One-electron density function

The one-electron density function is the first of a series of density functions which relate to clusters of any numbers of electrons. The second member of the series is... 

For every electronic wavefunction that is an eigenfunction of the electron spin operator S, the one-electron density function always comprises an spin part... 

According to the aspherical-atom formalism proposed by Stewart [12], the one-electron density function is represented by an expansion in terms of rigid pseudoatoms, each formed by a core-invariant part and a deformable valence part. Spherical surface harmonics (multipoles) are employed to describe the directional properties of the deformable part. Our model consisted of two monopole (three for the sulfur atom), three dipole, five quadrupole, and seven octopole functions for each non-H atom. The generalised scattering factors (GSF) for the monopoles of these species were computed from the Hartree-Fockatomic functions tabulated by Clementi [14]. 

If x = rs denotes the space-spin variable, we recall from first principles (Magnasco, 2007,2009a) that, for a normalized N-electronwavefunction satisfying the Pauli antisymmetry principle, the one-electron density function is defined as ... 

Density functional theory (DFT) is a form of quantum mechanics which uses the one-electron density function, p, instead of the more usual wave function, V , to describe a chemical system. Such a system is any collection of nuclei and electrons. It may be an atom, a molecule, an ion, a radical or several molecules in a state of interaction. 

R. F. Stewart, One-electron density functions and many-centered finite multipole expansions, Isr. J. Chem. 16, 124-131 (1977). 

Here S = 2ir(k—k0 )/X, where k and k0 are unit vectors in the direction of scattered and incident radiation, respectively Q is a vector of the nuclear coordinates and p(r,Q) is the one-electron density function for the molecule with a fixed Q. Note that Eq. (1) is a Fourier transform of a static charge density function and is generally valid for a radiation energy that is much larger than core electron ionization energies. The actual, observed intensity is a statistical average over the states for Q,... 

The valence density model represents the one-electron density function for a molecule in a crystal as... 

The discussion above is intended to emphasize that a systematic selection of density basis functions can be accomplished by projection analysis. Also note that these x-ray population coefficients are not the elements of a density matrix from a quantum chemical calculation as has been suggested.23 To compare the x-ray results, one can project the theoretical one-electron density function by least squares into the same basis functions that were used in the x-ray analysis. 

Recently there have been important new developments, both with respect to acid-base interactions and with the concept of hardness (1). These developments come from density functional theory (DFT). a branch of quantum mechanics which focuses on the one-electron density function of a molecule, rather than its wave function (2). 

The one-electron density function,/9(r), is given for this basis set as... 

The ELS method maintains spherically symmetrical atoms. By using a more complicated model, with more parameters, asphericity can be included. A one-electron density function can be described as... 

For most physical properties of interest, we need to know the probability for finding an electron in a three-dimensional volume element dv. Since the probability for finding an electron in cfu is the sum of the probabilities for finding each electron there, the one-electron density function p for the allyl radical is... 

---