Momentum Density Models

The kinetic energy is most easily computed in momentum space, where the classical expression 

Modern Theoretical Chemistry Electronic Structure and Reactivity 

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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. 

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],... 

Duncanson and Coulson [242,243] carried out early work on atoms. Since then, the momentum densities of aU the atoms in the periodic table have been studied within the framework of the Hartree-Fock model, and for some smaller atoms with electron-correlated wavefunctions. There have been several tabulations of Jo q), and asymptotic expansion coefficients for atoms [187,244—251] with Hartree-Fock-Roothaan wavefunctions. These tables have been superseded by purely numerical Hartree-Fock calculations that do not depend on basis sets [232,235,252,253]. There have also been several reports of electron-correlated calculations of momentum densities, Compton profiles, and momentum moments for He [236,240,254-257], Li [197,237,240,258], Be [238,240,258, 259], B through F [240,258,260], Ne [239,240,258,261], and Na through Ar [258]. Schmider et al. [262] studied the spin momentum density in the lithium atom. A review of Mendelsohn and Smith [12] remains a good source of information on comparison of the Compton profiles of the rare-gas atoms with experiment, and on relativistic effects. 

Let us briefly review the essential ingredients to this procedure (for more details of the method see [30] and for our model [42]). For a given system the hydrodynamic variables can be split up into two categories variables reflecting conserved quantities (e.g., the linear momentum density, the mass density, etc.) and variables due to spontaneously broken continuous symmetries (e.g., the nematic director or the layer displacements of the smectic layers). Additionally, in some cases non-hydrodynamic variables (e.g., the strength of the order parameter [48]) can show slow dynamics which can be described within this framework (see, e.g., [30,47]). 

Remarkably, no additional models are required to obtain consistent convection of the fluid-phase momentum density. Eor consistency, we can require that [Uf p = Uf so that the final term on the right-hand side of Eq. (4.99) is null however, due to conservation of momentum the final term will be cancelled out due to a contribution from (Apf2>i, as will be shown next. 

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),... 

Another manifestation of the reciprocity of densities in r- and p-space is provided by Fig. 19.2. It shows the radial electron number density D r) = Aiir pir) and radial momentum density /(p) = Aitp nip) for the ground state of the beryllium atom calculated within the Hartree-Fock model in which the Be ground state has a ls 2s configuration. Both densities show a peak arising from the Is core electrons and another from the 2s valence electrons. However, the origin of the peaks is reversed. The sharp,... 

Step 3. The next problem is to choose (by chance) the momentum of the nucleon (in the target) which has been struck. The uniform density model of the nucleus has a concomitant Fermi momentum sphere whose size is fixed by the number of particles in the nucleus and the radius of the nucleus. The momentum intervals must be equally subdivided taking into account the probability (i.e. cross-section) for a nucleon-nucleon collision for particles with various momenta with the incoming particle. 

Abstract A novel lattice-gas approach has been developed to model the effect of molecular interactions on dynamic interfacial structure and flows of liquid-vapor and liquid-liquid systems in microcapillaries, Within a mean-field approximation, discrete time evolution of species and momentum densities consists of alternating convective and diffusive steps subject to local conservation laws. Stick boundary conditions imposed during the convective step cause momentum transfer to lattice particles in contact... 

A comparison has been made of detailed CFD predictions, which have included all the aerodynamic processes involved in falling sprays, and a simple momentum conservation model which ignores the induced shear flow on the spray periphery. This has shown that for the scenarios considered here it is adequate to use the latter, simpler treatment, which is described in Annex 1. Typical results obtained using the simple momentum conservation model are shown in Figure 16. In overfilling incidents the mass flux density is likely to be in the range 1 to 10 kg/mVs. This corresponds to maximum droplet velocities of 10-13 m/s and vapour velocities of 4-6 m/s. 

In fact (98) is model independent to order u, since only the linear term in u c, contributes. To close the hydrodynamic equations for the mass and momentum densities [(74) and (75)] we need expressions for the pre-collision and post-collision momentum fluxes, and n . From (76) we can obtain an expression for in terms of the velocity gradient. 

We begin by describing the HPP model, which satisfies all of the above requirements except for the isotropy of the momentum flux density tensor. As we shall, however, this early model nonetheless has some very interesting and suggestive properties, despite not being able to reproduce Navier-Stokes-like behavior exactly. 

Experimental data on nitrogen obtained from spin-lattice relaxation time (Ti) in [71] also show that tj is monotonically reduced with condensation. Furthermore, when a gas turns into a liquid or when a liquid changes to the solid state, no breaks occur (Fig. 1.17). The change in density within the temperature interval under analysis is also shown in Fig. 1.17 for comparison. It cannot be ruled out that condensation of the medium results in increase in rotational relaxation rate primarily due to decrease in free volume. In the rigid sphere model used in [72] for nitrogen, this phenomenon is taken into account by introducing the factor g(ri) into the angular momentum relaxation rate... 

VOF or level-set models are used for stratified flows where the phases are separated and one objective is to calculate the location of the interface. In these models, the momentum equations are solved for the separated phases and only at the interface are additional models used. Additional variables, such as the volume fraction of each phase, are used to identify the phases. The simplest model uses a weight average of the viscosity and density in the computational cells that are shared between the phases. Very fine resolution is, however, required for systems when surface tension is important, since an accurate estimation of the curvature of the interface is required to calculate the normal force arising from the surface tension. Usually, VOF models simulate the surface position accurately, but the space resolution is not sufficient to simulate mass transfer in liquids. 

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