Molecular momentum density

The chemistry of momentum densities becomes more interesting when one goes over to molecules. There is an inherent stabilization when atoms bond together to form molecules. One of the first chemical interpretations of the EMD was presented by Coulson [11]. The simplest valence bond (VB) and molecular orbital (MO) wave functions for H2 molecule given in the following equations were used for this purpose ... 

Several review articles on the theoretical aspects of electron momentum densities of atoms and molecules were written in the 1970s by Benesch and Smith [9], Epstein [10,11], Mendelsohn and Smith [12], Epstein and Tanner [13], Lindner [14], and Kaijser and Smith [15]. Since that time (e,2e) spectroscopy and the momentum densities of Dyson orbitals have been reviewed very often [16-28]. However, to my knowledge, a review article on molecular electron momentum densities has not been written recently apart from one [29] devoted solely to the zero-momentum critical point. The purpose of this chapter is to survey what is known about the electron momentum density of atoms and molecules, and to provide an extensive, but not exhaustive, bibliography that should be sufficient to give a head start to a nonspecialist who wishes to enter the field. 

Values of the MacLaurin coefficients computed from good, self-consistent-field wavefunctions have been reported [355] for 125 linear molecules and molecular ions. Only type I and II momentum densities were found for these molecules, and they corresponded to negative and positive values of IIq(O), respectively. An analysis in terms of molecular orbital contributions was made, and periodic trends were examined [355]. The qualitative results of that work [355] are correct but recent, purely numerical, Hartree-Fock calculations [356] for 78 diatomic molecules have demonstrated that the highly regarded wavefunctions of Cade, Huo, and Wahl [357-359] are not accurate for IIo(O) and especially IIo(O). These problems can be traced to a lack of sufficiently diffuse functions in their large basis sets of Slater-type functions. 

Indices of molecular similarity and dissimilarity based on the electron momentum density have been found useful by Allan, Cooper and coworkers [388-393] and Ho et al. [394]. 

Bonds with r < dl < d[ become possible because of nuclear screening (increased bond order), which causes concentration of the bonding pair directly between the nuclei. The exclusion limit is reached at d = t and appears as a geometrical property of space. The distribution of molecular electron density is dictated by the local geometry of space-time. Model functions, such as VSEPR or minimum orbital angular momentum [65], that correctly describe this distribution, do so without dictating the result. The template is provided by the curvature of space-time which appears to be related to the three fundamental constants tt, t and e. 

Equation (2.16) consists of two contributions the molecular momentum flow tensor, it, and the convective momentum flow tensor, pvv. The term p8 represents the pressure effect, while the contribution t, for a Newtonian fluid, is related to the velocity gradient linearly through the viscosity. The convective momentum flow tensor pw contains the density and the products of the velocity components. A component of the combined momentum flow tensor of x-momentum across a surface normal to the x-direction is... 

Note that pip) = 0 whenever = (2n + 1) , for integer values of n. Consequently, there is a series of nodal planes perpendicular to the p -axis. Figure Ic shows the momentum density in the same plane as in Fig. lb, except that the view extends to larger values of p and the peak around p = 0 has been omitted. The planes which correspond to n = 0 can be seen in Fig. Ic at p = tc/Rab (i.e. Pj + 2.2 atomic units). These diffraction effects , with period IkIRab, contain the information about the molecular geometry. The cosine terms in Eqs. (14) and... 

It is seen in Eq. (17) that the effect of chemical bonding appears as the oscillation term cos (pR) except for the normalization factor. The momentum density vanishes if pR = (2n + l)ir, while it has relative maxima if pR = 2nn. Since the oscillation term is always unity in the momentum direction perpendicular to the bond axis, there is a greater probability of finding a given momentum in the perpendicular direction than in the parallel direction. The resultant molecular momentum distribution is an ellipsoid with its minor axis along the parallel axis, which should be compared to spherical distributions for atoms (see Fig. 16 for an example). 

Fig. 19.4. Surface plots of the electron momentum density of H2 illustrating a (3, — 3) maximum alp — Q. The left and right plots are in planes parallel and perpendicular to the molecular axis 0,pj and n p, py, 0),...

Cooper and Allan ° have used momentum density in several studies. A problem remains in obtaining the momentum space densities because most calculations are performed with position space wave functions. In a sense, working in momentum space is yet another way to reduce the overweighting of the core electron density. Most of the following discussions on, e.g., molecular alignment and quantum similarity indices, remain valid when we... 

Out of equilibrium there is no such rigorous principle. However, macroscopicaUy one can find a large variety of phenomenological equations for the time evolution which are based on macroscopic quantities alone, e.g., the diffusion equation, the heat transport equation, and the Navier-Stokes equations for hydrodynamics. A microscopic dynamical theory for the time evolution of slow variables such as the momentum density or the particle density with molecular spatial resolution is highly desirable. 

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... 

Daivis, R J., Travis, K. R, and Todd, B. D. 1996. A technique for the calculation of mass, energy, and momentum densities at planes in molecular dynamics simulations. J. Chem. Phys. 104 9651. 

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