations by the de nition (15) using equation ( 1) and the explicit expressions i e ?ik r i ; for higher-order dynamical variables, where _ v i Lv i = ? 1 m P N j(j6 =i) @' ij =@r i 1 m f i and _ w i Lw i = ? 1 J P N j(j6 =i) i @' ij =@ i 1 J K i denote the translational and rotational accelerations, respectively.In view of very complicated structures for L 3 M(k) and L 4 M(k), the highest-order static correlation functions G 6 (k) and G 8 (k) were evaluated numerically in terms of two-and four-fold time derivatives of the function G 4 (k; t) at t = 0, i.e., G 6 (k) = ?@ 2 G 4 (k; t)=@t 2 j t=0 and G 8 (k) = @ 4 G 4 (k; t)=@t 4 j t=0 .The evaluation of G 6 (k) and G 8 (k) was carried out with the help of a special procedure to reduce numerical errors to a minimum.
The obtained in such a way longitudinal g L 2s (k) and transverse g T 2s (k) components of the in nite-system functions g 2s (k) (as well as G L;T 2s (k)) are displayed in g. 1 at s = 0 4 in units of ?2s , where = LJ (m= LJ ) 1=2 .The compo- nents g L;T 2s (k), as autocorrelation static functions, are positively de ned at arbitrary wavenumbers.In the limit of great wavevectors they can be calculated analytically, namely, lim k!1 g L;T 2s (k) = (?1) s @ 2s g G (k; t) .@t 2s t!0 , where g G (k; t) = 1 3 exp(?ak 2 t 2 ) denotes the limiting Gaussian transition of g L;T (k; t) at k ! 1 and a = k B T=2m 26].In particular, g L;T (k) ! 1 3 , g L;T 2 (k) ! 2 3 ak 2 , g L;T 4 (k) !4a 2 k 4 , g L;T 6 (k) !40a 3 k 6 and g L;T 8 (k) !560a 4 k 8 .It is interesting to note that higher-order functions differ from lower-order ones considerably.This indicates the existence of dynamical processes in the system which correspond to essentially di erent scales of time.
The results presented in gure 1 allow one to check immediately our assumption about the possibility of expressing the correlation time in terms of static correlation functions.The corresponding calculations (22) of the correlation time (s) cor (k) performed in di erent approximations (s = 1 4), as well as the exact values (20) obtained by the MD method, are presented in gures 2a and 2b for the cases of longitudinal and transverse uctuations, respectively.As we can see from the gures, already the four-variable approximation reproduces the values of cor (k) not only qualitatively, but even quantitatively over the whole region of wavenumbers.
The generalized dipolar mode spectra in two-, three-, four-and ve-variable descriptions are shown in gure 3.In the case of transverse uctuations (subsets (a) and (b) of the gure), we can clearly identify the di usive mode D 1 (k) which is well separated from all the rest of the modes over a wide wavevector range.This mode  converges rapidly to its genuine value with increasing the order of the approximation, so that it is reproduced quantitatively at small wavenumbers already within the two-variable description.As far as the ve-variable description is used, we can talk about the quantitative reproducing at intermediate and great wavevectors as well.The appearance of the mode D 1 (k) is caused by the di usive mechanism of dielectric relaxation in polar systems and only this mechanism is considered in the well-known Debye theory.Neglecting the dipole-dipole interactions, as it was done originally by Debye, one can nd that in the in nite-wavelength limit lim k!0 D 1 (k) = 2D R , where D R is a rotational di usion coe cient.Applying the extended hydrodynamic approach, Bagchi and Chandra improved this result and obtained D 1 (k) = (2D R +D T k 2 )(1+ N V c(k)) where D T is the translational di usion coe cient and c(k) denotes a component of the spherical harmonic expansion of the two-particle direct correlation function 4].The latter result is valid not only for dilute systems but also for dense gases and liquids.However, it can be used at small wavevectors exclusively.Our scheme gives the possibility to de ne D 1 (k) in terms of the Kirkwood factor and its higher-order components at arbitrary values of wavenumber.
It is worth remarking that the di usive mode, as the mode with the lowest real part, gives the main contribution to the TCFs and dielectric quantities in almost the whole domain of k{space, especially at small wavevectors.That is why the single-relaxation-time approximation for dipole-moment uctuations, e ?D 1 (k) t , which is used in the Debye theory, can be applied here.This approximation works well in the overdamped limit of large times t and small frequencies !, where the inertial motions of the liquid molecules are not important.To describe properly the region of intermediate values of t and !, it is necessary to consider higher-order modes.The next two propagating modes P 2 (k) iW 2 (k) arise additionally begin- ning from the three-order approach.The three-, as well as higher-order descriptions include explicitly the free-motion e ects in terms of g 2 (k) and correlations due to interactions via the torque-torque h P i;j K i K j i and force-force h P i;j f i f j i (at k 6 = 0) uctuations in terms of g 4 (k) (see Eq. ( 31)).Within the four-order approximation the secondary di usive mode D 3 (k) appears at small k.It splits into two new propagating modes, P 3 (k) iW 3 (k), with increasing the order of the approximation to ve, whereas the previous two propagating modes are moderately corrected.The four-and ve-mode descriptions consider higher-order kinetic processes which are important at very small times (very high frequencies) and large wavevector values.The transverse propagating modes describe an oscillation behaviour of time polarization uctuations.However, it is hard to observe these oscillations because they damp signi cantly during their time periods, i.e., P 2 (k) W 2 (k) and P 3 (k) W 3 (k).
For the longitudinal uctuations (subsets (c), (d) in g. 3) the pattern is quite di erent.Here, we can easily distinguish two propagating modes, P 1 (k) iW 1 (k).Contrary to the case of transverse uctuations, these modes exhibit a quasiparticle feature at not very large wavenumbers, where P 1 (k) W 1 (k).They should be associated with dipolarons 30] (analogous to the well-known plasmons in Coulomb systems), where W 1 (k) and P 1 (k) de ne the frequency and damping of the dipolaron excitations, respectively.As one can see from the gures, the dipolaron mode is predicted already within the two-order description which includes the correct inertial short time behaviour of polarization uctuations.If the dipole-dipole interactions are neglected, the dipolaron frequency can be de ned approximately as lim k!0 W 1 (k) = 4 N 2 V J 35:2 ?1 that is very close to the values 27.8, 33.0, 35.5 and 31.7 ?1 obtained by us at S = 2, 3, 4 and 5, respectively.The secondary oscillation process is reproduced by the next two complex-conjugated modes P 2 (k) iW 2 (k) at S = 4. Finally, in the ve-order description the pure di usive mode D 3 (k) appears additionally.
The inequalities P 1 (k) < W 1 (k); P 2 (k); D 3 (k) can be considered as a condition of existing the dipolaron oscillations.This condition is satis ed as far as k k LJ < 4. With increasing wavevector values the di usive processes begin to dominate, especially at k 7, where D 3 (k) is much less than all the other modes.This feature is visible in all the orders of the approximations as well.For example, in the fourorder description the propagating modes P 2 (k) iW 2 (k) are separated into the two pure di usive modes D 2 (k) and D 3 (k) within a small region near k 7, where the longitudinal component g L (k) of the Kirkwood factor has a sharp maximum (see gure 1).
In view of the behaviour of dipolar modes, the whole region of wavevectors can be split into several characteristic intervals.In the rst one the lowest-lying dipolar modes are well separated from the rest of the modes and this separation is observed as long as k < 2 (the so-called extended hydrodynamic regime).In this interval the Debye-like theory can be applied to transverse dipole-moment uctuations, whereas the longitudinal component of the dielectric permittivity can be predicted by two complex-conjugated dipolaron modes.In the second range of intermediate wavenumbers, 2 < k < 12, all the modes are mixed in a very complicated manner (especially in the case of longitudinal uctuations).To describe the dynamical behaviour of dielectric quantities in this range, involving additional higher-order modes is necessary, excepting the subinterval 6 < k < 8, where the longitudinal di usive mode dominates over all the rest of the modes, similar to the behaviour of transverse modes in the extended hydrodynamic regime.Finally, in the so-called free-motion regime, k > 12, all the modes tend to their own linear asymptotes to reproduce the Gaussian time shape g G (k; t) of dipole-moment uctuations.
Examples of the normalized time correlation functions L;T (k; t) g L;T (k; t)= g L;T (k) obtained in two-, three-and ve-mode descriptions are presented and compared with the MD data 15] in gure 4. As we can see from the gure, the transverse component T (k; t) exhibits an almost pure damped feature over a wide range of wavenumbers.In the case of longitudinal uctuations this is valid for intermediate and great wavevector values only.At small wavevectors the longitudinal dipole-moment uctuations are described by strong dipolaron oscillations with a slight damping.Such a behaviour of L;T (k; t) is completely in line with the predictions of the generalized dipolar modes approach.The transverse TCFs are reproduced satisfactorily even within the three-mode description.The longitudinal oscillations are described in this case as well, but only qualitatively.At the same time, the approximated and genuine TCFs begin to be indistinguishable with increasing the order of the description to ve.
A similar pattern to that presented for TCFs is observed for the wavevectorand frequency-dependent dielectric permittivity " L;T (k; !)=" 0 L;T (k; !)?i" 00 L;T (k; !).Our calculations, carried out in one-, two-, three-and ve-order approximations for the longitudinal " L (k; !) and transverse " T (k; !) components are shown in gures 5 and 6, respectively, in comparison with the MD data of paper 15].We note that in the in nite-wavelength limit lim k!0 " L;T (k; !) = "(!).It can be easily seen that within the Debye-like theory (S = 1) the dielectric permittivity can be well reproduced in the hydrodynamic limit (low frequencies and wavenumbers).With increasing wavevector and frequency values this theory fails especially in the case of longitudinal uctuations.In the three-mode approximation we can talk about a qualitative description.Finally, within the ve-variable approach the entire frequency dependence of the dielectric permittivity is described quantitatively at arbitrary wavevectors (deviations from the MD data do not exceed a few per cent).This merely means that the ve variables constitute an almost complete set of slow quantities and the Markovian approximation begins to be almost exact.Therefore, the extended continued fractions (13) converge rapidly with increasing the order of the approximation and the hypothesis of an abbreviated description is in excellent accord.

Conclusion
It has been established that dielectric relaxation in a dipolar uid can be successfully studied within the generalized mode method.The proposed approach can   be considered as an extension of the three-variable theory of Madden and Kivelson 2] to arbitrary numbers of dynamical variables.Our scheme for the computation of dipolar modes is presented in such a form that is very convenient for actual applications.In particular, all the necessary input quantities are, in fact, static Kirkwood factors of di erent orders.The static factors can be determined by either equilibrium theories or direct computer simulations.This has allowed us both to avoid any tting procedures and to evaluate the generalized dipolar-mode spectra of a Stockmayer uid over the whole scale of wavelengths within up to the ve-order description for the rst time.It has been shown on the basis of direct calculations that the ve-variable theory enables one to de ne quantitatively the dielectric permittivity of a dipolar uid at arbitrary wavevector and frequency values.

Figure 1 .
Figure 1.Transverse (T) and longitudinal (L) components of the s-order static Kirkwood factors (s = 0 4) for a Stockmayer uid at n = 0:822 and T = 1:147.The MD data for the nite system are shown as dashed curves.The in nite-system Kirkwood factors are plotted by solid curves.Note that the transverse functions, corresponding to nite and in nite systems, are practically indistinguishable, excepting the case s = 0.

Figure 2 .
Figure 2. Transverse (a) and longitudinal (b) components of the correlation time for the Stockmayer uid.The results in one-, two-, three-and four-order approximations are plotted by long-dashed, long-short-dashed, short-dashed and solid curves, respectively.The exact values are presented as circles.

Figure 3 .
Figure 3. Generalized dipolar mode spectra of the Stockmayer uid: transverse ((a), (b)) and longitudinal ((c), (d)) modes in four-((a), (c)) and ve-((b), (d)) order descriptions.The pure di usive modes, real and imaginary parts of propagating modes are marked by the symbols D, P and W, respectively.For the purpose of comparison the results in two-and three-mode approaches are shown in (a), (c) and (b), (d) by the thinnest curves.

Figure 4 .
Figure 4.The normalized time autocorrelation functions of the dipole-moment uctuations for the Stockmayer uid at some xed values of wavenumber, where k min = 2 =V 1=3 = 0:927= LJ .The MD data for longitudinal and transverse components are shown as circles and squares.The results of two-, three-and ve-mode descriptions are plotted by the corresponding long-, short-dashed and solid curves, respectively.

Figure 5 .
Figure 5.The frequency-dependence of the longitudinal dielectric permittivity for the Stockmayer uid at in nite ((a), (b)) and nite ((c){(h)) wavelengths.The MD data are shown by circles.The results obtained within one-, two-, threeand ve-mode descriptions are plotted by long-short-, long-, short-dashed and solid curves, respectively.

Figure 6 .
Figure 6.The frequency-dependence of the transverse dielectric permittivity for the Stockmayer uid at nite wavelengths.Notations are as for g. 5.