Momentum density molecules

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. 

The outer most levels in C60 are due to rc orbitals . These are formed by 2p electrons which have their orbitals oriented along the radius of the molecule. The different environment inside and outside the spherical molecule causes the double-peaked structure in the momentum densities. In graphite the n band is formed by 2p orbitals oriented perpendicular to the sheets of carbon atoms. Using single-crystal graphite films we have a unique opportunity to study the effects of the orientation of these 2p orbitals in detail. 

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 ... 

C. Gatti and A. Famulari Interaction Energies and Densities. A Quantum Theory of Atom in Molecules insight on the Effect of Basis Set Superposition Error Removal , P.G. Mezey and B. Rohertson (Eds.), Understanding Chemical Reactivity Electron, Spin and Momentum Densities and Chemical Reactivity, Vol. 2, Kluwerhook series (1999). In press. 

THE MOMENTUM DENSITY PERSPECTIVE OE THE ELECTRONIC STRUCTURE OF ATOMS AND MOLECULES... 

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. 

Section III. Methods for obtaining momentum densities, both experimental and computational, are reviewed in Section IV. Only a sample of representative work on the electron momentum densities of atoms and molecules is summarized in Sections V and VI because the topic is now too vast for comprehensive coverage. Electron momentum densities in solids and other condensed phases are not considered at all. The literature on electron momentum spectroscopy and Dyson orbital momentum densities is not surveyed, either. Hartree atomic units are used throughout. 

For 5-state atoms, the momentum density is spherically symmetric and therefore n( p) = IIo(p). Nonspherical II( p) can occur in atomic states of higher angular momentum. However, these states are degenerate, and an ensemble average is required. This is equivalent to a spherical average for any one of the states in the degenerate manifold. The spherically averaged momentum density IIo(p) is also of interest for molecules because it is a quantity that can be obtained from gas-phase experiments. 

There are well-exploited connections between experimental phenomena and electron momentum densities. Inelastic scattering [167,181-184] of high-energy electrons, X rays, or y rays by electrons in a molecule allows us to measure the electron momentum density of the molecule. The observable is the intensity of the Compton scattering at wavelengths shifted, by a Doppler broadening-like... 

A vast number of directional Compton profiles have been measured for ionic and metallic solids, but none for free molecules. Nevertheless, several calculations of directional Compton profiles for molecules have been performed as another means of analyzing the momentum density. 

In binary (e,2e) or electron momentum spectroscopy, an incoming electron collides with a molecule and two electrons leave the molecule. The measured differential cross section is proportional to the spherically averaged momentum density of the pertinent Dyson orbital within the plane-wave impulse approximation. A Dyson orbital v[/ t is defined by... 

Subsequent work on graphical analysis of anisotropic momentum densities, directional Compton profiles, and their differences in diatomic molecules was reported by several groups, including Kaijser and Smith [195,316], Ramirez [317-319], Matcha, Pettit, Ramirez, and Mclntire [320-328], Leung and Brion [329,330], Simas et al. [331], Rozendaal and Baerends [332,333], Cooperand Allan [334], Anchell and Harriman [138], and Rerat et al. [335,336]. 

For most molecules, the small momentum expansion of the momentum density requires the full 3x3 Hessian matrix A of n( p) at p = 0. In Cartesian coordinates, this matrix has elements... 

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. 

It is apparent that only a trickle of work has been, and is currently being, done on momentum densities in comparison with the torrent of effort devoted to the position space electron density. Moreover, much of the early work on II( p) has suffered from an undue emphasis on linear molecules. Nevertheless, some useful insights into the electronic structure of molecules have been achieved by taking the electron momentum density viewpoint. The most recent phenomenal developments in computer hardware, quantum chemical methods and software for generating wavefunctions, and visualization software suggest that the time is ripe to mount a sustained effort to understand momentum densities from a chemical perspective. Readers of this chapter are urged to take part in this endeavor. 

The Momentum Density Perspective of the Electronic Structure of Atoms and Molecules By Ajit J. Thakkar... 

Thakkar and his group have completed many careful studies of intracules, extracules, Coulomb holes, and related topics.145 These position-space results are complemented by Thakkar s many studies of momentum densities and related quantities. He also has a long-standing interest in electron and X-ray scattering.146 His current interests include the relationship147 between the aromaticity of heterocyclic compounds and their polarizabilities, and the prediction of push-pull molecules that have a large nonlinear optical response and are thus candidates for materials to be used in optical computers. 

Equation (8.175) is a generalization of Ehrenfest s theorem (Ehrenfest 1927). This theorem relates the forces acting on a subsystem or atom in a molecule to the forces exerted on its surface and to the time derivative of the momentum density mJ(r). It constitutes the quantum analogue of Newton s equation of motion in classical mechanics expressed in terms of a vector current density and a stress tensor, both defined in real space. 

With increasing n, the peaks in the momentum density become more marked. This is as expected for delocalised bonding, since r and p are conjugate variables. In the limit n 00, p,[p) is only non-zero when Eq. (20) is satisfied. The diffraction term in Eq. (20) restricts the range of allowed momenta in the MOs much more than in those of smaller molecules, such as Ha-... 

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