One-electron Systems Research Articles

Solver for the Electronic V-Representation Problem of Time-Dependent Density Functional Theory.

One route to numerically propagating quantum systems is time-dependent density functional theory (TDDFT). The application of TDDFT to a particular system's time evolution is predicated on V-representability, which we have analyzed in a previous publication. Here, we describe a newly developed solver for the scalar time-dependent Kohn-Sham potential. We present and interpret the force-balance equation central to our numerical method, describe details of its implementation, and present illustrative numerical results for one- and two-electron systems in both one-dimensional and three-dimensional grids. Innovations of our numerical implementation include the use of preconditioning when inverting the force-balance matrix and an improved propagation method obtaining the Kohn-Sham potential self-consistently at each step of the propagation. A new characterization of V-representability for one-electron systems is also included, along with possible improvements and future directions.

Hyperfine structure in the H2+ and HD+ molecular ions at order mα6

A complete effective Hamiltonian for relativistic corrections at orders $m\alpha^6$ and $m\alpha^6(m/M)$ in a one-electron molecular system is derived from the NRQED Lagrangian. It includes spin-independent corrections to the energy levels and spin-spin scalar interactions contributing to the hyperfine splitting, both of which had been studied previously. In addition, corrections to electron spin-orbit and spin-spin tensor interactions are newly obtained. This allows improving the hyperfine structure theory in the hydrogen molecular ions. Improved values of the spin-orbit hyperfine coefficient are calculated for a few transitions of current experimental interest.

POLALMM: A program to compute polarizabilities for nominal one-electron systems using the Lagrange-mesh method

We present a program to compute polarizabilities of nominal one-electron systems using the Lagrange-mesh method (LMM) (Baye, 2015) [1], that was used by Filippin et al., (2018). A semiempirical-core-potential approach is implemented, ultimately solving a Dirac-like equation by diagonalizing the corresponding Hamiltonian matrix. In order to build the core potential, the core orbitals are obtained from independent calculations using the Grasp2018 package (Fischer et al., 2019). Therefore we provide an easy-to-use interface between the Grasp2018 package and the LMM complete finite basis, allowing to switch easily from one one-electron basis to the other. Program summaryProgram Title: POLALMMCPC Library link to program files:http://dx.doi.org/10.17632/6mw5gdwfkt.1Licensing provisions: MIT licenseProgramming language: Fortran90Nature of problem: Determination of the dipole and quadrupole polarizabilities.Solution method: We combine a semiempirical-core-potential approach with the numerical Lagrange-mesh method to solve a Dirac-like one-electron equation [2]. The building of the core potential requires the prior knowledge of core orbitals provided by Grasp [3]. Two free parameters are optimized by fitting the computed single-electron valence energies to their experimental reference value.

Erratum: Simple self-interaction correction to random-phase-approximation-like correlation energies [Phys. Rev. A 100, 022515 (2019)

The random phase approximation (RPA) is exact for the exchange energy of a many-electron ground state, but RPA makes the correlation energy too negative by about 0.5 eV/electron. That large short-range error, which tends to cancel out of iso-electronic energy differences, is largely corrected by an exchange-correlation kernel, or (as in RPA+) by an additive local or semilocal correction. RPA+ is by construction exact for the homogeneous electron gas, and it is also accurate for the jellium surface. RPA+ often gives realistic total energies for atoms or solids in which spin-polarization corrections are absent or small. RPA and RPA+ also yield realistic singlet binding energy curves for H2 and N2, and thus RPA+ yields correct total energies even for spin-unpolarized atoms with fractional spins and strong correlation, as in stretched H2 or N2. However, RPA and RPA+ can be very wrong for spin-polarized one-electron systems (especially for stretched H2+), and also for the spin-polarization energies of atoms. The spin-polarization energy is often a small part of the total energy of an atom, but important for ionization energies, electron affinities, and the atomization energies of molecules. Here we propose a computationally efficient generalized RPA+ (gRPA+) that changes RPA+ only for spin-polarized systems by making gRPA+ exact for all one-electron densities, in the same simple semilocal way that the correlation energy densities of many meta-generalized gradent approximations are made self-correlation free. By construction, gRPA+ does not degrade the exact RPA+ description of jellium. gRPA+ is found to greatly improve upon RPA and RPA+ for the ionization energies and electron affinities of light atoms. Many versions of RPA with an approximate exchange-correlation kernel fail to be exact for all one-electron densities, and they can also be self-interaction corrected in this way.

The one-electron self-interaction error in 74 density functional approximations: a case study on hydrogenic mono- and dinuclear systems.

The self-interaction error (SIE), i.e. unphysical interactions of electrons with themselves, has plagued developers and users of Density Functional Approximations (DFAs) since the inception of Density Functional Theory (DFT). Formally, it can be separated into the one-electron and many-electron SIE; herein we present one of the most comprehensive studies of the first. While we focus mostly on the total SIE, we also make use of two different decompositions. The first is a separation into functional and density-driven errors as championed by Sim, Burke and co-workers [J. Phys. Chem. Lett., 2018, 9, 6385-6392]; the second separates the error into exchange, correlation, and one-electron components, with the latter being a density error that has not been discussed in this form before. After investigating the familiar hydrogen atom and dihydrogen cation, we establish a relationship between the SIE and the nuclear charge with the help of a series of heavier hydrogenic analogues. For the mononuclear systems and the diatomics at the dissociation limit, this relationship is linear in nature with prominent exceptions, mostly belonging to the Minnesota and range-separated (double-)hybrid DFAs. For the first time, we also show how the magnitude of the SIE depends on the underlying atomic-orbital basis set and how DFAs that rely on a popular van-der-Waals DFT type London-dispersion term exhibit "self-dispersion". We find that range separation is not a panacea for solving the one-electron SIE. DFAs that have been developed to be one-electron SIE free for one system, such as the hydrogen atom, show larger errors for heavier hydrogenic systems. Often, one-electron SIE-free DFAs rely on fortuitous error cancellation between their exchange and correlation components. An analysis of the most robust methods for general applications to date reveals that they suffer moderately from the one-electron SIE, while DFAs that are nearly SIE-free do not perform well in applications. Implicit in the continued existence of the one-electron SIE is that well-performing DFAs continue to suffer insufficiencies at their fundamental levels that are being compensated for by the SIE. Our analysis includes more than 250 000 datapoints, resulting in multiple insights that may drive future developments of new DFAs or SIE corrections.

A new approach to estimate atomic energies.

A new approach to estimate atomic energies is introduced. The method is based on utilization of experimental ionization energies as well as conversion of "n-electron atomic systems" to n "one-electron systems." Sample detail calculations are presented with typical graphs to show the distribution of different types of energy within an atom. The breakdown of atomic energies into kinetic, electron-nucleus attraction, and electron-electron repulsion is shown within an atom as well the total of energies of each type for elements. Then in a following step, the variations in kinetic, electron-electron, and electron-nucleus interaction energies of electrons as evidence for atomic shell changes are presented. Furthermore, the article overviews the spatial gaps between orbitals as an added evidence for existence of electronic shells. The findings in this article have significant implications for the structure of atoms and the layout of periodic table. Graphical abstractElectronic energy variations of barium (Ba) for kinetic, electron-electron repulsion, and electron-nucleus energies versus electron number.

On the statistical properties of a hydrogenic atom broadened by linear Stark effect

Spectroscopy based on Stark broadening is a very efficient and successful method for diagnosing the physical conditions in laboratory or astrophysical plasmas. The studies are often focused on the lines of hydrogen or hydrogen-like ions due to the simple atomic structure. However, the Stark modeling may be difficult even for one-electron systems, in particular when radiative transitions with highly-degenerate Rydberg states are considered. Besides, the computations may become challenging for the modeling of line shapes in an optically thick plasma and accounting for non-local thermodynamical equilibrium (non-LTE) effects on the state populations. The statistical approach is a powerful technique which can be used to simplify complex coupled problems by replacing the numerous levels or lines by a single continuous distribution characterized by a reduced number of exact moments: area, center-of-gravity, variance, etc. In this paper, we study various statistical properties of a hydrogenic atom broadened by Stark effect. First, we derive the moments of level energies accounting for linear and quadratic Stark effects, electric-quadrupole and fine structure corrections. The resulting formulas allows one to estimate the relative importance of these corrections on the energies. We then derive the moments of the strength-weighted line energies for the Lyman and Balmer series in the framework of Linear-Stark effect. The moments are then used in a Gram–Charlier expansion series to represent the line shape, the resolution of the spectrum being controlled by the order at which the series is truncated. We show that the line shape of the transition can be expressed using universal functions of n and , independent of the atomic number and of the field intensity. This allows for fast Stark broadening calculations which can be used in more complex simulations (in opacity or emissivity codes, for radiative transfer simulations or non-LTE models).

Simple self-interaction correction to random-phase-approximation-like correlation energies

The random-phase approximation (RPA) is exact for the exchange energy of a many-electron ground state, but RPA makes the correlation energy too negative by about 0.5 eV/electron. That large short-range error, which tends to cancel out of isoelectronic energy differences, is largely corrected by an exchange-correlation kernel, or (as in RPA+) by an additive local or semilocal correction. RPA+ is by construction exact for the homogeneous electron gas, and it is also accurate for the jellium surface. RPA+ often gives realistic total energies for atoms or solids in which spin-polarization corrections are absent or small. RPA and RPA+ also yield realistic singlet binding energy curves for ${\mathrm{H}}_{2}$ and ${\mathrm{N}}_{2}$, and thus RPA+ yields correct total energies even for spin-unpolarized atoms with fractional spins and strong correlation, as in stretched ${\mathrm{H}}_{2}$ or ${\mathrm{N}}_{2}$. However, RPA and RPA+ can be very wrong for spin-polarized one-electron systems (especially for stretched ${{\mathrm{H}}_{2}}^{+}$), and also for the spin-polarization energies of atoms. The spin-polarization energy is often a small part of the total energy of an atom, but important for ionization energies, electron affinities, and the atomization energies of molecules. Here we propose a computationally efficient generalized RPA+ (gRPA+) that changes RPA+ only for spin-polarized systems by making gRPA+ exact for all one-electron densities, in the same simple semilocal way that the correlation energy densities of many metageneralized gradient approximations are made self-correlation free. By construction, gRPA+ does not degrade the exact RPA+ description of jellium. gRPA+ is found to greatly improve upon RPA and RPA+ for the ionization energies and electron affinities of light atoms. Many versions of RPA with an approximate exchange-correlation kernel fail to be exact for all one-electron densities, and they can also be self-interaction corrected in this way.

Distributed Gaussian orbitals for molecular calculations: application to simple systems

ABSTRACTIn this article, the possible use of sets of basis functions alternative with respect to the usual atom-centred orbitals sets is considered. The orbitals describing the inner part of the wavefunction (i.e. the region close to each nucleus) are still atomic Gaussian functions: tight Gaussian orbitals having different angular momenta and large exponential coefficients, centred on each nucleus. On the other hand, the outer part of the wavefunction is described through a set of s-type distributed Gaussian orbitals: s-type Gaussians having a unique fixed exponent, and whose fixed centres are placed on a uniform mesh of points evenly distributed in the region surrounding all the atoms of the molecule. The resulting basis sets are applied to various one-electron systems in order to assess the capability to describe different types of one-electron wavefunctions. Moreover, the hydrogen atom and the dihydrogen cation, for which accurate solutions exist, are also considered for comparison, to assess the effectiveness of the proposed approach. Preliminary results concerning the treatment of electron correlation, necessary for a quantitatively correct description of many-electron atoms and molecules, are also presented.

Spin-averaged effective Hamiltonian of orders mα6 and mα6(m/M) for hydrogen molecular ions

The spin-averaged effective Hamiltonian of orders $m{\ensuremath{\alpha}}^{6}$ and $m{\ensuremath{\alpha}}^{6}(m/M)$ for a one-electron two-center Coulombic system is derived by using the theory of nonrelativistic quantum electrodynamics (NRQED), without assuming the Born-Oppenheimer approximation. The separated singularities from the first- and second-order perturbations are shown to be canceled out analytically for both order $m{\ensuremath{\alpha}}^{6}$ and $m{\ensuremath{\alpha}}^{6}(m/M)$ corrections by regularizing the effective Hamiltonian. Our results can be used to perform high-precision spectroscopic calculations of hydrogen molecular ions.

Distributed Gaussian orbitals for the description of electrons in an external potential.

In this work, we demonstrate the viability of using distributed Gaussian orbitals as a basis set for the calculation of the properties of electrons subjected to an external potential. We validate our method by studying one-electron systems for which we can compare to exact analytical results. We highlight numerical aspects that require particular care when using a distributedGaussian basis set. In particular, we discuss the optimal choice for the distance between two neighboring Gaussian orbitals. Finally, we show how our approach can be applied to many-electron problems.

Range-Separated Double-Hybrid Functional from Nonempirical Constraints.

On the basis of our previous developments in the field of nonempirical double hybrids, we present here a new exchange-correlation functional based on a range-separated model for the exchange part and integrating a nonlocal perturbative correction to the electron correlation contribution. Named RSX-QIDH, the functional is free from any kind of empirical parametrization. Its range-separation parameter is set to recover the total energy of the hydrogen atom, thus eliminating the self-interaction error for this one-electron system. Subsequent tests on some relevant benchmark data sets confirm that the self-interaction error is particularly low for RSX-QIDH. This new functional provides also correct dissociation profiles for charged rare-gas dimers and very accurate ionization potentials directly from Kohn-Sham orbital energies. Above all, these good results are not obtained at the expense of other properties. Indeed, further tests on standard benchmarks show that RSX-QIDH is competitive with the more empirical ωB97X-2 double hybrid and outperforms the parent LC-PBE long-range corrected hybrid, thus underlining the important role of the nonlocal perturbative correlation.

Particle–hole symmetry in many-body theories of electron correlation

ABSTRACTSecond-quantised creation and annihilation operators for fermionic particles anticommute, but the same is true for the creation and annihilation operators for holes. This introduces a symmetry into the quantum theory of fermions that is absent for bosons. In ab initio electronic structure theory, it is common to classify methods by the number of electrons for which the method returns exact results: for example Hartree–Fock theory is exact for one-electron systems, whereas coupled-cluster theory with single and double excitations is exact for two-electron systems. Here, we discuss the generalisation: methods based on approximate wavefunctions that are exact for n-particle systems are also exact for n-hole systems. Novel electron correlation methods that attempt to improve on the coupled-cluster framework sometimes retain this property, and sometimes lose it. Here, we argue for retaining particle–hole symmetry as a desirable design criterion of approximate electron correlation methods. Dispensing with it might lead to loss of n-representability of density matrices, and this in turn can lead to spurious long-range behaviour in the correlation energy.

Universal lower bounds on the kinetic energy of electronic systems with noncollinear magnetism

The distribution of noncollinear magnetism in an electronic system provides information about the kinetic energy as well as some kinetic energy densities. Two different everywhere-positive kinetic densities related to the Schr\"odinger--Pauli Hamiltonian are considered. For one-electron systems described by a single Pauli spinor, the electron density, spin density and current density completely determines these kinetic energy densities. For many-electron systems, lower bounds on the kinetic energy densities are proved. These results generalize a lower bound due to von Weizs\"acker, which is based on the electron density alone and plays an important role in density functional theory. The results have applications in extensions of density functional theory that incorporate noncollinear spin densities and current densities.

Photon-momentum transfer in photoionization: From few photons to many

In most models and theoretical calculations describing multiphoton ionization by infrared light, the dipole approximation is used. This is equivalent to setting the very small photon momentum to zero. Using numerical solutions of the (nondipole) three-dimensional time-dependent Schr\odinger equation for one-electron (H-like) systems, we investigate the effect the photon-momentum transfer to the photoelectron in various regimes: from the few (one, two, three, eight) absorbed photons to multiphoton and tunneling regimes. We find that in all regimes the average electron momentum acquired from absorbed photons is a linear function of the average energy of the photoelectron, but the slope is different in the few-photon and multiphoton regimes. In the multiphoton regime, the photon-momentum signature in the photoelectron-momentum distributions (along the photon momentum) depends on the ellipticity of the laser polarization, with linear laser polarization spectra different from circular polarization spectra. For a given laser intensity the average electron-momentum gain from an absorbed photon is over two and a half times smaller for linear polarization than for circular polarization. However, for both polarizations, the average electron-momentum gain is nearly equal to the average electron energy divided by the speed of light.

One-loop vacuum polarization at mα7 and higher orders for three-body molecular systems

We present calculations of the one-loop vacuum polarization correction (Uehling potential) for the three-body problem in the NRQED formalism. The case of one-electron molecular systems is considered. Numerical results of the vacuum polarization contribution at m$\alpha$7 and higher orders for the fundamental transitions (v = 0, L = 0) $\rightarrow$ (v ' = 1, L ' = 0) in the H2+ and HD+ molecular ions are presented and compared with calculations performed in the adiabatic approximation. The residual uncertainty from this contribution on the transition frequencies is shown to be of a few tens of Hz.

Inapplicability of exact constraints and a minimal two-parameter generalization to the DFT+ U based correction of self-interaction error

In approximate density functional theory (DFT), the self-interaction error is an electron delocalization anomaly associated with underestimated insulating gaps. It exhibits a predominantly quadratic energy-density curve that is amenable to correction using efficient, constraint-resembling methods such as DFT + Hubbard $U$ (DFT+$U$). Constrained DFT (cDFT) enforces conditions on DFT exactly, by means of self-consistently optimized Lagrange multipliers, and while its use to automate error corrections is a compelling possibility, we show that it is limited by a fundamental incompatibility with constraints beyond linear order. We circumvent this problem by utilizing separate linear and quadratic correction terms, which may be interpreted either as distinct constraints, each with its own Hubbard $U$ type Lagrange multiplier, or as the components of a generalized DFT+$U$ functional. The latter approach prevails in our tests on a model one-electron system, $H_2^+$, in that it readily recovers the exact total-energy while symmetry-preserving pure constraints fail to do so. The generalized DFT+$U$ functional moreover enables the simultaneous correction of the total-energy and ionization potential or the correction of either together with the enforcement of Koopmans condition. For the latter case, we outline a practical, approximate scheme by which the required pair of Hubbard parameters, denoted as U1 and U2, may be calculated from first-principles.

Frequency-dependent polarizabilities and shielding factors for confined one-electron systems

Frequency-dependent dipole polarizabilities and shielding factors are calculated for the ground state of spherically symmetric screened one-electron systems embedded in an impenetrable spherical cavity. Coulomb, Yukawa, Hulthén and exponential cosine-screened Coulomb potentials are considered. In contrast to free systems, Dirichlet boundary conditions introduce a contribution to the shielding factor that results from an integral over the surface of the confining boundary. This is a fundamental difference between free and confined systems and results in unexpected modifications to some of the classic relations for free systems. The methods derived also give a simple expression for the polarizability of the confined harmonic oscillator as an example of extending the methods of this work to potentials beyond the four studied.

Solution of the Dirac equation using the Rayleigh-Ritz method: Flexible basis coupling large and small components. Results for one-electron systems.

An algebraic solution of the Dirac equation is reinvestigated. Slater-type spinor orbitals and their corresponding system of differential equations are defined in two- and four-component formalism. They describe the radial function in components of the wave function of the Dirac equation solution to high accuracy. They constitute the matrix elements arising in a generalized eigenvalue equation. These terms are evaluated through prolate spheroidal coordinates. The corresponding integrals are calculated by the numerical global-adaptive method taking into account the Gauss-Kronrod numerical integration extension. Sample calculations are performed using flexible basis sets generated with both signs of the relativistic angular momentum quantum number κ. Applications to one-electron atoms and diatomics are detailed. Variationally optimum values for orbital parameters are obtained at given nuclear separation. Methods discussed in this work are capable of yielding highly accurate relativistic two-center integrals for all ranges of orbital parameters. This work provides an efficient way to overcome the problems that arise in relativistic calculations.

Sum rules for the polarization correlations in photoionization and bremsstrahlung

The polarization correlations in doubly differential cross sections are investigated for photoionization and ordinary bremsstrahlung. These correlations describe the polarization transfer between incident light and ejected photoelectrons as well as between an incoming electron beam and bremsstrahlung light, respectively. They are characterized by a set of seven real parameters ${C}_{ij}$. We show that the squares of these parameters are connected by simple ``sum rules.'' These sum rules can be applied for both one-electron systems and also for atoms, if the latter are described within the independent particle approximation. In particular, they are exact in their simplest form (i) for the photoionization of $K$-, ${L}_{I,II}$-, and ${M}_{I,II}$-atomic shells, as well as (ii) for bremsstrahlung in which the electron is scattered into ${s}_{1/2}$ or ${p}_{1/2}$ states, as in the tip (bremsstrahlung) region. Detailed calculations are performed to verify the derived identities and to discuss their possible applications for the analysis of modern photoionization and bremsstrahlung experiments. In particular, we argue that the sum rules may help to determine the entire set of (significant) polarization correlations in the case when not all ${C}_{ij}$ are available for experimental observation.