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Physical review. B, Condensed matter

Two interacting electrons in a spherical box: An exact diagonalization study

David C. Thompson, Ali Alavi

Published Dec 31, 2002• 40 citations
Publisher

Abstract

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 $k)$ 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 ${r}_{s},$ the Wigner--Seitz density parameter) for cases where there is no background potential $(k=0)$ and for when $k\ensuremath{\ne}0.$ With $k=0$ and small ${r}_{s},$ 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 ${r}_{s}$ 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 ${r}_{s}$ and k.

DOI
10.1103/physrevb.66.235118

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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≠0.k\ensuremath{\ne}0.k\ensuremath=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 rs\ensuremath≈8a.u.,{r}_{s}\ensuremath{\approx}8\mathrm{a}.\mathrm{u}.,rs​\ensuremath≈8a.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.
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.
The problem of two Coulombically interacting electrons confined in a cylindrical potential box with impenetrable walls is solved by the method of full configuration interaction (CI) for states with zero axial component of the total orbital angular momentum, including the ground state ${}^{1}{\ensuremath{\Sigma}}_{g}^{+}$. The number of variables in the Schr\"odinger equation is reduced from six to five by separating the ${\mathrm{L\ifmmode \hat{}\else \^{}\fi{}}}_{z}$ operator from the complete Hamiltonian. Two-electron basis sets of symmetry-adapted configuration state functions are constructed from real eigenfunctions of the one-electron Hamiltonian for a cylindrical potential box. Kinetic energy integrals are evaluated analytically. Electron repulsion integrals are computed by developing a separable expression for the 1/r121/{r}_{12}1/r12​ operator in the form of a double Fourier-Bessel-Fourier-cosine series and using enough terms to achieve a relative accuracy of ${10}^{\ensuremath{-}10}$ for the eigenvalues. Energy-level ordering and electron densities of selected electronic states are reported for cylindrical cavities of varying size with a fixed 1:1 length-to-diameter ratio. A remarkable feature of this system is that convergence of the full CI energy with respect to the one-electron basis set varies dramatically with the extent of confinement: For small cavities, the rate of approaching the basis-set limit is similar to that for two-electron atoms but is significantly faster for large cavities. Formation of singlet and triplet Wigner molecules is observed in cylindrical cavities with radii greater than about 5 bohrs.
Two interacting electrons that are confined to the surface of a sphere have a uniform ground-state (surface) density. The Schr\"odinger equation of this helium-type two-electron system is solved here accurately for different values $\ensuremath{\alpha}∊\mathbb{R}$ of a constant that is multiplied to the electron-electron repulsion ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}}_{\mathit{ee}}$. The correlation structure in the resulting wave functions is analyzed for different values of $\ensuremath{\alpha}$. The asymptotic limits $\ensuremath{\alpha}\ensuremath{\rightarrow}0$ and $\ensuremath{\alpha}\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$ are treated analytically. Using these results, the ISI (interaction-strength interpolation) model for the density-functional ${E}_{\mathit{xc}}[\ensuremath{\rho}]$ of the exchange-correlation energy in the real system with $\ensuremath{\alpha}=1$ is tested against the exact adiabatic connection in density-functional theory.
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$\ensuremath{\rightarrow}$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 consider few-electron concentric double quantum rings with parabolic confining potential and compare the ground-state energies calculated by exact diagonalization of the Hamiltonian, accurate quantum Monte Carlo, and local spin-density functional approaches. Electronic localization in one of the rings and the formation of rotating Wigner molecules is shown, respectively, from the one-body and the two-body density operators. As the confinement strength of the outer ring is finely increased, the circularly symmetric electron density exhibits a radial crossover from this ring to the inner one without altering the angular character of the system.
Aiming to combine density functional theory (DFT) and wave-function theory, we study a mapping from the many-body interacting system to an effectively interacting Kohn-Sham system instead of a noninteracting Kohn-Sham system. Because a ground state of effectively interacting systems requires having a solution for the correlated many-body wave functions, this provides a natural framework to many-body wave-function theories such as the configuration interaction and the coupled-cluster method in the formal theoretical framework of DFT. Employing simple one-dimensional two-electron systems---namely, the one-dimensional helium atom, the hydrogen molecule, and the heteronuclear diatomic molecule---we investigate properties of many-body wave functions and exact exchange-correlation potentials of effectively interacting Kohn-Sham systems. As a result, we find that the asymptotic behavior of the exact exchange-correlation potential can be controlled by optimizing that of the effective interaction. Furthermore, the typical features of the exact noninteracting Kohn-Sham system, namely, a spiky feature and a step feature in the exchange-correlation potential for the molecular dissociation limit, can be suppressed by a proper choice of the effective interaction. These findings open a possibility to construct numerically robust and efficient exchange-correlation potentials and functionals based on the effectively interacting Kohn-Sham scheme.