Subshell-pair interelectronic angles of atoms in momentum space

Average angles between linear momenta of an electron in a subshell nl and another electron in a subshell nl are examined for the 102 atoms He through Lr in their ground states, where n and l are the principal and azimuthal quantum numbers, respectively. Congruency in the mathematical structures of the average interelectronic angles in position and momentum spaces leads to the theoretical results that with even |l–l| are exactly equal to 90°, while with odd |l–l| are always larger than 90°. Numerical analyses of 3,275 subshell-pair angles with odd |l–l| in the 102 atoms clarify that deviations of the total average interelectronic angles from 90° are mainly governed by subshell pairs with |n–n|1 and |l–l|=1, in contrast to the position-space results where only subshell pairs with n=n and |l–l|=1 are important.Acknowledgments. We thank Mr. T. Shimazaki for his assistance in the compilation of data. This work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education of Japan.     

Subshell-pair correlation coefficients of atoms in momentum space

Angular correlation coefficients τ ^{nl,n^′ l^′} [p] between linear momenta of an electron in a subshell nl and another electron in a subshell n′ l′ are studied for the 102 neutral atoms He through Lr in their ground states, where n and l are the principal and azimuthal quantum numbers, respectively. We theoretically find that electron momenta are negatively correlated or uncorrelated; τ ^{nl,n^′ l^′} [p] < 0 when |l − l′|=1, while τ ^{nl,n^′ l^′} [p]=0 when |l − l′| ≠ 1. Numerical examinations of the atoms show that except for the He–B atoms, negative correlations are largest between 1s and 2p subshells, which have the most diffuse electron distributions in momentum space.     

Radial electron-pair densities in momentum space and shell structure of atoms

 The radial electron-pair intracule (relative motion) H(u) and extracule (center-of-mass motion) D(R) densities in position space were known to reveal four types of maxima which are related to the four inner electron shells, K, L, M, and N, of atoms. The corresponding radial electron-pair intracule Hˉ(v) and extracule Dˉ(P) densities in momentum space are studied for the 102 atoms from He (atomic number Z=2) to Lr (Z=103). The densities Hˉ(v) and Dˉ(P) are found to have either one maximum or two maxima, and the numbers of maxima in Hˉ(v) and Dˉ(P) are the same for 98 atoms. For these atoms, the locations υ _max and P _max and the heights Hˉ _max and Dˉ _max of the corresponding maxima satisfy the approximate relations υ _max ≅ 2P _max and Hˉ _max ≅Dˉ _max /2. On the basis of their Z-dependence, the maxima in Hˉ(v) and Dˉ(P) of the 102 atoms are classified into five types. Shell-pair decompositions of the radial densities show that these maxima reflect five outer electron shells of atoms. Received: 24 January 2001 / Accepted: 12 March 2001 / Published online: 13 June 2001     

Approximate formulas for the characteristics of interelectronic angle densities

The accuracies of approximate formulas are examined for several characteristics of the interelectronic angle density A(₁₂), where ₁₂ (0₁₂) is the angle subtended by the position vectors r₁ and r₂ of two electrons. Numerical results for 102 atoms show that simple approximations have sufficient accuracies for the moments with n=1–4, the central moments _n with n=2, 4, and the kurtosis, when measured by the absolute and relative errors. For heavy atoms, however, the relative errors for the third central moment ₃ and the skewness are large.     

Electron correlation effects in position and momentum space: the atoms Li through Ar

Electron correlation effects on the electronic structure of atoms were investigated by means of a variety of position and momentum space related properties such as radial one-electron densities and radial electron momentum densities, Compton profiles and radial electron pair distributions. The results were obtained from MR-SDCI wavefunctions utilizing very large basis sets and are discussed in a comparative manner, analysing characteristic features and trends.     

Atomic electron-pair distances and subshell radii in position and momentum space

For the 53 neutral atoms from He to Xe in their ground states, the average distances < u> _{n l , n ′ l ′} in position space and < v> _{n l , n ′ l ′} in momentum space between an electron in a subshell nl and another electron in a subshell n ^′ l ^′ are studied, where n and l are the principal and azimuthal quantum numbers of an atomic subshell, respectively. Analysis of 1700 subshell pairs shows that the electron-pair distances < u> _{n l , n ′ l ′} in position space have an empirical but very accurate linear correlation with a one-electron quantity U _{n l , n ′ l ′}≡L _r +S _r ²/(3L _r ), where L _r and S _r are the larger and smaller of subshell radii < r> _{n l} and < r> _{n ′ l ′}, respectively. The correlation coefficients are never smaller than 0.999 for the 66 different combinations of two subshells appearing in the 53 atoms. The same is also true in momentum space, and the electron-pair momentum distances < > _{n l , n ′ l ′} have an accurate linear correlation with a one-electron momentum quantity V _{n l , n ′ l ′}≡L _p +S _p ²/(3L _p ), where L _p and S _p are the larger and smaller of average subshell momenta < p> _{n l} and < p> _{n ′ l ′}, respectively. Trends in the proportionality constants between < u> _{n l , n ′ l ′} and U _{n l , n ′ l ′} and between < > _{n l , n ′ l ′} and V _{n l , n ′ l ′} are discussed based on a hydrogenic model for the subshell radial functions. Received: 8 April 1998 / Accepted: 6 July 1998 / Published online: 18 September 1998     

Subshell-pair interelectronic angles of atoms

For the 102 atoms from He to Lr in their ground states, the average interelectronic angles <₁₂> _{nl, n'l'} between an electron in a subshellnl and another electron in a subshell n'l' are examined, where n and l are the principal and azimuthal quantum numbers, respectively. Theoretical study clarifies that <₁₂> _nl,n'l' are 90° precisely if l–l' are even, while they are larger than 90° if l–l' are odd. Numerical analysis of 3,275 subshell pairs with odd l–l' of the 102 atoms shows that the increases in the total average interelectronic angles <₁₂> from 90° are attributed predominantly to subshell pairs with n=n' and l–l'=1.     

Fukutome classes in momentum space

Summary Fukutome's group theoretical classification scheme for determinants, based on the transformation properties of the Fock-Dirac density matrix under spin rotations and time reversal, has been extended to momentum space. Particular attention is paid to the transformation properties of orbitals and density matrices under inversion in momentum space.     

Interelectronic angle densities of equivalent electrons in Hartree-Fock theory of atoms

The interelectronic angle density A(theta12) is the probability density function that the angle thetaij (0 < or = thetaij < or = pi) subtended by the vectors ri and rj of any two electrons i and j becomes theta12. For equivalent electrons in atoms, it is shown that the density A(theta12) in the Hartree-Fock theory is given by a simple polynomial of cos theta12. Detailed expressions are reported for all LS terms arising from s2, pN (N = 2-6), dN (N = 2-10), and f(N) (N = 2,12) electron configurations. With no modifications, the present results apply as well to the interelectronic angle density A(theta12) in momentum space, where theta12 is the angle between two electron momenta.     

Symmetric orthogonalisation in momentum space: A numerical study

Summary Symmetric orthogonalisation is favourable to perform in momentum space, as this article will show. We have used a model of a body centered cubic lattice with 1s- and 2s-Slater orbitals centered at each atom site. Computer programs have been written to calculate the eigenvalues of the overlap matrix which play an important role in constructing symmetrically orthogonalised wavefunctions.     

Atomic one- and two-electron subshell moments in position and momentum spaces

For 357 subshells of the 53 neutral atoms He through Xe in their ground states, the two-electron intracule (relative motion) <u ^k > _nl and extracule (center-of-mass motion) <R ^k > _nl subshell moments in position space are examined as well as their counterparts <v ^k > _nl and <P ^k > _nl in momentum space, where n and l are the principal and azimuthal quantum numbers of the atomic subshell, respectively. It is clarified that between the intracule and extracule moments the “2 ^k -rule” is strictly valid, which means <u ^k > _nl = 2 ^k <R ^k > _nl and <v ^k > _nl = 2 ^k <P ^k > _nl for any nl subshell. Theoretical analysis also proves that for a particular case of k = +2, two relations <u ²> _nl = (N _nl −1)<r ²> _nl and <v ²> _nl = (N _nl −1)<p ²> _nl hold exactly, where N _nl (≥2) is the number of electrons in the subshell nl, and <r ^k > _nl and <p ^k > _nl are the familiar one-electron subshell moments in position and momentum spaces, respectively. The latter equality establishes a new and rigorous relation between the second electron-pair moments in momentum space and the total energy of an atom through the virial theorem. For k=+1, −1, and −2, the numerical Hartree-Fock results for the 357 subshells show that there are approximate but accurate linear relations between <u ^k > _nl and <r ^k > _nl and between <v ^k > _nl and <p ^k > _nl , in which the proportionality constant in each space depends on n,l, and k. Received: 27 April 1998 / Accepted: 29 May 1998 / Published online: 28 August 1998     

Correlated electron-pair properties of the Li atom in momentum space

On the basis of multiconfiguration Hartree–Fock calculations, correlated electron-pair intracule (relative motion) and extracule (center-of-mass motion) properties are reported for the Li atom in momentum space. The present results are more accurate and consistent than those in the literature. Received: 10 September 2001 / Accepted: 11 December 2001 / Published online: 22 March 2002     

Energy-density relations in momentum space

The integrated Hellmann-Feynman theorem is used to derive a rigorous relation between the energy and the electron density in momentum space. Choosing the electron mass as a differential parameter, we obtain a formula corresponding to the Wilson-Frost formula in coordinate space. Analysing the mass-dependence of momentum density, we then show that the present formula is equivalent to one of the previous results deduced from the virial theorem. Use of the integral Hellmann-Feynman theorem is also discussed. Several illustrative examples are given for the calculation of energy from momentum density.     

Density matrices from position and momentum densities

Summary One-electron density matrices, which are representable in single-centers-orbital basis sets, have been investigated with respect to their reconstruction from densities. The maximum allowed dimension for reconstruction from a combination of position & momentum density dependent properties is only slightly bigger than the dimension in the case of position (or momentum) densities only. Since for a given one-particle basis of dimensionM, the number of one-matrix elements which can be determined is also of orderM only, while the total number of one-matrix elements is of orderM ², it is in general necessary to introduce severe constraints and restrictions. The accuracy demands on the data and algorithms increase exponentially for linearly increasing size of basis set.     

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.     

Isomorphism in electron-pair densities of atoms

The spherically averaged electron-pair intracule (relative motion) h(u) and extracule (center-of-mass motion) d(R) densities are a couple of densities which characterize the motion of electron pairs in atomic systems. We study a generalized electron-pair density (q; a, b) that represents the probability density function for the magnitude of two-electron vector a r _j +b r _k of any pair of electrons j and k to be q, where a and b are nonzero real numbers. In particular, h(u)=g(u;1, −1) and d(R) = . It is shown that the scaling property of the Dirac delta function and the inversion symmetry of orbitals in atoms due to the central force field generate several isomorphic relations in the electron-pair density (q; a, b) with respect to the two parameters a and b. The approximate isomorphism d(R)≅8h(2R) known in the literature between the intracule and extracule densities is a special case of the present results. Received: 24 May 2000 / Accepted: 18 July 2000 / Published online: 27 September 2000     

On numerical solution of the integral Hartree-Fock equations for molecules by the method of MO LCAO in the momentum space

To develop a numerical solution of mentioned equations the method of factorized projection of integral operator kernel is applied. All matrix elements of the method are calculated analytically, being expressed in terms of two types of standard integrals: the overlap integrals and one-electron Coulomb integrals. To calculate the integrals we used the O(4)-symmetry of hydrogen-like atomic orbitals as well as operational technique of differentiation with respect to scalar and vector parameters.     

Atoms-in-molecules in momentum space: a Hirshfeld partitioning of electron momentum densities

A direct application of the Hirshfeld atomic partitioning (HAP) scheme is implemented for molecular electron momentum densities (EMDs). The momentum density contributions of individual atoms in diverse molecular systems are analyzed along with their topographical features and the kinetic energies of the atomic partitions. The proposed p-space HAP-based charge scheme does seem to possess the desirable attributes expected of any atoms in molecules partitioning. In addition to this, the main strength of the p-space HAP is the exact knowledge of the kinetic energy functional and the inherent ease in computing the kinetic energy. The charges derived from HAP in momentum space are found to match chemical intuition and the generally known chemical characteristics such as electronegativity, etc.     

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.     

Two-electron atomic wave functions which satisfy proportionality relations between one-and two-electron moments

Based on momentum- and position-space analyses of the moment operators for two-electron atoms, it is shown that there exists a family of two-electron wave functions which satisfy a proportionality relation, r/ ₁ ^v /r ₁₂ ^v =p/ ₁ ^v /p ₁₂ ^v =2^–v/2, between the one and two-electron moments in position and momentum spaces, where v is an arbitrary number for which the moments are well-defined.