Ground state of two electrons on a sphere

Loos, Pierre‐François; Gill, Peter M. W.

doi:10.1103/physreva.79.062517

Cited by 61 publications

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“…We will use the term "spherium" to describe this system. In recent work [19], we examined various schemes and described a method for obtaining near-exact estimates of the 1 S ground state energy of spherium for any given R. Because the corresponding Hartree-Fock (HF) energies are also known exactly [19], this is now one of the most complete theoretical models for understanding electron correlation effects.In this Letter, we consider D-spherium, the generalization in which the two electrons are trapped on a D-sphere of radius R. We adopt the convention that a D-sphere is the surface of a (D + 1)-dimensional ball. (Thus, for example, the Berry system is 2-spherium.)…”

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Two Electrons on a Hypersphere: A Quasiexactly Solvable Model

2009

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We show that the exact wave function for two electrons, interacting through a Coulomb potential but constrained to remain on the surface of a D-sphere (D ≥ 1), is a polynomial in the interelectronic distance u for a countably infinite set of values of the radius R. A selection of these radii, and the associated energies, are reported for ground and excited states on the singlet and triplet manifolds. We conclude that the D = 3 model bears the greatest similarity to normal physical systems. Quantum mechanical models for which it is possible to solve explicitly for a finite portion of the energy spectrum are said to be quasi-exactly solvable [1]. They have ongoing value and are useful both for illuminating more complicated systems and for testing and developing theoretical approaches, such as density functional theory (DFT) [2][3][4] and explicitly correlated methods [5][6][7][8]. One of the most famous two-body models is the Hooke's law atom which consists of a pair of electrons, repelling Coulombically but trapped in a harmonic external potential with force constant k. This system was first considered nearly 50 years ago by Kestner and Sinanoglu [9], solved analytically in 1989 for one particular k value [10], and later for a countably infinite set of k values [11].A related system consists of two electrons trapped on the surface of a sphere of radius R. This has been used by Berry and collaborators [12][13][14][15] to understand both weakly and strongly correlated systems and to suggest an "alternating" version of Hund's rule [16]. Seidl utilized this system to develop new correlation functionals [17] within the adiabatic connection in DFT [18]. We will use the term "spherium" to describe this system. In recent work [19], we examined various schemes and described a method for obtaining near-exact estimates of the 1 S ground state energy of spherium for any given R. Because the corresponding Hartree-Fock (HF) energies are also known exactly [19], this is now one of the most complete theoretical models for understanding electron correlation effects.In this Letter, we consider D-spherium, the generalization in which the two electrons are trapped on a D-sphere of radius R. We adopt the convention that a D-sphere is the surface of a (D + 1)-dimensional ball. (Thus, for example, the Berry system is 2-spherium.) We show that the Schrödinger equation for the 1 S and the 3 P states can be solved exactly for a countably infinite set of R values and that the resulting wave functions are polynomials in the interelectronic distance u = |r 1 − r 2 |. Other spin and angular momentum states can be addressed in the same way using the ansatz derived by Breit [20].The electronic Hamiltonian, in atomic units, iŝand because each electron moves on a D-sphere, it is natural to adopt hyperspherical coordinates [21,22]. For 1 S states, it can be then shown [19] that the wave function S(u) satisfies the Schrödinger equation(2) By introducing the dimensionless variable x = u/2R, this becomes a Heun equation [23] with singular points at x...

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Two Electrons on a Hypersphere: A Quasiexactly Solvable Model

2009

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“…Taking (3) as a zeroth-order wave function, one can use standard perturbation theory methods [43] to show that the small-R (weak correlation) expansion of the ground- We have demonstrated that, for each value of the angular momentum J, one is able to obtain polynomial and irrational solutions for both the singlet and triplet manifolds. The latter are degenerate but exhibit different geometric (Berry) phase behavior.…”

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Exact Wave Functions of Two-Electron Quantum Rings

Loos

Gill

2012

Phys. Rev. Lett.

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We demonstrate that the Schrödinger equation for two electrons on a ring, which is the usual paradigm to model quantum rings, is solvable in closed form for particular values of the radius. We show that both polynomial and irrational solutions can be found for any value of the angular momentum and that the singlet and triplet manifolds, which are degenerate, have distinct geometric phases. We also study the nodal structure associated with these two-electron states. Quantum rings are usually modelled by electrons confined to a strict-or quasi-one-dimensional circular space interacting via a short-ranged or Coulomb operator. In this Letter, we focus on the simple system in which two electrons are confined to a ring of radius R and interact via a Coulomb operator. This choice has often been avoided in the literature due to the divergence of the Coulomb interaction at small interelectronic distances.Contrary to frequent claims, systems with two electrons do not inevitably have intractable Schrödinger equations and we show here that, for each electronic state of a two-electron QR, the Schrödinger equation can be solved exactly for a countably infinite set of R values, yielding both polynomial and irrational solutions in terms of the interelectronic distance. Quantum mechanical systems whose Schrödinger equations can be solved in this way, such as the Hooke's law [22] or spherium [23,24] atoms, have ongoing value both for illuminating more complicated systems [25,26] and for testing and developing theoretical approaches, such as DFT [27][28][29][30] and explicitly correlated methods [31].In atomic units ( = m = e = 1), the Hamiltonian of two electrons on a ring of radius R iŝ

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“…26,34 The non-interacting orbitals for an electron on a sphere of radius R are the normalized spherical harmonics Y m (Ω), where Ω = (θ, φ) are the polar and azimuthal angles respectively. We will label spherical harmonics with = 0, 1, 2, 3, 4, .…”

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Nodal surfaces and interdimensional degeneracies

Loos

Bressanini

2015

The Journal of Chemical Physics

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The aim of this paper is to shed light on the topology and properties of the nodes (i.e. the zeros of the wave function) in electronic systems. Using the "electrons on a sphere" model, we study the nodes of two-, threeand four-electron systems in various ferromagnetic configurations (sp, p 2 , sd, pd, p 3 , sp 2 and sp 3 ). In some particular cases (sp, p 2 , sd, pd and p 3 ), we rigorously prove that the non-interacting wave function has the same nodes as the exact (yet unknown) wave function. The number of atomic and molecular systems for which the exact nodes are known analytically is very limited and we show here that this peculiar feature can be attributed to interdimensional degeneracies. Although we have not been able to prove it rigorously, we conjecture that the nodes of the non-interacting wave function for the sp 3 configuration are exact.

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Ground state of two electrons on a sphere

Cited by 61 publications

References 51 publications

Two Electrons on a Hypersphere: A Quasiexactly Solvable Model

Two Electrons on a Hypersphere: A Quasiexactly Solvable Model

Exact Wave Functions of Two-Electron Quantum Rings

Nodal surfaces and interdimensional degeneracies

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