Coulomb correlation and information entropies in confined helium-like atoms

Abstract

The present work studies aspects of the electronic correlation in confined $\hbox {H}^{-}$, He and $\hbox {Li}^+$ atoms in their ground states using the informational entropies. In this way, different variational wavefunctions are employed in order of better take account of Coulomb correlation. The obtained values for the $S_r$, $S_p$ and $S_t$ entropies are sensitive in relation to Coulomb correlation effects. In the strong confinement regime, the effects of the Coulomb correlation are negligible and the employment of the models of independent particle and two non-interacting electrons confined by a impenetrable spherical cage gains importance in this regime. Lastly, energy values are obtained in good agreement with the results available in the literature.

Graphic abstract

Appendix A:

The $2e^{-}$ system consists of two non-interacting electrons confined in an impenetrable spherical cage. The non-relativistic Hamiltonian in atomic units for one electron confined in an impenetrable spherical cage, $e^{-}$, is

$$\begin{aligned} {\hat{H}} = -\frac{1}{2} \nabla ^2 + V_c (r) \ . \end{aligned}$$

(A.1)

The confining potential is defined as

$$\begin{aligned} V_c (r)= \left\{ \begin{array}{ccc} 0 &{} \mathrm {for} &{} 0< r < r_c \\ \infty &{} \mathrm {for} &{} r \ge r_c \end{array}\right. , \end{aligned}$$

(A.2)

being $r_c$ the confinement radius. The ground state wavefunction is

$$\begin{aligned} \phi (r) = A\left[ \frac{\sin \left( \pi r / r_c \right) }{r}\right] \left( \frac{1}{4 \pi } \right) \ , \end{aligned}$$

(A.3)

where A is a normalization constant. In this background, for the $2e^{-}$ system the expectation values of the energy, beyond the entropic quantities, are the double of the values obtained for the $e^{-}$ system. The $S_r$, $S_p$, $S_t$ and $\langle E \rangle$ values as a function of $r_c$ for the $2e^{-}$ system in ground state are available in Table S5 of the Supplementary Material.