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

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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.