One-electron momentum density

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. 

What does Fourier transformation of the momentum density yield This question has been considered [29] in some generality and detail. Here we merely summarize the outcome for the one-electron momentum density [28,29]. Consider the Fourier transform, or characteristic function in the terminology of probability theory, of 7T(p) ... 

The one-electron momentum density for bound states of atoms and molecules always has inversion symmetry as a consequence of time-reversal symmetry or the principle of microreversibility [42,43] ... 

In the Bom-Oppenheimer approximation, the nuclei are fixed and have zero momenta. So the momentum density IT(p) is an intrinsically one-centered function whereas p(r) is a multi-centered function. Thus one-center expansions in spherical harmonics work well for one-electron momentum densities [51-53]. The leading term of such an expansion is the spherically averaged momenmm density IIq p) defined by... 

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. 

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]... 

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] ... 

Typical electron momentum densities with (3, — 1) and (3, +1) saddle points at zero momentum are found in MgO and acetylene (HCCH), respectively. A momentum density with a zero momentum (3, — 1) critical point is shown for MgO in Fig. 19.6. In the vertical plane of symmetry /T(p, 0,pj has the structure of two hills separated by a ridge or col, and one sees two local (and global) maxima located symmetrically along the p axis. The plot in the horizontal symmetry plane has the structure of a hill. 

As the study of the electron momentum density and the Fermi surface by the positron annihilation techniques is rather indirect, it is important to perform the experimental investigations in close relation with a strong computational approach. The first valuable quantity to estimate is the positron density distribution in the lattice unit cell it will indicate which electronic states might be observed experimentally. One has then to evaluate the electron momentum density as seen by the positron. From this, the... 

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

Term wavefunctions describe the behaviour of several electrons in a free ion coupled together by the electrostatic Coulomb interactions. The angular parts of term wavefunctions are determined by the theory of angular momentum as are the angular parts of one-electron wavefunctions. In particular, the angular distributions of the electron densities of many-electron wavefunctions are intimately related to those for orbitals with the same orbital angular momentum quantum number that is. 

Secondly, correlations in the initial state can lead to experimental orbital momentum densities significantly different from the calculated Hartree-Fock ones. Figure 3 shows such a case for the outermost orbital of water, showing how electron-electron correlations enhance the density at low momentum. Since low momentum components correspond in the main to large r components in coordinate space, the importance of correlations to the chemically interesting long range part of the wave function is evident. 

Several experimental techniques such as Compton scattering, positron annihilation, angular correlation, etc., are used for measuring momentum densities. One of the most popular techniques involved in measuring momentum densities is termed as electron momentum spectroscopy (EMS) [29]. This involves directing an electron beam at the surface of the metal under study. Hence EMS techniques fall under what is classified as coincidence spectroscopy. 

We have recently introduced the Wigner intracule (2), a two-electron phase-space distribution. The Wigner intracule, W ( , v), is related to the probability of finding two electrons separated by a distance u and moving with relative momentum v. This reduced function provides a means to interpret the complexity of the wavefunction without removing all of the explicit multi-body information contained therein, as is the case in the one-electron density. 

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],... 

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... 

The electron and momentum densities are just marginal probability functions of the density matrix in the Wigner representation even though the latter, by the Heisenberg uncertainty principle, cannot be and is not a true joint position-momentum probability density. However, it is possible to project the Wigner density matrix onto a set of physically realizable states that optimally fulfill the uncertainty condition. One such representation is the Husimi function [122,133-135]. This seductive line of thought takes us too far away from the focus of this... 

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