Average inner product sums of electron linear momenta for atoms and ions

From the second moments of the electron-pair densities in momentum space, accurate Hartree–Fock values of the average inner product sum 〈∑ _i<j p _i ·p _j 〉 of electron linear momenta are evaluated for the 102 neutral atoms from He to Lr, the 53 singly charged cations from Li⁺ to Cs⁺, and the 43 stable anions from H⁻ to I⁻ in their experimental ground states. The present results are new for 38 species and improve the literature values for 68 species. Received: 18 July 2002 / Accepted: 4 September 2002 / Published online: 8 November 2002 Acknowledgement. This work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education of Japan. Correspondence to: H. Matsuyama e-mail: hisashi@mmm.muroran-it.ac.jp     

Average electron radii in many-electron atoms

In many-electron atoms, the average electron radius r represents the mean distance of a single electron from the nucleus when all the interelectronic interactions are averaged. If the electron-electron interaction is explicitly considered, the average radius r splits into two different radii, inner radius r(<) and outer radius r(>). For the 102 atoms He through Lr in their ground states, the radii r(<) and r(>) are systematically examined at the Hartree-Fock limit level. The effect of electron correlations on r(<) and r(>) is also discussed for the He atom and its isoelectronic ions.     

Multipole analysis of electron repulsion energies in many-electron atoms

The repulsion energy W between two electrons located at r ₁ and r ₂ can be expressed by the sum of the interaction energies W _k between an electron located at and linear electric multipoles located at the coordinate origin along the vector , where and are the vectors with smaller and larger moduli, respectively, of the two vectors r ₁ and r ₂. All the existing multipole contributions W _k to the Hartree–Fock electron repulsion energy W are examined for the 102 atoms He through Lr in their ground states. It is found that |W _k | decreases rapidly with increasing k, W ₀ > W, and W _k with k ≥ 1 work to reduce W ₀. The effect of electron correlation is also discussed for some helium-like atoms.     

Analytical Hartree–Fock electron densities for atoms He through Lr

The Hartree-Fock electron density has an important property that it is identical to the exact density to first order in the perturbation theory. For the neutral atoms from He (Z = 2) to Lr (Z = 103) in their ground state, we report an accurate analytical approximation F(r) to the spherically averaged electron densityρ(r) obtained by the numerical Hartree-Fock method. The present density functionF(r) is expressed by a linear combination of reasonable number (not more than 30) of basis functionsr ⁿⁱ exp(- ζ _i r), and has the following properties: (i)F(r) is nonnegative, (ii)F(tr) is normalized, (iii)F(r) reproduces the Hartree-Fock moments <r ^k > (k = −2 to +6), (iv)F(0) is equal toρ(0), (v)F′(0) satisfies the cusp condition, and (vi)F(r) has the correct exponential decay in the long-range asymptotic region.     

Inner and outer radial density functions in many-electron atoms

When any two electrons are considered simultaneously, the radial density function D(r) in many-electron atoms is shown to be rigorously separated into inner D _<(r) and outer D _>(r) radial densities. Accordingly, radial properties such as the electron–nucleus attraction energy V _en and the diamagnetic susceptibility χ _d are the sum of the inner and outer contributions. The electron–electron repulsion energy V _ee has an approximate relation with the minus first moment of the outer density D _>(r). For the 102 atoms He through Lr in their ground states, different characteristics of local maxima in the radial densities D _<(r), D _>(r), and D(r) are reported based on the numerical Hartree-Fock wave functions. Relative contributions of the inner and outer components to V _en and are also discussed for these atoms.     

Charge and radial correlations in helium and helium-like atoms

Summary For two-electron atoms, the method of a variable exponent, which treats the orbital exponent (or effective nuclear charge) of an electron as an explicit function of the radial coordinate of the other electron, is studied. The method is shown to improve the energy and other electronic properties remarkably. An incorporation of the variable exponent into the Kellner approximation for He, for example, gives the energy –2.872 606 1 a.u., which is lower than the original Kellner energy by 0.024 949 8 a.u. and exceeds the Hartree-Fock limit energy by 0.010 926 1 a.u. The improvement due to the variable exponent originates from the inclusion of the charge and radial correlations. Applications of the method to the Eckart and Hylleraas approximations are also presented.     

Study of electron impact excitation of atoms through lasers

We discuss three different experiments for studying electron-excitation of atoms where lasers have been used in combination. These are stepwise electron–photon excitation, superelastic electron scattering from laser excited atoms and excitation of atoms using spin polarized electrons produced by lasers. We present distorted wave calculations and compare our results with the recently reported such experimental measurements. In particular, the results for the alignment and orientation of the excited n ²P states of K ($n=4$) and Rb ($n=5$) atoms and the spin parameters for the lowest excited ¹P₁ and ³P_0,1,2 states of argon by polarized electrons are presented and discussed.     

Sizes of atoms and ions and covalency of bonding in molecules and crystals

A new system of atomic radii for the elements up to barium inclusive is constructed. Values of the radii are chosen so as the dependence between the dissociation energy of diatomic homonuclear molecules and a depth of atom overlapping is monotonous, and the scatter of data is minimal. The depth of overlapping is calculated as a difference between the sum of atomic radii and an experimental interatomic distance. Conclusions are made that: the radii of free atoms and ions are determined by the value of the electron density equal to 0.01 au; they considerably change in molecules and crystals only as a result of the charge transfer from cation to anion; covalent bonding is well described by the overlapping of free atoms (ions), confined by the surface of the given radius, and its energy depends upon the depth of overlapping of valence electron densities of atoms. A method of overlapping atoms is proposed for the approximate estimation of ionic sizes and charges in bound systems.     

Electronic energy inequalities for isoelectronic molecular systems

Within the Born-Oppenheimer approximation an inequality relation is derived between points of electronic energy hypersurfaces of pairs of isoelectronic molecules. The inequality is directly applicable to point pairs fulfilling a symmetry criterion for the nuclear frameworks and it may be extended to coordinate domains on both hypersurfaces. The result is applied to special examples of conformational problems, dissociation reactions and impurity —vacancy centres in solid clusters.     

Description of correlated densities for few‐electron atoms by simple functional forms

Simple analytical functional forms for the electron density of two‐ and three‐electron atoms which reproduce fairly the correlated (exact) values are presented. The procedure is based on the fitting of an auxiliary f(r) function which has adequate properties for this purpose and can be extended to more complex atoms. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 443–454, 1999     

Direct and exchange contributions to inner and outer radii in many-electron atoms

Within the Hartree-Fock framework, the spinless two-electron density function Γ (r ₁, r ₂) consists of direct Γ_di (r ₁, r ₂) and exchange Γ_ex (r ₁, r ₂) parts. Accordingly, the inner and outer radii in many-electron systems are rigorously separated into the direct and exchange contributions, i.e., and . It is generally shown that and , where is the usual average radius of an electron. Numerical examinations of the direct and exchange contributions for the 102 atoms from He to Lr in their ground states find that the electron exchange works to decrease and increase . However, the exchange parts are very small and the direct parts essentially govern the inner and outer radii.     

Correlation energy,correlated electron density,and exchange‐correlation potential in some spherically confined atoms

We report correlation energies, electron densities, and exchange‐correlation potentials obtained from configuration interaction and density functional calculations on spherically confined He, Be, Be²⁺, and Ne atoms. The variation of the correlation energy with the confinement radius R_c is relatively small for the He, Be²⁺, and Ne systems. Curiously, the Lee–Yang–Parr (LYP) functional works well for weak confinements but fails completely for small R_c. However, in the neutral beryllium atom the CI correlation energy increases markedly with decreasing R_c. This effect is less pronounced at the density‐functional theory level. The LYP functional performs very well for the unconfined Be atom, but fails badly for small R_c. The standard exchange‐correlation potentials exhibit significant deviation from the “exact” potential obtained by inversion of Kohn–Sham equation. The LYP correlation potential behaves erratically at strong confinements. © 2016 Wiley Periodicals, Inc.     

Fast density matrix-based partitioning of the energy over the atoms in a molecule consistent with the Hirshfeld-I partitioning of the electron density

For the Hirshfeld-I atom in the molecule (AIM) model, associated single-atom energies and interaction energies at the Hartree-Fock level are efficiently determined in one-electron Hilbert space. In contrast to most other approaches, the energy terms are fully consistent with the partitioning of the underlying one-electron density matrix (1DM). Starting from the Hirshfeld-I AIM model for the electron density, the molecular 1DM is partitioned with a previously introduced double-atom scheme (Vanfleteren et al., J Chem Phys 2010, 132, 164111). Single-atom density matrices are constructed from the atomic and bond contributions of the double-atom scheme. As the Hartree-Fock energy can be expressed solely in terms of the 1DM, the partitioning of the latter over the AIM naturally leads to a corresponding partitioning of the Hartree-Fock energy. When the size of the molecule or the molecular basis set does not grow too large, the method shows considerable computational advantages compared with other approaches that require cumbersome numerical integration of the molecular energy integrals weighted by atomic weight functions.     

Differential density matrix overlap: an index for assessment of electron correlation in atoms and molecules

Summary A new index, called the differential density matrix overlap (DDMO), is proposed for assessment of the electron correlation effects in atoms and molecules. DDMO can be easily calculated as the negative value of the correlation energy derivative with respect to the relative position of the occupied and virtual orbitals. DDMO is transparent to physical interpretation. It can serve as a tool for analyzing the accuracy of approximate electron correlation methods and the validity of the Hartree-Fock wavefunction as the zeroth-order approximation. The properties of DDMO are discussed using test calculations on 11 atoms and molecules as an example.     

Isoelectronic changes in energy of quark atoms and molecules via the Levy equation

Summary Z-transition state calculations based on the Levy equation suggest that the isoelectronic changes in energy of quark atoms,Q, (ordinary atoms with extra nuclear charge in units of ±1/3 and/or ±2/3) can be expressed quantitatively in terms of the electrostatic potential at the nucleus of an isoelectronic ordinary atom. Numerical tests within the local density functional theory are presented for the quark atoms of Li-F. Theab initio MO (molecular orbital) calculations using STO-5G basis on the C₂ molecule and its quark derivatives lead to similar conclusions.     

Electron‐density delocalization in many‐electron atoms confined by penetrable walls: A Hartree–Fock study

The electronic structure of several many‐electron atoms, confined within a penetrable spherical box, was studied using the Hartree–Fock (HF) method, coupling the Roothaan's approach with a new basis set to solve the corresponding one‐electron equations. The resulting HF wave‐function was employed to evaluate the Shannon entropy, , in configuration space. Confinements imposed by impenetrable walls induce decrements on when the confinement radius, R_c, is reduced and the electron‐density is localized. For confinements commanded by penetrable walls, exhibits an entirely different behavior, because when an atom starts to be confined, delivers values less than those observed for the free system, in the same way that the results presented by impenetrable walls. However, from a confinement radius, shows increments, and precisely in these regions, the spatial restrictions spread to the electron density. Thus, from results presented in this work, the Shannon entropy can be used as a tool to measure the electron density delocalization for many‐electron atoms, as the hydrogen atom confined in similar conditions.     

Chemical insight into electron density and wave functions: software developments and applications to crystals, molecular complexes and materials science

This paper overviews the work made by our group during the past 10–15 years on crystalline systems, semiconductor surfaces, molecular complexes and on materials of interest for technological applications, such as the defective silicon or the novel generation thermoelectric materials. Our main aim of extracting chemical insight into the analysis of electron densities and computed wave functions is illustrated through a number of examples. The recently proposed Source Function analysis is reviewed and a few of its more interesting applications are summarized. Software package developments, motivated by the need of a direct comparison with experiment or by the help these packages can provide for interpreting complex experimental outcomes, are described and future directions outlined. A particular emphasis is given to the TOPOND and TOPXD programs, which enable one to analyze theoretical and experimental crystalline densities using the rigorous framework of the Quantum Theory of Atoms in Molecules, due to Bader.     

Upper bounds to atomic electron densities in position and momentum spaces

Modified functions r ^–(r) and p ^–(p) of the spherically averaged electron densities (r) in position space and (p) in momentum space are found to be convex (i.e., the second derivatives are nonnegative everywhere) for all the 103 ground-state atoms from hydrogen (atomic number Z=1) to lawrencium (Z=103), if the parameters are chosen to be 0.6 and 1.4. The convex property of r ^–(r) and p ^–(p) is used to derive upper bounds to the density functions (r) and (p) in terms of their radial moments r ^s and p ^s or frequency moments ^t and ^t . In most cases, the present bounds are shown to be more general and more accurate than those reported in the literature.     

Second-order atomic Fukui indices from the electron-pair density in the framework of the atoms in molecules theory

Atomic Fukui indices, which are obtained from the electron density, have been previously shown to be useful in predicting which atoms in a molecule are most likely to suffer nucleophilic, electrophilic, or radicalary attacks. Here, we present a second-order generalization of these indices based on the electron pair density. We show how second-order atomic Fukui indices can be used to analyze the effects of electron loss or gain in several molecules from an electron pair point of view. Further, these indices also highlight which atoms or pairs of atoms are more likely to suffer nucleophilic, electrophilic, or radical attacks. In conclusion, second-order indices can complement first-order ones by affording relevant information on molecular reactivity from an electron pair perspective.