Momentum density properties

Spin densities determine many properties of radical species, and have an important effect on the chemical reactivity within the family of the most reactive substances containing free radicals. Momentum densities represent an alternative description of a microscopic many-particle system with emphasis placed on aspects different from those in the more conventional position space particle density model. In particular, momentum densities provide a description of molecules that, in some sense, turns the usual position space electron density model inside out , by reversing the relative emphasis of the peripheral and core regions of atomic neighborhoods. 

Such an approach is conceptually different from the continuum description of momentum transport in a fluid in terms of the NS equations. It can be demonstrated, however, that, with a proper choice of the lattice (viz. its symmetry properties), with the collision rules, and with the proper redistribution of particle mass over the (discrete) velocity directions, the NS equations are obeyed at least in the incompressible limit. It is all about translating the above characteristic LB features into the physical concepts momentum, density, and viscosity. The collision rules can be translated into the common variable viscosity, since colliding particles lead to viscous behavior indeed. The reader interested in more details is referred to Succi (2001). 

In addition to this, average properties like (r > or (/> ) play a special role in the formulation of bounds or approximations to different properties like the kinetic energy [4,5], the average of the radial and momentum densities [6,7] and p(0) itself [8,9,10] they also are the basic information required for the application of bounds to the radial electron density p(r), the momentum one density y(p), the form factor and related functions [11,12,13], Moreover they are required as input in some applications of the Maximum-entropy principle to modelize the electron radial and momentum densities [14,15],... 

D. Connections with Other Functions Properties of the Momentum Density... 

Density functional theory purists are apt to argue that the Hohenberg-Kohn theorem [1] ensures that the ground-state electron density p(r) determines all the properties of the ground state. In particular, the electron momenmm density n( ) is determined by the electron density. Although this is true in principle, there is no known direct route from p to IT. Thus, in practice, the electron density and momentum density offer complementary approaches to a qualitative understanding of electronic structure. 

Since momentum densities are unfamiliar to many. Section II outlines the connection between the position and momentum space representations of wavefunctions and reduced-density matrices, and the connections among one-electron density matrices, densities, and other functions such as the reciprocal form factor. General properties of momentum densities, including symmetry, expansion methods, asymptotic behavior, and moments, are described in... 

Fokker-Planck(F-P) equation for the distribution functional /[n(r),(] and discuss genered properties of the TD-DFT. By combining a number-conservation law, the DFT and theory of Brownian motion, we derived the following equations for the density n(r,t) and the momentum density g(r,t) ... 

The definition and properties of B(r) may be summarized as follows. For simplicity, we treat the spinless one-electron wave function assuming the independent-particle model or the natural orbital expansion (Lowdin, 1955 Benesch and Smith, 1971). Based on the three-dimensional momentum density p(p),... 

This article provides an introduction to the momentum perspective of the electronic structure of atoms and molecules. After an explanation of the genesis of momentum-space wave functions, relationships among one-electron position and momentum densities, density matrices, and form factors are traced. General properties of the momentum density are highlighted and contrasted with properties of the number (or charge) density. An outline is given of the experimental measurement of momentum densities and their computation. Several illustrative computations of momentum-space properties are summarized. 

Finally, we consider density functional theory (DFT) computations of p-space properties. A naive way of calculating p-space properties is to use the Kohn-Sham orbitals obtained from a DFT computation to form a one-electron, r-space density matrix Fourier transform / according to Eq. (14), and proceed further. This approach is incorrect because the Kohn-Sham density matrix F is not the true one and, in fact, corresponds to a fictitious non-interacting system with the same p(r) as the true system. On the other hand, Hamel and coworkers [112] have shown that if the exact Kohn-Sham exchange potential is used, then the spherically averaged momentum densities of the Kohn-Sham orbitals should be very close to those of the Hartree-Fock orbitals. Of course, in practical computations the exact Kohn-Sham exchange potential is not used since it is generally not known. 

What is needed for a correct computation of momentum-space properties from DPT is an accurate functional for approximating the exact first-order reduced density matrix r f f ), or failing that, good functionals for each of the p-space properties of interest. Of course, a sufficiently good functional for (p ) would obviate the necessity of using Kohn-Sham orbitals and enable the formulation of an orbital-free DFT. Unfortunately, a kinetic energy functional sufficiently accurate for chemical purposes remains an elusive goal [118,119]. 

The foregoing discussion was restricted to a consideration of a single variable, A. Many circumstances arise in which we will be interested in the time evolution of many coupled variables. For example, in hydrodynamics the mass density, momentum density, and energy density are coupled. It is possible to extend the previous analysis to the case of many variables Ai...Am] These properties can be represented by the col-... 

One-electron Properties Band Structure, Density of States, Electron Momentum Density... 

Spontaneous thermal fluctuations of the density, p r,t), the momentum density, g(r,t), and the energy density, e(r,t), are dynamically coupled, and an analysis of their dynamic correlations in the limit of small wave numbers and frequencies can be used to measure a fluid s transport coefficients. In particular, because it is easily measured in dynamic light scattering. X-ray, and neutron scattering experiments, the Fourier transform of the density-density correlation function - the dynamics structure factor - is one of the most widely used vehicles for probing the dynamic and transport properties of liquids [56]. 

To extract infomiation from the wavefimction about properties other than the probability density, additional postulates are needed. All of these rely upon the mathematical concepts of operators, eigenvalues and eigenfiinctions. An extensive discussion of these important elements of the fomialism of quantum mechanics is precluded by space limitations. For fiirther details, the reader is referred to the reading list supplied at the end of this chapter. In quantum mechanics, the classical notions of position, momentum, energy etc are replaced by mathematical operators that act upon the wavefunction to provide infomiation about the system. The third postulate relates to certain properties of these operators ... 

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