Nonrelativistic Level Research Articles

Dirac fermions in a spinning conical Gödel-type spacetime

AbstractIn this paper, we determine the relativistic and nonrelativistic energy levels for Dirac fermions in a spinning conical Gödel-type spacetime in(2+1)-dimensions, where we work with the curved Dirac equation in polar coordinates and we use the tetrads formalism. Solving a second-order differential equation for the two components of the Dirac spinor, we obtain a generalized Laguerre equation, and the relativistic energy levels of the fermion and antifermion, where such levels are quantized in terms of the radial and total magnetic quantum numbersnandmj, and explicitly depends on the spin parameters(describes the ‘spin’), spinorial parameteru(describes the two components of the spinor), curvature and rotation parametersαandβ(describes the conical curvature and the angular momentum of the spinning cosmic string), and on the vorticity parameter Ω (describes the Gödel-type spacetime). In particular, the quantization is a direct result of the existence of Ω (i.e. such quantity acts as a kind of ‘external field or potential’). We see that formj>0, the energy levels do not depend onsandu; however, depend onn,mj,α, andβ. In this case,αbreaks the degeneracy of the energy levels and such levels can increase infinitely in the limit4Ωβα→1. Already formj<0, we see that the energy levels depends ons,uandn; however, it no longer depends onmj,αandβ. In this case, it is as if the fermion/antifermion ‘lives only in a flat Gödel-type spacetime’. Besides, we also study the low-energy or nonrelativistic limit of the system. In both cases (relativistic and nonrelativistic), we graphically analyze the behavior of energy levels as a function of Ω,α, andβfor three different values ofn(ground state and the first two excited states).

Isotopic shifts in 3 P states of the carbon atom

ABSTRACT Isotope shifts in the transition energies between several lowest 3 P e states of the carbon atom are calculated at the nonrelativistic level of theory. We considered the 12 C, 13 C, and 14 C isotopes, for which we performed variational calculations by expanding the wave functions of the atomic states in terms of all-electron explicitly correlated Gaussian functions. The Born–Oppenheimer approximation was not assumed in the calculations. By combining the computed isotope shifts with the experimentally derived Ritz wavelengths for the natural mixture of the isotopes we make predictions of the positions of the spectral lines for specific carbon isotopes. These predictions may be useful for guiding future experimental measurements of these lines using high-resolution spectroscopy.

Equivalence principle, de-Sitter space, and cosmological twistors

In this paper, I discuss the impact of the positive cosmological constant on the interplay between the equivalence principle in general relativity, and the rules of quantum mechanics. At the nonrelativistic level, there is an ambiguity in the definition of a phase of a wave function measured by inertial and accelerating observes. This is the cosmological analogue of the Penrose effect, which can also be seen as a nonrelativistic limit of the Unruh effect. The symmetries of the associated Schrödinger equation are generated by the Newton–Hooke algebra, which arises from a nonrelativistic limit of a cosmological twistor space.

Binding energies, oscillator strengths of core/shell spherical quantum dots with or without on- or off-center donor impurities under an external magnetic field

Non-relativistic energy levels, binding energies and oscillator strengths of CdSe/ZnTe core/shell spherical quantum dots with or without impurities submitted to an external magnetic field have been investigated by using a B-spline based variational method, within the framework of the effective mass approximation. In the case where the system contains hydrogenic impurity, the effects of its off-center displacement combined to the height of the confining potential have also been studied. The dielectric constant as well as the effective mass are considered to be dependent on the radius. The modifications occurring due to the presence of the magnetic field have been analyzed. We have found that the electronic and optical properties are strongly affected by the magnetic field strength, the spatial confinement and the off-center displacement. The oscillator strengths and the binding energies increase with the magnetic field, but its effect on the binding energies (both ground and excited states) is dimmed by the reduction of the core-to-shell radii ratio and the increase of the off-center displacement to the vicinity of the shell. However, the oscillator strengths increase with the magnetic field when the off-center displacement increases towards the shell.

Spin-State Splittings in 3d Transition-Metal Complexes Revisited: Benchmarking Approximate Methods for Adiabatic Spin-State Energy Differences in Fe(II) Complexes.

The CASPT2+δMRCI composite approach reported in a companion paper has been extended and used to provide high-quality reference data for a series of adiabatic spin gaps (defined as ΔE = Equintet - Esinglet) of [FeIIL6]2+ complexes (L = CNH, CO, NCH, NH3, H2O), either at nonrelativistic level or including scalar relativistic effects. These highly accurate data have been used to evaluate the performance of various more approximate methods. Coupled-cluster theory with singles, doubles, and perturbative triples, CCSD(T), is found to agree well with the new reference data for Werner-type complexes but exhibits larger underestimates by up to 70 kJ/mol for the π-acceptor ligands, due to appreciable static correlation in the low-spin states of these systems. Widely used domain-based local CCSD(T) calculations, DLPNO-CCSD(T), are shown to depend very sensitively on the cutoff values used to construct the localized domains, and standard values are not sufficient. A large number of density functional approximations have been evaluated against the new reference data. The B2PLYP double hybrid gives the smallest deviations, but several functionals from different rungs of the usual ladder hierarchy give mean absolute deviations below 20 kJ/mol. This includes the B97-D semilocal functional, the PBE0* global hybrid with 15% exact-exchange admixture, as well as the local hybrids LH07s-SVWN and LH07t-SVWN. Several further functionals achieve mean absolute errors below 30 kJ/mol (M06L-D4, SSB-D, B97-1-D4, LC-ωPBE-D4, LH12ct-SsirPW92-D4, LH12ct-SsifPW92-D4, LH14t-calPBE-D4, LHJ-HFcal-D4, and several further double hybrids) and thereby also still overall outperform CCSD(T) or uncorrected CASPT2. While exact-exchange admixture is a crucial factor in favoring high-spin states, the present evaluations confirm that other aspects can be important as well. A number of the better-performing functionals underestimate the spin gaps for the π-acceptor ligands but overestimate them for L = NH3, H2O. In contrast to a previous suggestion, non-self-consistent density functional theory (DFT) computations on top of Hartree-Fock orbitals are not a promising path to produce accurate spin gaps in such complexes.

Constraining GUP models using limits on SME coefficients

Generalized uncertainty principles (GUP) and, independently, Lorentz symmetry violations are two common features in many candidate theories of quantum gravity. Despite that, the overlap between both has received limited attention so far. In this brief paper, we carry out further investigations on this topic. At the nonrelativistic level and in the realm of commutative spacetime coordinates, a large class of both isotropic and anisotropic GUP models is shown to produce signals experimentally indistinguishable from those predicted by the standard model extension (SME), the common framework for studying Lorentz-violating phenomena beyond the standard model. This identification is used to constrain GUP models using current limits on SME coefficients. In particular, bounds on isotropic GUP models are improved by a factor of 107 compared to current spectroscopic bounds and anisotropic models are constrained for the first time.

A theoretical and experimental analysis of the luminescent properties of Europium(III) complex sensitized by tryptophan

Basic luminescent and intramolecular energy transfer properties of the Trp-Eu(III) complex in water at different concentrations are reported. Theoretical and experimental results are described and compared, showing good agreement between them, demonstrating that the modeling proposed in this work is appropriate to describe its behavior at the molecular level. The emission spectrum of the Eu(III) ion in water has a very high intensity for the 5D0→7F1 transition, which is hypersensitive to the environment in which it is found. So that the excitation of the Trp at low concentrations at λ=295 nm intensifies the signal of this transition through energy transfer from the ligand (Trp) to the Eu(III) ion. In contrast, at high concentrations of Trp, that is 1 and 10 mM, the luminescence of the Eu(III) ion is quenched. The methods use the TD-DFT and (INDO/S-CIS) approximations, working at a non-relativistic quantum chemistry level. The B3LYP functional was selected in the first method and the second, the SPARKLE/RM1 model. Both yield complementary information on the absorption and emission processes. This calculation allows a more detailed analysis of the experimental results based on the molecular orbitals involved in electronic transitions and the possible energy transfer paths. This work contributes to the understanding of the behavior of Eu(III) ion as a probe to study the fluorescence of proteins constituted by biologically important molecules such as Trp in aqueous media.

Three-dimensional non-relativistic supergravity and torsion

In this paper we present a torsional non-relativi-stic Chern–Simons (super)gravity theory in three spacetime dimensions. We start by developing the non-relativistic limit of the purely bosonic relativistic teleparallel Chern–Simons formulation of gravity. On-shell the latter yields a non-Riemannian setup with non-vanishing torsion, which, at non-relativistic level, translates into a non-vanishing spatial torsion sourced by the cosmological constant. Then we consider the three-dimensional relativistic {mathcal {N}}=2 teleparallel Chern–Simons supergravity theory and obtain its non-relativistic counterpart by exploiting a Lie algebra expansion method. The non-relativistic supergravity theory is characterized, on-shell, by a non-vanishing spatial super-torsion, again sourced by the cosmological constant.

Relativistic and QED corrections for the ground state lithiumlike ionization energies

The ground state ionization energies of Z ⩽ 10 lithiumlike ions are calculated using fully correlated Gaussian wavefunctions. Leading-order relativistic corrections are evaluated, while QED corrections are established with small uncertainties by directly calculating the Araki–Sucher energy and expanding the three-electron Bethe logarithm in 1/Z. The non-relativistic α6 level shifts have also been calculated, and we have used these energies to recommend ionization energies, which include estimates of the influence of the relativistic portion of the α6 energy. The results emphasize the importance of the direct computation of the complete α6 correction, but also the need for new, higher accuracy experimental ionization limits.

Relativistic study of parity-violating nuclear spin-rotation tensors.

We present a four-component relativistic approach to describe the effects of the nuclear spin-dependent parity-violating (PV) weak nuclear forces on nuclear spin-rotation (NSR) tensors. The formalism is derived within the four-component polarization propagator theory based on the Dirac-Coulomb Hamiltonian. Such calculations are important for planning and interpretation of possible future experiments aimed at stringent tests of the standard model through the observation of PV effects in NSR spectroscopy. An exploratory application of this theory to the chiral molecules H2X2 (X = 17O, 33S, 77Se, 125Te, and 209Po) illustrates the dramatic effect of relativity on these contributions. In particular, spin-free and spin-orbit effects are even of opposite signs for some dihedral angles, and the latter fully dominate for the heavier nuclei. Relativistic four-component calculations of isotropic nuclear spin-rotation constants, including parity-violating electroweak interactions, give frequency differences of up to 4.2 mHz between the H2Po2 enantiomers; on the nonrelativistic level of theory, this energy difference is 0.1 mHz only.

Exclusively Relativistic: Periodic Trends in the Melting and Boiling Points of Group 12.

First‐principles simulations can advance our understanding of phase transitions but are often too costly for the heavier elements, which require a relativistic treatment. Addressing this challenge, we recently composed an indirect approach: A precise incremental calculation of absolute Gibbs energies for the solid and liquid with a relativistic Hamiltonian that enables an accurate determination of melting and boiling points (MPs and BPs). Here, we apply this approach to the Group 12 elements Zn, Cd, Hg, and Cn, whose MPs and BPs we calculate with a mean absolute deviation of only 5 % and 1 %, respectively, while we confirm the previously predicted liquid aggregate state of Cn. At a non‐relativistic level of theory, we obtain surprisingly similar MPs and BPs of 650±30 K and 1250±20 K, suggesting that periodic trends in this group are exclusively relativistic in nature. Ultimately, we discuss these results and their implication for Groups 11 and 14.

The Landau problem and nonclassicality

Exploring the concept of the extended Galilei group [Formula: see text]. Representations for field theories in a symplectic manifold have been derived in association with the method of the Wigner function. The representation is written in the light-cone of a de Sitter space–time in five dimensions. A Hilbert space is constructed, endowed with a symplectic structure, which is used as a representation space for the Lie algebra of [Formula: see text]. This representation gives rise to the spin-0 Schrödinger (Klein–Gordon-like) equation for the wave functions in phase space, such that the dependent variables have the content of position and linear momentum. This is a particular example of a conformal theory, such that the wave functions are associated with the Wigner function through the Moyal product. We construct the Pauli–Schrödinger (Dirac-like) equation in phase space in its explicitly covariant form. In addition, we analyze the gauge symmetry for spin-1/2 particles in phase space and show how implement the minimal coupling in this case. We applied to the problem of an electron in an external field, and we recovered the nonrelativistic Landau levels. Finally, we study the parameter of negativity associated with the nonclassicality of the system.

Non-relativistic limits and three-dimensional coadjoint Poincaré gravity

We show that a recently proposed action for three-dimensional non-relativistic gravity can be obtained by taking the limit of a relativistic Lagrangian that involves the coadjoint Poincaré algebra. We point out the similarity of our construction with the way that three-dimensional Galilei gravity and extended Bargmann gravity can be obtained by taking the limit of a relativistic Lagrangian that involves the Poincaré algebra. We extend our results to the anti-de Sitter case and we will see that there is a chiral decomposition at both the relativistic and non-relativistic level. We comment on possible further generalizations.

Nonrelativistic effective field theory for heavy exotic hadrons

We propose an effective field theory to describe hadrons with two heavy quarks without any assumption on the typical distance between the heavy quarks with respect to the typical hadronic scale. The construction is based on Non-Relativistic QCD and inspired in the strong coupling regime of Potential Non-Relativistic QCD. We construct the effective theory at leading and next-to-leading order in the inverse heavy-quark mass expansion for arbitrary quantum numbers of the light degrees of freedom. Hence our results hold for hybrids, tetraquarks, double heavy baryons and pentaquarks, for which we also present the corresponding operators at Non-Relativistic level. At leading order, the effective theory enjoys heavy quark spin symmetry and corresponds to the Born-Oppenheimer approximation. At next-to-leading order, spin and velocity dependent terms arise, which produce splittings in the heavy-quark spin symmetry multiplets. A concrete application to double heavy baryons is presented in an accompanying paper.

Four-component relativistic computational NMR study of ferrous, cobalt and nickel bisglycinates

The equilibrium structures of ferrous(II), cobalt(III) and nickel(II) bisglycinates were optimized at the DFT level using eight functionals, and their 1H, 13C, 15N and 17O NMR chemical shifts were evaluated at both non-relativistic and four-component relativistic levels. Essential deshielding relativistic corrections were observed for nitrogen and oxygen, while they were found to be small for carbons and protons. Solvent corrections for chemical shifts noticeably increased with the dielectric constant of a solvent for nitrogen and carbon, and they were negligibly small for protons.

NMR Spin-Spin Coupling Constants Derived from Relativistic Four-Component DFT Theory-Analysis and Visualization.

An unambiguous assignment of coupling pathways plays an important role in the description and rationalization of NMR indirect spin-spin coupling constants (SSCCs). Unfortunately, the SSCC analysis and visualization tools currently available to quantum chemists are restricted to nonrelativistic theory. Here, we present the theoretical foundation for novel relativistic SSCC visualization techniques based on analysis of the SSCC densities and the first-order current densities induced by the nuclear magnetic dipole moments. Details of the implementation of these techniques in the ReSpect program package are discussed. Numerical assessments are performed on through-space SSCCs, and we choose as our examples the heavy-atom Se-Se, Se-Te, and Te-Te coupling constants in three similar molecules for which experimental data are available. SSCCs were calculated at the nonrelativistic, scalar relativistic, and four-component relativistic density functional levels of theory. Furthermore, with the aid of different visualization methods, we discuss the interpretation of the relativistic effects, which are sizable for Se-Se, very significant for Se-Te, and cannot be neglected for Te-Te couplings. A substantial improvement of the theoretical SSCC values is obtained by also considering the molecular properties of a second conformation.

Hydrogenoid Spectra with Central Perturbations

Through the Kre\u{\i}n-Vi\v{s}ik-Birman extension scheme, unlike the classical analysis based on von Neumann's theory, we reproduce the construction and classification of all self-adjoint realisations of three-dimensional hydrogenoid-like Hamiltonians with singular perturbation supported at the Coulomb centre (the nucleus), as well as of Schr\"{o}dinger operators with Coulomb potentials on the half-line. These two problems are technically equivalent, albeit sometimes treated by their own in the the literature. Based on such scheme, we then recover the formula to determine the eigenvalues of each self-adjoint extension, as corrections of the non-relativistic hydrogenoid energy levels. We discuss in which respect the Kre\u{\i}n-Vi\v{s}ik-Birman scheme is somewhat more natural in yielding the typical boundary condition of self-adjointness at the centre of the perturbation.

Nonrelativistic energy levels of helium atoms

The nonrelativistic ionization energy levels of a helium atom are calculated for $S$, $P$, $D$ and $F$ states. The calculations are based on the variational method of "exponential" expansion. The convergence of the calculated energy levels is studied as a function of the number of basis functions $N$. This allows us to claim that the obtained energy values (including the values for the states with a nonzero angular momentum) are accurate up to 28-35 significant digits. Calculations of the nonrelativistic ionization energy of the negative hydrogen ion H$^-$ are also presented.

Nonrelativistic energy levels of HD.

Nonadiabatic exponential functions are employed to solve the four-body Schrödinger equation. Nonrelativistic bound energy levels of the HD molecule are calculated to the relative accuracy of 10-12-10-13, which is the first step toward highly accurate prediction of dissociation and transition energies. Such energies, in connection with equally accurate experimental data, will enable refinement of the physical constant and aid the search for deviations caused by yet unknown interactions at the atomic scale.

On the long-range relativistic effects in the 15 N NMR chemical shifts of halogenated azines.

Long-range β- and γ-relativistic effects of halogens in 15 N NMR chemical shifts of 20 halogenated azines (pyridines, pyrimidines, pyrazines, and 1,3,5-triazines) are shown to be unessential for fluoro-, chloro-, and bromo-derivatives (1-2ppm in average). However, for iodocontaining compounds, β- and γ-relativistic effects are important contributors to the accuracy of the 15 N calculation. Taking into account long-range relativistic effects slightly improves the agreement of calculation with experiment. Thus, mean average errors (MAE) of 15 N NMR chemical shifts of the title compounds calculated at the non-relativistic and full 4-component relativistic levels in gas phase are accordingly 7.8 and 5.5ppm for the range of about 150ppm. Taking into account solvent effects within the polarizable continuum model scheme marginally improves agreement of computational results with experiment decreasing MAEs from 7.8 to 7.4ppm and from 5.5 to 5.3ppm at the non-relativistic and relativistic levels, respectively. The best result (MAE: 5.3ppm) is achieved at the 4-component relativistic level using Keal and Tozer's KT3 functional used in combination with Dyall's relativistic basis set dyall.av3z with taking into account solvent effects within the polarizable continuum solvation model. The long-range relativistic effects play a major role (of up to dozen of parts per million) in 15 N NMR chemical shifts of halogenated nitrogen-containing heterocycles, which is especially crucial for iodine derivatives. This effect should apparently be taken into account for practical purposes.