When radiation is Compton-scattered the emerging beam is Doppler broadened because of the motion of the target electrons. An analysis of this broadened lineshape the Crompton profile, provides detailed information about the electron momentum distribution in the scatter. The technique is particularly sensitive to the behaviour of the slower moving outer electrons involved in bonding in condensed matter and can be used to test their quantum-mechanical description. The review begins with a brief survey of the historical development of the subject to within a decade of the present. The behaviour of quantum systems from a momentum viewpoint, is explained and the conditions under which the scattering experiment can be interpreted directly in terms of electron momentum density are discussed. The experimental techniques with gamma -rays, X-rays and electron beams are compared. Finally, recent results on insulators and conductors are surveyed and the extent to which they challenge conventional assumptions of band theory is critically reviewed.

https://iopscience.iop.org/article/10.1088/0034-4885/48/4/001/pdf

Compton scattering and electron momentum determination

Unique information about the motion and correlation of valence electrons in atoms, molecules and their ions is obtained from electron-impact ionization reactions near the Bethe ridge at total energies of the order of 1000 eV or higher. This is electron momentum spectroscopy. The history, theory and practice of the field are discussed and its value is shown by numerous examples.

https://iopscience.iop.org/article/10.1088/0034-4885/54/6/001/pdf

Electron momentum spectroscopy of atoms and molecules

Most of our knowledge of the electronic structure of atoms and molecules is derived from excitation energies and transition probabilities. These observable quantities are related to the electronic wave functions by integrals over unmeasured variables. Another observable more directly related to the wave function than energy or transition probability is the single-electron momentum density, the probability that an electron in a well-defined orbital has a given value of momentum. Over the last twenty years a technique has been developed for measuring momentum densities in atoms and molecules. The technique, ([ital e],2[ital e]) spectroscopy, is based on electron-impact ionization with complete determination of the momenta of both incoming and outgoing electrons. The conditions necessary to extract momentum-density information from the ionization experiments are examined and related to general theories of electron scattering. Different experimental arrangements are reviewed and momentum-density results from selected examples are discussed.

(e,2e) spectroscopy

The electron localizability indicator in momentum space is proposed as a functional of the same-spin momentum pair density This functional yields a discrete distribution of values, which are proportional to the charge needed to form a fixed very small fraction of a same-spin electron pair It resolves all atomic shells for the examined atoms (Li–Kr) with reasonable occupation numbers, especially in the valence region © 2006 Wiley Periodicals, Inc Int J Quantum Chem, 2006

Atomic Shells From the Electron Localizability in Momentum Space

CRYSCOR is a periodic post-Hartree–Fock program based on local functions in direct space, i.e., Wannier functions and projected atomic orbitals. It uses atom centered Gaussians as basis functions. The Hartree–Fock reference, as well as symmetry information, is provided by the CRYSTAL program. CRYSCOR presently features an efficient and parallel implementation of periodic local second order Moller–Plesset perturbation theory (MP2), which allows us to study 1D-, 2D- and 3D-periodic systems beyond 1000 basis functions per unit cell. Apart from the correlation energy also the MP2 density matrix, and from that the Compton profile, are available. Very recently, a new module for calculating excitonic band gaps at the uncorrelated Configuration-Interaction-Singles (CIS) level has been added. Other advancements include new extrapolation techniques for calculating surface adsorption on semi-infinite solids. In this paper the diverse features and recent advances of the present CRYSCOR version are illustrated by exemplary applications to various systems: the adsorption of an argon monolayer on the MgO (100) surface, the rolling energy of a boron nitride nanoscroll, the relative stability of different aluminosilicates, the inclusion energy of methane in methane–ice-clathrates, and the effect of electron correlation on charge and momentum density of α-quartz. Furthermore, we present some first tentative CIS results for excitonic band gaps of simple 3D-crystals, and their dependence on the diffuseness of the basis set.

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CRYSCOR: a program for the post-Hartree–Fock treatment of periodic systems

Recent work has shown that the one-dimensional projection of the electron momentum density, the Compton profile, can be usefully interpreted as a position space quantity. This has led to an examination of B(r), the Fourier transform of the momentum density. A number of theoretical results relating to this new observable are given. The wave-mechanical representation with (natural) orbitals is employed, and this forms the basis for the subsequent analysis of B(r). The relationship of B(r) to overlap integrals and more generally to other electron density functions is considered. Atomic wavefunctions for krypton are used to illustrate the potential of this new approach to the analysis of momentum density data. General expressions are derived for atoms and molecules, and the radial and angular dependence of B(r) for various orbitals is displayed. The possibility of extracting accurate bond lengths from B(r) is assessed, and an example is given using some recent theoretical data for the fluorine molecule.

W. Weyrich

Papers

Fourier analysis of the Compton profile: Atoms and molecules

Observation of ionic deformation and bonding from compton profiles