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Physical Review A

Two electrons in a cylindrical box: An exact configuration-interaction solution

Ilya G. Ryabinkin, Viktor N. Staroverov

Published Mar 10, 2010• 16 citations
Publisher

Abstract

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 $1/{r}_{12}$ 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.

DOI
10.1103/physreva.81.032509

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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\ensuremathΣg+{}^{1}{\ensuremath{\Sigma}}_{g}^{+}1\ensuremathΣg+​. The number of variables in the Schr"odinger equation is reduced from six to five by separating the L\ifmmode^\elseˆ\fiz{\mathrm{L\ifmmode \hat{}\else \^{}\fi{}}}_{z}L\ifmmode^\elseˆ\fiz​ 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{10}^{\ensuremath{-}10}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.
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.
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.
Two of the most iconic systems of quantum physics are the particle in a box and the Coulomb potential (the third is, of course, the harmonic oscillator). In this expository paper, we consider the quantum solution to the problem of an electron confined between the grounded planes of an infinite capacitor. The potential arises from the image charges that form in the grounded planes, along with the added condition that at x = 0, L, where L is the distance between the planes, the wavefunction must be zero. This effectively couples a hydrogen like system to a particle-in-a-box (PIB) based on L, the distance between the planes. The problem of finding the electrostatic potential of this infinite series of image charges is an old one, going back to at least 1929. Here, we give a short derivation for one of the limiting cases that yields a compact expression and show how the Kellogg infinite summation formula converges to that value. We note here that this potential is a symmetric double well potential, so there will be many familiar properties of its solutions. Then using that potential, we solve Schrödinger's equation using a spectral technique. The limiting forms of a particle in a box for small L (and high E), and that of a (degenerate) bound image charge at large L and small energy are recovered. We also discuss the tunneling level splitting that occurs in the transition from the large L to the small L regime.
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 potential energy curve and the electron density distribution for low-lying electronic states of the helium and lithium atoms confined by an anisotropic harmonic oscillator potential have been studied for different confinement parameters (ω x , ω y , ω z ) by using the quantum chemical configuration interaction (CI) method employing a Cartesian anisotropic Gaussian basis set supplemented by a quantum chemical standard Cartesian Gaussian basis set, respectively. In the case of the helium atom full CI wavefunctions have been used, and in the case of the lithium atom multi-reference CI wavefunctions have been employed. The results for the two confined quantum systems are compared to each other: parabolic-and non-parabolic-type potential energy curves are observed for states of both atoms in different type harmonic oscillator confinement. The non-parabolic potentials always appear in pairs, one with a double minimum and the other with a sharp edge at the minimum. They result from the interaction of energetically nearby states of the same symmetry. The electron density distributions of the states of the two atoms show deformation from spherical symmetry according to the symmetry and strength of the confining potential at the position of the atoms. The electron distribution of the states having non-parabolic potential energy curves is strongly deformed and occasionally shows curvilinear nodal planes of parabolic shape.
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.