Spin-orbit Interaction Operator Research Articles

Generalized Spin-Orbit Interaction and Its Manifestation in Two-Dimensional Electron Systems

In frame of Dirac quantum field theory that describes electrons and positrons as elementary excitations of the spinor field, the generalized operator of the spin-orbit interaction is obtained using non-relativistic approximation in the Hamilton operator of the spinor field taking into account the presence of an external potential. This operator is shown to contain a new term in addition to the known ones. By an example of a model potential in the form of a quantum well, it is demonstrated that the Schr¨odinger equation with the generalized spin-orbit interaction operator describes all spin states obtained directly from the Dirac equation. The dependence of the spin-orbit interaction on the spin states in quasi-two-dimensional systems of electrons localized in a quantum well is analyzed. It is demonstrated that the electric current in the quantum well layer induces the spin polarization of charge carriers near the boundary surfaces of the layer, with the polarization of the charge carriers being opposite at the different surfaces. This phenomenon appears due to the spin-orbit interaction and is known as the spin Hall effect, which was observed experimentally in heterostructures with the corresponding geometry.

On the theory of the Schrödinger equation with the full set of relativistic corrections

All relativistic corrections to the Scr{\"o}dinger equation which determine the interlink between spin and orbit of moving particles, are directly calculated from the Dirac equation using the spin invariant operators. It is shown that among the second order corrections there are not only the well-known Darwin and Thomas terms, but also the new ones. Only with the account of the latter corrections the energies found with the obtained spin-orbit interaction operator, coincide with the energies of the Dirac equation exact solution. The problem of electron spectrum in the quantum well type structures is studied in details and the physical reasons for the appearance of spin-orbit interaction operators in the Dresselhaus or Rashba form, are analyzed.

Excitation energies with spin-orbit couplings using equation-of-motion coupled-cluster singles and doubles eigenvectors.

A method of calculation of excited states with spin-orbit couplings, which utilizes left and right eigenvectors of equation-of-motion coupled-cluster singles and doubles model has been formulated and implemented. The spin-orbit interactions are introduced by using the spin-orbit mean field approximation of the Briet-Pauli Hamiltonian. In order to evaluate all the necessary matrix elements, a scheme based on the diagrammatic representation of the second-quantized form of the spin-orbit interaction operator and the standard rules of second-quantized algebra is presented. We posit that this scheme is general and much simpler to use than the often used rules derived for the configuration state functions by using the Wigner-Eckart theorem. We show that the spin-orbit coupled states (i.e., target relativistic states) must satisfy specific conditions in order to classify them according to the double group symmetry. This interrelation between the structure of the target relativistic states and its double group symmetry is discussed in detail. An algorithm to classify the target states according to the irreducible representation of the double group symmetry is offered and implemented. Numerical tests for several atoms and molecules show good agreement of predicted and experimental spin-orbit splittings of the target excited states.

Intersystem-crossing and phosphorescence rates in fac-Ir(III)(ppy)3: a theoretical study involving multi-reference configuration interaction wavefunctions.

We have employed combined density functional theory and multi-reference configuration interaction methods including spin-orbit coupling (SOC) effects to investigate the photophysics of the green phosphorescent emitter fac-tris-(2-phenylpyridine)iridium (fac-Ir(ppy)3). A critical evaluation of our quantum chemical approaches shows that a perturbational treatment of SOC is the method of choice for computing the UV/Vis spectrum of this heavy transition metal complex while multi-reference spin-orbit configuration interaction is preferable for calculating the phosphorescence rates. The particular choice of the spin-orbit interaction operator is found to be of minor importance. Intersystem crossing (ISC) rates have been determined by Fourier transformation of the time correlation function of the transition including Dushinsky rotations. In the electronic ground state, fac-Ir(ppy)3 is C3 symmetric. The calculated UV/Vis spectrum is in excellent agreement with experiment. The effect of SOC is particularly pronounced for the metal-to-ligand charge transfer (MLCT) band in the visible region of the absorption spectrum which does not only extend its spectral onset towards longer wavelengths but also experiences a blue shift of its maximum. Pseudo-Jahn-Teller interaction leads to asymmetric coordinate displacements in the lowest MLCT states. Substantial electronic SOC and a small energy gap make ISC an ultrafast process in fac-Ir(ppy)3. For the S1↝T1 non-radiative transition, we compute a rate constant of kISC = 6.9 × 10(12) s(-1) which exceeds the rate constant of radiative decay to the electronic ground state by more than six orders of magnitude, in agreement with the experimental observation of a subpicosecond ISC process and a triplet quantum yield close to unity. As a consequence of the geometric distortion in the T1 state, the T1 → S0 transition densities are localized on one of the phenylpyridyl moieties. In our best quantum chemical model, we obtain phosphorescence decay times of 264 μs, 13 μs, and 0.9 μs, respectively, for the T1,I, T1,II, and T1,III fine-structure levels in dichloromethane (DCM) solution. In addition to reproducing the correct orders of magnitude for the individual phosphorescence emission probabilities, our theoretical study gives insight into the underlying mechanisms. In terms of intensity borrowing from spin-allowed transitions, the low emission probability of the T1,I substate is caused by the mutual cancellation of contributions from several singlet states to the total transition dipole moment. Their contributions do not cancel but add up in case of the much faster T1,III → S0 emission while the T1,II → S0 emission is dominated by intensity borrowing from a single spin-allowed process, i.e., the S2 → S0 transition.

Spin–orbit DFT with analytic gradients and applications to heavy element compounds

We have implemented the unrestricted DFT approach with one-electron spin–orbit operators in the massively parallel NWChem program. Also implemented is the analytic gradient in the DFT approach with spin–orbit interactions. The current capabilities include single-point calculations and geometry optimization. Vibrational frequencies can be calculated numerically from the analytically calculated gradients. The implementation is based on the spin–orbit interaction operator derived from the effective core potential approach. The exchange functionals used in the implementation are functionals derived for non-spin–orbit calculations, including GGA as well as hybrid functionals. Spin–orbit Hartree–Fock calculations can also be carried out. We have applied the spin–orbit DFT methods to the Uranyl aqua complexes. We have optimized the structures and calculated the vibrational frequencies of both (UO2 2+)aq and (UO2 +)aq with and without spin–orbit effects. The effects of the spin–orbit interaction on the structures and frequencies of these two complexes are discussed. We also carried out calculations for Th2, and several low-lying electronic states are calculated. Our results indicate that, for open-shell systems, there are significant effects due to the spin–orbit effects and the electronic configurations with and without spin–orbit interactions could change due to the occupation of orbitals of larger spin–orbit interactions.

Approximate theoretical model for the five electronic states ( Ω = 5/2, 3/2, 3/2, 1/2, 1/2) arising from the ground 3d 9 configuration in nickel halide molecules and for rotational levels of the two Ω = 1/2 states in that manifold

In the first part of this paper an effective Hamiltonian for a non-rotating diatomic molecule containing only crystal-field and spin–orbit operators is set up to describe the energies of the five spin–orbit components that arise in the ground electronic configuration of the nickel monohalides. The model assumes that bonding in the nickel halides has the approximate form Ni +X −, with an electronic 3d 9 configuration plus closed shells on the Ni + moiety and a closed shell configuration on the X − moiety. From a crystal-field point of view, interactions of the positive d-hole with the cylindrically symmetrical electric charge distribution of the hypothetical NiX − closed-shell core can then be parameterized by three terms in a traditional expansion in spherical harmonics: C 0 + C 2 Y 20( θ, ϕ) + C 4 Y 40( θ, ϕ). Interaction of the hole with the magnetic field generated by its own orbital motion can be parameterized by a traditional spin–orbit interaction operator A L · S . The Hamiltonian matrix is set up in a basis set consisting of the 10 Hund’s case (a) basis functions | L, Λ; S , Σ〉 that arise when L = 2 and S = 1/2. Least-squares fits of the observed five spin–orbit components of the three lowest electronic states in NiF and NiCl are then carried out in terms of the four parameters C 0, C 2, C 4, and A which lead to good agreement, except for the two | Ω| = 1/2 states. The large equal and opposite residuals of the | Ω| = 1/2 states can be reduced to values comparable with those for the | Ω| = 3/2 and | Ω| = 5/2 states by fixing A to its value in Ni + and then introducing an empirical correction factor for one off-diagonal orbital matrix element. In the second part of this paper the usual effective Hamiltonian B( J – L – S ) 2 for a rotating diatomic molecule is used to derive expressions for the Ω-type doubling parameter p in the two | Ω| = 1/2 states. These expressions show (for certain sign conventions) that the sum of the two p values should be −2 B, but that their difference can vary between −10 B and +10 B. These theoretical results are in good agreement with the two observed p values for both NiF and NiCl. The present formalism should in principle be applicable to NiBr and NiI, and to the halides of palladium, since Pd + has a well isolated 4d 9 electronic ground configuration. Extension to metal halides having d n configurations with n < 9, or to platinum halides may present difficulties, since manifolds from the d n and d n −1 s configurations may be heavily mixed, thus requiring “too many” parameters in the electronic part of the problem. Application to linear triatomic molecules may also present problems because of the large number of vibronic perturbations made possible by their four vibrational degrees of freedom.

Effect of the vibrational pattern of out-of-plane vibrational modes on vibronically induced spin-orbit coupling between ππ* states involved in nonradiative intersystem crossing transitions

To describe nonradiative intersystem crossing between excited ππ* states, the matrix elements of the operator of spin-orbit interaction are calculated. Adiabatic electronic functions that depend on out-of-plane normal vibrational coordinates are used as a zero approximation. Using dibenzofuran and tetrachloro-substituted dioxin as examples and taking into account all the out-of-plane vibrational modes, the effects of the vibrational patterns of these modes on the matrix elements under consideration, as well as of the spin-orbit coupling in carbon, oxygen, and chlorine atoms of individual atomic groups of the molecule, are discussed.

Borrowing Intensity in Rare Earth Doped Materials; Magnetic Dipole Transitions

A theoretical model of magnetic dipole transitions in crystals doped by the lanthanide ions is based on the fourth-order perturbation theory applied for the Hamiltonian in which two perturbing mechanisms are taken into account. Namely, the impact due to the crystal field potential and also spin-orbit interaction operator is included in the transition amplitude. The analysis is performed in the language of Racah algebra applied for the formulation of the effective operators expressed in terms of unit double tensor operators. The radial integrals of the terms contributing at the fourth order are defined within the perturbed function approach, due to which their values are evaluated for the complete radial basis sets of one-electron states of given symmetry. The numerical analysis performed for the Eu3+ ion provides the information on the relative importance of various effective operators and allows one to establish a hierarchy of important terms contributing to the transition amplitude. The results of the analysis lead also to the verification of possible importance of magnetic dipole transitions in the description of 0 ↔ 0 and 0 ↔ 1 electric dipole transitions via the so-called borrowing mechanism introduced by Wybourne.

Ab initio study of AmCl(+): f-f spectroscopy and chemical binding.

A valence full configuration interaction study with a polarized double-zeta quality basis set has been carried out for the lowest 49 electronic states of AmCl(+). The calculations use a pseudopotential treatment for the core electrons and incorporate a one-electron spin-orbit interaction operator. Electrons in the valence s, p, d, and f subshells were included in the active space. The resulting electronic potential energy curves are largely repulsive. The chemical bonding is ionic in character with negligible participation of 5f electrons. The molecular f-f spectroscopy of AmCl(+) arises essentially from an in situ Am(2+) core with states slightly redshifted by the presence of chloride ion. Am(+)+Cl asymptotes which give rise to the few attractive potential energy curves can be predicted by analysis of the f-f spectroscopy of isolated Am(+) and Am(2+). The attractive curves have substantial binding energies, on the order of 75-80 kcal/mol, and are noticeably lower than recent indirect measurements on the isovalent EuCl(+). An independent empirical correlation supports the predicted reduction in AmCl(+) binding energy. The energies of the repulsive curves are strongly dependent on the selection of the underlying atomic orbitals while the energies of the attractive curves do not display this sensitivity. The calculations were carried out using our recently developed parallel spin-orbit configuration interaction software.

Three-center integrals of one-electron operator of a spin-orbit interaction

A computational algorithm for three-center integrals in matrix elements of a spin-orbit interaction operator is developed. The explicit form of the integrals is derived. Empirical and calculated atomic constants of the spin-orbit interaction are compared. The effect of interatomic spacing in the atom chain on the magnitude of integrals is considered. Distinctions between one-center and multicenter approximations as well as the range of application are analyzed using planar and nonplanar compounds as examples.

Internal and external heavy-atom effects on phosphorescence radiative lifetimes calculated using a mean-field spin–orbit Hamiltonian

Abstract We investigate the performance of an atomic mean-field approximation to the spin–orbit interaction Hamiltonian in calculations of phosphorescence radiative lifetimes. A comparison of the performance of the mean-field approximation with the full spin–orbit interaction operator is made for the phosphorescent radiative lifetimes of the 3 B 1u excited state of ethylene perturbed by an external halogen ion, and for the 3 A 1 and 3 B 1 excited states in mono- and di-halogen-substituted pyridines. The mean-field approximation is shown to give excellent results compared to calculations with the full one- and two-electron spin–orbit integrals.

Spin-orbit interaction in the field of an optical vortex of a few-mode fiber

A quantum-mechanical analog of the spin-orbit interaction operator \(\hat V\) is constructed for the fields of optical CV vortices and TE and TM modes in an optical fiber. It is shown that the polarization correction δβ to the propagation constant, which is the mean value of this operator, is a measure of the “level splitting” of the propagation constant \(\tilde \beta \) in the scalar case. The difference in the operation of the individual parts of the operator \(\hat V\) on the fields of CV vortices and on the fields of TM and TE modes is indicative of the presence of two different physical processes — circular and linear birefringence in the locally isotropic optical fiber. The conversion of the “scalar” field \(\tilde e\) to a vector field e1 as a consequence of the spin-orbit interaction operator can be regarded as resulting from a re-radiation of the additional field e1 by the vortex field \(\tilde e\), which rotates around the optical axis of the fiber. In this picture the additional field e1 can be regarded as being a “relativistic” correction to the vortex field for the distortions of the main field \(\tilde e\) and arising as a result of the rotation of the field of the optical vortex in the medium of the few-mode fiber.

Theoretical description of the spectroscopic properties of rare earth ions in crystals

In this article the present state of knowledge of the theory of one- and two-photon processes observed in rare earth ions in crystals is presented. The conclusions are based on the results of ab initio calculations performed for various ions across the lanthanide series. The model applied for the calculations is based on the Rayleigh–Schrödinger perturbation theory, and the amplitude of a certain electric dipole transition is expressed in terms of effective operators. The radial integrals of the effective operators are defined by the perturbed functions that contain the perturbing influence of single excitations from the 4f shell to all one-electron states of a given symmetry, discrete and continuum. In this approach the interactions between the 4f N and the excited configurations via the crystal field potential, the electron correlation operator and the spin–orbit interaction operator are discussed; it is believed that the presented theory contains the most important physical mechanisms responsible for the f↔f electric dipole transitions. Two alternative formulations of the theory of one-photon electric dipole transitions are presented. Consequently, the transition amplitude is defined within the standard theory based on the length formula and within a new approach which is based on the velocity form of the electric dipole radiation operator.

Spin-Orbit Coupling for f-Electrons in a Crystalline Field

The notion of a spin—orbit interaction arises from consideration of dynamics of multielectron atoms, i.e. systems of N electrons in a spherical potential. This notion is essentially a single-particle one. We sketch its origin as a second-order correction when Dirac four-component wave equations for an electron in external electromagnetic fields are simplified to the two-component Pauli spinors. The constraints in spinorial degrees of freedom consist, roughly speaking, in neglecting the small component of the electron four-function. The spin—orbit interaction term serves to compensate effects of the small component. The crystalline field induces some deviations from spherical symmetry of an isolated atom, which yields some modifications of the spherical form of the spin—orbit interaction operator. These modifications can be described in terms of a number of tensor operators adapted to appropriate chains of subgroups of the spherical symmetry group. We present a classification of independent tensor operators and discuss the relevant parameters for f-ions.

The second‐order energy contribution from the spin–orbit interaction operator to the potential energy curve of Cr2

AbstractThe second‐order Rayleigh–Schrödinger energy correction to the Born–Oppenheimer potential energy due to the spin–orbit interaction can be expressed as a linear response function evaluated at zero frequency. We have calculated this energy contribution to the Cr2 singlet ground state X1 Σ+g potential energy function for a multiconfiguration self‐consistent field (MCSCF) wave function. The calculations show that the effect of spin–orbit interaction is small and of the same magnitude for the whole potential energy curve.

Spin-orbit interaction in molecular photoelectron spectra An intermediate coupling approach

A new and general semiempirical method for calculating ionization energies of molecules containing heavy atoms is presented. The extended Hückel hamiltonian is amended with a phenomenological one-electron spin-orbit interaction operator, and ionization energies are equated to orbital energies according to Koopmans' theorem. Calculations are presented for molecules with Br and I as heavy atoms. The systems considered are the hydrogen halides, diatomic halogens and interhalogens, haloacetylenes, halomethanes, and boron trihalides. Good agreement with the observed spin-orbit splitting is obtained. New assignments are proposed for the trihalides considered.

Higher-order spin-orbit interaction in photoelectron spectra of mercury halides

Abstract Spin-orbit interaction is introduced into the semi-empirical Extended Huckel MO scheme by adding the spin-orbit interaction operator to the effective

Model for calculating spin-orbit interactions with applications to photoelectron spectroscopy

A semiempirical model has been developed for estimating spin-orbit interactions in molecular ion states derived by ionization of closed-shell neutral molecule states. A perturbation approach is used with an effective spin-orbit interaction operator Hso and approximate Mulliken-Wolfsberg-Helmholz molecular orbital (MO) wavefunctions. Effective spin-orbit parameters ζv were obtained by interpolation with respect to atomic charges derived from the MO calculations. The He I photoelectron spectra of the PX3(X = Cl, Br, I) and PYX3(X = Cl, Br; Y = O, S) series have been measured. Splittings observed in some of the ionization bands of this series have been successfully interpretated as spin-orbit splittings using the newly developed model.

Paramagnetic resonance of the Ag ++ ion in irradiated alkali chlorides

In the first part of this paper an effective Hamiltonian for a non-rotating diatomic molecule containing only crystal-field and spin–orbit operators is set up to describe the energies of the five spin–orbit components that arise in the ground electronic configuration of the nickel monohalides. The model assumes that bonding in the nickel halides has the approximate form Ni+X−, with an electronic 3d9 configuration plus closed shells on the Ni+ moiety and a closed shell configuration on the X− moiety. From a crystal-field point of view, interactions of the positive d-hole with the cylindrically symmetrical electric charge distribution of the hypothetical NiX− closed-shell core can then be parameterized by three terms in a traditional expansion in spherical harmonics: C0 + C2Y20(θ, ϕ) + C4Y40(θ, ϕ). Interaction of the hole with the magnetic field generated by its own orbital motion can be parameterized by a traditional spin–orbit interaction operator AL · S. The Hamiltonian matrix is set up in a basis set consisting of the 10 Hund’s case (a) basis functions |L, Λ;S ,Σ〉 that arise when L = 2 and S = 1/2. Least-squares fits of the observed five spin–orbit components of the three lowest electronic states in NiF and NiCl are then carried out in terms of the four parameters C0, C2, C4, and A which lead to good agreement, except for the two |Ω| = 1/2 states. The large equal and opposite residuals of the |Ω| = 1/2 states can be reduced to values comparable with those for the |Ω| = 3/2 and |Ω| = 5/2 states by fixing A to its value in Ni+ and then introducing an empirical correction factor for one off-diagonal orbital matrix element. In the second part of this paper the usual effective Hamiltonian B(J–L–S)2 for a rotating diatomic molecule is used to derive expressions for the Ω-type doubling parameter p in the two |Ω| = 1/2 states. These expressions show (for certain sign conventions) that the sum of the two p values should be −2B, but that their difference can vary between −10B and +10B. These theoretical results are in good agreement with the two observed p values for both NiF and NiCl. The present formalism should in principle be applicable to NiBr and NiI, and to the halides of palladium, since Pd+ has a well isolated 4d9 electronic ground configuration. Extension to metal halides having dn configurations with n < 9, or to platinum halides may present difficulties, since manifolds from the dn and dn−1s configurations may be heavily mixed, thus requiring “too many” parameters in the electronic part of the problem. Application to linear triatomic molecules may also present problems because of the large number of vibronic perturbations made possible by their four vibrational degrees of freedom.

Spin-orbit splittings in nuclei (II)

The j-doublet splittings of single-particle nuclear levels are calculated in terms of the effective two-body spin-orbit interaction operator - the reaction matrix. The recent result of Brueckner et al. is used for the odd-state K LS −-operator corresponding to Gammel-Thaler potentials. On the same basis the even-state K LS + is neglected. The exchange amplitudes of the K LS −-operator are taken into account and also the “higher derivative” terms. The self-consistent nuclear density of the Ca 40-nucleus and other forms of this density are used to calculate the doublt splittings of Ca 41. Satisfactory results are obtained. The doublet splittings of the nuclei Pb 209, Bi 209, O 17 and He 5 are discussed. A discussion is presented of the spin-orbit splittings in spheroidally deformed nuclei.