Solution of two-electron Schrödinger equations using a residual minimization method and one-dimensional basis functions

Abstract

Distinctive from conventional electronic structure methods, we solve the Schrödinger wave equations of the helium atom and its isoelectronic ions by employing one-dimensional basis functions to separate components. We use full two-electron six-dimensional operators and wavefunctions represented with real-space grids where the refinement of the latter is carried out using a residual minimization method. In contrast to the standard single-electron approach, the current approach results in exact treatment of repulsion energy and, hence, more accurate electron correlation within five centihartrees or better included, with moderate computational cost. A simple numerical convergence between the error to accurate results and the grid-spacing size is found. The obtained two-electron Schrödinger wavefunction that contains vast and elaborating information for the radial correlation function and common one-dimensional functions shows the electron correlation effect on one-electron distributions.

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

“…It is clear from these results that the increase in the repulsion energies was much less than the increase in the ionic energies as Z increases. This was caused by the fact that electrons are more localized near higher Z ions [30]. Therefore, the fact that our approximation does not include electron correlation function in the wavefunction has led to lower percentage errors in ground state energies for ions with larger atomic numbers has already been anticipated.…”

“…The results were reliable because as the atomic number increases, electrostatic interaction between nuclei and electrons becomes more dominant compared to mutual interaction between the two electrons [19]. Most recently, Rahman et al [30] calculated the electronelectron repulsion energies and Coulomb ionic energies of the He atom and some He-like ions (Li + and Be 2+ ) and found that as atomic number increases, the increase in the ionic energies of the ions was much more significant than that in the electron-electron repulsion energies. They found that the ionic energies of He, Li + and Be 2+ were -6.7533 a.u., -16.1275 a.u., and -29.5020 a.u., respectively, while the repulsion energies of the respective ions were 0.9458 a.u., 1.5677 a.u., and 2.1909 a.u.…”

An efficient finite difference approach to solutions of Schrödinger equations of atoms in non-linear coordinates

“…¥ The use of the Cartesianlike coordinates facilitates the multidimensional numerical integration [43][44][45][46], in contrast to other coordinates [58][59][60] which perform the integration analytically. The Jacobian is particularly…”

“…Second, the presence of the central difference formula represents the first and second derivative accurate to O h 2 ( ) where h is the uniform grids interval. To achieve the required accuracy in the atomic potentials [43][44][45][46][47][48][49][50][51], the use of very fine uniform grids is necessary, causing a large amount of computation to represent the Hamiltonian matrix. Except for the heavy computations, the convergence is also disappointing.…”

Density Functional Calculations on H2 Using 1s Slater Type Orbitals

“…Up to this point, this Journal has published an illustrative resource on DFT calculations by hand for the helium atom using the X-Alpha exchange functional on a single Gaussian orbital as the atomic orbital. However, because the test system chosen was the helium atom, the previous publication fails to capture the complexity of multicenter molecular system − where the calculations of the total energy components are more involved. Additionally, the application of the linear combination of atomic orbital (LCAO) framework to construct the molecular orbitals is missing. , Furthermore, DFT calculations on a molecular system will illustrate the notion of the potential energy surface and the Born–Oppenheimer approximation.…”