Four-component Wave Functions Research Articles

High-precision solution of the Dirac equation for the hydrogen molecular ion by an iterative method

The Dirac equation for H$_2^+$ is solved numerically using an iterative method proposed by Kutzelnigg [Z. Phys. 11, 15 (1989]. The four-component wavefunction is expanded in a newly introduced kinetically balanced exponential basis set. The ground-state relativistic energy is obtained with an accuracy of $10^{-20}$, which represents an improvement by several orders of magnitude, and is shown to be in good agreement with results obtained from perturbation theory.Highly accurate relativistic wavefunctions are obtained, which is a first step towards nonperturbative calculations of the one-loop self-energy correction in hydrogen molecular ions.

Electron chirality in amino acid molecules

We evaluated the total electron chirality in alanine, serine, and valine, which are molecules that have chiral structures. Previously, computations of the total electron chirality of molecules composed of only light elements are impossible within usual computational conditions of relativistic four-component wave functions. In this work, it is shown that the total electron chirality can be calculated if some diffuse functions are added to Gaussian basis sets. This is demonstrated for the H2O2 molecule. By adding diffuse Gaussian functions to basis sets, the total electron chirality of L-alanine, L-serine, and L-valine are evaluated. It is also shown that the total electron chirality is derived by the cancellation between large contributions from each orbital, and the total electron chirality in excited and ionized states is expected to be much larger than that of the ground state.

How the Klein-Nishina formula was derived: Based on the Sangokan Nishina Source Materials.

In 1928, Klein and Nishina investigated Compton scattering based on the Dirac equation just proposed in the same year, and derived the Klein–Nishina formula for the scattering cross section of a photon. At that time the Dirac equation had the following unsettled conceptual questions: the negative energy states, its four-component wave functions, and the spin states of an electron. Hence, during their investigation struggles, they encountered various difficulties. In this article, we describe their struggles to derive the formula using the “Sangokan Nishina Source Materials” retained in the the Nishina Memorial Foundation.

Variation of Spin Lifetime with Spin Injection Orientation in Strained Thin Silicon Films

The massive improvement of computational power of integrated circuits is supported by the continuous miniaturization of semiconductor devices' feature size. With scaling of the semiconductor devices apparently approaching its fundamental limits, new engineering solutions and innovations are required to further improve the performance of microelectronic components. Electron spin is promising for future nanoelectronic devices. Silicon is the main material of microelectronics, which possesses several properties attractive for spintronics: nuclei with predominantly zero spin and weak spin-orbit interaction. A long spin transfer distance of conduction electrons has already been shown experimentally [1]. As of the ongoing shift to FinFET/UTB SOI based devices from the 22nm to the 14nm node and beyond, enhancement of spin lifetime in such structures is a dominant issue under inspection. For the theoretical analysis of the spin lifetime we consider the effective 4x4 k.p Hamiltonian model, with the spin degree of freedom included, written at the vicinity of the X-point for the two relevant valleys [2] along the perpendicular axis in the Brillouin zone. This method accurately describes the electron wave functions in the presence of strain and spin-orbit interaction. The spin lifetime is determined by the spin-flip scattering between the valleys located on different crystallographic axes [2]. A large spin relaxation in gated silicon structures was observed [3], and a several orders of magnitude enhancement of the spin lifetime in (001) silicon films subjected to [110] uniaxial tensile stress was predicted [4], when spin is injected perpendicular to the film. Here we demonstrate that the spin relaxation rate can be further reduced and thus the spin lifetime will increase for spin injected in-plane. We begin with the subband four-component wave functions for two valleys and two spin projections. The up(down)-spin wave function consists of majority and minority components [5], and the absolute values depend on the shear strain ε xy . When a fixed ε xy and in-plane momentum (p x ,p y ) is maintained, we find that the absolute value of the majority (minority) component reduces (increases), when the spin injection direction is changed from perpendicular to in-plane. The surface roughness (SR) matrix scattering elements at opposite interfaces are assumed independent and proportional to the product of the corresponding subband wave functions' derivative at the interfaces [6]. For a film of thickness t = 1.36nm, where SR spin scattering is the prominent mechanism over the others, we notice that the intersubband relaxation matrix elements are reduced for an in-plane spin injection in OX-direction compared to the spin injection in perpendicular OZ-direction. As a consequence, the spin relaxation rate is reduced and the corresponding spin lifetime is increased once spin is injected in-plane over a large range of ε xy , ref: Fig. 1. The figure also shows that both of the inter- and intra-valley components of the spin relaxation time (τ) increase for in-plane spin injection. It is further observed that the momentum relaxation time stays nearly unaltered irrespective of the spin injection direction. In order to explain this behavior, we look at the spin relaxation hot spot [3] conditions as given by the equation: D ε xy – p x p y /M = 0, where D = 14eV is the shear deformation potential, M -1 ≈ m t -1 -m 0 -1 (m t : electron transversal effective mass, m 0 : electron rest mass). We consider the presence of the properly included spin-orbit field (SOF) which acts along (p x ,-p y ) direction. Spin precesses in the presence of SOF differently for spin injected in OX- or OZ-direction. For an in-plane injection along OX-direction, the spin projection expectation values at the hot spots can be written as: σ X = sin 2 (arctan p x /p y ), σ Y = -(p y /p x ) σ X , and σ Z = 0. On the contrary for spin injected perpendicular to the film (OZ-direction), σ X = σ Y = σ Z = σ = 0 at the spin hot spots, resulting in maximal spin randomization at any in-plane momentum (p x , p y ). Therefore, our results show that the spin relaxation rate will be strongest, when injected perpendicularly, which confirms that the spin relaxation time will increase for an in-plane spin injection. This work is supported by the European Research Council through the grant #247056 MOSILSPIN. REFERENCES 1. B. Huang et al., Phys. Rev. Lett. 99, 177209 (2007). 2. Y. Song and H. Dery, Phys. Rev. B 86, 085201, 2012. 3. J. Li and I. Appelbaum, Appl. Phys. Lett. 100, 162408, 2012. 4. D. Osintsev et al., Solid State Electron. 90, 34, 2013. 5. V. Sverdlov, Strain-Induced Effects in Advanced MOSFETs, Springer, 2011. 6. M. V. Fischetti et al., J. Appl. Phys. 94, 1079, 2003. Figure 1

A general, recursive, and open‐ended response code

We present a new implementation of a recent open-ended response theory formulation for time- and perturbation-dependent basis sets (Thorvaldsen et al., J. Chem. Phys. 2008, 129, 214108) at the Hartree-Fock and density functional levels of theory. A novel feature of the new implementation is the use of recursive programming techniques, making it possible to write highly compact code for the analytic calculation of any response property at any valid choice of rule for the order of perturbation at which to include perturbed density matrices. The formalism is expressed in terms of the density matrix in the atomic orbital basis, allowing the recursive scheme presented here to be used in linear-scaling formulations of response theory as well as with two- and four-component relativistic wave functions. To demonstrate the new code, we present calculations of the third geometrical derivatives of the frequency-dependent second hyperpolarizability for HSOH at the Hartree-Fock level of theory, a seventh-order energy derivative involving basis sets that are both time and perturbation dependent.

Quantum mechanical calculation of electron spin flip in a helical undulator

We derive a four-component spinor wave function for an electron in a helical undulator which, in the relativistic limit, successfully reproduces all of the results of the classical calculation for the radiation angular distribution, polarization, and energy spectrum. This wave function also allows the nonclassical calculation of the spin flip. For electron energies below the several hundred GeV range, the spin-flip probability is negligible, but for higher energies and high undulator strengths it cannot be neglected if beam polarization is to be preserved, even though nonflip radiation still greatly dominates the radiation intensity. The anomalous magnetic moment ${a}_{e}$ is seen to play a dominant role in the helical undulator spin-flip process. The probability of spin flip is shown to have a ${\ensuremath{\gamma}}^{5}$ dependence on electron energy. For high energy electrons, the direction of spin flip is independent of the handedness of the undulator. As a result, at sufficiently high energy, a polarized electron or positron beam rapidly depolarizes by spontaneous radiation in the undulator. Because of the high correlation between the direction of spin flip and the handedness of the spin-flip radiation, we conjecture that it may be possible to polarize electrons by using the intense circularly polarized photons in the helical undulator to stimulate spin flip.

Relativistic four-component calculations of Buckingham birefringence using London atomic orbitals

We present the first relativistic study of the electric-field-gradient induced birefringence (Buckingham birefringence), with application to the series of molecules CX2 (X = O, S, Se, Te). A recently developed atomic-orbital-driven scheme for the calculation of time-dependent molecular properties using one-, two- and four-component relativistic wave functions (Bast et al. in Chem Phys 356:177, 2009) is extended to first-order frequency-dependent magnetic-field perturbations, using London atomic orbitals to ensure gauge-origin independent results and to improve basis-set convergence. Calculations are presented at the Hartree–Fock and Kohn–Sham levels of theory and results for CO2 and CS2 are compared with previous high-level coupled-cluster calculations. Except for the heaviest member of the series, relativistic effects are small—in particular for the temperature-independent contribution to the birefringence. By contrast, the effects of electron correlation are significant. However, the reliability of standard exchange-correlation functionals in describing Buckingham birefringence remains unclear based on the comparison with high-level coupled-cluster singles-and-doubles calculations.

Atypical Fractional Quantum Hall Effect in Graphene at Filling Factor1/3

We study, with the help of exact-diagonalization calculations, a four-component trial wave function that may be relevant for the recently observed graphene fractional quantum Hall state at a filling factor ν(G) = 1/3. Although it is adiabatically connected to a 1/3 Laughlin state in the upper spin branch, with SU(2) valley-isospin ferromagnetic ordering and a completely filled lower spin branch, it reveals physical properties beyond such a state that is the natural ground state for a large Zeeman effect. Most saliently, it possesses at experimentally relevant values of the Zeeman gap low-energy spin-flip excitations that may be unveiled in inelastic light-scattering experiments.

Expectation values in two-component relativistic theories

We examined numerically the equivalence between the expectation values calculated by the four-component wave function and those calculated by the two-component wave functions generated by the infinite-order Douglas-Kroll (IODK) transformation. We showed the expectation values <r(-1)> and <delta(r-R)> in several closed-shell atoms using the so-called picture-changed 2x2 operators at some levels of approximation. The effect of the two-electron Coulomb term was also discussed. The numerical results indicated that the accuracy of <r(-1)> mainly depends on the level of the wave functions, while that of <delta(r-R)> is affected by the accuracy of both the wave functions and the picture-changed operators. As expected, the picture-changed operators and the IODK wave functions generate essentially equivalent expectation values in comparison with those calculated by the four-component Dirac-type method.

Combined theory of two- and four-component complete orthonormal sets of spinor wave functions and Slater type spinor orbitals in position, momentum and four-dimensional spaces

By the use of complete orthonormal sets of nonrelativistic scalar orbitals introduced by the author in previous papers the new complete orthonormal basis sets for two- and four-component spinor wave functions, and Slater spinor orbitals useful in the quantum-mechanical description of the spin-1/2 particles by the two- and four-component relativistic equations are established in position, momentum and four-dimensional spaces. These function sets are expressed through the corresponding nonrelativistic orbitals. The analytical formulas for overlap integrals over four-component relativistic Slater spinor orbitals with the same screening constants in position space are also derived. The relations obtained in this study can be useful in the study of different problems arising in the relativistic quantum mechanics when the position, momentum and four dimensional spaces are employed.

Atomic orbital-based cubic response theory for one-, two-, and four-component relativistic self-consistent field models

We present the first analytic calculations of the second hyperpolarizability in a relativistic framework. The calculations are made possible by our recent developments of a response theory built on a quasienergy formalism, in which the basis set may be both time and perturbation dependent. The approach is formulated for an arbitrary self-consistent field state in the atomic orbital basis. The implementation consists of a stand-alone code that only requires the unperturbed density in the atomic orbital basis as input, as well as a linear response solver by which we can determine the perturbed density matrices to different orders, at each new order solving equations that have the same structure as the linear response equation. Using these features of our formalism, we extend in this paper our approach to the relativistic domain, utilizing both two- and four-component relativistic wave functions. We apply the formalism to the calculation of the electronic and pure vibrational contributions to the second hyperpolarizability tensor for the hydrogen halides. Our results demonstrate that relativistic effects can be substantial for frequency-dependent second hyperpolarizabilities. Due to changes in the pole structure when going to the relativistic domain, the relativistic corrections to the hyperpolarizabilities are not transferable between different optical processes, except for very low frequencies.

In Search of the Electron Electric Dipole Moment: Relativistic Correlation Calculations of theP,T-Violating Effect in the Ground State ofHI+

We report the first results of ab initio relativistic correlation calculation of the effective electric field on the electron, E(eff), in the ground state of the HI+ cation. This value is required for interpretation of the suggested experiment on the search for the electron electric dipole moment. The generalized relativistic effective core potential, Fock-space relativistic coupled cluster with single and double cluster amplitudes (RCC-SD), and spin-orbit direct configuration interaction (SODCI) methods are used, followed by nonvariational one-center restoration of the four-component wave function in the iodine core. The RCC-SD value is E(eff) = 0.345 x 10(24) Hz/e cm and SODCI study gives E(eff) = 0.336 x 10(24) Hz/e cm (our final value). The structure of chemical bonding in HI+ is clarified, and a significant deviation of our value from that of Ravaine et al. [Phys. Rev. Lett. 94, 013001 (2005)] is explained.

Calculated optical and magnetic properties of hexafluorouranate (V) anion: UF- 6.

Our ab initio all-electron Dirac-Fock and the corresponding nonrelativistic limit calculations performed at four U-F bond distances yield for octahedral UF(6) (-) the optimized U-F bond distance of 2.091 and 2.088 A, respectively. We have also performed Dirac scattered wave calculations at the optimized U-F bond distances using the first-order pertubational procedure to obtain the Zeeman and hyperfine magnetic tensors for the octahedral anion UF(6) (-). The calculated isotropic Zeeman tensor of Deltag=-2.87 is in fairly good agreement with the value of Deltag=-2.78+/-0.10 obtained in electron spin resonance experiments on the H(3)O(+)UF(6) (-) adduct and the unpaired electron-spin spends approximately 2.5% of its time on the fluorine 2p(3/2) spinors. The calculated relativistic transition energies of the near-IR and visible absorption bands are also in good agreement with the experimental results. The octahedral uranium hexafluoride anion has a simple crystal field f(1) configuration; however, relativistic four-component wave functions are necessary to interpret correctly the available magnetic data, while a relativistic treatment taking into account double group symmetrized basis functions should suffice for the interpretation of the optical data.

On the performance of four-component relativistic density functional theory: Spectroscopic constants and dipole moments of the diatomics HX and XY (X,Y=F, Cl, Br, and I)

We have tested the performance of four-component relativistic density functional theory (DFT) by calculating spectroscopic constants (re, ωe, and ωexe) and dipole moments μ0 in the vibrational ground state for a selected set of 14 molecules: the hydrogen halides HX, the dihalogens X2, as well as the interhalogens XY (X,Y=F, Cl, Br, and I). These molecules have previously been studied by four-component relativistic wave function based methods by Visscher and co-workers [J. Chem. Phys. 108, 5177 (1998); 104, 9040 (1996); 105, 1987 (1996)]. We have used four different nonrelativistic functionals at the DZ and TZ basis set level. What is perhaps the most striking result of our study is the overall good performance of the local density approximation functional SVWN5; at the triple zeta basis set level it predicts bond lengths re, harmonic frequencies ωe, anharmonicities ωexe, and dipole moments μ0 with relative errors of 0.46%, 0.39%, −16.3%, and −0.74%, respectively. The corresponding values for the B3LYP hybrid functional are 1.27%, −2.10%, −20.4%, and 4.71%. The two generalized gradient approximation functionals PW86 and BLYP show a less convincing performance, characterized by a systematic overestimation of bond lengths and underestimation of harmonic frequencies. We show that only the constant term is modified in second-order vibrational perturbation theory upon the inclusion of a linear term, corresponding to the choice of a nonstationary reference geometry. Upon shifting the reference geometry from the optimized to the experimental geometry the calculated harmonic frequencies are significantly improved, whereas the anharmonicities are basically unchanged. Dipole moments calculated at the experimental geometry at the B3LYP/TZ level appear to be remarkably accurate with a mean relative error of −1.1% and a standard deviation of less than 4%. Our study reveals that anharmonicities are quite sensitive to the numerical integration scheme employed in the DFT calculations, and for the interhalogens we had to modify the Becke partitioning scheme by using atomic adjustments along the lines of the atom in molecules approach of Bader.

Correlation, Relativistic and Adiabatic Corrections to the Ground State Potential Curve of the Hydrogen Molecule

Correlation, relativistic and adiabatic corrections are computed for the electronic ground state of the hydrogen molecule for r ≤ 12 a.u. In contrast to previous calculations (based on the numerical solution of the Schrödinger equation, mainly done by Wolniewicz et al.), our results are based on the ordinary ab initio treatment using a four-component wave function with fully relativistically reoptimized basis sets and adiabatic corrections by the treatment developed in our laboratory. The calculated energies are fitted to the polynomial/exponential analytical function and the evaluated spectroscopic parameters are compared with those obtained in the same manner from the Wolniewicz data. The results presented show a good agreement with exact numerical calculations published previously.

THE ROLE OF RELATIVISTIC KINEMATICS IN DESCRIBING TWO-QUARK SYSTEMS

An attempt to incorporate relativistic kinematics in the description of light quark systems is made. It seems that the way of such incorporation along the suggestion expressed by R. Gaida and his collaborators is very promising. Comparison of these results with the experimental data concerning boson mass spectrum shows that this approach is among the best theoretical interpretation of the data. e-mail address: vrubish@univ.uzhgorod.ua e-mail address: kishs@univ.uzhgorod.ua e-mail address: shpenik@org.iep.uzhgorod.ua Surprising success of non-relativistic quark models with funnel or oscillator potential in describing hadron mass spectra for heavy quark systems had waken the hopes that at least some aspects of the strong interactions are understood after all. Good insight of previous works concerning this subject is given in [1, 2, 3]. But evidently the picture is not complete until it incorporates the light quarks u, d and s. But this means that the problem is shifting into quantum chromodynamical sector. In this respect the search of alternative possibilities of incorporating relativistic kinematics is quite actual. A very promising approach to the solution of relativistic problem of interaction of two particles was suggested recently by R. Gaida and his collaborators [4]-[7]. Their results can be directly applied to our problem of calculating the mass spectrum of quarkonium as a system of two-quarks. Somewhat different approach was simultaneously developed by I. Todorov and P. Bogolyubov [8, 9] and later by E. Predazzi et. al. [10]. In this work we shall investigate the results concerning the application of their findings to light-quark systems. Let us follow first the way, suggested by Predazzi et al [10]-[12]. Following the ideas expressed in [10, 11] and later developed in [12], let us start with the classical expression for the relativistic total energy of two particle system with masses m and M , respectively E = √ p1 +m 2 + √ p2 +M 2 (1) were p1 = −p2 ≡ p in the c.m. system. The main problem here is the mode of inclusion the interaction between two particles. For equal masses Lucha and Schoberl [13] write basing on (1) and passing to quantum mechanics ( 2 √ p2 +m2 + V ) Ψ (r) = EΨ (r) (2) which they call the spinless Bethe-Salpeter equation. They solve very cunningly this equation in the configurational momentum representation but obtain not very good results, which we shall discuss later. But in any case we can conclude that exactly the way of introducing the interaction is responsible for their failure. Therefore we propose here another way of introducing the interaction. Namely, following Predazzi et al [12] we linearize the expression ( E +m −M 2E )2 = p +m (3) which follows from (1) after simple algebraic transformations, and obtain ( E +m −M 2E ) Ψ (r) = (αcp+ βmc + Ṽ )Ψ (r) (4) where α and β are the usual Dirac matrices, Ψ is the fourcomponent wave function for which we shall use two-component representation Ψ (r) = ( φ (r) χ (r) ) . (5) In the general case we would have on this stage to decide what kind of Lorentz-transform properties we shall ascribe to the interaction.

Hawking thermal radiation of the dirac particle in spherically symmetric nonstatic space-time

The dynamical properties of Dirac spinor particles in a spherically symmetric nonstatic space-time are studied. The explicit representative of the four-component wave function of Dirac particles is obtained. The Dirac equation can be reduced to the standard form of the wave equation near the event horizon by the proper coordinate transformation. The event horizon location and Hawking radiation temperature are obtained.

HAWKING-UNRUH THERMAL EFFECT AND FOUR-COMPONENT WAVE FUNCTION OF DIRAC SPINOR PARTICLES IN NON-STATIC RINDLER SPACE-TIME

The dynamical properties of Dirac spinor particles in a non-static Rindler space-time is studied. The explicit representative of the four-component wave function of Dirac particles is obtained. A non-uniformly rectilinearly accelerating Rindler observer is accompanied by an event horizon depending on time. He explores some thermal radiation varying with time, whose temperature is proportional to the simultaneous acceleration. The existence of the non-static Rindler effect is confirmed.

Relativistic configuration-interaction theory for atomic systems.

The relativistic configuration-interaction method with analytical relativistic Hartree-Fock-Roothaan (RHFR) basis functions for atomic systems is presented. One-electron functions used for constructing configuration state functions (CSF's) are obtained with the RHFR method in which the large and small components of the radial part of a four-component wave function are expanded in terms of an analytical basis set consisting of Slater-type orbitals. Numerical application of the method to neonlike atomic systems is carried out. It is shown that calculated excitation energies with the method are in good agreement with experiment. The Z-dependent behavior of the optical oscillator strengths for various electric-dipole transitions from the ground state in the systems is also given.

Generation of mass and change of the neutrino helicity in a medium in the presence of right currents

In the symmetric SUL(2)⊗SUR(2)⊗U(1) model of electroweak interactions the Dirac neutrino possesses in a medium effective mass, even in the absence of vacuum rest mass (mvvac = 0). As a result the neutrino in such a model has always a four-component wave function in the medium. The left neutrino, produced in a nuclear reaction, becomes right in the material in elastic scattering from nuclei, and more efficiently than in the same process when taking into account the anomalous vacuum magnetic moment.