The Lagrange-mesh method is an approximate variational calculation which resembles a mesh calculation because of the use of a Gauss quadrature. The hydrogen atom confined in a sphere is studied with Lagrange-Legendre basis functions vanishing at the center and surface of the sphere. For various confinement radii, accurate energies and mean radii are obtained with small numbers of mesh points, as well as dynamic dipole polarizabilities. The wave functions satisfy the cusp condition with 11-digit accuracy.

The one- and two-electron Yukawa atoms, also referred to as the Debye-Hückel or screened Coulomb atoms, have been topics of considerable interest both for intrinsic reasons and because of their relevance to terrestrial and astrophysical plasmas. At sufficiently high screening the one-electron Yukawa atom ceases to be bound. Some calculations appeared to suggest that as the screening increases in the ground state of the two-electron Yukawa atom (in which both the one-particle attraction and the interparticle repulsion are screened) the two electrons are detached simultaneously, at the same screening constant at which the one-electron atom becomes unbound. Our results rule this scenario out, offering an alternative that is not less interesting. In particular, it is found that for $Z<1$ a mild amount of screening actually increases the binding energy of the second electron. At the nuclear charge $Z_c\approx 0.911028...$, at which the bare Coulomb two-electron atom becomes unbound, and even over a range of lower nuclear charges, an appropriate amount of screening gives rise to a bound two-electron system.

Energy eigenvalues of ground and singly excited $1sns$ (${}^{1,3}S$) ($n=2-5$, 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 $1sns$ (${}^{1,3}S$) ($n=2-5$) 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., ${\mathrm{He}}^{+}$ ($1s,{}^{2}S$) ion under the influence of the FO potential. The expectation value of different radial quantities $r_1, r_1^2, r_{12},r_{12}^2, r_{<}=min(r_1,r_2),r_{>}=max(r_1,r_2)$, angular quantities, e.g., interparticle angles ${\theta }_1, {\theta }_{12}$, one-electron delta function $\delta \left({\vec{r}}_1\right)$, and two-electron delta function $\delta \left({\vec{r}}_{12}\right)$ 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 ${\mathrm{He}}^{+}$ ($1s,{}^{2}S$) and He ($1s^2,{}^{1}S$ and $1s2s,{}^{1,3}S$) as well as between He ($1s2s,{}^{1}S$) and He ($1s2s,{}^{3}S$) 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.