Highly accurate solutions for the confined hydrogen atom

Abstract

ABSTRACT:The model of the confined hydrogen atom (CHA) was developed by Michels et al. [1] in the mid-1930s to study matter subject to extreme pressure. However, since the eigenvalues cannot be obtained analytically, even the most accurate calculations have yielded little more than 10 figure accuracy. In this work, we show that it is possible to obtain the CHA eigenvalues with extremely high accuracy (up to 100 decimal digits) and we do that using two completely different methods. The first is based on formal solution of the confluent hypergeometric function while the second uses a series method. We also compare radial expectation values obtained by both methods and conclude that the wave functions obtained by these two different approaches are of high quality. In addition, we compute the hyperfine splitting constant, magnetic screening constant, polarizability in the Kirkwood approximation, and pressure as a function of the box radius.

Temperature-dependent behavior of confined many-electron systems in the Hartree-Fock approximation

“…We have found a similar minimum at about the same volume by doing a Fermi-Dirac population of the high-precision eigenvalues of the spherically confined H atom given in Ref. 67. Figure 10 shows the contributions to the free energy for the single atom as a function of T for four cube volumes (L = 2, 3, 7, 15 bohr).…”

“…For bounding volumes, the circumscribed-sphere R = √ 3L/2 and inscribed-sphere R = L/2 results from Ref. 67 (also see Ref. 66) can be compared to our cube results.…”

Electron density and its derivatives at the nucleus for spherically confined hydrogen atom

“…Very recently, the energy values with 100 decimal digits accuracy have been reported [8] which surpasses the previously known computational accuracy [6,7] by an order of magnitude. The purpose of this work is to highlight the importance of the cusp condition on the wave function at the nuclear position r → 0.…”

“…In this work, we found the allowed energies by using the Maple computer algebra system to evaluate the hypergeometric function. The roots of this function were located as described in an earlier work [8]. For a specified value of E, the process can be reversed and the allowed values of R are found at the roots of Eq.…”

Finite-difference calculation of donor energy levels in a spherical quantum dot subject to a magnetic field

“…Most efforts have been devoted to the calculation of the electronic states, taking into account different shapes of the confining potential, the presence of inner and outer impurities [1], applied electric [2][3][4] and magnetic fields [5,6], excitonic states [7], electron-electron [8,9] and electron-phonon [10] interactions.…”