Abstract Accurate restricted Hartree‐Fock (RHF) wave‐functions are used to investigate information theoretic properties of the model problem of two interacting electrons confined within an infinite spherical potential of radius R . Benchmark quality calculations are performed to characterize this system via a range of information measures as a function of the tunable parameter R , across the full electron correlation regime (low to high correlation; small R to large R ). Both the Shannon information entropy and a statistical complexity measure provide quantitative insight regarding the onset of the formation of a “Wigner molecule” state for this system.
Abstract Accurate restricted Hartree‐Fock (RHF) wave‐functions are used to investigate information theoretic properties of the model problem of two interacting electrons confined within an infinite spherical potential of radius R . Benchmark quality calculations are performed to characterize this system via a range of information measures as a function of the tunable parameter R , across the full electron correlation regime (low to high correlation; small R to large R ). Both the Shannon information entropy and a statistical complexity measure provide quantitative insight regarding the onset of the formation of a “Wigner molecule” state for this system.
We study a model of N electrons confined to a hard spherical box of radius R, for N=3,4, and 5. Unrestricted Hartree--Fock and full configuration-interaction calculations have been performed for high, medium, and low density regimes. Remarkable changes in electronic structure can be induced by varying the effective Coulomb coupling. A low-spin$\ensuremath{\rightarrow}$high-spin transition occurs in the case of N=4 as R is increased. We demonstrate a close connection between the propensity to form hybrid orbitals and the onset of Wigner molecule formation.
We study the problem of N electrons confined to a hard three-dimensional spherical box, for N = 3,4 and 5. Full-configuration interaction and Hartree–Fock calculations have been performed. We report energies, densities, second-order density matrices and exchange–correlation holes. In the N = 4 system, we observe a spin transition as the radius R of the sphere is increased, changing the ground state from 3 P to 5 S. The N = 3 and 5 systems remain 2 P and 4 S, respectively, for all R investigated. Pair-correlation functions and physical exchange–correlation holes have been computed over a regime in which the systems transform from liquid droplets to Wigner molecules. These functions show marked, yet characteristic, changes in this regime. By computing | E c / E |, where E c is the correlation energy defined with respect to the unrestricted Hartree–Fock solution, we show that the maximum relative error in the UHF energies occurs in this crossover regime. These systems provide challenging benchmarks for post-Hartree–Fock and density functional theory methods.
Abstract The information‐theoretic measure of confined hydrogen atom has been investigated extensively in the literature. However, most of them were focused on the ground state and accurate values of information entropies, such as Shannon entropy, for confined hydrogen are still not determined. In this work, we establish the benchmark results of the Shannon entropy for confined hydrogen atom in a spherical impenetrable sphere, in both position and momentum spaces. This is done by examining the bound state energies, the normalization of wave functions, and the scaling property with respect to isoelectronic hydrogenic ions. The angular and radial parts of Shannon entropy in two conjugate spaces are provided in detail for both free and confined hydrogen atom in ground and several excited states. The entropies in position space decrease logarithmically with decreasing the size of confinement, while those in momentum space increase logarithmically. The Shannon entropy sum, however, approaches to finite values when the confinement radius closes to zero. It is also found that the Shannon entropy sum shares same trend for states with similar density distributions. Variations of entropy for nodeless bound states are significantly distinct form those owning nodes when changing the confinement radius.
In this work, we study the Wigner localization of interacting electrons that are confined to a quasi-one-dimensional harmonic potential using accurate quantum chemistry approaches. We demonstrate that the Wigner regime can be reached using small values of the confinement parameter. To obtain physical insight in our results, we analyze them with a semi-analytical model for two electrons. Thanks to electronic-structure properties such as the one-body density and the particle-hole entropy, we are able to define a path that connects the Wigner regime to the Fermi-gas regime by varying the confinement parameter. In particular, we show that the particle-hole entropy, as a function of the confinement parameter, smoothly connects the two regimes. Moreover, it exhibits a maximum that could be interpreted as the transition point between the localized and delocalized regimes.
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.
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.