In this work, we investigate several entropic and complexity measures, including the Shannon entropy, Fisher information, Tsallis entropy [Formula: see text] for [Formula: see text], and the disequilibrium D. In addition, we analyze the influence of the spherical confinement on the behavior of complexity measures, particularly, the Fisher–Shannon [Formula: see text] and López–Ruiz–Mancini–Calbet [Formula: see text] complexities. Within this framework, the ground state of the helium atom is studied under spherical confinement using the Variational Monte Carlo (VMC) method. One-electron expectation values of radial moments [Formula: see text] for [Formula: see text] as well as interelectronic moments [Formula: see text] over the same range of exponents are computed. The obtained results provide a detailed description of how information content, spatial distribution and structural complexity evolve as the confinement radius [Formula: see text] is varied, offering new insights into correlated confined quantum systems.
Abstract Despite extensive studies on the position-space electronic distributions for the ground state of the He atom trapped in an impenetrable spherical cavity, the corresponding momentum-space analysis has received little attention. The present article tries to bridge this gap by adopting an even-powered expansion of correlated coordinates in a Hylleraas basis consistent with the Dirichlet boundary condition. The corresponding momentum-space wavefunction is determined by the 3D Fourier transform of the explicitly correlated position-space wavefunctions. The analytic expression of the momentum-space wavefunction, and thereby the electronic density, allows us to examine the asymptotic limits of the same. With increased compression, the two-particle density distribution changes, and in the high compression regime, a crystalline-like structure in the contour map starts to appear. The corresponding topological complexity varies w.r.t. the cavity radius by maintaining symmetry along the equal momenta line. As a consequence, the one-particle momentum-space density and the derived Compton profile show oscillatory structure in the high-momentum regime. The evaluation of Shannon entropy provides further insights into the problem from the quantum information theoretic point of view. The estimation of position and momentum entanglement entropy measures is carried out here for the first time to exhibit the complete picture of the momentum correlation. Our investigation yields that despite showing all the contrasting differences in various density distributions and statistical information measures of the conjugate position and momentum spaces, the quantum correlation measure in terms of entanglement entropy remains invariant. Additionally, the Compton profiles are theoretically evaluated and subsequently compared with available theoretical and experimental results, demonstrating overall good agreement and consistency.
The effect of impenetrable confinement has been rigorously tested on the information-theoretic measures of Shannon information entropy, Fisher measure, Onicescu energy, and entanglement measures, such as von Neumann and linear entropy, corresponding to ${\mathrm{H}}^{\ensuremath{-}}$, He, and
Li+ in their ground electronic state. High-quality variational Hylleraas-like expansion as well as the accurate numerical Hartree-Fock wave functions have been employed for the chosen set of two-electron
Zee systems. The impenetrable confinement has been mimicked with a spherical cavity model potential of varying radius
R, which in turn, is used to quantify thermodynamic pressure within the adiabatic approximation. The structural properties such as total energy, energy components, and geometrical properties (expectation values electron-electron separation and electron-nuclear distance) have been estimated to test the accuracy and viability of the present calculation. Some intriguing scaling laws related to von Neumann, linear, and Shannon entropy as a function of pressure are proposed. The influence of impenetrable confinement on the transition probability of He $\ensuremath{\rightarrow}{\mathrm{Li}}^{+}$ during nuclear $\ensuremath{\beta}$ decay is quantitatively assessed.
A straightforward way to study the electronic structure and certain physical properties of atoms and molecules under high pressures is through the confinement (caging) model, in which the atom is placed at the center of a spherical box with impenetrable walls. The hydrogen atom confined inside penetrable and impenetrable spherical cavities is perhaps the most studied compressed system, and exact solutions exist for both types of confinement. The physical properties of the hydrogen atom confined within impenetrable-wall cavities (AHCI) have been widely analyzed; however, the same system embedded in penetrable spherical cavities (AHCP) has received less attention. A more physical model suggests using a spherical box with penetrable walls. Therefore, in this work, we compute via the variational method, the Shannon and Fisher information entropy in position space,
Srand
Ir, for the
2pand 3d states of a hydrogen atom subjected to spatial confinement by penetrable and impenetrable spherical cavities. Additionally, we calculate the tunneling probability for the penetrable confinement and seek to relate it to the overlaps of the radial probability densities as a function of the cavity radius and the height of the confining barrier. To model the penetrable confinement, we employ a logistic potential which, unlike earlier reports that used a discontinuous step potential, is a continuous function. The wavefunction is expanded in terms of Slater functions that satisfy appropriate boundary conditions; in some cases, we find notable differences with respect to the impenetrable potential.