Stability of a two-electron system under pressure confinement: structural and quantum information theoretical analysis
Santanu Mondal, Anjan Sadhukhan, K. D. Sen, Jayanta K. Saha
Abstract
Abstract The stability of a two-electron Zee system trapped inside an impenetrable spherical cavity is analyzed using an explicitly correlated multi-exponent Hylleraas-type basis set in the framework of the Ritz variational method. The wave function is considered to be consistent with Dirichlet’s boundary condition. Four different <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>Z</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mrow> <mml:mtext>c</mml:mtext> </mml:mrow> </mml:msub> <mml:mrow> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi>b</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mn>0.25</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0.0</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:math> values are considered, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mrow> <mml:mtext>c</mml:mtext> </mml:mrow> </mml:msub> <mml:mrow> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi>b</mml:mi> </mml:msub> </mml:mrow> </mml:math> denotes the critical nuclear charge beyond which the system is embedded in a discrete positive energy continuum due to the impenetrable nature of the cavity. The energy contribution due to the total correlation (in the presence of both radial and angular correlation) effect, the radial correlation limit, and the angular correlation limit are also studied in detail. The thermodynamic pressure felt by the two-electron Zee system inside the cavity is estimated, and a formula replicating the behavior between the pressure and volume of the cavity is deduced by a fitting procedure. Different geometrical properties, e.g. radial moments ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:msubsup> <mml:mi>r</mml:mi> <mml:mn>1</mml:mn> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>r</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:msubsup> <mml:mi>r</mml:mi> <mml:mrow> <mml:mn>12</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mo><</mml:mo> </mml:msub> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mo>></mml:mo> </mml:msub> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:math> ; p = <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mn>3</mml:mn> </mml:math> ), angular moments ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>〈</mml:mo> <mml:mi>cos</mml:mi> <mml:msub> <mml:mi>θ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>〉</mml:mo> </mml:mrow> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mo form="prefix">cos</mml:mo> <mml:msub> <mml:mi>θ</mml:mi> <mml:mrow> <mml:mn>12</mml:mn> </mml:mrow> </mml:msub> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:math> ), and related important physical quantities, are determined. The variation of Kirkwood and Buckingham polarizabilities w.r.t. the pressure felt by the two-electron Zee system is analyzed. A one-electron radial density is estimated for each pair of ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>Z</mml:mi> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> </mml:math> ), which has been employed to generate the electrostatic potential as well as different information theoretical measures like Shannon entropy, Fisher entropy, Rényi entropy, Tsallis entropy, and Onicescu informational energy. The chosen information theoretical measures have been found to be sensitive tools for describing changes in the electronic structure due to spatial confinement. An interesting interplay between the electronic and nuclear contribution to the classical electrostatic potential is observed, leading to a shift in the position of its characteristic minimum due to compression. Wherever possible, a comparison is made in order to ascertain the high accuracy of our numerical results. The procedure of analytic evaluation of the integrals needed to estimate the atomic properties under consideration is discussed in detail.