Spin Effects in sp 2 Nanocarbons in the Light of Unrestricted Hartree-Fock Approach and Spin-Orbit Coupling Theory

2017 ◽  
pp. 39-63
Author(s):  
Elena F. Sheka
Keyword(s):  
Orbit Coupling ◽  
Spin Orbit Coupling ◽  
Spin Orbit ◽  
Coupling Theory ◽  
Hartree Fock ◽  
Spin Effects

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.

Spin and Spin-orbit Coupling Effects in Nickel-based Superalloys: A First-principles Study on Ni3Al Doped with Ta/W/Re

2021 ◽  
Author(s):  
Liping Liu ◽  
Jin Cao ◽  
Wei Guo ◽  
Chongyu Wang
Keyword(s):  
Electronic Structures ◽  
Formation Energy ◽  
Orbit Coupling ◽  
Heavy Elements ◽  
Spin Orbit Coupling ◽  
Spin Orbit ◽  
Spin Effects ◽  
Element Doping ◽  
Spin Polarized ◽  
Necessary And Sufficient

Abstract Heavy elements (X = Ta/W/Re) play an important role in the performance of superalloys, which enhance the strength, anti-oxidation, creep resistance, and anti-corrosiveness of alloy materials in a high-temperature environment. In the present research, the heavy element doping effects in FCC-Ni (γ) and Ni3Al (γ') systems are investigated in terms of their thermodynamic and mechanical properties, as well as electronic structures. The lattice constant, bulk modulus, elastic constant, and dopant formation energy in non-spin, spin polarized, and spin-orbit coupling (SOC) calculations are compared. The results show that the SOC effects are important in accurate electronic structure calculations for alloys with heavy elements. We find that including spin for both γ and γ' phases is necessary and sufficient for most cases, but the dopant formation energy is sensitive to different spin effects, for instance, in the absence of SOC, even spin-polarized calculations give 1% to 9% variance in the dopant formation energy in our model. Electronic structures calculations indicate that spin polarization causes a split in the metal d states, and SOC introduces a variance in the spin-up and spin-down states of the d states of heavy metals and reduces the magnetic moment of the system.

Spin-orbit coupling from a two-component self-consistent approach. I. Generalized Hartree-Fock theory

2019 ◽  
Vol 151 (7) ◽  
pp. 074107
Author(s):  
Jacques K. Desmarais ◽  
Jean-Pierre Flament ◽  
Alessandro Erba
Keyword(s):  
Orbit Coupling ◽  
Spin Orbit Coupling ◽  
Spin Orbit ◽  
Hartree Fock ◽  
Two Component ◽  
Self Consistent ◽  
Consistent Approach

Hubbard Trimer with Spin–Orbit Coupling: Hartree–Fock Solutions, (Non)Collinearity, and Anisotropic Spin Hamiltonian

2019 ◽  
Vol 123 (12) ◽  
pp. 2361-2378 ◽  
Author(s):  
Shadan Ghassemi Tabrizi ◽  
Alexei V. Arbuznikov ◽  
Martin Kaupp
Keyword(s):  
Orbit Coupling ◽  
Spin Orbit Coupling ◽  
Spin Orbit ◽  
Spin Hamiltonian ◽  
Hartree Fock
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