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Energy eigenvalues of ground and singly excited 1sns (^1, 3S) (n=2-5, being the principal quantum number) states of a He atom in a quantum dot have been investigated in detail by incorporating explicitly correlated Hylleraas-type wave functions in the framework of the Ritz variational method. The quantum dot environment is simulated by considering the influence of the finite oscillator (FO) potential. We have examined the behavior of different energy components contributing to the total energy. In this regard, the Hund's spin multiplicity rule for 1sns (^1, 3S) (n=2-5) states of a He atom has been examined in depth in terms of observed unusual ordering of the electron repulsion and total energy. The energy contribution due to the total correlation (in the presence of both radial and angular correlation) effect, the radial correlation limit and angular correlation limit of a He atom under different strengths of the FO potential have been studied. As a quantitative replication of our results, we have introduced and verified the Hellmann-Feynman theorem and virial theorem for both a He atom and its first ionization threshold, i. e. , He^+ (1s, 0. 16em{0ex}^2S) ion under the influence of the FO potential. The expectation value of different radial quantities r₁, r₁^2, r₁₂, 0. 16em{0ex}r₁₂^2, r_=max (r₁, r₂), angular quantities, e. g. , interparticle angles ₁, ₁₂, one-electron delta function (P{r}₁), and two-electron delta function (P{r}₁₂) have been determined for different strengths of the FO potential. The effect of the FO potential on the quantum similarity index and dissimilarity between the He^+ (1s, 0. 16em{0ex}^2S) and He (1s^2, ^1S and 1s2s, ^1, 3S) as well as between He (1s2s, ^1S) and He (1s2s, ^3S) has been examined. The von Neumann, linear, and Shannon information entropy for the ground state He atom have been computed to ascertain the characteristic features of the electronic entanglement and the charge distribution under the FO potential.
Mondal et al. (Tue,) studied this question.