Dirac Coulomb Research Articles

Dirac–Coulomb operators with general charge distribution II. The lowest eigenvalue

Consider the Coulomb potential $-\mu\ast|x|^{-1}$ generated by a non-negative finite measure $\mu$. It is well known that the lowest eigenvalue of the corresponding Schr\"odinger operator $-\Delta/2-\mu\ast|x|^{-1}$ is minimized, at fixed mass $\mu(\mathbb{R}^3)=\nu$, when $\mu$ is proportional to a delta. In this paper we investigate the conjecture that the same holds for the Dirac operator $-i\alpha\cdot\nabla+\beta-\mu\ast|x|^{-1}$. In a previous work on the subject we proved that this operator is self-adjoint when $\mu$ has no atom of mass larger than or equal to 1, and that its eigenvalues are given by min-max formulas. Here we consider the critical mass $\nu_1$, below which the lowest eigenvalue does not dive into the lower continuum spectrum for all $\mu\geq0$ with $\mu(\mathbb{R}^3)<\nu_1$. We first show that $\nu_1$ is related to the best constant in a new scaling-invariant Hardy-type inequality. Our main result is that for all $0\leq\nu<\nu_1$, there exists an optimal measure $\mu\geq0$ giving the lowest possible eigenvalue at fixed mass $\mu(\mathbb{R}^3)=\nu$, which concentrates on a compact set of Lebesgue measure zero. The last property is shown using a new unique continuation principle for Dirac operators. The existence proof is based on the concentration-compactness principle.

An interacting fermion-antifermion pair in the spacetime background generated by static cosmic string

We consider a general relativistic fermion antifermion pair that they interact via an attractive Coulomb type interparticle interaction potential in the 2+1 dimensional spacetime background spanned by cosmic string. By performing an exact solution of the corresponding fully-covariant two body Dirac Coulomb type equation we obtain an energy spectrum that depends on angular deficit parameter of the static cosmic string spacetime background for such a composite system. We arrive that the influence of cosmic string spacetime topology on the binding energy of Positronium-like atoms can be seen in all order of the coupling strength constant, even in the well-known non-relativistic binding energy term (∝αc2). We obtain that the angular deficit of the static cosmic string spacetime background causes a screening effect. For a predicted value of angular deficit parameter, α∼1−10−6, we apply the obtained result to an ortho-positronium, which is an unstable atom formed by an electron and its antimatter counterpart a positron, and then we determine the shift in the ground state binding energy level as 27,2μeV. We also arrive that, in principle, the shift in ground state binding energy of ortho-positronium can be measured even for α∼1−10−11 value with current techniques in use today. Moreover, this also gives us an opportunity to determine the altered total annihilation energy transmitted by the annihilation photons. The yields also impose that the total lifetime of an ortho-positronium can be changed by the topological feature of static cosmic string spacetime background. In principle, we show that an ortho-positronium system has a potential to prove the existence of such a spacetime background.

Four-component full configuration interaction quantum Monte Carlo for relativistic correlated electron problems.

An adaptation of the full configuration interaction quantum Monte Carlo (FCIQMC) method is presented for correlated electron problems containing heavy elements and the presence of significant relativistic effects. The modified algorithm allows for the sampling of the four-component spinors of the Dirac-Coulomb(-Breit) Hamiltonian within the relativistic no-pair approximation. The loss of spin symmetry and the general requirement for complex-valued Hamiltonian matrix elements are the most immediate considerations in expanding the scope of FCIQMC into the relativistic domain, and the alternatives for their efficient implementation are motivated and demonstrated. For the canonical correlated four-component chemical benchmark application of thallium hydride, we show that the necessary modifications do not particularly adversely affect the convergence of the systematic (initiator) error to the exact correlation energy for FCIQMC calculations, which is primarily dictated by the sparsity of the wavefunction, allowing the computational effort to somewhat bypass the formal increases in Hilbert space dimension for these problems. We apply the method to the larger problem of the spectroscopic constants of tin oxide, correlating 28 electrons in 122 Kramers-paired spinors, finding good agreement with experimental and prior theoretical relativistic studies.

Determination of the dipole polarizability of the alkali-metal negative ions

We present electric dipole polarizabilities ($\alpha_d$) of the alkali-metal negative ions, from H$^-$ to Fr$^-$, by employing four-component relativistic many-body methods. Differences in the results are shown by considering Dirac-Coulomb (DC) Hamiltonian, DC Hamiltonian with the Breit interaction, and DC Hamiltonian with the lower-order quantum electrodynamics interactions. At first, these interactions are included self-consistently in the Dirac-Hartree-Fock (DHF) method, and then electron correlation effects are incorporated over the DHF wave functions in the second-order many-body perturbation theory, random phase approximation and coupled-cluster (CC) theory. Roles of electron correlation effects and relativistic corrections are analyzed using the above many-body methods with size of the ions. We finally quote precise values of $\alpha_d$ of the above negative ions by estimating uncertainties to the CC results, and compare them with other calculations wherever available.

Relativistic short-range exchange energy functionals beyond the local-density approximation.

We develop relativistic short-range exchange energy functionals for four-component relativistic range-separated density-functional theory using a Dirac-Coulomb Hamiltonian in the no-pair approximation. We show how to improve the short-range local-density approximation exchange functional for large range-separation parameters by using the on-top exchange pair density as a new variable. We also develop a relativistic short-range generalized-gradient approximation exchange functional that further increases the accuracy for small range-separation parameters. Tests on the helium, beryllium, neon, and argon isoelectronic series up to high nuclear charges show that the latter functional gives exchange energies with a maximal relative percentage error of 3%. The development of this exchange functional represents a step forward for the application of four-component relativistic range-separated density-functional theory to chemical compounds with heavy elements.

Relativistic double-ionization equation-of-motion coupled-cluster method: Application to low-lying doubly ionized states.

This article deals with the extension of the relativistic double-ionization equation-of-motion coupled-cluster (DI-EOMCC) method [H. Pathak et al. Phys. Rev. A 90, 010501(R) (2014)] for the molecular systems. The Dirac-Coulomb Hamiltonian with four-component spinors is considered to take care of the relativistic effects. The implemented method is employed to compute a few low-lying doubly ionized states of noble gas atoms (Ar, Kr, Xe, and Rn) and Cl2, Br2, HBr, and HI. Additionally, we presented results with two intermediate schemes in the four-component relativistic DI-EOMCC framework to understand the role of electron correlation. The computed double ionization spectra for the atomic systems are compared with the values from the non-relativistic DI-EOMCC method with spin-orbit coupling [Z. Wang et al. J. Chem. Phys. 142, 144109 (2015)] and the values from the National Institute of Science and Technology (NIST) database. Our atomic results are found to be in good agreement with the NIST values. Furthermore, the obtained results for the molecular systems agree well with the available experimental values.

Complex Rotated Relativistic Configuration-Interaction Calculations of 1s2l2l ' States in O5+ Ion

Relativistic calculations of the energies and Auger widths of the lowest autoionizing states of lithium-like oxygen are performed. All leading corrections to the energy, including the contribution of electronic correlations, nuclear recoil effect, and quantum electrodynamic (QED) corrections, are included. The calculations have been performed using the Dirac–Coulomb–Breit Hamiltonian by the method of configuration interaction with complex rotation. The results obtained have been compared with the known experimental and theoretical values.

SPECTROSCOPIC AND THERMOCHEMICAL PROPERTIES OF ACTINIDE-CONTAINING SPECIES FROM FIRST PRINCIPLES: THORIUM AND AMERICIUM MONOXIDE MOLECULES

A relativistic version of a composite ab initio treatment of molecular spectroscopy and thermochemistry is developed, focusing on high-accuracy description of the properties of actinide (An) containing species. It is based on combining the calculation results at levels of theory with sufficiently full account of electron correlation, e.g., at the CCSDT(Q) level, but tackling only scalar relativity, with those obtained from more rigorous four-component relativistic calculations with the Dirac–Coulomb Hamiltonian. High accuracy achievable via this approach is revealed taking the examples of thorium and americium monoxide molecules. The errors in ab initio values for the bond length re, vibrational frequency ωe, and atomization energy D0 of the ThO molecule did not exceed 0.001 Å, 2.5 cm–1, and 0.5 kcal/mol, respectively. The composite numerical values for the first ionization potentials of the AmO molecule and the Am atom deviate from the experimental data just by 0.03 eV and 1 cm–1, respectively. For the first time, the proposed approach enabled high-accuracy evaluation of the molecular constants re, ωe and D0 for AmO and AmO+, as well as the second and third ionization potentials of the Am atom. The calculation results are indicative of a minor actinide contraction of the An–O bonds on going through the molecular series ThO → UO → AmO: the bond length in AmO is by 0.0073 Å shorter than that in ThO. The re(An–O) value is shown to be linearly dependent on the actinide atomic number in the periodic table. The results obtained may be used as benchmarks for parametrizing and calibrating the DFT functionals designed for treating An-containing molecules.

Calculations of Relativistic, Correlation, Nuclear, and Quantum-Electrodynamics Corrections to Energy and Ionization Potential of the Ground State of Helium-Like Ions

Nonrelativistic and relativistic variational calculations of energies and ionization potentials of the ground state of helium-like ions with nuclear number Z = 2−20 have been performed. All leading corrections to the total energy have been calculated, including the contribution of electron correlations, relativistic and quantum-electrodynamics corrections, and contributions due to the finite nuclear size (field shift) and finite nuclear mass (nuclear recoil) effects. Relativistic wave functions were calculated using the Dirac–Coulomb–Breit Hamiltonian.

Quasibound states of the Dirac field in Schwarzschild and Reissner–Nordström black hole backgrounds

In this paper, we study the existence of (quasi)bound states in two spacetime geometries describing Schwarzschild and Reissner–Nordström black holes. For obtaining these types of states, we search for discrete quantum modes of the massive Dirac equation in the two geometries. After imposing the quantization condition, an analytical expression for the energy of the ground states is derived. The energy of higher states is then obtained numerically. For very small values of the black hole mass M, we compare the energy of the Reissner–Nordström black hole quasibound state with the Dirac–Coulomb energy and we have found the two to be in good agreement.

Photoionization of tungsten ions: experiment and theory for W5+

Experimental and theoretical cross sections are reported for single-photon single ionization of W5+ ions. Absolute measurements were conducted employing the photon–ion merged-beams technique. Detailed photon-energy scans were performed at (67 ± 10) meV resolution in the 20–160 eV range. In contrast to photoionization of tungsten ions in lower charge states, the cross section is dominated by narrow, densely-spaced resonances. Theoretical results were obtained from a Dirac–Coulomb R-matrix approach employing a basis set of 457 levels providing cross sections for photoionization of W5+ ions in the ground level as well as the and metastable excited levels. Considering the complexity of the electronic structure of tungsten ions in low charge states, the agreement between theory and experiment is satisfactory.

Isotope shifts of the transition energies in Be-like thorium and uranium

Precise calculations of the isotope shifts in berylliumlike thorium and uranium ions are presented. The main contributions to the field and mass shifts are calculated within the framework of the Dirac–Coulomb–Breit Hamiltonian employing the configuration-interaction Dirac–Fock–Sturm method. These calculations include the relativistic, electron–electron correlation, and Breit-interaction effects. The QED, nuclear deformation, and nuclear polarization corrections are also evaluated.

Beyond chemical accuracy in the heavy p-block: The first ionization potentials and electron affinities of Ga-Kr, In-Xe, and Tl-Rn.

A relativistic coupled-cluster version of the Feller-Peterson-Dixon composite method has been used to accurately calculate the first ionization potentials (IPs) and electron affinities (EAs) of the post-d, p-block elements Ga-Rn. Complete basis set extrapolations including outer-core correlation at the CCSD(T) level of theory were combined with contributions from higher order electron correlation up to CCSDTQ, quantum electrodynamic effects (Lamb shift), and spin-orbit (SO) coupling including the Gaunt contribution. Several methods for including SO were investigated, in which all involved the four-component (4c) Dirac-Coulomb (DC) Hamiltonian. The treatment of SO coupling was the contribution that limited the final accuracy of the present results. In the cases where 4c-DC-CCSD(T) could be reliably used for the SO contributions, the final composite IPs and EAs agreed with the available experimental values to within an unsigned average error of just 0.16 and 0.20 kcal/mol, respectively. In all cases, the final IPs and EAs were within 1 kcal/mol of the available experimental values, except for the EAs of the group 13 elements (Ga, In, and Tl), where the currently accepted experimental values appear to be too large by as much as 4 kcal/mol. The values predicted in this work, which have estimated uncertainties of ±0.5 kcal/mol, are 5.25 (Ga), 7.69 (In), and 7.39 (Tl) kcal/mol. For the EAs of Po and At, which do not have experimental values, the current calculations predict values of 34.2 and 55.8 kcal/mol with estimated uncertainties of ±0.6 and ±0.3 kcal/mol, respectively.

A Hardy-type inequality and some spectral characterizations for the Dirac–Coulomb operator

We prove a sharp Hardy-type inequality for the Dirac operator. We exploit this inequality to obtain spectral properties of the Dirac operator perturbed with Hermitian matrix-valued potentials \({\mathbf {V}}\) of Coulomb type: we characterise its eigenvalues in terms of the Birman–Schwinger principle and we bound its discrete spectrum from below, showing that the ground-state energy is reached if and only if \({\mathbf {V}}\) verifies some rigidity conditions. In the particular case of an electrostatic potential, these imply that \({\mathbf {V}}\) is the Coulomb potential.

Quasirelativistic Coulomb Sturmian basis sets for atomic physics problems: illustration of their applicability to two-photon decay rates in hydrogen-like atoms

The continuum and discrete quasirelativistic one-electron Sturmian basis sets associated with the second-order Dirac–Coulomb equation are constructed by solving appropriate Sturm–Liouville problems. The potential-weighted orthogonality and closure relations obeyed by the members of such sets are discussed. The special case where both quasirelativistic Coulomb Sturmian basis sets are used to obtain solutions to the Gell-Mann–Feynman equation is presented. It is proved that for the continuum quasirelativistic Sturm–Liouville problem, the spectrum of potential strengths is continuous and covers the whole real axis. We have also introduced the complete outgoing continuum quasirelativistic Coulomb wavefunctions and presented the discretization of its radial components. The compact representation of the fully relativistic and quasirelativistic Coulomb Green’s functions is outlined. By making use of the Sturmian expansion of the Coulomb Green’s functions, closed-form expressions of two-photon spontaneous decay rates are derived for arbitrary multipole channels. For the sake of assessing the domain of applicability of quasirelativistic basis sets, an illustrative application is then carried out for two-photon decay rates for hydrogen-like ions. A comparison of present results with reference values previously obtained within a fully relativistic scheme (Bona et al 2014) is undertaken.

Photoionization of Rb2+ ions in the valence energy region 37–44 eV

Absolute photoionization cross sections for the Rb2+ ion were recently measured at high resolution over the energy range 37.31–44.08 eV, with autoionizing Rydberg resonance series identified, using the photon-ion merged-beam setup at the Advanced Light Source (als) in Berkeley (Macaluso D A et al 2016 J. Phys. B: At. Mol. Opt. Phys. 49, 235002; 2017 50, 119501). Detailed photon-energy scans taken at 13.5 ± 2.5 meV bandwidth illustrated multiple Rydberg resonance series associated with the ground and metastable states. Here we present theoretical cross section results obtained using the Dirac–Coulomb R-matrix approximation with a detailed analysis of the resonances. The calculations were performed for the , ground state and the corresponding metastable level. Results from the large-scale calculations are benchmarked against the als high-resolution measurements and reproduce the dominant resonance features in the spectra, providing confidence in the theoretical work for astrophysical applications.

Analytic evaluation of first-order properties within the mean-field variant of spin-free exact two-component theory.

We present a scheme for the calculation of energies and analytic energy gradients within spin-free exact two-component (SFX2C) theory in its mean-field variant, which we refer to as SFX2C-mf. In the presented scheme, the Foldy-Wouthuysen transformation is carried out after the spin-free four-component Hartree-Fock treatment such that in electron-correlated calculations only the non-mean-field part of the two-electron interactions is handled in an untransformed manner. The formulation of analytic gradients requires some adjustments in comparison with the nonrelativistic case, i.e., the additional solution of the spin-free Dirac Coulomb coupled-perturbed Hartee-Fock equations together with a simplified treatment of orbital relaxation at the SFX2C-mf level. The improved accuracy of SFX2C-mf in comparison with SFX2C-1e is demonstrated in the calculation of energies, dipole moments, and electric-field gradients for the hydrogen halides HX, X = F-At. It is shown that the main contribution to the improvement stems from the elimination of the error at the Hartree-Fock (HF) level; however, the corresponding correlation contribution is also improved such that SFX2C-mf can be considered a suitable scheme for the treatment of heavy-element compounds for which the error of SFX2C-1e is rather substantial.

Single photoionization of the Kr-like Rb ii ion in the photon energy range 22–46.0 eV

Abstract Single photoionization cross-sections for Kr-like Rb+ ions are reported in the energy (wavelength) range from 22 (564 Å) to 46 eV (270 Å). Theoretical cross-section calculations for this trans-Fe element are compared with measurements from the ASTRID radiation facility in Aarhus, Denmark and from the dual laser plasma technique, at respectively 40 and 35 meV FWHM energy resolution. In the photon energy region 22–32 eV the spectrum is dominated by excitation auto-ionizing resonance states. Above 32 eV the cross-section exhibits classic Fano window resonances features, which are analysed and discussed. Large-scale theoretical photoionization cross-section calculations, performed using a Dirac Coulomb R-matrix approximation are benchmarked against these high-resolution experimental results. Comparison of the theoretical work with the experimental studies allowed the identification of resonance features and their parameters in the spectra in addition to contributions from excited metastable states of the Rb+ ions.

Essentially Nonperturbative Vacuum Polarization Effects in a Two-Dimensional Dirac–Coulomb System with Z &gt; Zcr: Vacuum Charge Density

For a planar Dirac–Coulomb system with a supercritical axially symmetric Coulomb source with the charge Z > Zcr,1 and radius R0, we consider essentially nonperturbative vacuum-polarization effects. Based on a special combination of analytic methods, computer algebra, and numerical calculations used in our previous papers to study analogous effects in the one-dimensional “hydrogen atom,” we study the behavior of both the vacuum density ρVP(r) and the total induced charge and also the vacuum-polarization energy EVP. We mainly focus on divergences of the theory and the corresponding renormalization, on the convergence of partial series for ρVP(r) and ɛVP, on the integer-valuedness of the total induced charge, and on the behavior of the vacuum energy in the overcritical region. In particular, we show that the renormalization via the fermion loop with two external legs turns out to be a universal method, which removes the divergence of the theory in the purely perturbative and essentially nonperturbative modes for ρVP and ɛVP. The most important result is that for Z ≫ Zcr,1 in such a system, the vacuum energy becomes a rapidly decreasing function of the source charge Z, which reaches large negative values and whose behavior is estimated from below (in absolute value) as ~ −|ηeffZ3|/R0. We also study the dependence of polarization effects on the cutoff of the Coulomb asymptotic form of the external field. We show that screening the asymptotic value significantly changes the structure and properties of the first partial channels with mj = ±1/2,±3/2. We consider the nonperturbative calculation technique and the behavior of the induced density and the integral induced charge QVP in the overcritical region in detail.

Relativistic two-dimensional hydrogen-like atom in a weak magnetic field

A two-dimensional (2D) hydrogen-like atom with a relativistic Dirac electron, placed in a weak, static, uniform magnetic field perpendicular to the atomic plane, is considered. Closed forms of the first- and second-order Zeeman corrections to energy levels are calculated analytically, within the framework of the Rayleigh–Schrödinger perturbation theory, for an arbitrary electronic bound state. The second-order calculations are carried out with the use of the Sturmian expansion of the two-dimensional generalized radial Dirac–Coulomb Green function derived in the paper. It is found that, in contrast to the case of the three-dimensional atom (Stefańska, 2015), in two spatial dimensions atomic magnetizabilities (magnetic susceptibilities) are expressible in terms of elementary algebraic functions of a nuclear charge and electron quantum numbers. The problem considered here is related to the Coulomb impurity problem for graphene in a weak magnetic field.