Spin–orbit and spin–other–orbit interactions for f4 electron configuration

1969 ◽  
Vol 47 (17) ◽  
pp. 1829-1862 ◽  
Author(s):  
K. M. S. Saxena ◽  
Gulzari Malli
Keyword(s):  
Electron Configuration ◽  
Spin Orbit ◽  
Matrix Elements ◽  
Atomic Systems ◽  
Hartree Fock ◽  
The Matrix

General expressions for the reduced matrix elements of spin–orbit and spin–other–orbit interactions are evaluated for all the states arising from f4 electron configuration. These are used to calculate the Hartree–Fock values of the matrix elements of the above-mentioned interactions for Nd(4f4), Dy(4f10), and Ho3+ (4f10) atomic systems.

Orbit–orbit interaction inn fN electron configurations

1969 ◽  
Vol 47 (17) ◽  
pp. 1885-1888 ◽  
Author(s):  
K. M. S. Saxena ◽  
G. Malli
Keyword(s):  
Orbit Interaction ◽  
Matrix Elements ◽  
Atomic Systems ◽  
Hartree Fock ◽  
The Matrix

The expressions of the matrix elements of the orbit–orbit interaction for various fN electron configurations are computed and tabulated for general usage. These expressions are used to evaluate the Hartree–Fock values of the orbit–orbit interaction in all the states for a large number of fN electron atomic systems.

Theory of spin-orbit coupling in atoms I. Derivation of the spin-orbit coupling constant

1962 ◽  
Vol 270 (1340) ◽  
pp. 127-143 ◽  
Keyword(s):  
Exact Expression ◽  
Orbit Coupling ◽  
Tensor Operator ◽  
Spin Orbit Coupling ◽  
Spin Orbit ◽  
Matrix Elements ◽  
Coupling Constant ◽  
Hartree Fock ◽  
Operator Form ◽  
Hartree Potential

An exact expression for the spin-orbit coupling constant is derived within the Hartree-Fock description of the atom by considering the two body mutual spin-orbit interaction between electrons. The interaction is rewritten in tensor operator form and the contribution of outer electron-core interactions to the coupling constant is calculated. We find that the usual expression < 3F/r8r > where V is the Hartree potential is only approximate, and that certain exchange type terms, which arise because we are dealing with a two-body interaction and determinantal wave function, must also be included. These exchange terms are not simply related to the ordinary electrostatic exchange. The resulting expression for the spin-orbit coupling constant is given in terms of radial integrals which can be calculated using Hartree or Hartree—Fock wave functions. We also discuss the effective magnetic Hamiltonian to be used for the calculation of matrix elements within an atomic configuration.

scholarly journals DIAGONAL AND NONDIAGONAL MATRIX ELEMENTS OF THE EFFECTIVE HAMILTONIAN OF ELECTRONS IN A SEMICONDUCTOR (TAKING INTO ACCOUNT SPIN-ORBIT INTERACTION)

2020 ◽  
pp. 120-127
Author(s):  
Voxob Rustamovich Rasulov ◽  
Rustam Yavkachovich Rasulov ◽  
Akhmedov Bahodir Bahromovich ◽  
Ravshan Rustamovich Sultanov
Keyword(s):  
Wave Function ◽  
Matrix Element ◽  
Valence Band ◽  
Heavy Hole ◽  
Orbit Interaction ◽  
Effective Hamiltonian ◽  
Spin Orbit ◽  
Matrix Elements ◽  
The Matrix ◽  
Split Band

The matrix elements of the effective Hamiltonian of current carriers are calculated as in the Kane approximation, where the conduction band, the valence band consisting of light and heavy hole subbands, and the spin-split band, as well as in the Luttinger-Kohn model, are considered. KEYWORDS: matrix element, effective Hamiltonian, current carriers, wave function.

Spin–spin interaction in f5 electron configurations

1969 ◽  
Vol 47 (24) ◽  
pp. 2779-2783 ◽  
Author(s):  
Gulzari Malli ◽  
K. M. S. Saxena
Keyword(s):  
Order Perturbation ◽  
Spin Splitting ◽  
Spin Interaction ◽  
Electron Configuration ◽  
Matrix Elements ◽  
First Order ◽  
Hartree Fock ◽  
Splitting Factor ◽  
Complete Matrix ◽  
First Order Perturbation

The complete matrix of the spin–spin interaction for the f5 electron configuration is evaluated for general usage. The results of the diagonal matrix elements are tabulated here in terms of the corresponding spin–spin splitting factor for usage in the first-order perturbation calculations. These are used to evaluate the Hartree–Fock values of the splitting factors for Pm(4f5), Sm3+ (4f5), Tb(4f9), and Dy3+ (4f9).

Fine Structure Intervals in Transition Elements

1975 ◽  
Vol 53 (21) ◽  
pp. 2421-2427 ◽  
Author(s):  
Jacek Karwowski ◽  
K. M. S. Saxena ◽  
Serafin Fraga
Keyword(s):  
Fine Structure ◽  
Ground State ◽  
Orbit Interaction ◽  
Matrix Elements ◽  
Transition Elements ◽  
Hartree Fock ◽  
Positive Ions ◽  
Transition Series ◽  
The Matrix ◽  
New Formulation

A new formulation for the evaluation of the matrix elements of the spin-own orbit interaction in many-electron atoms has been applied to the evaluation of the interaction matrices for pN, dN, and fN configurations, using functions that are simultaneous eigenfunctions of the operators J2, L2,S2, and.Jz; the complete results are available as indicated in the text. Using this formulation, the fine structure intervals for the ground states of the neutral atoms and the first three positive ions of the elements of the three transition series have been calculated within the framework of the monoconfigurational approximation, including the electrostatic and spin-own orbit interaction between the states arising from the configuration under consideration. In each case, the spin–orbit parameter and the set of Slater–Condon integrals, obtained from the numerical Hartree–Fock function for the ground state, were used.

Electron Spin–Spin Contact Interaction in Atoms with fn Configurations

1971 ◽  
Vol 49 (15) ◽  
pp. 2031-2032 ◽  
Author(s):  
K. M. S. Saxena ◽  
B. W. N. Lo ◽  
S. Fraga
Keyword(s):  
Contact Interaction ◽  
Electron Spin ◽  
Numerical Calculations ◽  
Matrix Elements ◽  
Hartree Fock ◽  
The Matrix ◽  
Lanthanide Atoms

The expressions of the matrix elements of the electron spin–spin contact interaction have been tabulated for all the states arising from fn configurations. Numerical calculations have been carried out for a large number of lanthanide atoms and ions using accurate numerical Hartree–Fock functions.

ON THE SPIN–ORBIT INTERACTION IN DIATOMIC MOLECULES. I

1958 ◽  
Vol 36 (3) ◽  
pp. 309-328 ◽  
Author(s):  
I. Kovács
Keyword(s):  
Diatomic Molecules ◽  
Electron System ◽  
Orbit Interaction ◽  
Spin Orbit ◽  
Matrix Elements ◽  
Molecular Constants ◽  
Spin Orbit Interaction ◽  
The Matrix

The purpose of the present paper is to give further details of an investigation of the spin–orbit interaction in diatomic molecules. The first part of the paper deals with perturbations between states of odd multiplicity in a two-electron system. With the aid of the matrix elements which have been obtained, a comparison of the perturbations in the various components of a given state and in some cases a comparison of perturbations in two different states is possible. Also for certain cases it has been possible to establish relationships between the matrix elements which give the magnitude of the perturbations and experimentally measurable molecular constants of the perturbed states.

Derivation and testing of a molecular orbital description of ligand field spectra

1974 ◽  
Vol 276 (1258) ◽  
pp. 277-339 ◽  
Keyword(s):  
Least Squares ◽  
Molecular Orbital ◽  
Energy Separation ◽  
Ligand Field ◽  
Spin Orbit ◽  
Matrix Elements ◽  
Orbital Parameters ◽  
Orbital Basis ◽  
The Matrix ◽  
Ligand Field Spectra

A model for describing the ligand-field spectra of cubic chromophores in a molecular orbital basis is derived and tested by carrying out least-squares-fits of the spectra of the tetrahedral ions CoX 2- 4 (X = Cl Br). Independent assignments of the ligand-field spectra of the latter ions are obtained by analysing the site-group splittings and vibrational fine-structure in the polarized spectra of Cs 3 CoCl 5 and Cs 3 CoBr 5 at 4.2 K. General expressions are calculated for the electron repulsion matrix elements of pairs of electrons occupying orbitals transforming as e, t 1 and t 2 in Oh and Td point groups, without further assumption about the functional form of the orbitals. Tanabe & Sugano’s (1954) matrices for d 1 ~d 5 configurations are rewritten in the more general form. The two-electron reduced matrix elements appearing in these expressions are further approximated in terms of orbital overlap populations (Mulliken 1955), one- and two-centre coulomb integrals and one-centre exchange integrals. An analogous reduction of the matrix elements of the molecular spin-orbit operator within a molecular orbital basis is also described. The energies of the spin-orbit baricentres of the ligand-field transitions are then calculated as functions of three orbital population parameters and the energy separation between the e and t 2 molecular orbitals. From the least-squares analysis of the independently assigned experimental spectra, empirical values of the four parameters are extracted, and their relation to the parameters of conventional ligand-field theory is discussed. The experimentally based molecular orbital parameters are further used to calculate spinorbit splittings of the cubic Russell-Saunders ligand-field states, and the relative dipole strengths of transitions to those which are formally spin-forbidden from the ground state.

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