Confinement effects on the electronic structure of M-shell atoms: A study with explicitly correlated wave functions

The electronic structure of several many-electron atoms, confined within a penetrable spherical box, was studied using the Hartree-Fock (HF) method, coupling the Roothaan's approach with a new basis set to solve the corresponding one-electron equations. The resulting HF wave-function was employed to evaluate the Shannon entropy, S q , in configuration space. Confinements imposed by impenetrable walls induce decrements on S q when the confinement radius, R c , is reduced and the electron-density is localized. For confinements commanded by penetrable walls, S q exhibits an entirely different behavior, because when an atom starts to be confined, S q delivers values less than those observed for the free system, in the same way that the results presented by impenetrable walls. However, from a confinement radius, S q shows increments, and precisely in these regions, the spatial restrictions spread to the electron density. Thus, from results presented in this work, the Shannon entropy can be used as a tool to measure the electron density delocalization for many-electron atoms, as the hydrogen atom confined in similar conditions.confined atoms, correct asymptotic behavior, Hartree-Fock, Shannon entropy, wave-function 1 | I N TR ODU C TI ON The confined atoms model is an important topic in physics and chemistry since atoms under spatial restrictions exhibit unusual characteristics, which differ to those on the same systems without such constraints. For example, effects over atoms confined by the fullerene have been analyzed using a model with a radial potential similar to a shell with width and size fitted to reproduce experimental information. [1][2][3] The simulation of the hydrogen atom submitted to high external pressures is another example of confined atoms. [4][5][6][7][8][9] In this case, the hydrogen atom was clamped at the center of one sphere, of radius R c , with impenetrable walls and consequently, wave-function or electron-density satisfy Dirichlet's boundary conditions. On this line, walls with infinite potential have been applied to study many-electron atoms, using wave-function techniques [10][11][12][13][14][15][16] or the density functional theory, [17,18] and some of these results have been contrasted with experimental values to predict sd electronic transitions observed when some metals are submitted to high pressures. [19,20] In early stages of the quantum mechanics, Sommerfeld proposed that positive eigenvalues found under this model indicate an electron delocalization. [5,21] Naturally, it is not easy to probe such a delocalization in systems where spatial restrictions trap electrons instead of the nuclear attraction. [22][23][24][25][26][27][28] The confinement by impenetrable boxes overestimates the response of physical observables, and, therefore, penetrable walls are more convenient to contrast the corresponding results with an experimental counterpart. [29][30][31][32][33] Such a model was proposed by Gorecki and Byers-Brown to reproduce the behavior of the helium atom submitted to high pressur...

Section: Discussionmentioning

confidence: 81%

Electron‐density delocalization in many‐electron atoms confined by penetrable walls: A Hartree–Fock study

Rodriguez‐Bautista

Vargas

Aquino

et al. 2017

“…Similar to the confinement in two‐electron atoms, they were undertaken in other atoms in periodic table as well. Thus energy spectra of ground and excited states, as well as filling of shells in multi‐electron atoms was investigated by means of a RHF‐type calculation, 13,36 configuration average HF, 37 HF, 38 variational Monte Carlo, 33 parameterized optimized effective potential (POEP) method using an appropriate cut‐off factor, 39 B‐spline random phase approximation with exchange, 40 explicitly correlated and multi‐configuration variational method 41 and so on. The effect of confinement on correlation energy in case of atoms of first few rows were discussed 9,10,41–43 …”

Section: Introductionmentioning

confidence: 99%

Density functional study of atoms spatially confined inside a hard sphere

Majumdar

Roy

2021

An atom placed inside a cavity of finite dimension offers many interesting features, and thus has been a topic of great current activity. This work proposes a density functional approach to pursue both ground and excited states of a multi‐electron atom under a spherically impenetrable enclosure. The radial Kohn‐Sham (KS) equation has been solved by invoking a physically motivated work‐function‐based exchange potential, which offers near‐Hartree‐Fock‐quality results. Accurate numerical eigenfunctions and eigenvalues are obtained through a generalized pseudospectral method (GPS) fulfilling the Dirichlet boundary condition. Two correlation functionals, viz., (i) simple, parametrized local Wigner‐type, and (ii) gradient‐ and Laplacian‐dependent non‐local Lee‐Yang‐Parr (LYP) functionals are adopted to analyze the electron correlation effects. Preliminary exploratory results are offered for ground states of He‐isoelectronic series (Z = 2 − 4), as well as Li and Be atom. Several low‐lying singly excited states of He atom are also reported. These are compared with available literature results–which offers excellent agreement. Radial densities as well as expectation values are also provided. The performance of correlation energy functionals are discussed critically. In essence, this presents a simple, accurate scheme for studying atomic systems inside a hard spherical box within the rubric of KS density functional theory.

“…Although the study of confined atoms was started many decades before, [1,2] it has received immense attention in recent years. [3][4][5][6][7][8][9][10][11][12][13] An atom with spatial restriction may be modeled for describing the effect of pressure on the system. [3,8] This pressure is responsible for modification of atomic energy levels and for continuum lowering.…”

mentioning

confidence: 99%

Explicitly correlated variational estimates of the energy levels of negative hydrogen ion under spatial confinement

Chandra¹,

Dutta²,

Saha

et al. 2018

Comprehensive investigations on the structural modifications of negative hydrogen ion within an impenetrable spherical domain has been performed in the framework of Ritz variational method. Electron correlation plays a major role in the formation of H– ion. The Hylleraas‐type basis set expansion of wave function considered here incorporates the effect of electron correlation in an explicit manner. Energy values of 1s2 (1normalSnormale), 2p2 (3normalPnormale) and 1sn p (1,3normalPnormalo) states (n=2−5) of H– ion within confined domain have been calculated. Although the singly excited states do not exist for a “free” H– ion, well converged energy values of such states have been found within a wide range of confinement radius. The thermodynamic pressure felt by the ion inside the sphere is also estimated. The general trend shows successive destabilization of the excited energy levels with increase of pressure. The contribution of angular correlation in the energy values have been estimated. Evolution of 1s2 (1normalSnormale) and 2p2 (3normalPnormale) energy levels of H– ion as quasi‐bound states are being reported.