Aufbau principle and singlet-triplet gap in spherical Hooke atoms

Singlet and triplet spin state energies for three-dimensional Hooke atoms, that is, electrons in a quadratic confinement, with even number of electrons (2, 4, 6, 8, 10) is discussed using Full-CI and CASSCF type wavefunctions with a variety of basis sets and considering perturbative corrections up to second order. The effect of the screening of the electron-electron interaction is also discussed by using a Yukawa-type potential with different values of the Yukawa screening parameter (lambda(ee) = 0.2, 0.4, 0.6, 0.8, 1.0). Our results show that the singlet state is the ground state for two and eight electron Hooke atoms, whereas the triplet is the ground spin state for 4-, 6-, and 10-electron systems. This suggests the following Aufbau structure 1s < 1p < 1d with singlet ground spin states for systems in which the generation of the triplet implies an inter-shell one-electron promotion, and triplet ground states in cases when there is a partial filling of electrons of a given shell. It is also observed that the screening of electron-electron interactions has a sizable quantitative effect on the relative energies of both spin states, specially in the case of two- and eight-electron systems, favoring the singlet state over the triplet. However, the screening of the electron-electron interaction does not provoke a change in the nature of the ground spin state of these systems. By analyzing the different components of the energy, we have gained a deeper understanding of the effects of the kinetic, confinement and electron-electron interaction components of the energy.

Automated summary

In this study, the singlet and triplet spin state energies for three-dimensional Hooke atoms, specifically with an even number of electrons, were investigated using Full-CI and CASSCF wavefunctions and various basis sets. The screening effect of electron-electron interaction was also examined using a Yukawa-type potential. The results reveal that the ground state is a singlet for two and eight electron Hooke atoms, while it is a triplet for 4, 6, and 10-electron systems. This study provides insights into the energy components and effects of confinement and electron-electron interaction.