Average electron momenta in many-electron atoms

In many-electron atoms, the average electron momentum p represents the mean momentum of a single electron when all the electron motions are averaged. If any two electrons are considered simultaneously, however, the average momentum p splits into two different momenta, low momentum p_< and high momentum p_>. For the 102 atoms He through Lr in their ground states, the momenta p_< and p_> are systematically examined at the Hartree–Fock limit level. It is also shown that the sum p_>+p_< and the difference p_>–p_< of the two momenta constitute upper and lower bounds to the electron-pair relative momentum p₁₂=|p₁–p₂| and to the electron-pair center-of-mass momentum P=|p₁+p₂|/2. The tightness of the bounds is discussed for the 102 atoms.     

Bounds to electron-pair relative and center-of-mass radii in many-electron atoms

When the electron-electron interaction is explicitly considered in many-electron atoms, the average electron radius (r) splits into the inner (r<) and outer (r>) radii. It is shown that the sum and difference of these radii constitute upper and lower bounds, respectively, to the electron-pair relative distance r(12)=(/r1-r2/). An analogous result is also derived for the electron-pair center-of-mass radius (R)=(/r1+r2//2). For the 102 atoms He through Lr in their ground states, the tightness of these bounds is numerically examined at the Hartree-Fock limit level. Good linear correlations are observed between (r12) and (r>), and between (R) and (r>)/2.     

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.     

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.     

Multiphoton ionization of many-electron atoms

We review the main experimental results obtained in the multielectron multiphoton ionization of rare gases. Multiple ionization is interpretated as a sequential process. We study the influence of many-electron effects in multiphoton ionization within the framework of the random phase approximation. This is applied to a calculation of the two-photon ionization cross section of xenon. Finally, the role of screening effects in the multiphoton stripping of heavy atoms (Xe) is estimated.     

Complete spin-orbit interaction in many-electron atoms

The complete spin-orbit coupling constants have been evaluated, including both the spin-own orbit and the spin-other orbit terms, for all the positive ions, neutral systems, and negative ions from He to Kr, as well as for various isoelectronic series.

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Zusammenfassung Für alle positiven und negativen Ionen und für die neutralen Atome von He bis Kr sowie für verschiedene isoelektronische Serien werden die kompletten Spin-Bahn-Kopplungskonstanten einschließlich der Spin-Bahn-Wechselwirkung verschiedener Elektronen berechnet.

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This work has been supported in part by the National Research Council of Canada.     

Nuclear mass effect in many-electron atoms

The total nuclear mass effect, including the normal mass and the mass polarization corrections, have been evaluated, from existing analytical Hartree-Fock functions, for the positive ions, neutral systems, and negative ions of all the atoms from He through Kr. The formulation for the mass polarization effect is presented in detail.

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Zusammenfassung Mit Hilfe existierender analytischer Hartree-Fock-Funktionen wurde der totale Kernmasseneffekt einschließlich der Korrekturen für normale Massen und Massenpolarisation für positive und negative Ionen und neutrale Systeme für alle Atome von He bis Kr berechnet. Der Formalismus für den Massenpolarisationseffekt wird im einzelnen dargelegt.

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This work has been supported in part by the National Research Council of Canada.     

Electron spin-spin contact interaction in many-electron atoms

A general formulation has been developed for the evaluation of the total electron spin-spin contact interaction in many-electron atoms, including both the intra- and inter-shell contributions. Calculations have been carried out, using existing analytical Hartree-Fock functions, for the positive ions, neutral systems, and negative ions for all the atoms from He to Kr.

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Zusammenfassung Es wird ein allgemeiner Formalismus zur Berechnung der gesamten Elektronen-Spin-Spin-Kontakt-Wechselwirkung in Vielelektronen-Atomen angegeben, der sowohl die Anteile der Wechselwirkung innerhalb einer Schale als auch zwischen einzelnen Schalen berücksichtigt. Berechnungen wurden unter Benutzung vorhandener analytischer Hartree-Fock-Funktionen für die positiven Ionen, die neutralen Systeme und die negativen Ionen aller Atome von He bis Kr durchgeführt.

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This work has been supported in part by the National Research Council of Canada.     

Spatial symmetry holes in many-electron atoms and molecules

When a many-electron system has a spatial symmetry, it is shown that there exist spatial symmetry holes, which imply that two or more electrons are prohibited from being at certain spatial positions simultaneously. Inversion holes, rotation holes, and reflection holes, which result from inversion, twofold rotation, and reflection symmetries, respectively, are discussed in detail. The electron-electron counterbalance hole reported in literature is a particular case of the inversion hole. The spatial symmetry holes are illustrated for simple atoms and molecules.     

Mass-variation and Darwin relativistic corrections in many-electron atoms

The formulation for the evaluation of the matrix elements over the relativisitic mass-variation and Darwin operators is presented. Calculations of these corrections have been carried out, using available analytical Hartree-Fock functions, for the positive ions, neutral systems, and negative ions for all the atoms from He to Kr.     

Kinetically balanced calculations on relativistic many-electron atoms

The kinetically balanced calculations proposed by Ishikawa, Grant, Stanton et al. have been performed on Be, C, Ni, Ag, and Hg. The results show that variational collapse does not occur.     

Generalized spherical functions in the theory of many-electron atoms

A new method for finding non-relativistic and relativistic wave-functions of an electron moving in the field of a nuclear charge in the jj coupling scheme is proposed. It is based on the usage of generalized spherical functions. The mathematical apparatus necessary to find the expressions for matrix elements of the non-relativistic and relativistic energy or electron transition operators is developed. The formulas obtained for these matrix elements are more convenient than those usually used in jj coupling scheme; only their radial integrals and some phase multipliers depend on orbital quantum numbers.     

Radial subshell splittings and double-zeta functions in many-electron atoms

When the electron-electron interaction is explicitly considered in many-electron atoms, the average subshell radiusnl splits into two different radii, inner radius nl and outer radius >nl, where n and l are the principal and azimuthal quantum numbers. For the 102 atoms He through Lr in their ground states, the radii nl and nl are systematically examined at the Hartree-Fock limit level. For a subshell nl, two exponents zetanl(est) estimated from these radii have good linear correlations with variationally determined exponents zetanl(var) of double-zeta Slater-type functions.     

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.     

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     

New method for the direct calculation of electron density in many-electron systems. I. Application to closed-shell atoms

Analytical properties of hydrogen-like atomic orbitals (HAO ) that are used in the MOLCAO approach to the quantum theory of molecules have been studied. Addition and expansion theorems for HAO have been proved, both in coordinate and momentum representations. A close relation has been established between HAO and the reduced Bessel functions of half-integer indices. New methods are suggested to calculate integrals for atomic and molecular form factors, and multicenter integrals, for the HAO basis in the MO LCAO theory.     

X-ray and electron scattering factors for many-electron atomic system

Analytical expressions are developed for the x-ray and electron scattering factors for a many-electron atomic system when the single configuration wave function of the system is written as a sum of Slater determinants of spin orbitals. The radial part of the orbital is expanded in terms of Slater-type orbitals (STO 's). The expressions so developed have been used to calculate the coherent and incoherent x-ray and electron scattering factors and intensities for all the neutral atoms up to krypton (Z = 36) and for some positive and negative ions of chemical interest. The results obtained are used to test the value of Hartree–Fock wave functions for the evaluation of “one-electron properties” of many-electron atomic systems.