One and two body densities for excited states of the helium confined atom

Abstract

A study of the first excited states of the helium atom confined under impenetrable spherical walls is carried out. Both single particle and two body, intracule and extracule, densities are constructed. Crossing levels and Hund's rule are analyzed in terms of the contribution to the total energy from kinetic, electron-nucleus, and electron-electron energies. A study about the behavior of the single particle and two body densities is carried out. The Multiconfiguration Parameterized Optimized Effective Potential method is employed with a cut-off factor to account for Dirichlet boundary conditions. Single particle density is analytically constructed whereas the Monte Carlo algorithm is used to calculate two body densities.confined atoms, one and two-body densities | INTRODUCTIONAtoms under pressure modify considerably their properties as compared to the free species. In particular the ordering of single particle levels is different when the atoms are spatially confined, leading to a filling of the shells that differs from the well-known one of the free atoms. [1][2][3] A simple model to describe atomic confinement is to place an atom inside an impenetrable cavity of adjustable radius. In this model, the electrostatic Hamiltonian is modified by adding a confining potential of radius r c . The hard-wall confinement model is very useful to study the role of spatial limitation on the properties of atoms and molecules. [4][5][6][7][8][9] An important empirical result for atomic and molecular systems are the so-called Hund's rules. According to the first of them, for the states arising from a same configuration, the one with the highest value of the spin is the most bound, that is, it has the lowest energy. For free atoms, this result was erroneously attributed to the interelectronic energy that, for the excited states of the helium atom should be greater in the singlet than in the triplet. The first correct justification of the Hund's rule for atoms was given by Davidson [10] who showed, for the first excited states of He, that the electron-electron repulsion energy, V ee , is greater in the triplet than in the singlet and the same happens with the kinetic energy due to the virial theorem. Thus, the lower total energy of the triplet is due to a much larger decrease due to the nuclear attraction potential energy, V en , that compensates for the energy increase in both electron-electron repulsion potential and kinetic energies. Thus, the electrons in the triplet are, on average, closer to the nucleus than in the singlet, and also the average distance between the electrons is smaller in the triplet. This property, V ee greater for the triplet than for the singlet, has been observed for a great number of neutral atoms. [11,12] But the study of atomic isoelectronic sequences shows that the interelectronic repulsion reaches greater values for the singlet states than for the corresponding triplet ones as the nuclear charge of the ion increases. [13,14] This same happens when the atoms are confined, that is, the interelectronic ...

Density functional study of atoms spatially confined inside a hard sphere

“…However, for 1 P, current energies remain slightly higher when r c < 5, but steadily lowers as we approach the free‐atom unconfined limit. These reference results also fully match with, 83 for all available r c . Absolute deviation relative to 84 lies in the range of 0.001%–0.08% and 0.07%–4.47% for 3 P, 1 P. The correlated 3 P energies are found to be lower for both functionals throughout entire strength with respect to the interpolation approach of 85 and variational Scheme 18 .…”

“…This large discrepancy is rather unexpected and unusual, as none of our X‐only results in preceding (for ground state) and any of the future tables (as will be evident from discussion later) produces this kind of differences when compared to other accurate theoretical methods. Moreover, very recently, a multi‐configuration parametrized optimized effective potential (POEP) method 83 has been proposed for both these states for r c ≥ 2, which offers excellent agreement with X‐only energies recording 0.001%–0.03% and 0.11%–0.88% of absolute deviations; in a few instances, two energies completely merge. Thus, on the light of above facts, we are confident that present energies are better than those of 31 ; more careful and sophisticated calculations in future will hopefully resolve this issue.…”

Density functional study of atoms spatially confined inside a hard sphere

“…However, for 1 P, current energies remain slightly higher when r c < 5, but steadily lowers as we approach the free-atom unconfined limit. These reference results also fully match with [71], for all available r c . Absolute deviation relative to [72] lies in the range of 0.001-0.08% and 0.07-4.47% for 3 P, 1 P. The correlated 3 P energies are found to be lower for both functionals throughout entire strength with respect to the interpolation approach of [73] and variational scheme [13].…”

“…This large discrepancy is rather unexpected and unusual, as none of our X-only results in preceding (for ground state) and any of the future tables (as will be evident from discussion later) produces this kind of differences when compared to other accurate theoretical methods. Moreover, very recently, a multi-configuration parametrized optimized effective potential (POEP) method [71] has been proposed for both these states for r c ≥ 2, e Ref. [73].…”

Applications of the Variational Quantum Monte Carlo Method to the Two-Electron Atoms

“…Also, they studied the two-electron quantum dot confined by either a harmonic potential or by an infinite spherical well. Furthermore, Álvaro Luzón et.al [24] studied the first excited states of the helium atom confined under impenetrable spherical walls. Also, Michael-Adán et.al [25] studied spatial confinements that induce localization or delocalization on the electron density in atoms and molecules.…”