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arXiv:1702.03960 · Physical Review A

Two-electron atom with a screened interaction

Charles Andrew Downing

Published Feb 6, 2017• 17 citations
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Abstract

We present analytical solutions to a quantum-mechanical three-body problem in three dimensions, which describes a heliumlike two-electron atom. Similarly to Hooke's atom, the Coulombic electron-nucleus interaction potentials are replaced by harmonic potentials. The electron-electron interaction potential is taken to be both screened (decaying faster than the inverse of the interparticle separation) and regularized (in the limit of zero separation). We reveal the exactly solvable few-electron ground state, which explicitly includes electron correlation, for certain values of the harmonic containment.

DOI
10.1103/physreva.95.022105

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We present analytical solutions to a quantum-mechanical three-body problem in three dimensions, which describes a heliumlike two-electron atom. Similarly to Hooke's atom, the Coulombic electron-nucleus interaction potentials are replaced by harmonic potentials. The electron-electron interaction potential is taken to be both screened (decaying faster than the inverse of the interparticle separation) and regularized (in the limit of zero separation). We reveal the exactly solvable few-electron ground state, which explicitly includes electron correlation, for certain values of the harmonic containment.
The Hooke's-law atom (hookium) provides an exactly soluble model for a two-electron atom in which the nuclear-electron Coulombic attraction has been replaced by a harmonic one. Starting from the known exact position-space wave function for the ground state of hookium, we present the momentum-space wave function. We also look at the intracules, two-electron probability distributions, for hookium in position, momentum, and phase space. These are compared with the Hartree-Fock results and the Coulomb holes (the difference between the exact and Hartree-Fock intracules) in position, momentum, and phase space are examined. We then compare these results with analogous results for the ground state of helium using a simple, explicitly correlated wave function.
We study the properties of the Hooke's law correlation energy (Ec{E}_{\mathrm{c}}Ec​), defined as the correlation energy when two electrons interact via a harmonic potential in a DDD-dimensional space. More precisely, we investigate the 1S{}^{1}S1S ground-state properties of two model systems: the Moshinsky atom (in which the electrons move in a quadratic potential) and the spherium model (in which they move on the surface of a sphere). A comparison with their Coulombic counterparts is made that highlights the main differences of the Ec{E}_{\mathrm{c}}Ec​ in both the weakly and strongly correlated limits. Moreover, we show that the Schr\"odinger equation of the spherium model is exactly solvable for two values of the dimension (D=1and3D=1 \text{and} 3D=1and3) and that the exact wave function is based on Mathieu functions.
We discuss a many-body Hamiltonian with two- and three-body interactions in two dimensions introduced recently by Murthy, Bhaduri and Sen. Apart from an analysis of some exact solutions in the many-body system, we analyse in detail the two-body problem which is completely solvable. We show that the solution of the two-body problem reduces to solving a known differential equation due to Heun. We show that the two-body spectrum becomes remarkably simple for large interaction strengths and the level structure resembles that of the Landau levels. We also clarify the 'ultraviolet' regularization which is needed to define an inverse-square potential properly and discuss its implications for our model.
We discuss a many-body Hamiltonian with two- and three-body interactions in two dimensions introduced recently by Murthy, Bhaduri and Sen. Apart from an analysis of some exact solutions in the many-body system, we analyse in detail the two-body problem which is completely solvable. We show that the solution of the two-body problem reduces to solving a known differential equation due to Heun. We show that the two-body spectrum becomes remarkably simple for large interaction strengths and the level structure resembles that of the Landau levels. We also clarify the `ultraviolet' regularization which is needed to define an inverse-square potential properly and discuss its implications for our model.
In this paper,due to the effect of positively-charged screening holes,the Coulomb potential energy 1/r is modified to be 1/rp.Using many-body quantum perturbation theory,we obtain a simple analytic representation of the ground state energy for a two-dimensional electron gas.It is found that the ground-state energy E and equilibrium radius reps are dependent on the interaction parameter p.Our results agree with those obtained by the numerical simulations and semi-analytic methods at low density limit.