N-electron Atom Research Articles

Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening

Atomic structure of N-electron atoms is often determined by solving the Hartree-Fock equations, which are a set of integro-differential equations. The integral part of the Hartree-Fock equations treats electron exchange, but the Hartree-Fock equations are not often treated as an integro-differential equation. The exchange term is often approximated as an inhomogeneous or an effective potential so that the Hartree-Fock equations become a set of ordinary differential equations (which can be solved using the usual shooting methods). Because the Hartree-Fock equations are an iterative-refinement method, the inhomogeneous term relies on the previous guess of the wavefunction. In addition, there are numerical complications associated with solving inhomogeneous differential equations. This work uses matrix methods to solve the Hartree-Fock equations as an integro-differential equation. It is well known that a derivative operator can be expressed as a matrix made of finite-difference coefficients; energy eigenvalues and eigenvectors can be obtained by using linear-algebra packages. The integral (exchange) part of the Hartree-Fock equation can be approximated as a sum and written as a matrix. The Hartree-Fock equations can be solved as a matrix that is the sum of the differential and integral matrices. We compare calculations using this method against experiment and standard atomic structure calculations. This matrix method can also be used to solve for free-electron wavefunctions, thus improving how the atoms and free electrons interact. This technique is important for spectral line broadening in two ways: it improves the atomic structure calculations, and it improves the motion of the plasma electrons that collide with the atom.

Regularized perturbative series for the ionization potential of atomic ions

We study N-electron atoms with nuclear charge Z. It is well known that, in the cationic (Z > N) large-Z region, the atom behaves as a weakly interacting system. The anionic (Z < N) regime, on the other hand, is characterized by an instability threshold at [Formula: see text], below which the atom spontaneously emits an electron. We construct a regularized perturbative series (RPS) for the ionization potential that is based on the behaviors for large Z and Z near Zc. The large-Z expansion coefficients are analytically computed from perturbation theory, whereas the slope of the energy curve at Z = N − 1 is computed from a kind of zero-range forces theory that uses as input the electron affinity and the covalent radius of the neutral atom with N − 1 electrons. Relativistic effects in the one-particle Hamiltonian are considered at the level of first-order perturbation theory. Our RPS formula is to be used to check the consistency of the ionization potential values for atomic ions contained in the NIST database.

Note on radial correlation factors of atoms

A peculiar property is reported for the pth order (2 ≤ p ≤ N) radial correlation factor of an N-electron atom (N ⩾ 2). When the pth electron goes away from the nucleus, the correlation factor reduces, apart from the constant unity, to the ratio between the (p − 1)th order reduced radial densities of the (N − 1)- and N-electron atoms, where both atoms are considered to be in their non-degenerate ground states. The result implies that an ‘electron annihilation function’ for the reduced densities is defined by a particular case of the correlation factor. A numerical illustration is given for a simple system.

EXISTENCE OF GROUND STATES FOR NEGATIVE IONS AT THE BINDING THRESHOLD

As the nuclear charge Z is continuously decreased an N-electron atom undergoes a binding-unbinding transition. We investigate whether the electrons remain bound and whether the radius of the system stays finite as the critical value Zc is approached. Existence of a ground state at Zc is shown under the condition Zc < N - K, where K is the maximal number of electrons that can be removed at Zc without changing the energy.

Regularity for Eigenfunctions of Schrödinger Operators

We prove a regularity result in weighted Sobolev spaces (or Babuska--Kondratiev spaces) for the eigenfunctions of a Schr\"odinger operator. More precisely, let K_{a}^{m}(\mathbb{R}^{3N}) be the weighted Sobolev space obtained by blowing up the set of singular points of the Coulomb type potential V(x) = \sum_{1 \le j \le N} \frac{b_j}{|x_j|} + \sum_{1 \le i < j \le N} \frac{c_{ij}}{|x_i-x_j|}, x in \mathbb{R}^{3N}, b_j, c_{ij} in \mathbb{R}. If u in L^2(\mathbb{R}^{3N}) satisfies (-\Delta + V) u = \lambda u in distribution sense, then u belongs to K_{a}^{m} for all m \in \mathbb{Z}_+ and all a \le 0. Our result extends to the case when b_j and c_{ij} are suitable bounded functions on the blown-up space. In the single-electron, multi-nuclei case, we obtain the same result for all a<3/2.

The effect of removal of “equivalence” restrictions on the electronic energies of atoms

Equivalence restrictions usually imposed on the Hartree-Fock approximation are removed by letting one of the 2p radial functions be different from the others. Two energy minima are found both with a single Slater-type atomic orbital basis and with a more extended basis. One minimum corresponds to the one diffuse-electron model and the other to the one tight-electron model. Although two wave functions based on these two models look different, the natural orbital analysis shows they are close to each other as a many-electron wavefunction. The calculation is further extended by using N radial functions for an N-electron atom. The results with the single Slater-type atomic orbital basis are presented and are discussed.

Dirac R-matrix with pseudo-states

We describe the development of the Dirac R-matrix with pseudo-states (DRMPS) method for electron and photon collisions with arbitrary atoms and ions. An N-electron atom is represented by an anti-symmetrized product of single-particle spinors comprising the usual four-component Dirac spinors as well as paired two-component Laguerre spinors (L-spinors). The convergent L-spinor basis forms a discretization of the electron and positron continua. This representation has been implemented quite generally within the general relativistic atomic structure package (grasp) specifically grasp0. An (N + 1)th ‘scattering’ electron is represented by the exact same L-spinor basis plus the usual R-matrix box-state spinors which are chosen so as to form a combined complete finite linearly independent orthogonal basis. The (non-diagonal) Buttle correction is determined consistently. This representation has been implemented quite generally within the Dirac atomic R-matrix code (DARC). Results of some simple model problems are presented which demonstrate the correctness of the implementation.

Ab initio asymptotic-expansion coefficients for pair energies in MP2 perturbation theory for atoms

The leading asymptotic expansion (AE) coefficients for two partial-wave (PW) expansions in powers of (l+1/2)−1 are derived for the second-order symmetry-adapted pair (SAP) energies defining the second-order Møller–Plesset (MP2) correlation energies for closed-shell N-electron atoms. In the main expansion, the PW increments (PW/m) are directly defined by the pair of angular momenta of the orbitals used in their computation. In the auxiliary expansion the PW increments (PW/a) are defined by the order, l, of the Legendre polynomial Pl (cos ϑ12) in the expansion of the pair function. Our approach differs in several respects from that put forward a decade ago by Kutzelnigg and Morgan, who derived the leading AE coefficients in the PW/a expansion for all states of He-like ions within the framework of the 1/Z expansion perturbation theory. However, we use their results concerning many technical details. It is shown that the expressions for the AE coefficients of the main expansion can be obtained from those for the He-like ions if some terms are multiplied by simple rational factors and the radial integrals are expressed in terms of Hartree–Fock orbitals defining the pair instead of hydrogen-like wave functions. To obtain the formulae for the AE coefficients for the main expansion, we have derived expressions relating the PW/m and PW/a energy increments. The AE coefficients for the main expansion are calculated for the Ne atom and compared with their counterparts determined from computed MP2 energy increments. Very close agreement is found.

Many-electron phenomena in the ionization of ions

This brief review of many-electron phenomena in the single and multiple ionization of ions tries to elucidate connections between ion–atom collisions and interactions of single free electrons and photons with neutral or charged atomic species. With the capability of using multiply charged ions in collisions with charged particles or photons, conventional studies on neutral targets can be extended to a new dimension: the target charge state. This is especially interesting for studies of selected effects along specific isoelectronic sequences, i.e. of n-electron atoms with variable nuclear charge. The necessary colliding-beams and merged-beams techniques are technically demanding but can provide access, for example, to electron correlation effects in collisions under the influence of increasingly strong electron–nucleus interactions.

Quantum Entanglement in Relativistic Three-Particle Systems

The relativistic three-particle systems are studied within the framework of Relativistic Schrodinger Theory (RST), with emphasis on the determination of the energy functional for the stationary bound states. The phenomenon of entanglement shows up here in form of the exchange energy which is a significant part of the relativistic field energy. The electromagnetic interactions become unified with the exchange interactions into a relativistic U(N) gauge theory, which has the Hartree–Fock equations as its non-relativistic limit. This yields a general framework for treating entangled states of relativistic many-particle systems, e.g., the N-electron atoms.

N identical particles under quantum confinement: a many-body dimensional perturbation theory approach

Systems that involve N identical interacting particles under quantum confinement appear throughout many areas of physics, including chemical, condensed matter, and atomic physics. In this paper, we present the methods of dimensional perturbation theory, a powerful set of tools that uses symmetry to yield simple results for studying such many-body systems. We present a detailed discussion of the dimensional continuation of the N-particle Schrödinger equation, the spatial dimension D→∞ equilibrium ( D 0) structure, and the normal-mode ( D −1) structure. We use the FG matrix method to derive general, analytical expressions for the many-body normal-mode vibrational frequencies, and we give specific analytical results for three confined N-body quantum systems: the N-electron atom, N-electron quantum dot, and N-atom inhomogeneous Bose–Einstein condensate with a repulsive hard-core potential.

Finite-size scaling method for the stability of atomic and molecular ions

Phase transitions at absolute zero temperature can take place as some parameter in the Hamiltonian of the system is varied. For the Hamiltonian of N-electron atoms, this parameter is taken to be the nuclear charge. As the nuclear charge reaches a critical point, the quantum ground state changes its characters from being bound to being degenerate or absorbed by a continuum. We describe the large-dimension approximation and the finite-size scaling method to calculate the critical nuclear charge for which an atom can bind an extra electron to form a stable negative ion. Results show that, at most, only one electron can be added to a free atom in the gas phase. The existence of doubly charged atomic negative ions in a strong magnetic field will be discussed.

Analytical properties of the Kohn–Sham theory exchange and correlation energy and potential via quantal density functional theory

In this study we derive analytical properties of the Kohn–Sham (KS) theory exchange Ex[ρ] and correlation \documentclass{article}\pagestyle{empty}\begin{document}$E_{c}^{\mathrm{KS}}[\rho]$\end{document} energy functionals of the density ρ(r) and of their respective functional derivatives vx(r) and vc(r). These properties are derived via quantal-density functional theory (Q-DFT) in terms of the different electron correlations present by application of adiabatic coupling constant (λ) perturbation theory. The results are: (i) The exchange energy Ex[ρ] and potential vx(r) are representative of electron correlations due to the Pauli exclusion principle as well as the lowest order O(λ) correlation-kinetic effects. While the contribution of the latter to vx(r) is explicit, their contribution to the energy Ex[ρ] is indirect via the orbitals of the noninteracting fermion or s-system. (ii) To leading order O(λ2), the correlation potential vc,2(r) has contributions from both Coulomb correlations and correlation-kinetic effects. However, at this order, the energy \documentclass{article}\pagestyle{empty}\begin{document}$E_{c,2}^{\mathrm{KS}}[\rho]$\end{document} is due entirely to correlation-kinetic effects. At higher order, both these correlations contribute to the correlation energy and potential. In a second component to this work we derive properties of the asymptotic structure of vx(r) and vc(r) via Q-DFT for systems for which the N-electron atom and (N−1)-electron ion are orbitally nondegenerate. The results are: (iii) There is no correlation-kinetic contribution to the potential vx(r) asymptotically. The asymptotic structure of vx(r), to exponential accuracy, is due entirely to Pauli correlations and is the work done Wx(r) to move an electron in the field of the Fermi hole, decaying as −1/r. (iv) The lowest order O(λ2) Coulomb correlations do not contribute asymptotically to vc(r). (v) The leading order O(λ2) correlation potential vc,2(r) is due entirely to correlation-kinetic effects decaying as 8κ0χs/5r5, where \documentclass{article}\pagestyle{empty}\begin{document}$\kappa_{0}^{2}/2$\end{document} is the ionization potential, and χs an expectation value of the s-system ion. (vi) To O(1/r5), the O(λ3) correlation potential vc,3(r) to leading order is entirely due to second-order Coulomb correlations and decays as −αs/2r4, where αs is the polarizability of the s-system ion. There are no Coulomb correlation contributions to vc,3(r) and thus vc(r) to O(1/r5). (vii) For systems for which the N-electron atom is orbitally degenerate, the asymptotic structure of vx(r) contains in addition terms of O(1/r3), O(1/r5), etc. The other remaining asymptotic properties of vc(r), however, remain unchanged. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 80: 555–566, 2000

Analytical properties of the Kohn-Sham theory exchange and correlation energy and potential via quantal density functional theory

In this study we derive analytical properties of the Kohn–Sham (KS) theory exchange Ex[ρ] and correlation \documentclass{article}\pagestyle{empty}\begin{document}$E_{c}^{\mathrm{KS}}[\rho]$\end{document} energy functionals of the density ρ(r) and of their respective functional derivatives vx(r) and vc(r). These properties are derived via quantal-density functional theory (Q-DFT) in terms of the different electron correlations present by application of adiabatic coupling constant (λ) perturbation theory. The results are: (i) The exchange energy Ex[ρ] and potential vx(r) are representative of electron correlations due to the Pauli exclusion principle as well as the lowest order O(λ) correlation-kinetic effects. While the contribution of the latter to vx(r) is explicit, their contribution to the energy Ex[ρ] is indirect via the orbitals of the noninteracting fermion or s-system. (ii) To leading order O(λ2), the correlation potential vc,2(r) has contributions from both Coulomb correlations and correlation-kinetic effects. However, at this order, the energy \documentclass{article}\pagestyle{empty}\begin{document}$E_{c,2}^{\mathrm{KS}}[\rho]$\end{document} is due entirely to correlation-kinetic effects. At higher order, both these correlations contribute to the correlation energy and potential. In a second component to this work we derive properties of the asymptotic structure of vx(r) and vc(r) via Q-DFT for systems for which the N-electron atom and (N−1)-electron ion are orbitally nondegenerate. The results are: (iii) There is no correlation-kinetic contribution to the potential vx(r) asymptotically. The asymptotic structure of vx(r), to exponential accuracy, is due entirely to Pauli correlations and is the work done Wx(r) to move an electron in the field of the Fermi hole, decaying as −1/r. (iv) The lowest order O(λ2) Coulomb correlations do not contribute asymptotically to vc(r). (v) The leading order O(λ2) correlation potential vc,2(r) is due entirely to correlation-kinetic effects decaying as 8κ0χs/5r5, where \documentclass{article}\pagestyle{empty}\begin{document}$\kappa_{0}^{2}/2$\end{document} is the ionization potential, and χs an expectation value of the s-system ion. (vi) To O(1/r5), the O(λ3) correlation potential vc,3(r) to leading order is entirely due to second-order Coulomb correlations and decays as −αs/2r4, where αs is the polarizability of the s-system ion. There are no Coulomb correlation contributions to vc,3(r) and thus vc(r) to O(1/r5). (vii) For systems for which the N-electron atom is orbitally degenerate, the asymptotic structure of vx(r) contains in addition terms of O(1/r3), O(1/r5), etc. The other remaining asymptotic properties of vc(r), however, remain unchanged. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 80: 555–566, 2000

Quantum critical phenomena and stability of atomic and molecular ions

In this review we discuss quantum phase transitions and the mapping between symmetry breaking of electronic structure configurations at the large-dimension limit and mean-field theory of phase transitions. We show that the finite size scaling method can be used for the calculations of the critical parameters of the few-body Schrodinger equation. In this approach, the finite size corresponds to the number of elements in a complete basis set used to expand the exact eigenfunction of a given Hamiltonian. The critical parameters such as the critical nuclear charges will be used to explain and predict the stability of atomic and molecular negative ions. For N-electron atoms with 2 N 86, results show that, at most, only one electron can be added to a free atom in the gas phase. However, doubly charged atomic negative ions might exist in a strong magnetic field.

Phase transitions and the stability of atomic and molecular ions

Quantum phase transitions at absolute zero temperature can take place as some parameter in the Hamiltonian of the system is varied. For such transitions, crossing the phase boundary means that the quantum ground state changes in some fundamental way. For the Hamiltonian of N-electron atoms, this parameter is taken to be the nuclear charge. As the nuclear charge reaches a critical point, the quantum ground state changes its characters from being bound to being degenerate or absorbed by a continuum. We describe here a method to calculate the critical nuclear charge for which an atom can bind an extra electron to form a stable negative ion. The estimate of the critical nuclear charge will be used to explain and predict the stability of atomic negative ions. The method can be generalized to predict the stability of molecular negative ions. A detailed calculation for the critical parameters for two center molecular ions is also included.

Phase transitions for N-electron atoms at the large-dimension limit

Symmetry breaking of electronic structure configurations for N-electron atoms in weak magnetic and electric fields at the large-dimension limit is described in terms of standard phase transitions. This symmetry breaking, which leads to ionization, is completely analogous to phase transitions and critical phenomena in statistical mechanics. This analogy is shown by allowing the nuclear charge to play a role analogous to temperature in statistical mechanics. For the exact solution of N-electron atoms at the large-dimension limit, the symmetry breaking is shown to be a first-order phase transition. For the special case of two-electron atoms, the first-order transition shows a triple point where three phases with different symmetry coexist. Treatment of the Hartree-Fock solution reveals a different kind of symmetry breaking where second-order phase transitions exist for N=2. We show that Hartree-Fock two-electron atoms in a weak external electric field exhibit a critical point with mean field critical exponents (\ensuremath{\mathrm{B}}=, \ensuremath{\alpha}=${0}_{\mathrm{dis}}$, \ensuremath{\delta}=3, and \ensuremath{\gamma}=1).

A new hyperfine operator in the Dirac equation

A recently found additional hyperfine interaction in the Dirac equation extends the validity of this equation to comparable masses of its two spin- 1/2 particles. For equal masses as in positronium, it ensures CP conservation. The new operator simplifies the hyperfine mixing of states with different j. Anomalous magnetic moments are incorporated, and an application to helium-like Rydberg states is mentioned. CPT invariance is demonstrated and discussed, including the extension to n-electron atoms.

Radiative Corrections in the Many-Particle and Dirac-Fock Approximations: Two Branches from a Common Origin

Starting from a common origin, the Bethe-Salpeter equation for an N-electron atom, expressions are derived for the energy through O(α 5mc 2) in the many-particle approach on the one hand and the Green′s function approach, leading to the Dirac-Fock equation in zeroth order on the other hand. For the sake of specificity, the case of He- and Li-like atoms is considered. The principal thrust of this, at first glance familiar, analysis is to emphasize the need for a careful treatment of the α 5mc 2 energy contributions, in order to avoid overlaps and double counting of contributions. Numerical examples are provided to illustrate the problem. The variation in the final results between correct and incorrect expressions is small, but serves to emphasize the need for more precise future calculations and more accurate experiments.

Charge renormalization at the large-D limit for N-electron atoms and weakly bound systems

We develop a systematic way to determine an effective nuclear charge ZRD such that the Hartree–Fock results will be significantly closer to the exact energies by utilizing the analytically known large-D limit energies. This method yields an expansion for the effective nuclear charge in powers of (1/D), which we have evaluated to the first order. This first order approximation to the desired effective nuclear charge has been applied to two-electron atoms with Z=2–20, and weakly bound systems such as H−. The errors for the two-electron atoms when compared with exact results were reduced from ∼0.2% for Z=2 to ∼0.002% for large Z. Although usual Hartree–Fock calculations for H− show this to be unstable, our results reduce the percent error of the Hartree–Fock energy from 7.6% to 1.86% and predicts the anion to be stable. For N-electron atoms (N=3–18, Z=3–28), using only the zeroth order approximation for the effective charge significantly reduces the error of Hartree–Fock calculations and recovers more than 80% of the correlation energy.