Absorption Spectrum of the NO Molecule. X. The 3d Rydberg Complex, its Vibrational Structure, Spin–Orbit Coupling, and Interactions with Non-Rydberg States

1971 ◽  
Vol 49 (18) ◽  
pp. 2350-2365 ◽  
Author(s):  
E. Miescher
Keyword(s):  
Vacuum Ultraviolet ◽  
Rydberg States ◽  
Vibrational Structure ◽  
Spin Orbit Coupling ◽  
Spin Orbit ◽  
Matrix Elements ◽  
Absorption Bands ◽  
Unusual Structure ◽  
Vibrational Levels ◽  
The Matrix

The vibrational levels of the 3dσ,π Rydberg complexes of the NO molecule are studied. They give rise to absorption bands 3d–X in the vacuum ultraviolet, most of which are severely overlapped by other bands, and, in addition, to Rydberg–Rydberg emission bands in the visible (3d–3s) and in the infrared (3d–3p). Both types of spectra have been photographed at high resolution, the absorption spectra for three different isotopes.Interactions with non-Rydberg levels and predissociations are observed. A detailed analysis, including spin–orbit effects, is given of the crossing between the 3dσ,π(ν = 1) and B2Π(ν = 23) rotational levels. Off-diagonal matrix elements are determined and precise vibrational and rotational constants are calculated. Largely responsible for the unusual structure of the bands is the matrix element [Formula: see text] which is shown to be independent of ν.

Matrix Elements of the Spin–Orbit Coupling for an ln Configuration in a Crystalline Electric Field

1972 ◽  
Vol 50 (2) ◽  
pp. 78-83 ◽  
Author(s):  
H. A. Buckmaster ◽  
R. Chatterjee ◽  
Y. H. Shing
Keyword(s):  
Electric Field ◽  
General Expression ◽  
Ground State ◽  
Orbit Coupling ◽  
Spin Orbit Coupling ◽  
Spin Orbit ◽  
Matrix Elements ◽  
Crystalline Electric Field ◽  
State Splitting ◽  
The Matrix

A general expression for the matrix elements of the spin–orbit coupling for an ln configuration in a crystalline electric field of arbitrary symmetry is derived using Racah formalism. This calculation is an extension of Lulek's treatment of this problem for an l1 configuration. This general expression is used to calculate the contribution to the ground-state splitting for the S-state lanthanide ion Gd3+ (4f7; 8S7/2)in an axial crystalline electric field of a second-order perturbation mechanism involving the matrix element of the spin–orbit coupling. It is shown that this mechanism, which was proposed by Lulek is incapable of explaining the observed ground-state splitting.

Magneto-optical properties of defect centres in alkali halide crystals

1967 ◽  
Vol 300 (1460) ◽  
pp. 78-93 ◽  
Keyword(s):  
Optical Absorption ◽  
Silver Atom ◽  
Alkali Halides ◽  
Spin Orbit Coupling ◽  
Spin Orbit ◽  
Coupling Constant ◽  
Absorption Bands ◽  
Optical Effects ◽  
Alkali Halide Crystals ◽  
Defect Centres

Zeeman spectroscopy is not practicable for the investigation of the structure of electronic conventional states which give rise to broad optical absorption bands in solids. We have investigated the application of Faraday rotation and circular dichroism techniques to absorption bands of neutral silver atoms and F centres in alkali halides. These centres give rise to optical absorption bands due to transitions of the type 2 S → 2 P which are 2000 to 6000 cm -1 in width because, in part, of strong coupling to lattice phonons. A discussion is given of information which may be obtained concerning the electonic states involved in the 2 S → 2 P transition by analysis of the magneto-optical effects by the method of moments. It is shown, for example, that the spin-orbit coupling constant of the 2 P state of the silver atom is reduced from 613 cm -1 in the free state to 365 cm -1 in KCl, to 102 cm -1 in KBr and to an unmeasurably small value in KI. This cancellation of spin-orbit interaction of the silver atom is assigned to symmetry allowed admixtures of lattice ion wavefunctions into the 2 P state.

scholarly journals Spin-Orbit Matrix Elements for a Combined Spin-Flip and IP/EA Approach

2020 ◽  
Author(s):  
Oinam Meitei ◽  
Shannon Houck ◽  
Nicholas Mayhall
Keyword(s):  
Mean Field ◽  
Coupling Constants ◽  
Basis Set ◽  
Reduced Density Matrix ◽  
Spin Orbit Coupling ◽  
Spin Orbit ◽  
Matrix Elements ◽  
Dynamical Correlation ◽  
The Core ◽  
Reduced Density

We present a practical approach for computing the Breit-Pauli spin-orbit matrix elements of multiconfigurational systems with both spin and spatial degeneracies based on our recently developed RAS-nSF-IP/EA method (JCTC, 15,<br>2278, 2019). The spin-orbit matrix elements over all the multiplet components are computed using a single one-particle reduced density matrix as a result of the Wigner-Eckart theorem. A mean field spin-orbit approximation was used to account for the two-electron contributions. Basis set dependence as well as the effect of including additional excitations is presented. The effect of correlating the core and semi-core orbitals is also examined. Surprisingly accurate results are obtained for spin-orbit coupling constants, despite the fact that the efficient wavefunction approximations we explore neglect the bulk of dynamical correlation.<br>

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.

On the Phosphorescence of Benzologues of Furan , Thiophene , Selenophene , and Tellurophene . A Systematic Study of the Intra -annular Internal Heavy-atom Effect

1989 ◽  
Vol 44 (3) ◽  
pp. 205-209 ◽  
Author(s):  
M. Zander ◽  
G. Kirsch
Keyword(s):  
Systematic Study ◽  
Linear Correlation ◽  
Heavy Atom ◽  
Vibrational Structure ◽  
Spin Orbit Coupling ◽  
Spin Orbit ◽  
Coupling Constant ◽  
Atom Effect ◽  
Internal Perturbation ◽  
Phosphorescence Spectra

Phosphorescence in ethanol and n-pentane at 77 K of the benzologues 1-8 of furan, thiophene, selenophene and tellurophene has been investigated. The rate constants of both the radiative (kPT) and non-radiative (kGT) deactivation of the lowest triplet state correlate linearly with where ck denotes the Hückel AO coefficients in the HOMO of the carbon atoms bound to the hetero-atom and ζ is the spin-orbit coupling constant of the hetero-atom present. - The linear correlation observed between kPT and kGT is an example for the Orlandi-Siebrand rule. - The influence of an external heavy-atom perturber (methyl iodide) on phosphorescence lifetimes and the vibrational structure of phosphorescence spectra is the more efficient the less efficient is the internal perturbation caused by the intra-annular heavy-atom.

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.

Infrared and Visible Emission Spectrum of the NO Molecule. The 2Δ–2Π Bands of the 3d–3p and 4d–3p Transitions between Rydberg Complexes

1971 ◽  
Vol 49 (1) ◽  
pp. 76-89 ◽  
Author(s):  
F. Ackermann
Keyword(s):  
High Resolution ◽  
Emission Spectrum ◽  
Rydberg State ◽  
Energy Levels ◽  
Rydberg States ◽  
Spin Orbit Coupling ◽  
Spin Orbit ◽  
Rotational Analysis ◽  
Mixed States ◽  
Doublet Splitting

The two mutually related bands B′2Δ–C2Π (7,0) → N2Δ–C2Π (0,0) and N2Δ–C2Π (0,0) → B′2Δ–C2Π (7,0) are observed with high resolution between 6620 and 6520 Å in the emission spectrum of the NO molecule. They are the 2Δ–2Π part of the 4d–3p transitions between the two Rydberg states N2Δ(4dδ) and C2Π (3pπ) of the molecule. A rotational analysis is carried out for both bands, and the very close similarity of the structure of these bands with the structure of the corresponding 2Δ–2Π bands of the 3d–3p transitions, observed in the infrared, is demonstrated. The two upper levels in these nd–3p transitions represent examples of mixed states showing complete changeover with increasing rotation from the Rydberg type with no spin–orbit coupling (AR = 0.00 ± 0.05 cm−1) to an inverted valence type and vice versa. The behavior of the doublet splitting is studied with regard to this changeover. The lower levels of the Rydberg state C2Π also are mixtures with levels of a valence state. The mixing with B2Π (ν = 7) is comparatively small in the C2Π (ν = 0) level, but it strongly affects the energy levels with the lowest J values. The beginning of one of the two bands observed in the visible, therefore, forms the (7,7) band of the system B′2ΔB2Π. Constants of the states involved are determined.

Spin-Orbit Coupling in Tetragonal d3 Systems

1981 ◽  
Vol 36 (12) ◽  
pp. 1276-1288 ◽  
Author(s):  
Patrick E. Hoggard
Keyword(s):  
Metal Ions ◽  
Orbit Coupling ◽  
Ligand Field ◽  
Spin Orbit Coupling ◽  
Spin Orbit ◽  
Matrix Elements ◽  
Angular Overlap Model ◽  
Forbidden Transitions ◽  
Complete Set ◽  
Overlap Model

Abstract The complete set of energy matrices for tetragonal d3 systems, including spin-orbit coupling, has been constructed within the framework of the Angular Overlap Model. Examples are presented of the variation in energy of the intraconfigurational (t32g) doublets as a function of spin-orbit coupling (appropriate to first row metal ions) and ligand field asymmetry. It is seen that asymmetry in Dq values is much less important than the relative partitioning of Dq into eσ and eπ. The use of spin-orbit matrix elements in the calculation of intensities of spin-forbidden transitions is also illustrated.

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