Electron-electron Repulsion Energy Research Articles

Density functional theory from spherically symmetric densities: Ground and excited states of Coulomb systems.

Recently, Theophilou [J. Chem. Phys. 149, 074104 (2018)] proposed a peculiar version of the density functional theory by showing that the set of spherical averages of the density around the nuclei determines uniquely the external potential in atoms, molecules, and solids. Here, this novel theory is extended to individual excited states. The generalization is based on the method developed in the series of papers by Ayers, Levy, and Nagy [Phys. Rev. A 85, 042518 (2012)]. Generalized Hohenberg-Kohn theorems are proved to the set of spherically symmetric densities using constrained search. A universal variational functional for the sum of the kinetic and electron-electron repulsion energies is constructed. The functional is appropriate for the ground state and all bound excited states. Euler equations and Kohn-Sham equations for the set are derived. The Euler equations can be rewritten as Schrödinger-like equations for the square root of the radial densities, and the effective potentials in them can be expressed in terms of wave function expectation values. The Hartree plus exchange-correlation potentials can be given by the difference of the interacting and the non-interacting effective potentials.

Relativistic effects on electronic pair densities: A perspective from the radial intracule and extracule probability densities.

While the effect of relativity in the electronic density has been widely studied, the effect on the pair probability, intracule, and extracule densities has not been studied before. Thus, in this work, we unveil new insights related to changes in the electronic structure caused by relativistic effects. Our numerical results suggest that the mean inter-electronic distance is reduced (mostly) due to scalar-relativistic effects. As a consequence, an increase in the electron-electron repulsion energy is observed. Preliminary results suggest that this observation is also valid when electronic correlation effects are considered.

One-Electron Reduced Density Matrix Functional Theory of Spin-Polarized Systems.

The formulation of the density matrix functional theory (DMFT) in which the correlation component U of the electron-electron repulsion energy is expressed in terms of a model two-electron density cumulant matrix (a.k.a. the 2-cumulant) [Formula: see text] and the tensor of two-electron integrals g is critically analyzed. The dependence of [Formula: see text] on both the vector of occupation numbers n and g derived from the corresponding vector [Formula: see text](1) of natural spinorbitals, and its concomitant invariance to the nature of the spin-independent interparticle interaction potential that enters the definition of g are emphasized. In the case of spin-polarized systems, [Formula: see text] (and thus U) is found to be a function of not only n and g, but also of the vector [Formula: see text] and the matrix [Formula: see text], where [Formula: see text] is the spin-flip operator. The presence of spin polarization imposes additional constraints upon [Formula: see text], including a simple condition that, when satisfied, assures the underlying wavefunction being an eigenstate of both the [Formula: see text] and [Formula: see text] operators with the eigenvalues [Formula: see text] and [Formula: see text], respectively. This feature allows targeting electronic states with definite multiplicities, which is virtually impossible in the case of the Kohn-Sham implementation of density functional theory. Among the four possible pairing schemes for the natural spinorbitals that give rise to approximations employing [Formula: see text] with only two independent indices, three are found to result in unphysical constraints even for spin-unpolarized systems, whereas the failure of the fourth one turns out to be precipitated by the presence of spin polarization. Consequently, any implementation of DMFT based upon "two-index" [Formula: see text] is shown to be generally unsuitable for spin-polarized systems (and incapable of yielding the spin-parallel components of U for the spin-unpolarized ones). A clear distinction is made between the genuine 1-matrix functionals that are defined for arbitrary N-representable 1-matrices and general energy expressions that depend on auxiliary quantities playing the role of fictitious 1-matrices subject to additional (often unphysical) constraints.

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.

Bilinear Constraints upon the Correlation Contribution to the Electron-Electron Repulsion Energy as a Functional of the One-Electron Reduced Density Matrix.

Perturbative analysis of the functional U[n, ψ] that yields the correlation component U of the electron-electron repulsion energy in terms of the vectors ψ(1) and n of the natural spinorbitals and their occupation numbers (the 1-matrix functional) facilitates examination of the flaws inherent to the present implementations of the density matrix functional theory. Recognizing that the practical usefulness of any approximate 1-matrix functional hinges upon its capability of exactly reproducing the leading contribution to U at the limit of vanishing electron-electron interactions gives rise to asymptotic bilinear constraints for the (exact or model) 2-cumulant that enters the expression for U. The asymptotic behavior of certain blocks of is found to be equally important. These identities, which are obtained for both the single-determinantal and a model multideterminantal cases, take precedence over the linear constraints commonly enforced in the course of approximate construction of such functionals. This observation reveals the futility of designing sophisticated approximations tailored for the second-order contribution to while neglecting proper formulation of the respective first-order contribution that in the case of the so-called JKL-only functionals requires abandoning the JK-dependence altogether. It has its repercussions not only for the functionals of the PNOF family but also for the expressions involving only the L-type two-electron repulsion integrals (in the guise of their exchange counterparts) that account only for the correlation effects due to electrons with antiparallel spins and are well-defined only for spin-unpolarized and high-spin systems (yielding vanishing U for the latter).

Five- and six-electron harmonium atoms: Highly accurate electronic properties and their application to benchmarking of approximate 1-matrix functionals.

Electronic properties of several states of the five- and six-electron harmonium atoms are obtained from large-scale calculations employing explicitly correlated basis functions. The high accuracy of the computed energies (including their components), natural spinorbitals, and their occupation numbers makes them suitable for testing, calibration, and benchmarking of approximate formalisms of quantum chemistry and solid state physics. In the case of the five-electron species, the availability of the new data for a wide range of the confinement strengths ω allows for confirmation and generalization of the previously reached conclusions concerning the performance of the presently known approximations for the electron-electron repulsion energy in terms of the 1-matrix that are at heart of the density matrix functional theory (DMFT). On the other hand, the properties of the three low-lying states of the six-electron harmonium atom, computed at ω = 500 and ω = 1000, uncover deficiencies of the 1-matrix functionals not revealed by previous studies. In general, the previously published assessment of the present implementations of DMFT being of poor accuracy is found to hold. Extending the present work to harmonically confined systems with even more electrons is most likely counterproductive as the steep increase in computational cost required to maintain sufficient accuracy of the calculated properties is not expected to be matched by the benefits of additional information gathered from the resulting benchmarks.

Robust validation of approximate 1-matrix functionals with few-electron harmonium atoms

A simple comparison between the exact and approximate correlation components U of the electron-electron repulsion energy of several states of few-electron harmonium atoms with varying confinement strengths provides a stringent validation tool for 1-matrix functionals. The robustness of this tool is clearly demonstrated in a survey of 14 known functionals, which reveals their substandard performance within different electron correlation regimes. Unlike spot-testing that employs dissociation curves of diatomic molecules or more extensive benchmarking against experimental atomization energies of molecules comprising some standard set, the present approach not only uncovers the flaws and patent failures of the functionals but, even more importantly, also allows for pinpointing their root causes. Since the approximate values of U are computed at exact 1-densities, the testing requires minimal programming and thus is particularly suitable for rapid screening of new functionals.

Construction of exchange-correlation functionals through interpolation between the non-interacting and the strong-correlation limit.

Drawing on the adiabatic connection of density functional theory, exchange-correlation functionals of Kohn-Sham density functional theory are constructed which interpolate between the extreme limits of the electron-electron interaction strength. The first limit is the non-interacting one, where there is only exchange. The second limit is the strong correlated one, characterized as the minimum of the electron-electron repulsion energy. The exchange-correlation energy in the strong-correlation limit is approximated through a model for the exchange-correlation hole that is referred to as nonlocal-radius model [L. O. Wagner and P. Gori-Giorgi, Phys. Rev. A 90, 052512 (2014)]. Using the non-interacting and strong-correlated extremes, various interpolation schemes are presented that yield new approximations to the adiabatic connection and thus to the exchange-correlation energy. Some of them rely on empiricism while others do not. Several of the proposed approximations yield the exact exchange-correlation energy for one-electron systems where local and semi-local approximations often fail badly. Other proposed approximations generalize existing global hybrids by using a fraction of the exchange-correlation energy in the strong-correlation limit to replace an equal fraction of the semi-local approximation to the exchange-correlation energy in the strong-correlation limit. The performance of the proposed approximations is evaluated for molecular atomization energies, total atomic energies, and ionization potentials.

The Coulomb, exchange, and correlation components of the electron-electron repulsion in harmonium atoms.

Highly accurate Coulomb, exchange, and correlation components of the electron-electron repulsion energies of the three-electron harmonium atoms in the (2)P- and (4)P+ states are obtained for 19 values of the confinement strength ω ranging from 10(-3) to 10(3). The computed data are consistent with their ω → 0 and ω → ∞ asymptotics that are given by closed-form algebraic expressions. Robust approximants that accurately reproduce the actual values of the energy components while strictly conforming to these limits are constructed, opening an avenue to stringent tests capable of predicting the performance of electronic structure methods for systems with varying extents of the dynamical and nondynamical electron correlation. The values of the correlation components, paired with the computed 1-matrices are expected to be particularly useful in the context of benchmarking of approximate density matrix functionals.

Magnetotransport in Aharonov-Bohm interferometers: Exact numerical simulations

The linear conductance of a two-terminal Aharonov-Bohm interferometer is an even function of the applied magnetic flux, as dictated by the Onsager-Casimir symmetry. Away from linear response this symmetry may be broken when many-body interactions are in effect. Using a numerically exact simulation tool, we study the dynamics and the steady-state behavior of the out-of-equilibrium double-dot Aharonov-Bohm interferometer, while considering different types of interactions: Model I includes a closed interferometer with an interdot electron-electron repulsion energy. In model II the interferometer is interacting with a dissipative environment, possibly driven away from equilibrium. In both cases we show that depending on the (horizontal, vertical) mirror symmetries of the setup, nonlinear transport coefficients obey certain magnetosymmetries. We compare numerically exact simulations to phenomenological approaches and special limits: The behavior of model I is compared to self-consistent mean-field calculations and master equation results in the Coulomb blockade regime. Model II, allowing heat dissipation to a thermal bath, is mimicked by an Aharonov-Bohm junction with a voltage probe. In both cases we find that phenomenological treatments capture the relevant transport symmetries, yet significant deviations in magnitude may show up.

Kinetic and electron-electron energies for convex sums of ground state densities with degeneracies and fractional electron number.

Properties of exact density functionals provide useful constraints for the development of new approximate functionals. This paper focuses on convex sums of ground-level densities. It is observed that the electronic kinetic energy of a convex sum of degenerate ground-level densities is equal to the convex sum of the kinetic energies of the individual degenerate densities. (The same type of relationship holds also for the electron-electron repulsion energy.) This extends a known property of the Levy-Valone Ensemble Constrained-Search and the Lieb Legendre-Transform refomulations of the Hohenberg-Kohn functional to the individual components of the functional. Moreover, we observe that the kinetic and electron-repulsion results also apply to densities with fractional electron number (even if there are no degeneracies), and we close with an analogous point-wise property involving the external potential. Examples where different degenerate states have different kinetic energy and electron-nuclear attraction energy are given; consequently, individual components of the ground state electronic energy can change abruptly when the molecular geometry changes. These discontinuities are predicted to be ubiquitous at conical intersections, complicating the development of universally applicable density-functional approximations.

Unified interpretation of Hund’s first and second rules for 2p and 3p atoms

A unified interpretation of Hund's first and second rules for 2p (C, N, O) and 3p (Si, P, S) atoms is given by Hartree-Fock (HF) and multiconfiguration Hartree-Fock (MCHF) methods. Both methods exactly satisfy the virial theorem, in principle, which enables one to analyze individual components of the total energy E(=T+V(en)+V(ee)), where T, V(en), and V(ee) are the kinetic, the electron-nucleus attraction, and the electron-electron repulsion energies, respectively. The correct interpretation for each of the two rules can only be achieved under the condition of the virial theorem 2T+V=0 by investigating how V(en) and V(ee) interplay to attain the lower total potential energy V(=V(en)+V(ee)). The stabilization of the more stable states for all the 2p and 3p atoms is ascribed to a greater V(en) that is caused by contraction of the valence orbitals accompanied with slight expansion of the core orbitals. The contraction of the valence orbitals for the two rules is a consequence of reducing the Hartree screening of the nucleus at short interelectronic distances. The reduced screening in the first rule is due to a greater amount of Fermi hole contributions in the state with the highest total spin-angular momentum S. The reduced screening in the second rule is due to the fact that two valence electrons are more likely to be on opposite sides of the nucleus in the state with the highest total orbital-angular momentum L. For each of the two rules, the inclusion of correlation does not qualitatively change the HF interpretation, but HF overestimates the energy difference ∣ΔE∣ between two levels being compared. The magnitude of the correlation energy is significantly larger for the lower L states than for the higher L states since two valence electrons in the lower L states are less likely to be on opposite sides of the nucleus. The MCHF evaluation of ∣ΔE∣ is in excellent agreement with experiment. The present HF and MCHF calculations demonstrate the above statements that were originally given by Katriel [Theor. Chem. Acta 23, 309 (1972); 26, 163 (1972)]. We have, for the first time, analyzed the correlation-induced changes in the radial density distribution for the excited LS terms of the 2p and 3p atoms as well as for the ground LS term.

Interesting properties of Thomas–Fermi kinetic and Parr electron–electron‐repulsion DFT energy functional generated compact one‐electron density approximation for ground‐state electronic energy of molecular systems

The reduction of the electronic Schrodinger equation or its calculating algorithm from 4N-dimensions to a (nonlinear, approximate) density functional of three spatial dimension one-electron density for an N-electron system, which is tractable in the practice, is a long desired goal in electronic structure calculation. If the Thomas-Fermi kinetic energy ( approximately integralrho(5/3)dr(1)) and Parr electron-electron repulsion energy ( approximately integralrho(4/3)dr(1)) main-term functionals are accepted, and they should, the later described, compact one-electron density approximation for calculating ground state electronic energy from the 2nd Hohenberg-Kohn theorem is also noticeable, because it is a certain consequence of the aforementioned two basic functionals. Its two parameters have been fitted to neutral and ionic atoms, which are transferable to molecules when one uses it for estimating ground-state electronic energy. The convergence is proportional to the number of nuclei (M) needing low disc space usage and numerical integration. Its properties are discussed and compared with known ab initio methods, and for energy differences (here atomic ionization potentials) it is comparable or sometimes gives better result than those. It does not reach the chemical accuracy for total electronic energy, but beside its amusing simplicity, it is interesting in theoretical point of view, and can serve as generator function for more accurate one-electron density models.

The influence of correlation on the interpretation of Hund’s multiplicity rule: A quantum Monte Carlo study

A systematic quantum Monte Carlo study of 2p atoms (C, N, O) and 3p atoms (Si, P, S) is performed to investigate the influence of correlation on the interpretation of Hund's multiplicity rule, which is an extension of our previous study of the carbon atom [J. Chem. Phys. 121, 7144 (2004)] to heavier atoms. The accuracy in the present study is significantly improved as compared with the previous study. A detailed analysis of the correlation contribution to individual energy components of the total energy is given beyond the self-consistent Hartree-Fock calculation. The stability of the highest spin-multiplicity state of all the atoms is ascribed to the greater electron-nucleus attraction energy that is gained at the cost of increasing the electron-electron repulsion energy as well as the kinetic energy. The present study demonstrates that correlation does not change the above conclusion due to the Hartree-Fock theory to support Boyd's less screening mechanism.

New constraints upon the electron-electron repulsion energy functional of the one-electron reduced density matrix

Three strict constraints upon the electron-electron repulsion energy functional of the one-electron reduced density matrix (the 1-matrix) are obtained by combining its invariance and stationary properties with the extended Koopmans' theorem. The constraints relate the quantities derived from the functional pertaining to an N-electron system with those of its (N-1)-electron congener. Together with the N-representability requirement for the 1-matrix of the congener, identities involving the electron-electron repulsion energies of the two systems and their derivatives with respect to the 1-matrices seriously narrow down the choices for potential approximate density-matrix functionals. This fact is well illustrated in the case of two-electron systems, where the validity of the new constraints is confirmed and found to originate from a nontrivial cancellation among different terms. Thus, the constraints provide a new tool for the construction and testing of new functionals that complements the previously known conditions such as the reproduction of the homogeneous gas energies and momentum distributions, convexity, and the N-representability of the associated 2-matrices.

Size versus volume extensivity of a new class of density matrix functionals

Despite being size-extensive, the "second-generation" 1-matrix functionals for the electron-electron repulsion energy V(ee) yield vanishing correlation energy for the homogeneous electron gas. This failure is directly related to the idempotency condition imposed upon an auxiliary matrix that enters the expression for V(ee). In particular, the recently proposed Kollmar-Hess functional is not volume-extensive and thus is incapable of properly describing any delocalized system at its thermodynamic limit.

Systematic construction of approximate one-matrix functionals for the electron-electron repulsion energy

The Legendre transform of an (approximate) expression for the ground-state energy E0(η,g) of an N-electron system yields the one-matrix functional Vee[Γ(x′,x)] for the electron-electron repulsion energy that is given by the function Vee(n;g) of the occupation numbers n pertaining to Γ(x′,x) and the two-electron repulsion integrals g computed in the basis of the corresponding natural spinorbitals. Extremization of the electronic energy functional, which is a sum of Vee[Γ(x′,x)] and the contraction of Γ(x′,x) with the core Hamiltonian, produces the (approximate) ground-state energy even if E0(η,g) itself is not variational. Thanks to this property, any electron correlation formalism can be reformulated in the language of the density matrix functional theory. Ten conditions that have to be satisfied by Vee(n;g) uncover several characteristics of Vee[Γ(x′,x)]. In particular, when applied in conjunction with the homogeneity property, the condition of volume extensivity imposes stringent constraints upon the possible dependence of Vee(n;g) on g.

On the theory of ferromagnetism of hexagonal diborides

The electronic structure and the magnetic properties of the MnB2 and CrB2 compounds with hexagonal AlB2-type lattices were studied. The problem was treated in terms of the generalized Hubbard model with an infinite electron-electron repulsion energy in the same atom. Equations for spin magnetic susceptibility were derived and used to determine the conditions of ferromagnetic instability and construct the phase diagram of the existence of ferromagnetic ordering.

Description of a homogeneous electron gas with simple functionals of the one-particle density matrix

The momentum distributions of a homogeneous electron gas, arising from a simple one-matrix functional for the electron-electron repulsion energy that generalizes the Hartree-Fock and Goedecker-Umrigar approximations, are analyzed in detail. Their properties are found to depend strongly on the exponent \ensuremath{\beta} that enters the functional. Smooth momentum distributions that, together with the corresponding energies per particle, exhibit simple scaling behavior with respect to the electron density \ensuremath{\rho}, are obtained only for $\frac{4}{5}<\ensuremath{\beta}<\frac{4}{3}$ and $\ensuremath{\rho}<~{\ensuremath{\rho}}_{\mathrm{crit}}(\ensuremath{\beta}).$

Use of electron-electron repulsion energy as a molecular descriptor in QSAR and QSPR studies.

Electron-electron repulsion energy (<Vee>) is presented as a new molecular descriptor to be employed in QSAR and QSPR studies. Here it is shown that this electronic energy parameter is connected to molecular quantum similarity measures (MQSM), and as a consequence can be considered as a complement to steric and electronic parameters in description of molecular properties and biological responses of organic compounds. The present strategy considers the molecule as a whole, thus there is no need to employ contributions of isolated fragments as in many calculations of molecular descriptors, like log P or the Free-Wilson analysis. The procedure has been tested in a widespread set of molecules: alcohols, alkanamides, indole derivatives and 1-alkylimidazoles. Molecular properties, as well as toxicity, are correlated using <Vee> as a parameter, and extensions to the method are given for handling difficult systems. In almost all studied cases, satisfactory linear relationships were finally obtained.