Many-electron Atoms Research Articles

Four-Dimensional Scaling of Dipole Polarizability: From Single-Particle Models to Atoms and Molecules.

Scaling laws enable the determination of physicochemical properties of molecules and materials as a function of their size, density, number of electrons or other easily accessible descriptors. Such relations can be counterintuitive and nonlinear, and ultimately yield much needed insight into quantum mechanics of many-particle systems. In this work, we show on the basis of single-particle models, multielectron atoms and molecules that the dipole polarizability of quantum systems is generally proportional to the fourth power of a characteristic length, computed from the ground-state wave function. This four-dimensional (4D) scaling is independent of the ratio of bound-to-bound and bound-to-continuum electronic transitions and applies to many-electron atoms when a correlated length metric is used. Finally, this scaling law is applied to predict the polarizability of molecules by electrostatically coupled atoms-in-molecules approach, obtaining approximately 8% absolute and relative accuracy with respect to hybrid density functional theory (DFT) on the QM7-X data set of organic molecules, providing an efficient and scalable model for the anisotropic polarizability tensors of extended (bio)molecules.

Going Beyond the GW Approximation Using the Time-Dependent Hartree-Fock Vertex.

The time-dependent Hartree-Fock (TDHF) vertex of many-body perturbation theory (MBPT) makes it possible to extend TDHF theory to charged excitations. Here we assess its performance by applying it to spherical atoms in their neutral electronic configuration. On a theoretical level, we recast the TDHF vertex as a reducible vertex, highlighting the emergence of a self-energy expansion purely in orders of the bare Coulomb interaction; then, on a numerical level, we present results for polarizabilities, ionization energies (IEs), and photoemission satellites. We confirm the superiority of THDF over simpler methods such as the random phase approximation for the prediction of atomic polarizabilities. We then find that the TDHF vertex reliably provides better IEs than GW and low-order self-energies do in the light-atom, few-electron regime; its performance degrades in heavier, many-electron atoms instead, where an expansion in orders of an unscreened Coulomb interaction becomes less justified. New relevant features are introduced in the satellite spectrum by the TDHF vertex, but the experimental spectra are not fully reproduced due to a missing account of nonlinear effects connected to hole relaxation. We also explore various truncations of the self-energy given by the TDHF vertex, but do not find them to be more convenient than low-order approximations such as GW and second Born (2B), suggesting that vertex corrections should be carried out consistently both in the self-energy and in the polarizability.

О расширенном пространстве Гильберта

The construction of the extended Hilbert space is presented in the form of a direct sum of the spaces of vectors of finite and infinite norms as the main space in the mathematical formalism of quantum mechanics of a many-electron atom. On the example of constructing the analytical structure of the probability amplitude 1s-3p of photoexcitation of a neon atom, the implementation of the EHS-construct for solving the equations of the self-consistent Hartree-Fock field is given.

A nuclear configuration interaction approach to study nuclear spin effects: an application to ortho- and para-3 He2 @C60.

We introduce a non-orthogonal configuration interaction approach to investigate nuclear quantum effects on energies and densities of confined fermionic nuclei. The Hamiltonian employed draws parallels between confined systems and many-electron atoms, where effective non-Coulombic potentials represent the interactions of the trapped particles. One advantage of this method is its generality, as it offers the potential to study the nuclear quantum effects of various confined species affected by effective isotropic or anisotropic potentials. As a first application, we analyze the quantum states of two 3 He atoms encapsulated in C60 . At the Hartree-Fock level, we observe the breaking of spin and spatial symmetries. To ensure wavefunctions with the correct symmetries, we mix the broken-symmetry Hartree-Fock states within the non-orthogonal configuration interaction expansion. Our proposed approach predicts singly and triply degenerate ground states for the singlet (para-3 He2 @C60 ) and triplet (ortho-3 He2 @C60 ) nuclear spin configurations, respectively. The ortho-3 He2 @C60 ground state is 5.69 cm-1 higher in energy than the para-3 He2 @C60 ground state. The nuclear densities obtained for these states exhibit the icosahedral symmetry of the C60 embedding potential. Importantly, our calculated energies for the lowest 85 states are in close agreement with perturbation theory results based on a harmonic oscillator plus rigid rotor model of 3 He2 @C60 .

Basis Set Calculations of Heavy Atoms

Most modern calculations of many-electron atoms use basis sets of atomic orbitals. An accurate account for electronic correlations in heavy atoms is a very difficult computational problem, and an optimization of the basis sets can reduce computational costs and increase final accuracy. Here, we propose a simple differential ansatz to form virtual orbitals from the Dirac–Fock orbitals of the core and valence electrons. We use basis sets with such orbitals to calculate different properties in Cs including hyperfine structure constants and QED corrections to the valence energies and to the E1 transition amplitudes.

An efficient finite difference approach to solutions of Schrödinger equations of atoms in non-linear coordinates

We present a transformed-coordinates method to solve the Schrödinger equation for H-like, He-like, and Li-like systems. Each Cartesian axes of the original Schrödinger equation is transformed to another coordinate system with the square root transformation The resulting Hamiltonian contains the first and the second derivative for the kinetic energy part and with the potential proportional to the power of four, decaying faster than the original Coulomb potential. The total energies, their components, and the virial ratio are superior to those of the untransformed coordinates due to the considerably many data-points obtained and long-range sampling. Furthermore, a five-times or better computational efficiency is demonstrated in comparison to the standard method with much-improved accuracy. In agreement with the accurate method, the obtained wavefunction includes not only the radial but also the angular electron correlation of many-electron ions or atoms.

Analytical and Parameter-Free Hückel Theory Made Possible for Symmetric Hx Clusters.

It is widely accepted that energetics of chemical bond breaking and formation can be described with simple mathematical forms only at the expense of extensive parameterization. In this work, the discovery of a simple tight-binding-type mathematical framework that can accurately predict the relative energetics of regular Hx polygons (2 ≤ x ≤ 15) in the ground states with their respective spin multiplicities using no parameters has been reported. The framework recasts Hückel theory in a density functional theory form by making use of Anderson and Adams-Gilbert theories of localized orbitals. For the systems examined, the method exhibits mean absolute errors of ∼0.02 Å (edge lengths) and ∼0.15 eV/atom (energy minima) relative to correlated-electron quantum chemistry calculations. Its accuracy is found to be comparable to the generalized gradient approximation and superior to standard parameterized tight binding and reactive potentials applied to Hx structures. Generalization of the theoretical framework to systems of many-electron atoms is presented, along with the comparison of the method to existing semiempirical tight binding and bond order potential approaches.

Finite element method as an alternative to study the electronic structure of confined atoms.

The finite element method (FEM) based on a nonregular mesh is used to solve Hartree-Fock and Kohn-Sham equationsfor three atoms (hydrogen, helium, and beryllium) confined by finite and infinite potentials, defined in terms of piecewise functions or functions with a well-defined first derivative. This approach's reliability is shown when contrasted with Roothaan's approach, which depends on a basis set. Therefore, its exponents must be optimized for each confinement imposed over each atom, which is a monumental task. The comparison between our numerical approach and Roothaan's approach is made by using total and orbitals energies from the Hartree-Fock method, where there are several comparison sources. Regarding the Kohn-Sham method, there are few published data and consequently the results reported here can be used as a benchmark for future comparisons. The way to solve Hartree-Fock or Kohn-Sham equationsby the FEM is entirely appropriate to study confined atoms with any form of confinement potential. This article represents a step toward developing a fully numerical quantum chemistry code free of basis sets to obtain the electronic structure of many-electron atoms confined by arbitrary confinement.

A simple data-driven level finding method of many-electron atoms and heavy nuclei based on statistical outlier detection *

We report a simple and pure data-driven method to find new energy levels of quantum many-body systems only from observed line wavelengths. In our method, all the possible combinations are computed from known energy levels and wavelengths of unidentified lines. As each excited state exhibits many transition lines to different lower levels, the true levels should be reconstructed coincidentally from many level-line combinations, while the wrong combinations distribute randomly. Such a coincidence can be easily detected statistically. We demonstrate this statistical method by finding new levels for various atomic and nuclear systems from unidentified line lists available online.

Hellmann–Feynman theorem and internal pressure for atoms, molecules and plasmas under pressure

Since the beginning of the twentieth century, confinement of atoms, molecules and plasma inside impenetrable sharp (hard) and smooth (soft) cavities has been studied with utmost interest. Physically, such situations introduce the trapping of a quantum system under high pressure. The internal pressure () of the system under multi-megabar external pressure has not yet been explored. Also, there is a lack of Hellmann–Feynman theorem (HFT) in such a scenario, which is essential to derive a concrete analytical expression of under stressed condition. Here using a scaling concept we provide a general HFT in terms of expectation values of total energy, potential and kinetic energy of a given confined system. A change in boundary condition (from smooth to sharp and vice versa) does not affect this. Applying this proposed HFT, a closed-form expression of and has been obtained for confined and shell-confined quantum systems. Based on this , several virial-like equations are also modeled. This HFT and are demonstrated for one- and many-electron atoms, plasmas and molecules using both analytical and numerical methods. Their applicability in several other confined systems is discussed from simple arguments.

Partial-wave representation of the strong-field approximation. II. Coulomb asymmetry in the photoelectron angular distribution of many-electron atoms

Strong-field ionization experiments are routinely performed with different atomic targets. While these experiments helped reveal many details about the light-matter interaction in intense laser fields, the target atoms are often modeled in a rather crude manner by neglecting most of their electronic structure. Until the present, therefore, many above-threshold ionization measurements had been understood only qualitatively, and this especially applies to the ionization with elliptically polarized laser pulses, for which the photoelectron angular distributions are known to exhibit a quite strong (Coulomb) asymmetry. This asymmetry arises from the Coulomb and short-range forces between the photoion and the outgoing field-dressed electron in the continuum. We demonstrate here how the strong-field approximation (SFA) can be combined with atomic structure theory for modeling such target-resolved observations. Using a partial-wave representation of the SFA, we show that this combination reproduces the Coulomb asymmetry for argon and xenon targets in good agreement with previous experiments and especially if a distorted-wave Volkov continuum is applied for the active electron. We therefore conclude that a better account of the initial and final electron waves in strong-field theories will enhance our understanding of ionization phenomena in intense fields.

Model-QED-operator approach to relativistic calculations of the nuclear recoil effect in many-electron atoms and ions

A model-operator approach to fully relativistic calculations of the nuclear recoil effect on energy levels in many-electron atomic systems is worked out. The one-electron part of the model operator for treating the normal mass shift beyond the Breit approximation is represented by a sum of semilocal and nonlocal potentials. The latter ones are constructed by employing the diagonal and off-diagonal matrix elements rigorously evaluated for hydrogenlike ions to first order in the electron-to-nucleus mass ratio. The specific mass shift beyond the lowest-order relativistic approximation has a form which can be directly employed in calculations. The capabilities of the method are probed by comparison of its predictions with the results of ab initio QED calculations. The proposed operator can be easily incorporated into any relativistic calculation based on the Dirac-Coulomb-Breit Hamiltonian.

Variational Methods for Atoms and the Virial Theorem

In the case of the one-electron Dirac equation with a point nucleus, the virial theorem (VT) states that the ratio of the kinetic energy to potential energy is exactly −1, a ratio that can be an independent test of the accuracy of a computed solution. This paper studies the virial theorem for subshells of equivalent electrons and their interactions in many-electron atoms. This shows that the linear scaling of the dilation is achieved through the balancing of the contributions to the potential of an electron from inner and outer regions that some Slater integrals impose conditions on a single subshell, but others impose conditions between subshells. The latter slows the rate of convergence of the self-consistent field process in which radial functions are updated one at a time. Several cases are considered. Results are also extended to the nonrelativistic case.

Bohr-Weisskopf effect: From hydrogenlike-ion experiments to heavy-atom calculations of the hyperfine structure

In this paper we study the influence of electron screening on the Bohr-Weisskopf (BW) effect in many-electron atoms. The BW effect gives the finite-nucleus magnetization contribution to the hyperfine structure. Relativistic atomic many-body calculations are performed for s and p_1/2 states of several systems of interest for studies of atomic parity violation and time-reversal-violating electric dipole moments -- Rb, Cs, Fr, Ba+, Ra+, and Tl. For s states, electron screening effects are small, and the relative BW correction for hydrogenlike ions and neutral atoms is approximately the same. We relate the ground-state BW effect in H-like ions, which may be cleanly extracted from experiments, to the BW effect in s and p_1/2 states of neutral and near neutral atoms through an electronic screening factor. This allows the BW effect extracted from measurements with H-like ions to be used, with screening factors, in atomic calculations without recourse to modelled nuclear structure input. It opens the way for unprecedented accuracy in accounting for the BW effect in heavy atoms. The efficacy of this approach is demonstrated using available experimental data for H-like and neutral Tl-203 and Tl-205.

Chemists Without Knowing It?

This paper considers aspects of the chemistry-physics relationship from a historical perspective and with a focus on the entrance of quantum mechanics in twentieth-century chemistry. Traditionally, theoretical physics was widely regarded as epistemically superior to chemistry if also, from the chemists’ point of view, of little practical relevance. With the emergence of quantum chemistry in about 1930, the gulf widened as it seemed that the new discipline was more physics than chemistry. One way of investigating theoretically many-electron atoms was by means of the Hartree-Fock approximation method. The Møller-Plesset perturbation theory introduced in 1934 by a Danish and an American physicist was a refinement to the Hartree-Fock method. Although the Møller-Plesset theory was initially neglected – and is still neglected in the historiography of quantum chemistry – it came to play a most important role in later studies. Indeed, it is a prime example of what in sociological studies of science is known as a “sleeping beauty.” The paper discusses the historical context of the Møller-Plesset theory, concluding that, in a sense, its originators were “chemists without knowing it.”

Near-exact non-relativistic ionization energies for many-electron atoms.

Electron–electron interactions and correlations form the basis of difficulties encountered in the theoretical solution of problems dealing with multi-electron systems. Accurate treatment of the electron–electron problem is likely to unravel some nice physical properties of matter embedded in the interaction. In an effort to tackle this many-body problem, a symmetry-dependent all-electron potential generalized for an n-electron atom is suggested in this study. The symmetry dependence in the proposed potential hinges on an empirically determined angular momentum-dependent partitioning fraction for the electron–electron interaction. With the potential, all atoms are treated in the same way regardless of whether they are open or closed shell using their system specific information. The non-relativistic ground-state ionization potentials for atoms with up to 103 electrons generated using the all-electron potential are in reasonable agreement with the existing experimental and theoretical data. The effects of higher-order non-relativistic interactions as well as the finite nuclear mass of the atoms are also analysed.

Approximate Atomic Green Functions.

In atomic and many-particle physics, Green functions often occur as propagators to formally represent the (integration over the) complete spectrum of the underlying Hamiltonian. However, while these functions are very crucial to describing many second- and higher-order perturbation processes, they have hardly been considered and classified for complex atoms. Here, we show how relativistic (many-electron) Green functions can be approximated and systematically improved for few- and many-electron atoms and ions. The representation of these functions is based on classes of virtual excitations, or so-called excitation schemes, with regard to given bound-state reference configurations, and by applying a multi-configuration Dirac-Hartree-Fock expansion of all atomic states involved. A first implementation of these approximate Green functions has been realized in the framework of Jac, the Jena Atomic Calculator, and will facilitate the study of various multi-photon and/or multiple electron (emission) processes.

Ionization of many-electron atoms by the action of two plasma models

The Hartree-Fock equations for many-electron atoms embedded in a plasma medium are solved using two different plasma models: (a) Debye-Hückel screening (DHS) potential and (b) exponential cosine screened Coulomb (ECSC) potential. Roothaan's approach is implemented for these models after solving the inherent difficulties to evaluate integrals where screening appears explicitly. A corresponding computer code was developed using the method of global basis sets (GBS). The reliability of this approach was verified by solving the Hartree-Fock equations through implementation of the finite-differences and finite-element grid methods and applied to two-electron atoms, yielding excellent agreement with the Roothaan-GBS (RGBS) method. The RGBS method was used to study the energy evolution and ionization threshold of several closed- and open-shell many-electron atoms embedded either in weak or strong DHS or ECSC plasma conditions. In all cases, a critical value of the screening length is obtained for which ionization is achieved, being systematically larger for DHS conditions, indicating the effect of a more repulsive ECSC potential. For He-like atoms in the ground state, we report a comprehensive set of accurate total energy data as a function of the screening constant using the Lagrange mesh method, which includes the electron correlation effects. The electron correlation energy is estimated using this data with reference to the RGBS estimates of energy as the Hartree-Fock energy. The variation of correlation energy as a function of screening constant under the different plasma potentials is rationalized in terms of a conjectured comparison theorem. Finally, a discussion on the effect of plasma strength on localization or delocalization of the electronic density derived from the RGBS method is presented in terms of changes in the Shannon entropy, yielding consistent results for delocalization close to the ionization threshold.

Many-electron atom confinement by a penetrable prolate spheroidal cavity

Confinement effects on the ground-state energy of many-electron atoms located inside a penetrable prolate spheroidal cavity are studied within a variational treatment of the Thomas–Fermi–Dirac–Weizsacker density functional scheme. Given a confining cavity size and shape as well as barrier height, isotropic and anisotropic confinement effects on the energy evolution, ionization potentials and pressure are discussed as well as their different conditions for electron escape. Comparison of the spheroidal box results for the energy evolution with corresponding ab initio calculations for endohedral confinement of Ne within the supermolecule approach $$\hbox {Ne@}\hbox {Ne}_{\mathrm {10}}$$ ( $$\hbox {He}_{\mathrm {10}})$$ suggests that reasonable agreement between both types of calculation is achieved provided the atom-in-a-box model incorporates the mean size of surrounding atoms with nuclei positioned at the spheroidal baseline of the cavity and a realistic choice of the mean confining barrier height.

On the Biot–Savart law of electromagnetism applied to the atomic circulation current *

The law of Biot and Savart, derived by observation of the magnetic field produced by electric current flowing in a macroscopic conductor, is shown to yield the correct expression for the nuclear hyperfine interaction, when applied to the electron circulation current derived for hydrogenic atoms from the theories of Pauli and Dirac, as further developed independently by Darwin, Gordon and Hartree. The law thus applies also on the microscopic scale to the magnetic field generated by the electron probability flux of the various atomic eigenstates, and may clearly be generalised formally for application to many-electron atoms, as outlined by Hartree. The present, fully relativistic, treatment is advantageous in avoiding the usual artificial splitting of the hyperfine interaction into a magnetic dipole–dipole contribution and a separate contribution arising from electron orbital motion. The formula obtained forms the basis for the inclusion of higher order (quantum electrodynamic and nuclear recoil) corrections. Interest in this subject is enhanced by observations of ‘strongly forbidden’ (magnetic dipole) atomic transitions arising from the nuclear magnetic interaction, notably the 21 cm line of hydrogen, which has played an important role in astrophysics, in both measurements of galactic rotation and studies of the early Universe.