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

Abstract

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...

Electronic structure of first and second row atoms under harmonic confinement

“…Interestingly, our ionization pressures agree with the prediction of Hoffmann et al that no element of the second period ionizes for pressures below than 500 GPa but lithium, which does below 80 GPa. We also estimate the ionization pressure of Li from recent HF calculations with hard walls done by Rodriguez et al Despite the restricted data, we found that the ionization pressure (30 GPa) agrees very well with our HF value (27 GPa). This nice agreement strongly supports our scheme to translate values of ω into pressure.…”

“…Other kind of works have been done by Garza et al They also began with the study of confined atoms in a hard spherical cavity. However, they interpreted the results in terms of pressure by using the thermodynamic relation and they also derived an equation to calculate the Gibbs free energy.…”

Electron Density Analysis for the H2+ System Confined by Hard Walls: The Chemical Bond Under Extreme Conditions

“…Recently, Rodriguez-Bautista et al showed that the Shannon entropy in the configuration space, [19] S ρ = drρ(r) ln ρ(r) (23) gave information of the delocalization (or localization) of an atomic system when spherical cavities trapped atoms with infinite or finite potential. In specific, these authors showed that this quantity decreased when hard walls confined the system.…”

“…[15][16][17][18][19] For this confinement, some atoms delocalize their electron density when the barrier height, associated to a constant potential, and R c are small. [15][16][17][18][19] For this confinement, some atoms delocalize their electron density when the barrier height, associated to a constant potential, and R c are small.…”

“…An atom confined by a sphere with penetrable walls is another example where spatial restrictions produce exciting results. [15][16][17][18][19] For this confinement, some atoms delocalize their electron density when the barrier height, associated to a constant potential, and R c are small. Some people argue that this potential is realistic and its results can be compared with experimental data.…”