Precise estimation of the energy levels of two-electron atoms under spherical confinement

Bhattacharyya, S.; Saha, Jayanta K.; Mukherjee, P. K.; Mukherjee, Tapan Kumar

doi:10.1088/0031-8949/87/06/065305

Cited by 45 publications

(85 citation statements)

References 41 publications

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“…Hence the variation produces a 'knee' around some particular value of R. For higher excited states, this 'knee' appears at a higher value of R. All other ions also show same features. Similar behaviour of energy values of He-like ions inside a spherical impenetrable box (referred to as 'Coulombic sphere' hereinafter) was reported in a recent publication [41] where the potential inside the box was purely Coulombic. Within the ion-sphere, energy value of the positively charged ion is modified for two factors:…”

Section: Resultssupporting

confidence: 80%

“…Laughlin and Chu [39] made an effort to remove this difficulty and extended the basis size up to 95 parameters where they have to compromise with the flexibility of the non-linear parameters. Recently, for 1 S e states of He-like systems under spherical confinement, the present authors have calculated the energy values [41] by using standard Hylleraas basis set of dimension 161 and the results have been confirmed by Montgomery and Pupyshev [42]. In the present work, a successful effort has been made to develop a general methodology in Hylleraas basis for both S and P states of He-like systems.…”

Section: Introductionsupporting

confidence: 64%

“…However,for excited states having finite lifetime, this approach is not valid as the equilibrium criteria is not maintained. Under an adiabatic approximation, the pressure on the ions in the ground state can be expressed as [41] …”

Section: Methodsmentioning

confidence: 99%

See 2 more Smart Citations

Nonrelativistic structure calculations of two-electron ions in a strongly coupled plasma environment

Bhattacharyya¹,

Saha

Mukherjee³

2015

Phys. Rev. A

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In this work, the controversy between the interpretations of recent measurements on dense aluminum plasma created with Linac coherent light sources (LCLS) X-ray free electron laser (FEL) and Orion laser has been addressed. In both kind of experiments, helium-like and hydrogen-like spectral lines are used for plasma diagnostics . However, there exist no precise theoretical calculations for He-like ions within dense plasma environment. The strong need for an accurate theoretical estimates for spectral properties of He-like ions in strongly coupled plasma environment leads us to perform ab initio calculations in the framework of Rayleigh-Ritz variation principle in Hylleraas coordinates where ion-sphere potential is used. An approach to resolve the long-drawn problem of numerical instability for evaluating two-electron integrals with extended basis inside a finite domain is presented here. The present values of electron densities corresponding to disappearance of different spectral lines obtained within the framework of ion-sphere potential show excellent agreement with Orion laser experiments in Al plasma and with recent theories. Moreover, this method is extended to predict the critical plasma densities at which the spectral lines of H-like and He-like carbon and argon ions disappear. Incidental degeneracy and level-crossing phenomena are being reported for the first time for two-electron ions embedded in strongly coupled plasma. Thermodynamic pressure experienced by the ions in their respective ground states inside the ion-spheres are also reported.

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Section: Resultssupporting

confidence: 80%

Section: Introductionsupporting

confidence: 64%

See 1 more Smart Citation

Nonrelativistic structure calculations of two-electron ions in a strongly coupled plasma environment

Bhattacharyya¹,

Saha

Mukherjee³

2015

Phys. Rev. A

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“…Contrary to the free helium atom, the confined helium has not been described with a very high accuracy until recently, when we have developed a Lagrange-mesh method to study this system [1]. This method improves by several order of magnitudes the accuracy of previous approaches [2][3][4].The outlines of the model are the following. The assumed-infinite-mass nucleus is fixed and the electrons are characterized by coordinates r 1 and r 2 with respect to this nucleus.…”

mentioning

confidence: 99%

“…Ground-state energies and mean interparticle distances for several confinement radii R are given in table 1. The pressure acting on the confined helium atom is also given in calculated from the radius dependence of the energies E(R) by the formula [2][3][4],…”

mentioning

confidence: 99%

Helium atom under pressure

Dohet-Eraly

Baye

2016

EPJ Web of Conferences

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Abstract. Hard-sphere confinement is used to study helium atoms under pressure. The confined-helium Schrödinger equation is solved with a high accuracy by a Lagrangemesh method.The effects of high pressure on a helium gas can be estimated by studying the helium atom in a hard confinement, i.e. confined at the centre of an impenetrable spherical cavity, for different cavity radii. Contrary to the free helium atom, the confined helium has not been described with a very high accuracy until recently, when we have developed a Lagrange-mesh method to study this system [1]. This method improves by several order of magnitudes the accuracy of previous approaches [2][3][4].The outlines of the model are the following. The assumed-infinite-mass nucleus is fixed and the electrons are characterized by coordinates r 1 and r 2 with respect to this nucleus. In atomic units, the Hamiltonian of the helium atom readswhere r 12 = r 1 − r 2 and Δ 1 and Δ 2 are the Laplacians with respect to r 1 and r 2 . The confinement is introduced by forcing the wave function into some spherical cavity of radius R (r 1 , r 2 ≤ R). The wave function ψ(r 1 , r 2 , r 12 ) of an S state must thus verify the Schrödinger equationand vanishes at r 1 = R and r 2 = R. The coordinates (r 1 , r 2 , r 12 ) are advantageously replaced by the coordinates (u, v, w) defined over [0,1] byThe confinement implies that the wave function ψ(u, v, w) vanishes at u = 1, v = 1, and w = 1. The Schrödinger equation is solved by the Lagrange-mesh method [5][6][7], an approximate variational approach taking the form of a system of mesh equations by computing the Hamiltonian and overlap matrix elements with a Gauss quadrature. Using the coordinates (u, v, w) is essential for an easy treatment of the confinement and an high accuracy of the Gauss quadrature. The ground-state energy and wave function are obtained by diagonalizing a rather large (matrix dimension≈ 10 3 ∼ 2×10 4 ) but sparse matrix. Ground-state energies and mean interparticle distances for several confinement radii R are given in table 1. The pressure acting on the confined helium atom is also given in table 1. It is a

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Ritz variational calculation for the singly excited states of compressed two‐electron atoms

Saha

Bhattacharyya²,

Mukherjee³

2016

Int J of Quantum Chemistry

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A detailed analysis on the effect of spherical impenetrable confinement on the structural properties of two-electron ions in S2 states has been performed. The energy values of 1sns [n5224] ( 3 S e ) states of helium-like ions (Z5125) are estimated within the framework of Ritz variational method using explicitly correlated Hylleraas-type basis sets. The correlated wave functions used here are consistent with the finite boundary conditions due to spherical confinement. A comparative study between the singlet and triplet states originating from a particular electronic configuration shows incidental degeneracy and the subsequent level-crossing phenomena. The thermodynamic pressure felt by the ion inside the sphere pushes the energy levels toward continuum. Critical pressures for the transition to strong confinement regime (where the singly excited two-electron energy levels cross the corresponding one-electron threshold) as well as for the complete destabilization are also estimated.correlation, pressure confinement, two-electron atom, variational method | I N T R O D U C T I O NStudies on the subject of confined atomic systems have started soon after the advent of quantum mechanics. In recent years, a paradigm shift in the facility for doing controlled experiment on confined atom and modern high-speed computational resources open up a new horizon for this subject. The modifications of the structural and spectral properties of the atom or the molecule depending on the number and degree of confining parameters is the interesting aspect of the problem. Comprehensive reviews [1][2][3] are now available on this topic. Theoretically, different types of confinement models can be realized for different physical situations, for example, atoms under plasma environment, [4,5] endohedrally confined atoms and molecules in fullerene cages, [6] impurities in quantum dots or nano crystals, [7,8] matter under extreme pressure in zeolite sieves, [9] or in the walls of nuclear reactors, [10] and so forth. Moreover, the confined atom models assume contemporary significance in understanding the cores of Jovian planets such as Jupiter and Saturn. [11] Nonspherical confinement models [12][13][14][15] are also assumed to deal with important quantum mechanical problems. Vast applicability of the outcomes from such studies in different branches of science and technology makes the subject a topic of immense interest in the recent times. [16][17][18] Hydrogen atom confined in impenetrable spherical cavity was first studied by Michels et al. [19] to model the effects of pressure on its energy and polarizability. Subsequently, many workers [20][21][22][23][24][25][26] have addressed this problem from different angles to develop a comprehensive understanding about such confined systems. Helium atom confined by such spherical cavity is a much complicated numerical problem as compared to the hydrogen-like ions for obvious reasons. For ions having two or more electrons, the effect of electron correlation plays an important role in the str...

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Precise estimation of the energy levels of two-electron atoms under spherical confinement

Cited by 45 publications

References 41 publications

Nonrelativistic structure calculations of two-electron ions in a strongly coupled plasma environment

Nonrelativistic structure calculations of two-electron ions in a strongly coupled plasma environment

Helium atom under pressure

Ritz variational calculation for the singly excited states of compressed two‐electron atoms

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