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Physical Review B

Electron correlation in a hard spherical external potential: Wigner molecule formation and hybridization

David C. Thompson, Ali Alavi

Published May 12, 2004• 18 citations
Publisher

Abstract

We study a model of N electrons confined to a hard spherical box of radius R, for $N=3, 4,$ and 5. Unrestricted Hartree--Fock and full configuration-interaction calculations have been performed for high, medium, and low density regimes. Remarkable changes in electronic structure can be induced by varying the effective Coulomb coupling. A low-spin$\ensuremath{\rightarrow}$high-spin transition occurs in the case of $N=4$ as R is increased. We demonstrate a close connection between the propensity to form hybrid orbitals and the onset of Wigner molecule formation.

DOI
10.1103/physrevb.69.201302

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We study a model of N electrons confined to a hard spherical box of radius R, for N=3,4,N=3, 4,N=3,4, and 5. Unrestricted Hartree--Fock and full configuration-interaction calculations have been performed for high, medium, and low density regimes. Remarkable changes in electronic structure can be induced by varying the effective Coulomb coupling. A low-spin\ensuremath→\ensuremath{\rightarrow}\ensuremath→high-spin transition occurs in the case of N=4N=4N=4 as R is increased. We demonstrate a close connection between the propensity to form hybrid orbitals and the onset of Wigner molecule formation.
We study the problem of N electrons confined to a hard three-dimensional spherical box, for N = 3,4 and 5. Full-configuration interaction and Hartree–Fock calculations have been performed. We report energies, densities, second-order density matrices and exchange–correlation holes. In the N = 4 system, we observe a spin transition as the radius R of the sphere is increased, changing the ground state from 3 P to 5 S. The N = 3 and 5 systems remain 2 P and 4 S, respectively, for all R investigated. Pair-correlation functions and physical exchange–correlation holes have been computed over a regime in which the systems transform from liquid droplets to Wigner molecules. These functions show marked, yet characteristic, changes in this regime. By computing | E c / E |, where E c is the correlation energy defined with respect to the unrestricted Hartree–Fock solution, we show that the maximum relative error in the UHF energies occurs in this crossover regime. These systems provide challenging benchmarks for post-Hartree–Fock and density functional theory methods.
We study a system of two electrons interacting with a Coulomb potential in a sphere of radius R, bounded by an infinite wall using exact diagonalization. We have also investigated the influence of an additional parabolic potential (of strength k)k)k) arising from a uniform background smeared throughout the sphere. The convergence of the ground state energy of the singlet spin state of the system is investigated as a function of sphere size (essentially rs,{r}_{s},rs​, the Wigner--Seitz density parameter) for cases where there is no background potential (k=0)(k=0)(k=0) and for when $k\ensuremath{\ne}0.$ With k=0k=0k=0 and small rs,{r}_{s},rs​, we observe a maximum in the ground state density at the origin of the sphere. At ${r}_{s}\ensuremath{\approx}8\mathrm{a}.\mathrm{u}.,$ the ground state density acquires a minimum at the origin. For this and larger systems we identify the formation of a ``Wigner'' molecule state. We further investigate the ground state density as a function of k and also the correlation hole density as a function of rs{r}_{s}rs​ and k. We invert the Kohn--Sham equation for a two electron system and calculate the local effective potential and correlation potential (to within an additive constant) as functions of the radial coordinate for a number of values of rs{r}_{s}rs​ and k.
Abstract Accurate restricted Hartree‐Fock (RHF) wave‐functions are used to investigate information theoretic properties of the model problem of two interacting electrons confined within an infinite spherical potential of radius R . Benchmark quality calculations are performed to characterize this system via a range of information measures as a function of the tunable parameter R , across the full electron correlation regime (low to high correlation; small R to large R ). Both the Shannon information entropy and a statistical complexity measure provide quantitative insight regarding the onset of the formation of a “Wigner molecule” state for this system.
We propose a model describing the interaction of an electron confined in a spherical box with a polar molecule in the center of the box, including an inner repulsive potential. As a preparation to its investigation, we calculate the average radius of a dipole-bound nonconfined electron, for which a closed form expression is found. The angular dependence of the wave function is also discussed, with implications for the choice of basis functions for ab initio calculations. The Schr\"odinger equation in the spherical box is then solved. The negative energy eigenstates of the system and the dependence of the ground-state energy on the radius of the confining sphere are obtained. Interestingly, a very simple expression is found for the total number of negative energy levels for a given dipole strength and confining sphere radius.
Quantum dot structures confine electrons in a small region of space. Some properties of semiconductor quantum dots, such as the discrete energy levels and shell filling effects visible in addition spectra, have analogies to those of atoms and indeed dots are sometimes referred to as 'artificial atoms'. However, atoms and dots show some fundamental differences due to electron correlations. For real atoms, the kinetic energy of electrons dominates over their mutual Coulomb repulsion energy and for this reason the independent electron approximation works well. For quantum dots the confining potential may be shallower than that of real atoms leading to lower electron densities and a dominance of mutual Coulomb repulsion over kinetic energy. In this strongly correlated regime the independent electron picture leads to qualitatively incorrect results. This thesis concentrates on few-electron quantum dots in the strongly correlated regime both for quasi-one-dimensional and two-dimensional dots in a square confining potential. In this so-called 'Wigner' regime the ground-state electronic charge density is localised near positions of classical electrostatic minima and the interacting electronic spectrum consists of well separated spin multiplets. In the strongly correlated regime the structure of low-energy multiplets is explained by mapping onto lattice models with extended-Hubbard and Heisenberg effective Hamiltonians. The parameters for these effective models are calculated within a Hartree approximation and are shown to reproduce well the exact results obtained by numerical diagonalisation of the full interacting Hamiltonian. Comparison is made between square dots and quantum rings with full rotational symmetry. In the very low-density regime, direct diagonalisation becomes impractical due to excessive computer time for convergence. In this regime a numerical renormalisation group method is applied to one-dimensional dots, enabling effective spin-interactions to be pr
Strong correlations effects, which are often associated to the approach to a Mott insulating state, in some cases may be observed even far from half filling. This typically happens whenever the intersite Coulomb repulsion induces a tendency toward charge ordering, an effect that confines the electrons, and in turn favors local moment formation, i.e., Mott localization. A distinct intermediate regime then emerges as a precursor of such a Wigner-Mott transition, which is characterized by both charge and spin correlations, displaying large mass enhancements and strong renormalizations of other Fermi-liquid parameters. Here we present a careful study of a quarter-filled extended Hubbard model---a simple example where such physics can be studied in detail, and discuss its relevance for the understanding of the phenomenology of low-density two-dimensional electron gases.