Matrix Elements Of Coulomb Interaction Research Articles

Generation and Verification of Algorithms for Symbolic-Numeric Processing

Some large scale physical computations require algorithms performing symbolic computations with a particular class of algebraic formulas in a numerical code. Developing and implementing such algorithms in a numerical programming language is a tedious and error prone task. The algorithms can be developed in a computer algebra system and their correctness can be checked by comparison with build-in facilities of the system so that the system is used as an advanced debugging tool. After that a numerical code for the algorithms is automatically generated from the same source code. The proposed methodology is explained in detail on a simple example. Real applications to calculation of matrix elements of Coulomb interaction and two-centre exchange integrals needed in atomic collision codes, are described. The method makes the developing and debugging of such algorithms easier and faster.

Coincident two-electron emission from surfaces by low-energy electrons

The simultaneous ejection of two electrons from non-magnetic surfaces due to the collision of incident low-energy electrons with valence electrons is treated in a relativistic distorted-wave Born approximation including exchange. The primary electron and the two emitted electrons are described by quasi-particle multiple scattering states. The valence electron is represented by linear combinations of Bloch waves matched at the surface. Screened Coulomb interaction matrix elements between these four states are evaluated. Numerical results for W(0 0 1) are presented and compared with one-dimensional bulk densities of states. Energy-integrated spectra and general features of the 2 e-distribution are in good agreement with recent experimental data.

Inelastic scattering of polarized electrons from nonmagnetic metals via electron-hole excitation

Inelastic scattering of low-energy electrons from nonmagnetic metals due to electron-hole pair excitation is treated in a distorted-wave first Born approximation including exchange. Screened Coulomb interaction matrix elements between bulk Bloch states and plane waves (including elastic scattering in a simple approximation) are evaluated to yield, for a spin-polarized primary beam, the intensities of electrons scattered with and without spin flip. Some basic insight is obtained by calculations for a homogeneous electron gas. Numerical results for Cu(001) are found to agree fairly well with experimental data and provide clear evidence for the importance of elastic scattering processes in electron energy loss spectroscopy.

Coulomb interaction in the supermultiplet basis

The approximate expression for matrix elements of Coulomb interaction in the supermultiplet basis, taking into account the orbitally non-symmetric terms, is obtained. The quantum numbers for the orbital part of the interaction are not specified. The expression can be used in nuclear models employing the supermultiplet scheme. The obtained expression has been used to calculate the Coulomb energy of light (4<or=A<or=40) and heavy (158<or=A<or=196) nuclei in the framework of a strictly restricted dynamics model. The results are in sufficient agreement with the experimental data for light mirror nuclei and with the results of semi-empirical calculations for heavy nuclei. The contribution of the orbitally non-symmetric part to the full Coulomb energy matrix element has been estimated to be less than 5%.

Quantum subband interference effect on the exciton spectra in semiconductor quantum wells.

The set of eigenequations for determining the energy levels and wave functions of Wannier excitons in a semiconductor quantum well form a matrix equation labeled by the subband indices of the quantum well. The nondiagonal matrix elements of the electron-hole Coulomb interaction give rise to the quantum subband interference effect between excitons of different subband branches. We solve this coupled matrix eigenequation by an accurate numerical method. The calculated values of exciton binding energies are significantly enhanced by including this effect.

A reduce package for exact Coulomb interaction matrix elements

Title of program: R12 interaction matrix elements Catalogue number: ABBN Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland (see application form in this issue) Computer: IBM compatible PC/ Cray 1 M Operating system: MS DOS 3.x/ COS 1.15 Programming language used: Reduce 3.2 High speed storage required: 512 KB for PC/ 290 Kwords for Cray No. of bits in a word: 64 No. of lines in combined program and test deck: 500

The Mossbauer effect and electron configuration for tin metal

The electron contact densities for the various electron configurations of tin and its oxidation states have been computed using the relativistic multi-configuration Dirac-Fock program of Desclaux (1975). The interactions between the configurational states obtained using jj-coupling and the seniority scheme have been investigated using Grant's program (1976) MCP75. All the off-diagonal matrix elements of the inter-electronic Coulomb interaction between the configurational states are found to be vanishingly small. The dependence of the computed electron contact density on the valence shell occupation numbers has been investigated and a phenomenological formula obtained. Using the formula and the measured Mossbauer isomer shift, the electron configuration of beta -tin has been found to be 5s1.785p2.22 in good agreement with the literature value.

Laporte-Platt degeneracies and delta-function interactions

The Laporte-Platt degeneracies for the spectroscopic terms of the configurations lN are equivalent, when l=2, to the repeated eigenvalues found by diagonalising a delta-function interaction delta (ri-rj) for electrons i and j. For l>2 a limited generalisation is possible, since terms of maximum multiplicity remain degenerate while much of the degeneracy of the other terms is lost. This is put in the context of group theory, the techniques of which are used to account for the invariably null matrix elements of delta (rh-ri) delta (ri-rj). Configurations involving inequivalent electrons lead to generalisations of the Laporte-Platt degeneracies, and the use of the interaction Sigma delta (ri-rj) provides checks on the matrix elements of the Coulomb interaction.

Matrix elements in configuration interaction calculations

AbstractA method is proposed for the calculation of matrix elements among various states of atoms. A set of tensor operators is the only entity in the formalism, and all formulas involve merely the vacuum expectation values of these tensor operators and the recoupling transformation coefficients. Some numerical examples are given for the Coulomb interaction matrix elements.

Superconductivity in Nontransition Metals. I

The theoretical calculation of superconducting-state parameters $〈{V}_{\mathrm{ph}}〉$, $〈{V}_{\mathrm{C}}〉$, and ${Z}_{0}$ [$〈{V}_{\mathrm{ph}}〉$ estimates the phonon-mediated attractive electron-electron interaction in the metal; $〈{V}_{\mathrm{C}}〉$, the Coulomb pseudo-potential, measures the effectiveness of the Coulomb repulsion in inhibiting superconductivity; and ${Z}_{0}$, the quasiparticle mass (or energy) renormalization parameter, takes into account the many-body renormalization effects due to electron-phonon and Coulomb interactions] is undertaken for both nonsuperconductors and simple-metal (nontransition-metal) superconductors. The two questions faced are: why non-superconductors (alkali, alkaline-earth, and noble metals) are nonsuperconductors (this question arises from the startling prediction by Morel and Anderson that all metals should be superconductors); and whether better agreement can be achieved between theoretically estimated superconducting-state parameters and empirical parameters---better, e.g., than those of Pines and of Morel and Anderson. Our calculation is differentiated from that of Morel and Anderson first in using electron-phonon and Coulomb-interaction matrix elements between Bloch electron states, secondly in using for the electron-phonon matrix element the form suggested by Harrison for the orthogonalized-plane-wave form factor, and lastly in including the renormalization effects due to electron-phonon and Coulomb interactions. The renormalization due to Coulomb interaction is included by taking the values from the calculations by Rice. The Bloch electron states are calculated in a model in which the ionic core is replaced by a three-dimensional constant repulsive potential, and the wave function is determined in the spirit of the Wigner-Seitz approximation and first-order perturbation theory. The theoretically estimated parameters $〈{V}_{\mathrm{ph}}〉$, $〈{V}_{\mathrm{C}}〉$, and ${Z}_{0}$ are compared with the empirically estimated parameters. These are also compared with the parameters estimated by Garland. A good comparison---better than that of Morel and Anderson---is found between theoretical and empirical parameters. It is pointed out that if renormalization effects are included in the calculation by Morel and Anderson, then their apparently good comparison with empirical parameters does not remain so good. In the case of nonsuperconductors, we find that the phonon-mediated attractive interaction fails to dominate over the Coulomb repulsion, contrary to the conclusions of Morel and Anderson, and the effective interaction strength $\frac{(〈{V}_{\mathrm{ph}}〉\ensuremath{-}〈{V}_{\mathrm{C}}〉)}{{Z}_{0}}$, which is analogous to the BCS parameter ${N}_{0}V$, is repulsive or very slightly attractive. The nonsuperconducting status is thus restored to alkali and alkaline-earth metals.

H-H interaction potentials

Direct Coulomb interaction matrix elements between hydrogen atoms in ground states calculated, presenting interaction potential

Electrostatic Interaction of dn Systems in Tetragonal Fields

Energy levels of n d transition metal ions in a ligand field of the tetragonal symmetry are studied in the strong field approximation. In this field, d level is split into four sub-levels differing in energy, of symmetry b 1 , a 1 , e and b 2 respectively. Starting from the configurations containing these orbitals, Coulomb interaction between electrons is introduced, which gives rise to energy levels, each characterized by \(S\varGamma\). Matrix elements of Coulomb interaction are expressed in terms of Racah's parameters A , B and C with the ligand field parameters X , Y and Z which give the intervals between orbital energies. These matrix elements are shown for d 2 , d 3 , d 4 and d 5 .