Spin-orbit interactions from self consistent field wavefunctions

Effects of the spin-orbit Hamiltonian H_LS , including both the spin-same orbit and spin-other orbit terms, are studied at the self consistent field (SCF) level of theory. Separate calculations are carried out for each state considered, and biorthogonal orbitals are constructed for the evaluation of matrix elements. Doublet-doublet and singlet-triplet interactions are discussed. The evaluation of the Gaussian basis function integrals is described in a Cartesian component representation, these integrals being directly related to one and two electron second derivative integrals. This new SCF spin-orbit code is used to (i) determine the spatial dependence of the spin-orbit parameters for the Renner-Teller ²B₁, ²A₁ states of H₂O⁺, (ii) determine the spin-orbit splitting of the ²Π states of OH as a function of bond-length and (iii) calculate the radiative lifetime of the a ³Σ⁺ state of NO⁺. In each case these calculations are compared with more sophisticated configuration interaction studies, but it is found that the evaluation of these effects near equilibrium geometries at the SCF level is sufficiently accurate, provided that a large basis set is used.     

A new basis set for molecular wavefunctions

A new basis set is proposed for molecular self-consistent field and configuration interaction calculations. Expansion functions are proposed in the form sin (r _A · p) exp (-ar_A ²) and cos (r _A · p) exp (-ar _A ²) and it is suggested that they are used as an alternative to the gaussian basis set x_A ⁱy_A ^jz_A ^k exp (-ar _A ²).

It is shown how the new basis can have the same effects, for suitable vectors p, as the gaussian basis. It is further shown that all the one and two-electron integrals in the new basis are evaluable in terms of exponentials, square roots and the complex error function erf (z). First calculations, using the new basis, are presented in LiH, HF and H₂O.     

Spin-orbit matrix elements for internally contracted multireference configuration interaction wavefunctions

An efficient method for the calculation of Breit-Pauli spin-orbit matrix elements for internally contracted multireference configuration interaction wavefunctions is presented. Instead of taking all two-electron contributions of the wavefunction explicitly into account, the most important two-electron contributions of the spin-orbit operator are incorporated by means of an effective one-electron Fock operator. As a further refinement, explicit two-electron contributions can be reinstated for the dominant all-internal parts of the wavefunctions.     

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. Siberian Physicotekhnical Institute at Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 68–78, July 2000.     

Spectroscopy of spin-orbit quantum bits in indium antimonide nanowires

A double quantum dot in the few-electron regime is achieved using local gating in an InSb nanowire. The spectrum of two-electron eigenstates is investigated using electric dipole spin resonance. Singlet-triplet level repulsion caused by spin-orbit interaction is observed. The size and the anisotropy of singlet-triplet repulsion are used to determine the magnitude and the orientation of the spin-orbit effective field in an InSb nanowire double dot. The obtained results are confirmed using spin blockade leakage current anisotropy and transport spectroscopy of individual quantum dots.     

Field distributions at57Fe in Zn2-Y and their fitting with the transmission integral

Conclusions Introduction of lorentzian or gaussian distributions of fields in the exact transmission integral leeds to integrals which can be solved. With this more general function it was possible to interpret even such broadened lines as in the spectra of Zn₂-Y. Because of the detected broadening the Zn₂-Y is a good polarizator. After transition through such a crystal even the wings of a unpolarized unbroadened linie are polarized. So this ferrite is a good candidate for a fast switch in linear polarisation experiments.     

Construction of the two-electron contribution to the Fock matrix by numerical integration

A novel method to numerically calculate the Fock matrix is presented. The Coulomb operator is re-expressed as an integral identity, which is discretized. The discretization of the auxiliary t dimension separates the x, y, and z dependencies transforming the two-electron Coulomb integrals of Gaussian-type orbitals (GTO) to a linear sum of products of two-dimensional integrals. The s-type integrals are calculated analytically and integrals of the higher angular-momentum functions are obtained using recursion formulae. The contributions to the two-body Coulomb integrals obtained for each discrete t value can be evaluated independently. The two-body Fock matrix elements can be integrated numerically, using common sets of quadrature points and weights. The aim is to calculate Fock matrices of enough accuracy for electronic structure calculations. Preliminary calculations indicate that it is possible to achieve an overall accuracy of at least 10^?12 E _h using the numerical approach.     

An analytical procedure to evaluate electronic integrals for molecular quantum mechanical calculations

The main contribution of this paper is to provide an alternative strategy to reduce the number of bi-electronic integrals in ab initio quantum mechanics calculations. An analytical procedure to evaluate the energy of a molecule as well as two-electron integrals is proposed. This approach is based on the generalized exponential function (q-exponential) and is particularly advantageous because it reduces substantially the CPU time in quantum mechanical calculations. Some examples of the effectiveness of this methodology are presented. It is important to point out that the new methodology is applicable to any kind of molecular system including relatively large molecular systems in the context of the Hartree–Fock (HF) and density functional theories (DFT).     

Ab initio calculation of spin-orbit coupling constants for gaussian lobe and gaussian-type wave functions

The spin-orbit coupling constants of a series of diatomic compounds containing first row atoms have been calculated using ab initio molecular wave functions with gaussian lobe and gaussian-type basis sets. In most cases, fair agreement with the experimental values has been achieved. It is pointed out that multicentre terms have almost no effect on the spin-orbit coupling constant. Furthermore, suggestions for a one-electron approximation using effective nuclear charges are presented.     

Determination of consistent semiempirical one-centre integrals based on coupled-cluster theory

ABSTRACT

We examine the one-centre integrals used in semiempirical molecular orbital theory, for the elements H–Ne. The currently used parameters do not provide good estimates for the relative energies of ionised states of atoms. Directly calculating the one-electron integrals U _ss and U _pp with coupled-cluster theory and fitting the two-electron repulsion integrals G _ss and G _pp to accurate coupled-cluster ionisation curves improves this behaviour. Since all the remaining parameters can be derived from these, the number of fitted variables is reduced from seven to two. The two-parameter model provides qualitative agreement with coupled-cluster theory for all ionisation potentials (IPs) and the principal electron affinity (EA). To obtain quantitative agreement for the principal IP and EA, U _ss and U _pp are included as variables in a four-parameter model. We discuss the new parameters and implications for the development of new, consistent semiempirical Hamiltonians.     

2-Matrix versus Complex Matrix Model, Integrals over the Unitary Group as Triangular Integrals

We prove that the 2-hermitian matrix model and the complex-matrix model obey the same loop equations, and as a byproduct, we find a formula for Itzykzon-Zuber type integrals over the unitary group. Integrals over U(n) are rewritten as gaussian integrals over triangular matrices and then computed explicitly. That formula is an efficient alternative to the former Shatashvili's formula. An erratum to this article is available at .     

Magnetic bloch states and carrier transport in two-dimensional semiconductor lattice structures with spin-orbit interaction

Quantization rules have been obtained for the Hall conductance of fully occupied Landau subbands of the two-dimensional electron gas with the Dresselhaus spin-orbit interaction in a periodic electrostatic field of a superlattice and a transverse magnetic field. The spin-orbit interaction mixes states of different magnetic subbands and changes the quantization rule for the Hall conductance compared to spinless particles. The calculations have been performed for the two-dimensional electrons in the structures with both a weak (AlGaAs/GaAs) and sufficiently strong (GaAs/In_0.23Ga_0.77As) spin-orbit interaction and Zeeman splitting. It has been found that the distribution of the Hall conductance among the magnetic subbands depends on the geometric parameters of the superlattices and promptly changes upon the touching of the adjacent subbands in the spectrum. The quantization rule for the Hall conductance in real semiconductor structures with relatively strong spin-orbit interaction has been shown to differ from that calculated by Thouless et al. [Phys. Rev. Lett. 49, 405 (1982)] for the systems without the spin-orbit interaction and Zeeman effect.     

Some investigations in the theory of open-shell ions

The use of V, W and X coefficients in calculations of matrix elements is shown. General formulae for all two-electron spin-orbit coupling matrices in an octahedral strong-field coupling scheme are deduced. The parametrization of the electrostatic energy in ligand field theory is discussed and the number of parameters required is given in various cases. The configuration f ² in an octahedral strong-field scheme is treated in some detail and reasons for the anomalously low paramagnetism of PuF₆ are exposed, largely in contradiction to earlier views [13, 14]. Finally it is shown that expressions involving recoupling matrix elements, which occur in a scheme for calculating matrix elements of one-electron operators given by Tanabe and Kamimura [3], are proportional to W and X coefficients, thus considerably extending the scope of the latter work.     

Upper critical field of a superconductor with arbitrary spin-orbit scattering: CeCu2Si2 and UBe13

The upper critical field is determined for an even-parity singlet pairing state in the presence of arbitrary spin-orbit scattering. Comparison with critical field experiments suggests that superconductivity in CeCu₂Si₂ is a singlet pairing state, and in UBe₁₃ is either a triplet pairing state or is a singlet state with restrictive conditions that the pair orbital be nearly isotropic and that strong spin-orbit scattering increase strongly as the field increases.     

Intramolecular heavy atom effect on the polarization of phosphorescence of phenylacetylene

Phosphorescence spectra, both unpolarized quasilinear in a Shpolskii matrix and polarized in a rigid glass matrix, have been measured for seven monohalogen ring substituted phenylacetylenes. The vibrational analyses indicate the presence of two subspectra, labelled I and II. Subspectrum I involves transitions from T₁ to totally symmetric vibrational levels of S₀ while Subspectrum II refers to transitions from T₁ to b₁ (out-of- plane) vibrational levels of S₀. In the parent phenylacetylene, 90% of the intensity is attributed to subspectrum I; in the halo derivatives, the percentage of subspectrum II increases with mass of halogen. The polarization results show an enhancement of 0,0 in plane polarization contributions with heavy atom. Subspectrum I's out-of-plane polarized contribution is accounted for by T₁ mixing with σπ states, while the in plane polarized part is rationalized by direct spin-orbit coupling with ππ¹ states via three-center integrals and by 2nd order Herzberg-Teller spin-orbit coupling. Subspectrum II arises from 1st order Herzberg-Teller spin-orbit coupling.     

Dirac-Fock studies of some electronic properties of actinide ions

Results of accurate Dirac-Fock studies of some U, Np, Pu and Am ions are reported; these include one-electron eigen-values, spin-orbit splittings, 〈rⁿ〉 values for the 5f electrons, Slater F^k and G^k integrals, Mössbauer isomer shift electronic densities and the 〈j_i〉 integrals needed for neutron magnetic form factors.     

Two-electron coulomb integrals with minimal group theory. Direct analytic solution and computer program

A general closed-form analytic solution is developed for two-electron Coulomb integrals such as those that appear in perturbative treatments of multielectron atoms. In developing this solution, no recourse is necessary to group theory, Clebsch-Gordan coefficients, or specialized coordinate transformations: only familiar properties of hydrogenic wavefunctions and standard integrals are employed. One constraint from group theory is adopted for the purpose of establishing an upper limit on an index of summation for computational purposes. An accurate and efficient computer program for evaluating such integrals is made available.     

Direct measurement of the spin-orbit interaction in a two-electron InAs nanowire quantum dot

We demonstrate control of the electron number down to the last electron in tunable few-electron quantum dots defined in catalytically grown InAs nanowires. Using low temperature transport spectroscopy in the Coulomb blockade regime, we propose a method to directly determine the magnitude of the spin-orbit interaction in a two-electron artificial atom with strong spin-orbit coupling. Because of a large effective g factor |g(*)|=8+/-1, the transition from a singlet S to a triplet T+ ground state with increasing magnetic field is dominated by the Zeeman energy rather than by orbital effects. We find that the spin-orbit coupling mixes the T+ and S states and thus induces an avoided crossing with magnitude Delta(SO)=0.25+/-0.05 meV. This allows us to calculate the spin-orbit length lambda(SO) approximately 127 nm in such systems using a simple model.     

Electron energy state spin-splitting in 3D cylindrical semiconductor quantum dots

In this article we study the impact of the spin-orbit interaction on the electron quantum confinement for narrow gap semiconductor quantum dots. The model formulation includes: (1) the effective one-band Hamiltonian approximation; (2) the position- and energy-dependent quasi-particle effective mass approximation; (3) the finite hard wall confinement potential; and (4) the spin-dependent Ben Daniel-Duke boundary conditions. The Hartree-Fock approximation is also utilized for evaluating the characteristics of a two-electron quantum dot system. In our calculation, we describe the spin-orbit interaction which comes from both the spin-dependent boundary conditions and the Rashba term (for two-electron quantum dot system). It can significantly modify the electron energy spectrum for InAs semiconductor quantum dots built in the GaAs matrix. The energy state spin-splitting is strongly dependent on the dot size and reaches an experimentally measurable magnitude for relatively small dots. In addition, we have found the Coulomb interaction and the spin-splitting are suppressed in quantum dots with small height. Received 15 May 2001 / Received in final form 14 May 2002 Published online 13 August 2002