Electron momentum density

In the ideal case being performed at X-ray energy transfers much higher than the characteristic energies of the scattering system, the impulse approximation [14] is applicable. In this case, the dynamical structure factor is directly connected with the electron momentum density p(p) ... 

Figure 1. The electron momentum density for atomic hydrogen measured by EMS for the indicated energies compared with the square of Schrodinger wave function (solid curve) [4].

Three-dimensional reconstruction of electron momentum densities and occupation number densities of Cu and CuAl alloys... 

The reconstruction of the electron momentum densities and the occupation number functions of Cu and Cuq.953A10 047 could not produce results on an equal profound base as those based on the results of Li and LiMg reconstructions. This would need approximately 100 times the number of counts per spectrum which was not achieved. 

Mijnarends, P.E., (1979) Electron momentum densities in metals and alloys. In Positrons in Solids, (Ed.) Hautojarvi, P., Springer. 

Electronic structure is often visualized with the help of the electron density p(r), which tells us where the electrons are likely to be found. A different perspective of electronic structure is provided by the electron momentum density IT(p) because II( ) Ap is proportional to the probability of finding an electron with... 

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. 

We restrict ourselves to the clamped-nucleus or Born-Oppenheimer approximation [30,31] because essentially all the work done to date on electron momentum densities has relied on it. Therefore we focus on purely electronic wavefunctions and the electron densities that they lead to. 

B( ) is variously called the reciprocal form factor, the p-space form factor, and the internally folded density. B(s) is the basis of a method for reconstructing momentum densities from experimental data [145,146], and it is useful for the r-space analysis of Compton profiles [147-151]. The B(s) function probably first arose in an examination of the connection between form factors and the electron momentum density [129]. The B f) function has been rediscovered by Howard et al. [152]. 

The principle of microreversibility [154,155] applies to all bound states corresponding to real Hamiltonian operators. Hence the A-electron momentum density before and after time reversal are related by... 

Eq. (5.34). However, it is possible to construct approximate wavefunctions that lead to electron momentum densities that do not have inversion symmetry. Within the Born-Oppenheimer approximation, the total electronic system must be at rest the at-rest condition... 

Within the Born-Oppenheimer approximation, the nuclei are at rest and have zero momentum. So the electron momentum density is an intrinsically one-center function that can be expressed usefully in spherical polar coordinates and expanded as follows [162,163]... 

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... 

Figure 5.5. Types of electron momentum densities no(p) >n atoms. Solid lines are used for the total density, whereas dotted lines and crosses indicate the contribution from the outermost s and p orbitals, respectively. Top left a type I density for the potassium atom. Top right a typical type II density for the argon atom. Bottom left a typical type III density for the silver atom. Bottom right a closeup of rio(p) for the silver atom showing the minimum and secondary maximum. Adapted from Thakkar [29].

Obviously, graphical techniques are equally important in the study of electron momentum densities. Coulson [2,4] made the pioneering effort in this direction. Other early work was published by Henneker and Cade [305] Epstein and Lipscomb [306-308] Kaijser, Lindner, and coworkers [309-311] Tanner and Epstein [312] and Tawil and Langhoff [313,314]. A synthesis of these studies was made by Epstein and Tanner [13], who abstracted some principles that they hoped would be generally applicable to chemical bonding. One of their abstractions pinned down an observation about the anisotropy of n(p) that had been made in several of the earlier studies. Epstein and Tanner called it the bond directional principle, and they stated it as follows [13] ... 

Although the principle has turned out to have many exceptions [315], as discussed later, it has served as an inspiration for many researchers interested in uncovering the links between the reorganization of electron momentum densities and chemical bonding. 

Figure 5.6. Surface plots of the electron momentum density of N2 illustrating a (3, —3) maximum at — 0. Left Right n(/)j,/)y,0). Adapted from Thakkar [29,351].

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]. 

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. 

In summary, structure calculations can obtain 1 or 2% agreement with accurate optical data. A broader perspective is given in chapter 11 by electron momentum spectroscopy. Hartree—Fock calculations agree with one-electron momentum densities within experimental error, but configuration-interaction calculations agree only qualitatively with detailed data on correlations. 

M. MARANGOLO, J. MOSCOVICI, G. LOUPIAS, S. RABII, S.C. ERWIN, C. HEROLD, J.F. MARECHE and P. LAGRANGE, Experimental and theoretical study of the electron momentum density of KeCeo and comparison to pristine Ceo- Phys. Rev., B58, 7593 (1998). 

The lowest 24 bands were expanded into 531 PWs, cut off at iGl — 7.3/ao, with a norm in equation (3) above 0.85 Vuc for the A -shell electrons (excitations from which were excluded in the subsequent calculations of sqqi and dE/dx anyway), and a norm above 0.98Vuc for the remaining 22 bands. The first result is the all-electron momentum density (EMD)... 

Fig. 23. Perspective plots of electron momentum densities for the

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