The order of three lowest-energy states of the six-electron harmonium at small force constant

Long-Range Configuration Interaction with an Ab Initio Short-Range Correction and an Asymptotic Lower Bound†

Short-range corrections to long-range selected configuration interaction calculations are derived from perturbation theory considerations and applied to harmonium (with two to six electrons for some low-lying states). No fitting to reference data is used, and the method is applicable to ground and excited states. The formulas derived are rigorous when the physical interaction is approached. In this regime, the second-order expression provides a lower bound to the long-range full configuration interaction energy. A long-range/short-range separation of the interaction between electrons at a distance of the order of one atomic unit provides total energies within chemical accuracy, and, for the systems studied, provide better results than short-range density functional approximations.

Modified Expression for the Hamiltonian Expectation Value Exploiting the Short-Range Behavior of the Wave Function

The expectation value of the Hamiltonian using a model wave function is widely used to estimate the eigenvalues of electronic Hamiltonians. We explore here a modified formula for models based on a long-range interaction. It scales differently the singlet and triplet components of the repulsion between electrons not present in the model (its short-range part). The scaling factors depend uniquely on the parameter used in defining the model interaction and are constructed using only exact properties. We show results for the ground states and low-lying excited states of Harmonium with two to six electrons. We obtain important improvements for the estimation of the exact energy, not only over the model energy but also over the expectation value of the Hamiltonian.

Harmonium atoms at weak confinements: The formation of the Wigner molecules

The formation of the Wigner molecules in three-dimensional assemblies of equicharged particles upon weakening of harmonic confinements with spherical symmetry is investigated using five electronic states of the four-electron harmonium atom as a representative example. Numerical results of accurate explicitly-correlated calculations, carried out for a wide range of confinement strengths ω, reveal nonmonotonic convergence of the total energies and one-electron densities to those predicted by the formalism asymptotically exact at the ω→0 limit. The convergence rates are strongly state- and property-dependent, which implies the nonexistence of a single measure quantifying the extent of the formation of the Wigner molecule in a particular state of the harmonium atom. In light of these observations, it is concluded that, like in two dimensions, the emergence of the Wigner molecules in Coulombic systems confined by spherically symmetric harmonic potentials is a complex and gradual process that takes place over a range of confinement strengths spanning several orders of magnitude.