Moment expansion

The rotationally resolved differential cross-section are subsequently smooth-ened by the moments expansion (M) in cosines [77-79] ... 

Differential cross-sections for particular final rotational states (f) of a particular vibrational state (v ) are usually smoothened by the moment expansion (M) in cosine functions mentioned in Eq, (38). Rotational state distributions for the final vibrational state v = 0 and 1 are presented in [88]. In each case, with or without GP results are shown. The peak position of the rotational state distribution for v = 0 is slightly left shifted due to the GP effect, on the contrary for v = 1, these peaks are at the same position. But both these figures clearly indicate that the absolute numbers in each case (with or without GP) are different. 

It is possible to perform a systematic decoupling of this moment expansion using the superoperator formalism (34,35). An infinite dimensional operator vector space defined by a basis of field operators Xj which supports the scalar product (or metric)... 

The reason for file choice of this specific algdira can be justified by the correspondence between the Heisenberg and SchrOdinger representations (129). Applying this algebra to the moment expansion yields the px gators in the resolvent form,... 

The second method is more elegant, because it only involves the numerical computation of moments (cf. Sect. 1.3) of the smeared CLDg2 (rn) followed by moment arithmetics [200], The first step is the computation of the Mellin transform102 of the analytical function gc (rn) which we have selected to describe the needle diameter shape. This is readily accomplished by Mathematica [205], Because the Mellin transform is just a generalized moment expansion, we retrieve for the moments of the normalized chord distribution of the unit-disc103... 

Nadler, W. Schulten, K., Generalized moment expansion for Brownian relaxation processes, J. Chem. Phys. 1985, 82, 151-160... 

When a large nurWber of spins interact, the numerous lines of the splitting pattern overlap and merge into a continuous lineshape whose functional form cannot be obtained from theory. Yet this shape contains useful geometrical information. One means of representing this dipolar lineshape g(w) is by a moment expansion... 

To our knowledge, the first paper devoted to obtaining characteristic time scales of different observables governed by the Fokker-Planck equation in systems having steady states was written by Nadler and Schulten [30]. Their approach is based on the generalized moment expansion of observables and, thus, called the generalized moment approximation (GMA). 

Like the first Hohenberg-Kohn theorem, the preceding theorems are existence theorems they say that the shape function is enough but they do not provide any guidance for evaluating properties based on the shape function alone. Once one knows that shape functionals exist, however, there are systematic ways to construct them using, for example, the moment expansion technique [48-51]. For atomic... 

J. Cioslowski, Connected-moments expansion—a new tool for quantum many-body theory. Phys. Rev. Lett. 58, 83 (1987). 

If the flow rate is increased so that Peclet number Pe l, then there is a timescale at which transversal molecular diffusion smears the contact discontinuity into a plug. In Taylor (1993), Taylor found an effective long-time axial diffusivity proportional to the square of the transversal Peclet number and occurring in addition to the molecular diffusivity. After this pioneering work of Taylor, a vast literature on the subject developed, with over 2000 citations to date. The most notable references are the article (Aris, 1956) by Aris, where Taylor s intuitive approach was explained through moments expansion and the lecture notes (Caflisch and Rubinstein, 1984), where a probabilistic justification of Taylor s dispersion is given. In addition to these results, addressing the tube flow with a dominant Peclet number and in the absence of chemical reactions, there is... 

Table 11.2 AGep values (kcal moD ) for trans 1.2-dichloroethane as a function of the truncation point in the multipole moment expansion"...

However, Eq. (2.21) is not very convenient in the context of intramolecular electrostatic interactions. In a protein, for instance, how can one derive the electrostatic interactions between spatially adjacent amide groups (which have large local electrical moments) In principle, one could attempt to define moment expansions for functional groups that recur with high frequency in molecules, but such an approach poses several difficulties. First, there is no good experimental way in which to measure (or even define) such local moments, making parameterization difficult at best. Furthermore, such an approach would be computationally quite intensive, as evaluation of the moment potentials is tedious. Finally, the convergence of Eq. (2.20) at short distances can be quite slow with respect to the point of truncation in the electrical moments. 

Note that G(t) is also the reduced autocorrelation function.10 The above series, however, converges very slowly. In practice it is therefore necessary to use a modified moment expansion that converges more rapidly. In order to obtain an explicit expression for G(t), we consider a single crystal containing only one species of nuclei with spin such that the nuclei are located at the points of a lattice. We further assume that the only interaction present is the nuclear dipole-dipole interaction. It can be shown that10... 

THE MULTIPOLE MOMENT EXPANSION SOLVENT CONTINUUM MODEL A BRIEF REVIEW... 

Abstract The multipole moment expansion solvent continuum model is being developed by our... 

The Multipole Moment Expansion Solvent Continuum Model... 

Thanks to efficient recurrence formulae, multipole moments and multipole moment derivatives can be calculated at very high order with a low computational cost. The calculation of reaction field factors, however, may become computationally expensive at high order due to the increasing number of linear equations to be solved. Thus, in practice, the multipole moment expansion is cut off at a maximum value of f (/max), usually taken around 6. In order to get an order of magnitude of the error introduced by the truncation, let us consider Kirkwood s equations [5] for the free energy of a charge distribution of charges q, and r, in a spherical cavity of radius a ... 

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