Spin-orbit interaction electronic Hamiltonian

The expressions for the rotational energy levels (i.e., also involving the end-over-end rotations, not considered in the previous works) of linear triatomic molecules in doublet and triplet II electronic states that take into account a spin orbit interaction and a vibronic coupling were derived in two milestone studies by Hougen [72,32]. In them, the isomorfic Hamiltonian was inboduced, which has later been widely used in treating linear molecules (see, e.g., [55]). 

The spin magnetic moment Ms of an electron interacts with its orbital magnetic moment to produce an additional term in the Hamiltonian operator and, therefore, in the energy. In this section, we derive the mathematical expression for this spin-orbit interaction and apply it to the hydrogen atom. 

An indirect mode of anisotropic hyperfine interaction arises as a result of strong spin-orbit interaction (174)- Nuclear and electron spin magnetic moments are coupled to each other because both are coupled to the orbital magnetic moment. The Hamiltonian is... 

Electronic Hamiltonian, conical intersections, spin-orbit interaction, 559 Electronic states ... 

The main effect of taking spin-orbit interaction into account will be an admixture of singlet character to triplet states and triplet character to singlet states. The spin-orbit coupling Hamiltonian can to a good approximation be described by an effective one-electron operator Hso ... 

Consideration of the spin-orbit interaction and the effect of an external magnetic field on the electronic ground state of an ion in a CF allows evaluation of the various terms in the spin-Hamiltonian of Equation (31). In addition, the interaction of the nucleus of the paramagnetic ion and ligand nuclei with the d-electron cloud must be considered. In this way the experimentally determined terms of the spin-Hamiltonian may be related to such parameters as the energy differences between levels of the ion in the CF and amount of charge transfer between d-electrons and ligands. 

Let us emphasize that in single-configurational approach the terms of the Hamiltonian describing kinetic and potential energies of the electrons as well as one-electron relativistic corrections, contribute only to average energy and, therefore, are not contained in, which in the non-relativistic approximation consists only of the operators of electrostatic interaction e and the one-electron part of the spin-orbit interaction so, i.e. 

Hamiltonian Eq. (1.24) is formally equivalent to that describing the dipolar interaction between two spins s and S2 whose sum is S. Actually, in organic radicals where spin-orbit interactions are negligibly small, the dipolar interaction between the two electron spins in an S = 1 system causes ZFS. 

There is a general statement [17] that spin-orbit interaction in ID systems with Aharonov-Bohm geometry produces additional reduction factors in the Fourier expansion of thermodynamic or transport quantities. This statement holds for spin-orbit Hamiltonians for which the transfer matrix is factorized into spin-orbit and spatial parts. In a pure ID case the spin-orbit interaction is represented by the Hamiltonian //= a so)pxaz, which is the product of spin-dependent and spatial operators, and thus it satisfies the above described requirements. However, as was shown by direct calculation in Ref. [4], spin-orbit interaction of electrons in ID quantum wires formed in 2DEG by an in-plane confinement potential can not be reduced to the Hamiltonian H s. Instead, a violation of left-right symmetry of ID electron transport, characterized by a dispersion asymmetry parameter Aa, appears. We show now that in quantum wires with broken chiral symmetry the spin-orbit interaction enhances persistent current. 

In quantum wires formed in a two-dimensional electron gas (2DEG) by lateral confinement the Rashba spin-orbit interaction is not reduced to a pure ID Hamiltonian H[s = asopxaz. As was shown in Ref. [4] the presence of an inplane confinement potential qualitatively modifies the energy spectrum of the ID electrons so that a dispersion asymmetry appears. As a result the chiral symmetry is broken in quantum wires with Rashba coupling. Although the effect was shown [4] not to be numerically large, the breakdown of symmetry leads to qualitatively novel predictions. 

Since in many cases the Coulomb interaction between the electrons dominates their spin-orbit interaction (see the Hamiltonian in equ. (1.1)),... 

The first term in eq. (1) Ho represents the spherical part of a free ion Hamiltonian and can be omitted without lack of generality. F s are the Slater parameters and ff is the spin-orbit interaction constant /<- and A so are the angular parts of electrostatic and spin-orbit interactions, respectively. Two-body correction terms (including Trees correction) are described by the fourth, fifth and sixth terms, correspondingly, whereas three-particle interactions (for ions with three or more equivalent f electrons) are represented by the seventh term. Finally, magnetic interactions (spin-spin and spin-other orbit corrections) are described by the terms with operators m and p/. Matrix elements of all operators entering eq. (1) can be taken from the book by Nielsen and Koster (1963) or from the Argonne National Laboratory s web site (Hannah Crosswhite s datafiles) http //chemistry.anl.gov/downloads/index.html. In what follows, the Hamiltonian (1) without Hcf will be referred to as the free ion Hamiltonian. 

Most of the variational treatments of spin-orbit interaction utilize one-component MOs as the one-particle basis. The SOC is then introduced at the Cl level. A so-called SOCI can be realized either as a one- or two-step procedure. Evidently, one-step methods determine spin-orbit coupling and electron correlation simultaneously. In two-step procedures, typically different matrix representations of the electrostatic and magnetic Hamiltonians are chosen. 

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