Two Interacting Particles in a Parabolic Well: Harmonium and Related Systems*

Karwowski, Jacek; Cyrnek, Lech

doi:10.12921/cmst.2003.09.01.67-78

Cited by 11 publications

(4 citation statements)

References 18 publications

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“…Spectral properties are studied as a function of Debye screening length d . We have also verified our results with the those of Karwowski et al [37] (not shown here) that the comparison ensures the effectiveness of our method. In Table 1 energy eigenvalues of harmonium without Debye effect are shown.…”

Section: Resultssupporting

confidence: 90%

Spectra of electron pair under harmonic and Debye potential

Munjal

Prasad

2017

Contributions to Plasma Physics

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Two electron systems confined by harmonic potential is known as harmonium. Such a system has been studied for many reasons in the literature. In this work we study harmonium under Debye potential. We use higher order finite difference method for the solution of Schrodinger equation. Complete energy spectrum of harmonium and harmonium under Debye potential is studied. Debye screening length shows considerable effect on the energy levels and the radial matrix elements. The results are analysed in the light of existing results and the comparison with available results shows remarkable agreement. KEYWORDSconfined harmonium, Debye potential, radial matrix element INTRODUCTIONThe study of confined atoms was initiated by Michels et al. [1] in 1937 in order to study the effect of high pressure on atomic hydrogen. Since then much of the work has been devoted to study confined atoms. Confined quantum systems have attracted many studies owing to their importance in the physics of quantum heterostructures, where, off late, many new structures are being proposed. [2][3][4][5][6][7] A few structures in various environments are studied using approximate potentials such as power series potentials and screened coulomb Debye potential. [8] The confinement and atoms under confinement have been studied in various interdisciplinary studies. [9][10][11][12][13][14][15][16][17] The study of such systems has gained importance because of technological importance of atoms, molecules, and ions in cavity and excitons in semiconductor heterostructures. [18,19] In addition, it is also a very well-known fact that the chemical reactivity of confined systems shows drastic change with change of confinement, hence it has led to many applications. Even atoms with high atomic numbers have been studied in different confinements. Jiao et al. [20] studied spectral and structural data of He atom under screened coulomb potential. Ghoshal and Ho [21] studied the ground state of two-electron system in generalized screened potential. Jiang et al. [22] studied this system under Lorentzian astrophysical plasma. Two interacting electrons with or without harmonic potential form a system that attracts a lot of interest. Two interacting electron system is a model system which is used to explore the physical aspects of many phenomena involving several electron systems. Two electrons confined in an infinite spherical well has been well studied theoretically. [23][24][25][26][27][28] Study of harmonium atom can be helpful in density functional theory and to study the optical properties of two-electron quantum dot. Even harmonium atom with more than two electrons [29][30][31][32] is also well studied in the literature due to its usefulness in solid state physics and quantum chemistry. Here, we study in detail the properties of harmonium in Debye environment. We solve this problem using eighth order finite difference method to describe the relative motion of the system. The effectiveness of our method is shown by comparing our results with those of available data for ...

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Section: Resultssupporting

confidence: 90%

Spectra of electron pair under harmonic and Debye potential

Munjal

Prasad

2017

Contributions to Plasma Physics

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“…The excited states can be obtained accurately. 99,100 One excited state is treated here, the first of the same symmetry as the ground state, E(µ = ∞) = 2.9401169 . .…”

Section: System: Harmoniummentioning

confidence: 99%

Models and corrections: Range separation for electronic interaction—Lessons from density functional theory

Savin

2020

The Journal of Chemical Physics

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a) Model Hamiltonians with long-range interaction yield energies that are corrected taking into account the universal behavior of the electron-electron interaction at short range. Although the intention of the paper is to explore the foundations of using density functionals combined with range separation, the approximations presented can be used without them, as illustrated by a calculation on Harmonium. In the regime when the model system approaches the Coulomb system, they allow the calculation of ground states, excited states, and properties, without making use of the Hohenberg-Kohn theorem. Asymptotically, the technique is improvable, allows for error estimates that can validate the results. Some considerations for correcting the errors of finite basis sets in this spirit are also presented. Being related to the present understanding of density functional approximations, the results are comparable to those obtained with the latter, as long as these are accurate.

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“…To explore whether the model potentials can be used to describe repulsive interactions, we consider a system of two interacting electrons confined in a harmonic oscillator potential, called Harmonium [21,22]. The Hamiltonian of Harmonium is: the center of mass and the relative coordinates…”

Section: Harmoniummentioning

confidence: 99%

Smooth models for the Coulomb potential

et al. 2016

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Smooth model potentials with parameters selected to reproduce the spectrum of one-electron atoms are used to approximate the singular Coulomb potential. Even when the potentials do not mimic the Coulomb singularity, much of the spectrum is reproduced within the chemical accuracy. For the Hydrogen atom, the smooth approximations to the Coulomb potential are more accurate for higher angular momentum states. The transferability of the model potentials from an attractive interaction (Hydrogen atom) to a repulsive one (Harmonium and the uniform electron gas) is discussed. I. PHILOSOPHYHow feasible is it to find a model for the Coulomb interaction that is easier to evaluate but still reproduces key properties of the physical interaction? A logical starting point is a system with no interaction, as in KohnSham density functional theory (DFT) [1]. The KohnSham (KS) approximation starts from a non-interacting system, described as the sum of the individual electrons' contributions to the energy:In order to ameliorate the effect of omitting the Coulomb repulsion between the electrons, an extra term, the exchange-correlation functional E xc [ρ], is introduced into the energy expression. For a given external potential v(r) In principle, the KS solutions are exact when E xc is exact, and the KS orbitals yield the exact density of the system with N electrons in the external potential v(r). The accuracy of KS density functional approximations (DFA) depends on the approximation one uses for E xc .The simplest approximation is the local density approximation (LDA) [3][4][5]. In LDA it is assumed that the exchange-correlation functional is local,where the exchange-correlation energy density xc (ρ(r)) at r is taken from the uniform electron gas with density ρ(r).To accurately recover the effect of omitting the interaction between the electrons, one constructs an adiabatic connection that links the KS non-interacting system with the physical interacting system. Traditionally, this adiabatic connection is written as a function of the strength of the interaction, using a simple multiplicative factor λ arXiv:1610.08730v1 [physics.chem-ph]

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Two Interacting Particles in a Parabolic Well: Harmonium and Related Systems*

Abstract: The quasi-exactly solvable problem of two interacting electrons confined by a parabolic potential (harmonium) has been generalized for the case of two arbitrary particles. Several new features of the analytical solutions are presented.

Cited by 11 publications

References 18 publications

Spectra of electron pair under harmonic and Debye potential

Spectra of electron pair under harmonic and Debye potential

Models and corrections: Range separation for electronic interaction—Lessons from density functional theory

Smooth models for the Coulomb potential

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