The electrostatic potential V(r) that is created by a system of nuclei and electrons is formulated directly from Coulomb's law and is a physical observable, which can be determined both experimentally and computationally. When V(r) is evaluated in the outer regions of a molecule, it shows how the latter is ‘seen’ by an approaching reactant, and thus is a useful guide to the molecule's reactive behavior, especially in noncovalent interactions. However, V(r) is a fundamental property of a system, the significance of which goes beyond its role in reactivity. For example, the energy of an atom or molecule can be expressed rigorously in terms of the electrostatic potentials at its nuclei. These and other features of V(r) are discussed in this overview. © 2011 John Wiley & Sons, Ltd. WIREs Comput Mol Sci 2011 1 153-163 DOI: 10.1002/wcms.19

The electrostatic potential: an overview

(1 < p ≤ ∞) [LS87f]. (2) [HR88a]. (2m− 2) [KL88]. (A0, A1)θ1 [Xu87a]. (α, β) [Pie88a, Fin88a]. (d ≥ 1) [Wsc85a]. (λ) [DM85b]. (Z/2) [Car86b]. (nα) [Sch85h]. (φ)2 [BM89c]. (τ − λ)u = f [Wei87r]. (x, t) [Lum87, Lum89]. (X1 −X3, X2 −X3) [SW87]. 0 [Caz88, Kas86, Pro87]. 0 < p < 1 [Cle87]. 1 [Bak85a, DD85, Drm87, Eli88, FT88a, Gek86d, HN88, Kos86a, LT89, Pet89a, Pro87, Tan87, vdG89]. 1/4 [KS86e]. 1 ≤ q < 2 [Gue86]. 2 [BPPS87, Cam88, Cat85a, ES87b, Gan85e, Gol86a, HRL89g, Hei85, Hua86, Kan89, KB86, Li86, LT89, Mil87b, Mur85a, Qui85b, SP89, Shi85, Spe86, Wal85b, Wan86]. 2m− 2 [Kos88b]. 2m− 3 [Kos88b]. 2m− 4 [Kos88b]. 2× 2 [Vog88]. 3 [Aso89, BPPS87, BW85c, BG88b, Che86d, Fis86, Gab85, Gu87a, HLM85b, Kam89b, Kir89c, Lev85c, Mil85c, Néd86, Pet86, Ron86, Sch85b, ST88, Tur88b, Wan86, Wen85, tDP89, vdW86]. 4 [Bau88a, Don85a, FKV88, Kha88, Kir89l, SS86, Seg85b, Wal85b]. 5 [Ito89, Kir89e, SV85]. 5(4) [Cas86]. 5819539783680 [KSX87]. 6 [PH89, Žub88].

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A Complete Bibliography of Lecture Notes in Mathematics (1985{1989)

An efficient procedure and computer program are outlined for fitting numerical X-ray and electron scattering factors with the correct inclusion of all physical constraints. The numerical electron scattering factors have been parameterized using five analytic non-relativistic hydrogen electron scattering factors as basis functions for 103 neutral atoms of the periodic table. The inclusion of the correct physical constraints in the electron scattering factor and its derived quantities allows the use of the new parameterization in different fields. In terms of quality of the fit, the proposed parameterization of the electron scattering factor is one order of magnitude better than the previous analytic fittings.

An accurate parameterization for scattering factors, electron densities and electrostatic potentials for neutral atoms that obey all physical constraints

Progress on high precision calculations for the ground state of atomic lithium is reviewed. The following properties are considered: upper and lower bounds to the nonrelativistic ground state energy, the specific mass shift, the transition isotope shift, relativistic corrections to the ground state energy, the Lamb shift, the ionization potential, the electron affinity, the hyperfine coupling constant, the nuclear magnetic shielding constant, the diamagnetic susceptibility, several polarizability factors, shielding constants, oscillator strength sums, the electron density and spin density, intracule functions, moments 〈rin〉 and 〈rijn〉 and form factors. A discussion is also given on some convergence considerations as they apply to high precision calculations on the lithium atom.

Progress on high precision calculations for the ground state of atomic lithium

Weizsäcker energy of many-electron systems

Starting from any two finite expectation values, rigorous bounds to quantities of D-dimensional many-particle systems, depending on the single-particle density \ensuremath{\rho}(r) in the form F${\ensuremath{\rho}}^{n}$${d}^{D}$r, n\ensuremath{\in}${\mathrm{openR}}^{+}$ (the set of positive real numbers), are explicitly given. Similar bounds are also valid in momentum space. The resulting expressions are used to rigorously bound the kinetic (${T}_{0}$,D) and Dirac exchange (${K}_{0}$,D) energies of fermionic systems in the plane-wave approximation (e.g., the Thomas-Fermi approach). As a numerical illustration, very accurate upper bounds to the Dirac exchange energy of atomic systems are found by means of 〈${r}^{\mathrm{\ensuremath{-}}1}$〉 and 〈${r}^{\mathrm{\ensuremath{-}}2}$〉. The limit of large dimensionality is also considered; in particular, it is found that ${T}_{0}$,D\ensuremath{\ge}${\mathrm{D}}^{2}$〈${\mathrm{r}}^{\mathrm{\ensuremath{-}}2}$〉/2${\mathrm{e}}^{2}$ and ${K}_{0}$,D\ensuremath{\le}-2〈${\mathrm{r}}^{\mathrm{\ensuremath{-}}1}$〉/\ensuremath{\pi}e, where e=2.718 28. In addition rigorous lower bounds to the exact kinetic energy of real (D=3) fermionic systems are given.

Bounds to density-dependent quantities of D-dimensional many-particle systems in position and momentum spaces: Applications to atomic systems.

A self-consistent study of the function f(r)=\ensuremath{\rho}(r)+\ensuremath{\rho}'(r)/(2Z), where \ensuremath{\rho}(r) is the spherically averaged electron density of the ground state of an atom, shows that f(r)\ensuremath{\ge}0 everywhere. This allows us to obtain several inequalities relating the atomic electronic density at the nucleus \ensuremath{\rho}(0) and atomic radial expectation values 〈${\mathit{r}}^{\mathrm{\ensuremath{\alpha}}}$〉. In particular, we generalize upper bound found by Hoffmann-Ostenhof, Hoffmann-Ostenhof, and Thirring [J. Phys. B 11, L571 (1978)] to \ensuremath{\rho}(0). We also obtain accurate inequalities between the radial expectation values 〈${\mathit{r}}^{\mathrm{\ensuremath{\alpha}}}$〉 of atomic systems.

Atomic charge density at the nucleus and inequalities among radial expectation values

The distribution of zeros of the polynomial eigenfunctions of ordinary differential operators of arbitrary order with polynomial coefficients is calculated via its moments directly in terms of the parameters which characterize the operators. Some results of K.M. Case and the authors are extended. In particular, the restriction for the degree of the polynomial coefficient of the ith-derivative to be not greater than i is relaxed. Applications to the Heine polynomials, the Generalized Hermite polynomials and the Bessel type orthogonal polynomials (which no trivial properties of zeros were known of) are shown.

The distribution of zeros of the polynomial eigenfunctions of ordinary differential operators of arbitrary order

The convexity of the spherically averaged electron density \ensuremath{\rho}(r) of neutral atoms with Z\ensuremath{\le}54 is numerically studied in a Hartree-Fock framework. It is found that \ensuremath{\rho}(r) is convex for Z=1, 2, 7--15, and 33--44, while for the rest of the atoms it presents a small ``nonconvex'' region. Second, rigorous relationships between the values of the single-particle density and its first derivative at the origin and the radial expectation values 〈${\mathit{r}}^{\mathit{k}}$〉 for a convex \ensuremath{\rho}(r) are derived. These conditions are valid for any many-body system whose spherically averaged single-particle density is convex. Finally, for atomic systems, these determinantal conditions together with the electron-nucleus cusp condition are used to obtain novel upper and lower bounds to \ensuremath{\rho}(0) by means of two or more expectation values. The quality of the new lower bounds is better than that of the corresponding ones known up to now. In particular, the convexity bound (1/6\ensuremath{\pi})(〈${\mathit{r}}^{\mathrm{\ensuremath{-}}2}$${\mathrm{〉}}^{2}$/〈${\mathit{r}}^{\mathrm{\ensuremath{-}}1}$〉) to \ensuremath{\rho}(0) improves by a factor of 4/3 the accuracy of the corresponding bound obtained from the property of monotonic decrease of \ensuremath{\rho}(r).

Atomic-charge convexity and the electron density at the nucleus

Two functions built from the spherically averaged atomic density, i.e., f(r)=\ensuremath{\rho}(r)+\ensuremath{\rho}'(r)/(2Z) and g(r)=${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$(r)+\ensuremath{\rho}'(r)/(2Z), are studied in the Hartree-Fock framework for atoms in the range Z=2--54 and in their ground state. These studies and previous ones show that f(r) is a positive function for all these atoms, and g(r) is also a positive function for all atoms whose least-bounded electron is s type. For these atoms the upper bound \ensuremath{\rho}(0)\ensuremath{\le}Z〈${\mathit{r}}^{\mathrm{\ensuremath{-}}2}$${\mathrm{〉}}_{\mathit{s}}$/2\ensuremath{\pi}, which improves the accuracy of the bound of Hoffmann-Ostenhof, Hoffmann-Ostenhof, and Thirring [J. Phys. B 11, L571 (1978)], is found. In addition we obtain sharp lower bounds to \ensuremath{\rho}(0) and inequalities between total and s-state radial expectation values, 〈${\mathit{r}}^{\mathrm{\ensuremath{\alpha}}}$〉 and 〈${\mathit{r}}^{\mathrm{\ensuremath{\alpha}}}$${\mathrm{〉}}_{\mathit{s}}$, respectively.

F. J. Galvez

Papers

Bounds to density-dependent quantities of D-dimensional many-particle systems in position and momentum spaces: Applications to atomic systems.

Atomic charge density at the nucleus and inequalities among radial expectation values

The distribution of zeros of the polynomial eigenfunctions of ordinary differential operators of arbitrary order

Atomic-charge convexity and the electron density at the nucleus

Spatial generalizations of Kato's cusp condition for atoms: Applications.