Spin-orbit hamiltonian

In this work, relativistic effects are included in the no-pah or large component only approximation [13]. The total electronic Hamiltonian is H (r R) = H (r R) + H (r R), where H (r R) is the nom-elativistic Coulomb Hamiltonian and R) is a spin-orbit Hamiltonian. The relativistic (nomelativistic) eigenstates, are eigenfunctions of R)(H (r R)). Lower (upper)... 

The corresponding spin-orbit Hamiltonian operator is, then,... 

The spin-orbit Hamiltonian (HB0) requires some explanation. The energy of interaction between the magnetic moment M and the magnetic field caused by the orbital motion of an electron can be derived as(134)... 

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. 

The most renowned one- and two-electron spin-orbit Hamiltonian is given by... 

The Breit-Pauli spin-orbit Hamiltonian is found in many different forms in the literature. In expressions [101] and [102], we have chosen a form in which the connection to the Coulomb potential and the symmetry in the particle indices is apparent. Mostly s written in a short form where spin-same- and spin-other-orbit parts of the two-electron Hamiltonian have been contracted to a single term, either as... 

As may be seen by comparing Eqs. [103] and [105], the no-pair spin-orbit Hamiltonian has exactly the same structure as the Breit-Pauli spin-orbit Hamiltonian. It differs from the Breit-Pauli operator only by kinematical factors that damp the 1/rfj and l/r singularities. 

Since spin-orbit coupling is very important in heavy element compounds and the structure of the full microscopic Hamiltonians is rather complicated, several attempts have been made to develop approximate one-electron spin-orbit Hamiltonians. The application of an (effective) one-electron spin-orbit Hamiltonian has several computational advantages in spin-orbit Cl or perturbation calculations (1) all integrals may be kept in central memory, (2) there is no need for a summation over common indices in singly excited Slater determinants, and (3) matrix elements coupling doubly excited configurations do not occur. In many approximate schemes, even the tedious four-index transformation of two-electron integrals ceases to apply. The central question that comes up in this context deals with the accuracy of such an approximation, of course. 

Most common among the approximate spin-orbit Hamiltonians are those derived from relativistic effective core potentials (RECPs).35-38 Spin-orbit coupling operators for pseudo-potentials were developed in the 1970s.39 40 In the meantime, different schools have devised different procedures for tailoring such operators. All these procedures to parameterize the spin-orbit interaction for pseudo-potentials have one thing in common The predominant action of the spin-orbit operator has to be transferred from... 

Tike all effective one-electron approaches, the mean-field approximation considerably quickens the calculation of spin-orbit coupling matrix elements. Nevertheless, the fact that the construction of the molecular mean-field necessitates the evaluation of two-electron spin-orbit integrals in the complete AO basis represents a serious bottleneck in large applications. An enormous speedup can be achieved if a further approximation is introduced and the molecular mean field is replaced by a sum of atomic mean fields. In this case, only two-electron integrals for basis functions located at the same center have to be evaluated. This idea is based on two observations first, the spin-orbit Hamiltonian exhibits a 1/r3 radial dependence and falls off much faster... 

Spin-Orbit Coupling For the derivation of selection rules, it is sufficient to employ a simplified Hamiltonian. To this end, we rewrite each term in the microscopic spin-orbit Hamiltonians in form of a scalar product between an appropriately chosen spatial angular momentum 2 and a spin angular momentum S... 

The operator [157] is a phenomenological spin-orbit operator. In addition to being useful for symmetry considerations, Eq. [157] can be utilized for setting up a connection between theoretically and experimentally determined fine-structure splittings via the so-called spin-orbit parameter Aso (see the later section on first-order spin-orbit splitting). In terms of its tensor components, the phenomenological spin-orbit Hamiltonian reads... 

The phenomenological spin-orbit Hamiltonian ought not to be used for computing spin-orbit matrix elements, though. An example for a failure of such a procedure will be discussed in detail in the later subsection on a word of caution. 

Given a molecule that possesses C2p symmetry, let us try to figure out how to calculate ( Ai ffsol Bi) from wave functions with Ms = 1. The coupling of an Ai and a B state requires a spatial angular momentum operator of B2 symmetry. From Table 11, we read that this is just the x component of It. A direct computation of (3A2, Ms = 1 t x spin tensor. This is the only nonzero matrix element for the given wave functions. 

To understand these seemingly opposite facts, we have to leave the global S expression and rather write the spin-orbit Hamiltonian as a sum of one-particle operators... 

The no-pair spin-orbit Hamiltonian [105] differs from the corresponding BP terms [103] by momentum dependent factors of the type Ai/(Ej + me2) or (AjAj)/ (Ej+ mec2), where E, and A,- or A - have been defined in [106] and [107], respectively. There are essentially two ways of taking these factors into account. 

Only spatially degenerate states exhibit a first-order zero-field splitting. This condition restricts the phenomenon to atoms, diatomics, and highly symmetric polyatomic molecules. For a comparison with experiment, computed matrix elements of one or the other microscopic spin-orbit Hamiltonian have to be equated with those of a phenomenological operator. One has to be aware of the fact, however, that experimentally determined parameters are effective ones and may contain second-order contributions. Second-order SOC may be large, particularly in heavy element compounds. As discussed in the next section, it is not always distinguishable from first-order effects. 

A phenomenological spin-orbit Hamiltonian, formulated in terms of tensor operators, was presented already in the subsection on tensor operators. Few experimentalists utilize an effective Hamiltonian of this form (see Eq. [159]). Instead, shift operators are used to represent space and spin angular... 

Approximate Spin-Orbit Hamiltonians in Light Conjugated Molecules The Fine-Structure Splitting of HC6H+, NC5H+, and NC4N+. 

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