Debye time assembly

Here the summation of charges times position vectors is replaced by the integral over the total wavefunction T (the square of the wavefunction is a measure of charge) of the dipole moment operator (the summation over all electrons of the product of an electronic charge and the position vectors of the electrons). To perform an ab initio calculation of the dipole moment of a molecule we want an expression for the moment in terms of the basis functions r/j, their coefficients c, and the geometry (for a molecule of specified charge and multiplicity these are the only variables in an ab initio calculation). The Hartree-Fock total wavefunction T is composed of those component orbitals i// which are occupied, assembled into a Slater determinant (Section 5.2.3.1), and the i// s are composed of basis functions and their coefficients (Sections 5.3). Equation (5.206), with the inclusion of the contribution of the nuclei to the dipole moment, leads to the dipole moment in Debyes as (ref. [lg], p. 41)... 

Chapter 8 by W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, entitled Fractional Rotational Diffusion and Anomalous Dielectric Relaxation in Dipole Systems, provides an introduction to the theory of fractional rotational Brownian motion and microscopic models for dielectric relaxation in disordered systems. The authors indicate how anomalous relaxation has its origins in anomalous diffusion and that a physical explanation of anomalous diffusion may be given via the continuous time random walk model. It is demonstrated how this model may be used to justify the fractional diffusion equation. In particular, the Debye theory of dielectric relaxation of an assembly of polar molecules is reformulated using a fractional noninertial Fokker-Planck equation for the purpose of extending that theory to explain anomalous dielectric relaxation. Thus, the authors show how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended via the continuous-time random walk to yield the empirical Cole-Cole, Cole-Davidson, and Havriliak-Negami equations of anomalous dielectric relaxation from a microscopic model based on a... 

The next issue to concern us will be anomalous relaxation in which the smearing out of a relaxation spectrum (i.e., the deviation of complex susceptibility from its Debye form) is associated with the concept of a relaxation time distribution. As is well known, this concept implies an assembly of dipoles with a continuous distribution of relaxation times of Eq. (379). 

Thus the Debye equation [Eq. (1)] may be satisfactorily explained in terms of the thermal fluctuations of an assembly of dipoles embedded in a heat bath giving rise to rotational Brownian motion described by the Fokker-Planck or Langevin equations. The advantage of a formulation in terms of the Brownian motion is that the kinetic equations of that theory may be used to extend the Debye calculation to more complicated situations [8] involving the inertial effects of the molecules and interactions between the molecules. Moreover, the microscopic mechanisms underlying the Debye behavior may be clearly understood in terms of the diffusion limit of a discrete time random walk on the surface of the unit sphere. 

Any assembly of molecules initially oriented along some direction, say uo, behaves such that each molecule follows a different trajectory—made up of small steps—on the surface of the unit sphere. Initially this assembly is represented by a cloud of points which is very intense in the direction uo, but as time progresses, molecules reorient and the cloud spreads out, finally covering the unit sphere uniformly. It is the basic assumption of the Debye theory that the cloud of points simply diffuses on the surface of the unit sphere. We begin, therefore, with the diffusion equation, and ask the question how particles diffuse if they are constrained to remain on the sphere of unit radius The equation that governs this motion is the diffusion equation... 

Tbe form of the response function may be derived from molecular models or from other considerations. The simplest Ametion. introduced by Debye, is r) 1 exp(—(frX where the parameter t is referred to as tbe relaxation time of the assembly of dipoles. 

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