A tale of two electrons: Correlation at high density

Hartree-Fock critical nuclear charge in two-electron atoms

Electron correlation effects play a key role in stabilizing two-electron atoms near the critical nuclear charge, representing the smallest charge required to bind two electrons. However, deciphering the importance of these effects relies on fully understanding the uncorrelated Hartree-Fock description. We investigate the properties of the ground state wave function in the small nuclear charge limit using various symmetry-restricted Hartree-Fock formalisms. We identify the nuclear charge where spin-symmetry breaking occurs to give an unrestricted wave function that predicts an inner and outer electron. We also identify closed-shell and unrestricted critical nuclear charges where the highest occupied orbital energy becomes zero and the electron density detaches from the nucleus. Finally, we identify the importance of fractional spin errors and static correlation for small nuclear charges.

Green Functions and Self-Consistency: Insights From the Spherium Model

We report an exhaustive study of the performance of different variants of Green function methods for the spherium model in which two electrons are confined to the surface of a sphere and interact via a genuine long-range Coulomb operator. We show that the spherium model provides a unique paradigm to study electronic correlation effects from the weakly correlated regime to the strongly correlated regime, since the mathematics are simple while the physics is rich. We compare perturbative GW, partially self-consistent GW and second-order Green function (GF2) methods for the computation of ionization potentials, electron affinities, energy gaps, correlation energies as well as singlet and triplet neutral excitations by solving the Bethe-Salpeter equation (BSE). We discuss the problem of self-screening in GW and show that it can be partially solved with a second-order screened exchange correction (SOSEX). We find that, in general, self-consistency deteriorates the results with respect to those obtained within perturbative approaches with a Hartree-Fock starting point. Finally, we unveil an important problem of partial self-consistency in GW: in the weakly correlated regime, it can produce artificial discontinuities in the self-energy caused by satellite resonances with large weights.

Comprehensive benchmarking of density matrix functional approximations

The energy usually serves as a yardstick in assessing the performance of approximate methods in computational chemistry. After all, these methods are mostly used for the calculation of the electronic energy of chemical systems. However, computational methods should be also aimed at reproducing other properties, such strategy leading to more robust approximations with a wider range of applicability. In this study, we suggest a battery of ten tests with the aim to analyze density matrix functional approximations (DMFAs), including several properties that the exact functional should satisfy. The tests are performed on a model system with varying electron correlation, carrying a very small computational effort. Our results not only put forward a complete and exhaustive benchmark test for DMFAs, currently lacking, but also reveal serious deficiencies of existing approximations that lead to important clues in the construction of more robust DMFAs.

Exploring Unorthodox Dimensions for Two-Electron Atoms

Melding quantum and classical mechanics is an abiding quest of physical chemists who strive for heuristic insights and useful tools. We present a surprisingly simple and accurate treatment of ground-state two-electron atoms. It makes use of only the dimensional dependence of a hydrogen atom, together with the exactly known first-order perturbation value of the electron-electron interaction, both quintessentially quantum, and the D → ∞ limit, entirely classical. The result is an analytic formula for D-dimensional two-electron atoms with Z ≥ 2. For D = 3 helium, it gives accuracy better than 2 millihartrees, which is better than current density functional theory. A kindred explicit formula for correlation energy exploits interpolation between D → ∞ and D = 1 or 2; when added to the Hartree-Fock energy, it improves accuracy for D = 3 helium to better than 0.1 millihartrees.

Chemistry in one dimension

Orbital basis functions in a one-dimensional triatomic molecule ABC.