V. One‐Electron Density Functions and Many‐Centered Finite Multipole Expansions

Abstract

The one‐electron density function, ρ(r), (in principle deduced from elastically scattered X‐ray intensities) is the probability distribution function of an electron, averaged over the positions of all other electrons. A partitioning of ρ(r) into constituent parts is an intellectual exercise that does not lend itself to unique measurement from elastic X‐ray scattering experiments. It is shown that in the limit of perfect data and an infinite Ewald sphere, a least‐squares fit with a many‐centered finite multipole expansion of the charge density about the N nuclei will necessarily satisfy the q‐centered multipoles of the molecule for q = 1, 2,…, N. This means that a large number of static‐charge physical properties (averages over ρ(r)) are correctly given. Several expressions for averages of ρ(r) or over FH are given. It is shown that outer moments, such as atomic charges, dipole moments and quadrupole moments, always depend on a shape function. On the other hand, inner moments such as potentials, electric forces, and electric field gradients, may be represented by direct Fourier analysis of FH (obs) (suitably phased, of course). Nuclear vibrations have been neglected throughout the discussion.

Electric field gradients and charge density in corundum, α-Al2O3

“…The zero-field condition of the HellmannFeynman electrostatic theorem is thus obeyed within 3.5 e.s.d.'s. Since these values are similar in all refinements, we conclude that properties of the charge density with different angular symmetries are only weakly correlated, in agreement with Stewart (1977).…”

A model study of the κ-refinement procedure for fitting valence electron densities

“…The dipole moment determined from a K refinement for formamide agrees well with values obtained by other experimental methods (Coppens et al, 1979), and the individual atomic charges agree extremely well with those calculated from an ab initio wavefunction by fitting to the electrostatic potential (Chirlian & Francl, 1987). Stewart (1977) has demonstrated that the model electron-density function which best fits an observed electron density is the same model function which best fits the electrostatic potential, at least for data of unlimited resolution. It is because of this that the atomic charges obtained from a K refinement based on X-ray diffraction data are physically reasonable, and hence they can sensibly be used to calculate the electrostatic potential within and around a molecule or crystal for use in modelling intermolecular interactions.…”

Single-crystal structures and electron density distributions of ethane, ethylene and acetylene. II. Single-crystal X-ray structure determination of acetylene at 141 K

“…The np values taken are n(H) = 0, 1, 2 for monopole (l = 0), dipole (l = 1) and quadrupole (l = 2) respectively and n(C) = 2 for the monopole up to the quadrupole and n(C) = 3 for the octopole function. The quality of the (H like) STO type function (9) has been discussed by Bentley & Stewart (1976) and by Stewart (1977).…”