Relativistic Hydrogen-like Atom Research Articles

Effective quantum electrodynamics: One-dimensional model of the relativistic hydrogen-like atom.

We consider a one-dimensional effective quantum electrodynamics (QED) model of the relativistic hydrogen-like atom using delta-potential interactions. We discuss the general exact theory and the Hartree-Fock approximation. The present one-dimensional effective QED model shares the essential physical feature of the three-dimensional theory: the nuclear charge polarizes the vacuum state (creation of electron-positron pairs), which results in a QED Lamb-type shift of the bound-state energy. Yet, this 1D effective QED model eliminates some of the most serious technical difficulties of the three-dimensional theory coming from renormalization. We show how to calculate the vacuum-polarization density at zeroth order in the two-particle interaction and the QED Lamb-type shift of the bound-state energy at first order in the two-particle interaction. The present work may be considered a step toward the development of a quantum-chemistry effective QED theory of atoms and molecules.

Dia- and paramagnetic contributions to magnetizabilities of relativistic hydrogenlike atoms in some low-lying discrete energy eigenstates

In this paper we present tabulated data for relative diamagnetic and paramagneticcontributions to the magnetizability (χ) of the relativistic hydrogenlike atoms with a pointlike, motionless and spinless nucleus of charge Ze. Utilizing general analytical formulas for the diamagnetic (χd) and paramagnetic (χp) components of χ, recently derived by us (P. Stefańska, 2020) with the aid of the Gordon decomposition technique, valid for an arbitrary discrete energy state, we have computed the numerical values of χd∕χ and χp∕χ for the ground state and for the first and the second set of excited states (i.e.: 2s1∕2, 2p1∕2, 2p3∕2, 3s1∕2, 3p1∕2, 3p3∕2, 3d3∕2, and 3d5∕2) of the hydrogen atom (Z=1) and for hydrogenic ions with 2⩽Z⩽137. We compare also the numerical values of the total magnetizabilities for the ground state 1s1∕2 and for each state belonging to the first set of excited states of selected hydrogenlike atoms, obtained with the use of two different values of the fine-structure constant, i.e.: α−1=137.035999139 (from CODATA 2014) and α−1=137.035999084 (from CODATA 2018).

Static magnetic multipole susceptibilities of the relativistic hydrogenlike atom in the ground state: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

Static magnetic multipole susceptibilities of the relativistic hydrogenlike atom in the ground state: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

Transformation of bound states of relativistic hydrogenlike atoms into a two-component form

A single-step Eriksen transformation of~$1S_{1/2}$,~$2P_{1/2}$ and~$2P_{3/2}$ states of the relativistic hydrogen-like atom is performed exactly by expressing each transformed function (TF) as a linear combination of eigenstates of the Dirac Hamiltonian. The transformed functions, which are four-component spinors with vanishing two lower components, are calculated numerically and have the same symmetries as the initial states. For all nuclear charges~$Z \in [1\ldots 92]$ a contribution of the initial state to TFs exceeds 86\% of the total probability density. Next large contribution to TFs comes from continuum states with negative energies close to~$-m_0c^2-E_b$, where~$E_b$ is the binding energy of initial state. Contribution of other states to TFs is less than~$0.1\%$ of the total probability density. Other components of TFs are nearly zero which confirms both validity of the Eriksen transformation and accuracy of the numerical calculations. The TFs of~$1S_{1/2}$ and~$2P_{1/2}$ states are close to~$1s$ and~$2p$ states of the nonrelativistic hydrogen-like atom, respectively, but the TF of~$2P_{3/2}$ state differs qualitatively from the~$2p$ state. Functions calculated with use of a linearized Eriksen transformation, being equivalent to the second order Foldy-Wouthuysen transformation, are compared with corresponding functions obtained by Eriksen transformation. A very good agreement between both results is obtained.

Closed-form expression for the magnetic shielding constant of the relativistic hydrogenlike atom in an arbitrary discrete energy eigenstate: Application of the Sturmian expansion of the generalized Dirac–Coulomb Green function

We present analytical derivation of the closed-form expression for the dipole magnetic shielding constant of a Dirac one-electron atom being in an arbitrary discrete energy eigenstate. The external magnetic field, by which the atomic state is perturbed, is assumed to be weak, uniform, and time independent. With respect to the atomic nucleus we assume that it is pointlike, spinless, motionless, and of charge $Ze$. Calculations are based on the Sturmian expansion of the generalized Dirac--Coulomb Green function [R. Szmytkowski, J. Phys. B 30, 825 (1997); erratum R. Szmytkowski, J. Phys. B 30, 2747(E) (1997)], combined with the theory of hypergeometric functions. The final result is of an elementary form and agrees with corresponding formulas obtained earlier by other authors for some particular states of the atom.

Static electric multipole susceptibilities of the relativistic hydrogenlike atom in the ground state: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

The ground state of the Dirac one-electron atom, placed in a weak, static electric field of definite $2^{L}$-polarity, is studied within the framework of the first-order perturbation theory. The Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30 (1997) 825, erratum: 30 (1997) 2747] is used to derive closed-form analytical expressions for various far-field and near-nucleus static electric multipole susceptibilities of the atom. The far-field multipole susceptibilities --- the polarizabilities $\alpha_{L}$, electric-to-magnetic cross-susceptibilities $\alpha_{\mathrm{E}L\to\mathrm{M}(L\mp1)}$ and electric-to-toroidal-magnetic cross-susceptibilities $\alpha_{\mathrm{E}L\to\mathrm{T}L}$ --- are found to be expressible in terms of one or two non-terminating generalized hypergeometric functions ${}_{3}F_{2}$ with the unit argument. Counterpart formulas for the near-nucleus multipole susceptibilities --- the electric nuclear shielding constants $\sigma_{\mathrm{E}L\to\mathrm{E}L}$, near-nucleus electric-to-magnetic cross-susceptibilities $\sigma_{\mathrm{E}L\to\mathrm{M}(L\mp1)}$ and near-nucleus electric-to-toroidal-magnetic cross-susceptibilities $\sigma_{\mathrm{E}L\to\mathrm{T}L}$ --- involve terminating ${}_{3}F_{2}(1)$ series and for each $L$ may be rewritten in terms of elementary functions. Exact numerical values of the far-field dipole, quadrupole, octupole and hexadecapole susceptibilities are provided for selected hydrogenic ions. Analytical quasi-relativistic approximations, valid to the second order in $\alpha Z$, where $\alpha$ is the fine-structure constant and $Z$ is the nuclear charge number, are derived for all types of the far-field and near-nucleus susceptibilities considered in the paper.

Magnetic-dipole-to-electric-quadrupole cross-susceptibilities for relativistic hydrogenlike atoms in some low-lying discrete energy eigenstates

In this paper we present tabulated data for magnetic-dipole-to-electric-quadrupole cross-susceptibilities (χM1→E2) for Dirac one-electron atoms with a pointlike, spinless and motionless nucleus of charge Ze. Numerical values of this susceptibility for the hydrogen atom (Z=1) and for hydrogenic ions with 2⩽Z⩽137 are computed from the general analytical formula, recently derived by us (Stefanska, 2016), valid for an arbitrary discrete energy eigenstate. In this work we provide 30 tables with the values of χM1→E2 for the ground state, and also for the first, the second and the third set of excited states (i.e.: 2s1/2, 2p1/2, 2p3/2, 3s1/2, 3p1/2, 3p3/2, 3d3/2, 3d5/2, 4s1/2, 4p1/2, 4p3/2, 4d3/2, 4d5/2, 4f5/2 and 4f7/2) of the relativistic hydrogenlike atoms. The value of the inverse of the fine-structure constant used in the calculations is α−1=137.035999139, and was taken from CODATA 2014.

Magnetic-field-induced electric quadrupole moments for relativistic hydrogenlike atoms: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

We consider a Dirac one-electron atom placed in a weak, static, uniform magnetic field. We show that, to the first order in the strength $B$ of the external field, the only electric multipole moments, which are induced by the perturbation in the atom, are those of an even order. Using the Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30, 825 (1997); J. Phys. B 30, 2747 (1997)], We derive a closed-form expression for the electric quadrupole moment induced in the atom in an arbitrary discrete energy eigenstate. The result, which has the form of a double finite sum involving the generalized hypergeometric functions ${}_{3}{F}_{2}$ of the unit argument, agrees with the earlier relativistic formula for that quantity, obtained by us for the ground state of the atom.

Magnetizabilities of relativistic hydrogenlike atoms in some arbitrary discrete energy eigenstates

We present the results of numerical calculations of magnetizability (χ) of the relativistic one-electron atoms with a pointlike, spinless and motionless nuclei of charge Ze. Exploiting the analytical formula for χ recently derived by us Stefańska (2015), valid for an arbitrary discrete energy eigenstate, we have found the values of the magnetizability for the ground state and for the first and the second set of excited states (i.e.: 2s1/2, 2p1/2, 2p3/2, 3s1/2, 3p1/2, 3p3/2, 3d3/2, and 3d5/2) of the Dirac one-electron atom. The results for ions with the atomic number 1⩽Z⩽137 are given in 14 tables. The comparison of the numerical values of magnetizabilities for the ground state and for each state belonging to the first set of excited states of selected hydrogenlike ions, obtained with the use of two different values of the fine-structure constant, i.e.: α−1=137.035999139 (CODATA 2014) and α−1=137.035999074 (CODATA 2010), is also presented.

Magnetizability of the relativistic hydrogenlike atom in an arbitrary discrete energy eigenstate: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

The Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30, 825 (1997); R. Szmytkowski, J. Phys. B 30, 2747(E) (1997)] is exploited to derive a closed-form expression for the magnetizability of the relativistic one-electron atom in an arbitrary discrete state, with a pointlike, spinless, and motionless nucleus of charge $Ze$. The result has the form of a double finite sum involving the generalized hypergeometric functions ${}_{3}{F}_{2}$ of the unit argument. Our general expression agrees with formulas obtained analytically earlier by other authors for some particular states of the atom. We present also numerical values of the magnetizability for some excited states of selected hydrogenlike ions with $1\ensuremath{\le}Z\ensuremath{\le}137$ and compare them with data available in the literature.

Electric-field-induced magnetic quadrupole moment in the ground state of the relativistic hydrogenlike atom: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

We consider a Dirac one-electron atom placed in a weak, static, uniform electric field. We show that, to the first order in the strength of the perturbing field, the only magnetic multipole moment induced in the ground state of the atom is the quadrupole one. The tensorial structure of that moment is resolved. Using the Sturmian expansion of the generalized Dirac-Coulomb Green function [Szmytkowski, J. Phys. B 30, 825 (1997); Szmytkowski, J. Phys. B 30, 2747(E) (1997)], we derive a closed-form expression for an $E1\ensuremath{\rightarrow}M2$ cross-susceptibility of the atom in the ground state.

Magnetic-field-induced electric quadrupole moment in the ground state of the relativistic hydrogenlike atom: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

We consider a Dirac one-electron atom placed in a weak, static, uniform magnetic field. We show that, to the first order in the strength $B$ of the perturbing field, the only electric multipole moment induced by the field in the ground state of the atom is the quadrupole one. Using the Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30 (1997) 825; erratum 30 (1997) 2747], we derive a closed-form expression for the induced electric quadrupole moment. The result contains the generalized hypergeometric function $_{3}F_{2}$ of the unit argument. Earlier calculations by other authors, based on the non-relativistic model of the atom, predicted in the low-field region the quadratic dependence of the induced electric quadrupole moment on $B$.

Electric and magnetic dipole shielding constants for the ground state of the relativistic hydrogen-like atom: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

The Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30 (1997) 825; erratum 30 (1997) 2747] is exploited to derive closed-form expressions for electric ($\sigma_{\mathrm{E}}$) and magnetic ($\sigma_{\mathrm{M}}$) dipole shielding constants for the ground state of the relativistic hydrogen-like atom with a point-like and spinless nucleus of charge $Ze$. It is found that $\sigma_{\mathrm{E}}=Z^{-1}$ (as it should be) and $$\sigma_{\mathrm{M}}=-(2Z\alpha^{2}/27)(4\gamma_{1}^{3}+6\gamma_{1}^{2}-7\gamma_{1}-12) /[\gamma_{1}(\gamma_{1}+1)(2\gamma_{1}-1)],$$ where $\gamma_{1}=\sqrt{1-(Z\alpha)^{2}}$ ($\alpha$ is the fine-structure constant). This expression for $\sigma_{\mathrm{M}}$ agrees with earlier findings of several other authors, obtained with the use of other analytical techniques, and is elementary compared to an alternative one presented recently by Cheng \emph{et al.} [J. Chem. Phys. 130 (2009) 144102], which involves an infinite series of ratios of the Euler's gamma functions.

Adiabatic spectrum for relativistic hydrogen in a strong homogeneous magnetic field

We study the bound states of relativistic hydrogen-like atoms coupled to strong homogeneous magnetic fields, assuming a fixed, infinitely heavy nucleus. Working in the adiabatic approximation in which the electron is confined to the lowest Landau level, we show that the corresponding Dirac Hamiltonian always has an infinite discrete spectrum accumulating at mc 2, m being the electron mass, and that, as the field strength increases, its eigenvalues successively descend into the lower part of the continuous spectrum, (−∞, −mc 2]. This phenomenon is for large B roughly periodical in log B.

Relativistic effects on information measures for hydrogen-like atoms

Position and momentum information measures are evaluated for the ground state of the relativistic hydrogen-like atoms. Consequences of the fact that the radial momentum operator is not self-adjoint are explicitly studied, exhibiting fundamental shortcomings of the conventional uncertainty measures in terms of the radial position and momentum variances. The Shannon and Rényi entropies, the Fisher information measure, as well as several related information measures, are considered as viable alternatives. Detailed results on the onset of relativistic effects for low nuclear charges, and on the extreme relativistic limit, are presented. The relativistic position density decays exponentially at large r , but is singular at the origin. Correspondingly, the momentum density decays as an inverse power of p . Both features yield divergent Rényi entropies away from a finite vicinity of the Shannon entropy. While the position space information measures can be evaluated analytically for both the nonrelativistic and the relativistic hydrogen atom, this is not the case for the relativistic momentum space. Some of the results allow interesting insight into the significance of recently evaluated Dirac–Fock vs. Hartree–Fock complexity measures for many-electron neutral atoms.

Static dipole magnetic susceptibilities of relativistic hydrogenlike atoms: A semianalytical approach

The binding energies and magnetic susceptibilities for states evolving from 1s{sub 1} at {sub {approx}}{sub sol{approx}} at {sub 2}, 2s{sub 1} at {sub {approx}}{sub sol{approx}} at {sub 2}, 2p{sub 1} at {sub {approx}}{sub sol{approx}} at {sub 2}, 2p{sub 3} at {sub {approx}}{sub sol{approx}} at {sub 2}, 3s{sub 1} at {sub {approx}}{sub sol{approx}} at {sub 2}, 3d{sub 3} at {sub {approx}}{sub sol{approx}} at {sub 2}, and 3d{sub 5} at {sub {approx}}{sub sol{approx}} at {sub 2} are calculated using power-series solutions of the Dirac equation for hydrogenic atoms in static and uniform magnetic B. The accuracy of the binding energies for low and medium magnetic fields exceeds that of previous variational calculations. In the low-magnetic-field limit the highly accurate values of energies are used to determine the relativistic Paschen-Back effect and relativistic magnetic susceptibilities by expansion of the fully relativistic energy into power series of the parameter B/Z{sup 2}. The linear term of this series is related to the relativistic Paschen-Back effect and the square term is proportional to the relativistic dipole magnetic susceptibility of the atom.

Publisher's Note: Stark-induced magnetic anapole moment in the ground state of the relativistic hydrogenlike atom: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function [Phys. Rev. A73, 022511 (2005)

Publisher's Note: Stark-induced magnetic anapole moment in the ground state of the relativistic hydrogenlike atom: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function [Phys. Rev. A<b>73</b>, 022511 (2005)

Stark-induced magnetic anapole moment in the ground state of the relativistic hydrogenlike atom: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

The Sturmian expansion of the generalized Dirac-Coulomb Green function R. Szmytkowski, J. Phys. B 30, 825 1997; 30, 2747E1997 is used to derive an analytical formula for the static magnetic anapole toroidal dipole moment induced in the ground state of the relativistic hydrogenlike atom by a weak, spatially uniform, static electric field. An expression for the anapole polarizability for the system in question is found. This expression contains a single generalized hypergeometric series 3 F 2 with the unit argument. In the non- relativistic limit our result agrees with that of Lewis and Blinder Phys. Rev. A 52, 4439 1995. Over the past three decades, in atomic and molecular physics an interest arose in effects associated with magnetic anapole moments. These effects may be divided into two principal categories. The first category comprises those atomic and molecular phenomena in which a nuclear mag- netic anapole moment, resulting from parity nonconservation in nuclear forces, manifests itself cf., e.g., Refs. 1-6. The second category includes those effects which evince the magnetic anapole moment of an atomic or a molecular elec- tronic cloud 7-22. In this paper, we shall be concerned with an effect falling into the second of the aforementioned categories. Specifi- cally, we shall be interested in the anapole moment induced in the ground state of the one-electron Dirac atom by a weak, spatially uniform, static electric field. This problem was al- ready considered a decade ago, both nonrelativistically and relativistically, by Lewis and Blinder 16; the reason for which we have decided to revisit it is that the relativistic calculations carried out in Ref. 16 were approximate in character. Exploiting the Sturmian expansion of the general- ized Dirac-Coulomb Green function 23-27, in the present paper we shall calculate the Stark-induced anapole moment exactly. The structure of the paper is as follows. In Sec. II we define the anapole moment. In Sec. III we consider the one- electron Dirac atom in the weak, spatially uniform, static electric field and apply the first-order Rayleigh-Schrodinger perturbation theory, together with the Green functions tech- nique, to derive an approximate expression for a perturbed electronic wave function. Next, in Sec. IV, this approximate wave function is used to determine an induced electric cur- rent density in the atom. Subsequently, the anapole moment of the resulting i.e., unperturbed plus induced current dis- tribution is considered. It is found that only the induced cur- rent contributes and the form of this contribution is calcu- lated analytically exploiting the Sturmian functions technique. The paper ends with conclusions, constituting Sec. V, and with several appendixes containing supplemen- tary material.

Relationship between the static dipole polarizability and the static dipole magnetizability of the relativistic hydrogen-like atom

We show that the ground-state static dipole polarizability α d( Z) and the ground-state static dipole magnetizability χ d( Z) of the relativistic hydrogen-like atom, with a spinless and point-like nucleus of charge + Ze, are related through the formula α d ( Z ) = - 3 ( γ 1 - 2 ) 2 ( α Z ) - 2 χ d ( Z ) - a 0 3 Z - 4 ( γ 1 + 1 ) ( 2 γ 1 + 1 ) ( 14 γ 1 3 - 63 γ 1 2 + 55 γ 1 - 24 ) / 36 , where α and a 0 are, respectively, the Sommerfeld fine-structure constant and the Bohr radius, while γ 1 = 1 - ( α Z ) 2 .

Gordon decomposition of the static dipole polarizability of the relativistic hydrogen-like atom: application of the Sturmian expansion of the first-order Dirac–Coulomb Green function

The Sturmian expansion of the generalized first-order Dirac–Coulomb Green function (Szmytkowski 1997 J. Phys. B: At. Mol. Opt. Phys. 30 825, 2747) is used to derive an analytical formula for the static dipole polarizability of the relativistic hydrogen-like atom in the ground state. The formula contains a generalized hypergeometric series 3F2 with the unit argument. It is identical with one found recently, in a completely different way, by Yakhontov (2003 Phys. Rev. Lett. 91 093001) and is the most compact among all known analytical expressions for the polarizability. Partitioning of the polarizability into convection and spin-polarization components, resulting from the Gordon decomposition of an induced electronic charge density, is carried out. It appears that the spin-polarization component of the polarizability vanishes.