Ground State Energies of Helium-Like Ions Using a Simple Parameter-Free Matrix Method

Abstract

This study aims to use hydrogenic orbitals within an analytic and numeric parameter-free truncated-matrix method to solve the projected Schrödinger equation of some Helium-like ions (3 ≤ Z ≤ 10). We also derived a new analytical expression of the ion ground state energies, which was simple and accurate and improved the accuracy of the analytic calculation, numerically using Mathematica. The standard matrix method was applied, where the wave function of the ions was expanded in a finite number of eigenvectors comprising hydrogenic orbitals. The Hamiltonian of the systems was calculated using the wave function and diagonalized to obtain their ground state energies. The results showed that a simple analytic expression of the ground state energies of He-like ions was successfully derived. Although the analytic expression was derived without involving any variational parameter, it was reasonably accurate with a 0.12% error for Ne8+ ion. From this method, the accuracy of the analytic energies was also numerically improved to 0.10% error for Ne8+ ion. The results clearly showed that the energies obtained using this method were more accurate than the hydrogenic perturbation theory and the uncertainty principle-variational approach. In addition, for Z > 4, our results were more accurate than those from the geometrical model.

Lead-free Perovskites TlGeClxBr3-x (x=0,1,2,3) as Promising Materials for Solar Cell Application: a DFT Study

“…This causes the electrons in the outermost shells of the perovskites to be less tightly bound to the atoms they belong to, and as a result, less energy is needed to make the electrons move more freely in the materials as conducting electrons. In other words, the energy gap of the compounds decreases [41].…”

“…These preliminary studies suggest that the material is a large bandgap semiconductor. Furthermore, it is expected that its isoelectronic compound of the form of TlGeCl3 and TlGeBr3 would have lower electronic band gaps as it was found in several studies that the replacement of halogen anions in perovskites by their counterparts with higher electronegativity leads to a decrease in the bandgap energy [39][40][41][42][43][44]. Körbel et al [45] reported the band gap energy of around 1.28 eV was found for TlGeBr3 using the more accurate HSE hybrid functional and that TlGeCl3 has a slightly larger electronic band gap.…”

“…The variation in the composition of the anions is predicted to affect the structural and electronic behavior of the perovskites. The DFT is used as it is widely known to be more powerful than some simple approximations, such as the semi-classical approximations and the simple matrix methods neglecting electron-electron correlation [46][47][48][49][50][51][52][53][54]. The DFT has been widely applied to various materials and has proven accurate [55][56][57][58][59][60][61][62][63][64].…”

Approximate Analytical Solution of the Ground State Problem of He and He-like Ions using Symmetrized-Hydrogenic States

“…In our previous studies [16][17], it was proven that with simple analytical matrix calculation involving a 3×3 matrix, low-lying state energies of helium and helium-like ions were obtained with good accuracy. In the studies, unsymmetrized hydrogenic basis states were used in the wavefunction expansion.…”

“…For example, a simple analytical calculation using the uncertainty principle was successfully applied to estimate the ground state energy of helium and helium-like atoms [12]. Some other methods include the geometrical model [13], the matrix method with logarithmic terms [14], the variational principle [15], and the matrix method with hydrogenic basis states [16,17].…”