A scheme of numerical solution for three‐dimensional isoelectronic series of hydrogen atom using one‐dimensional basis functions

Abstract

The solution of three-dimensional Schrödinger wave equations of the hydrogen atoms and their isoelectronic ions (Z = 1 − 4) are obtained from the linear combination of one-dimensional hydrogen wave functions. The use of one-dimensional basis functions facilitates easy numericalintegrations. An iteration technique is used to obtain accurate wave functions and energy levels.The obtained ground state energy level for the hydrogen atom converges stably to −0.498 a.u. The result shows that the novel approach is efficient for the three-dimensional solution of the wave equation, extendable to the numerical solution of general many-body problems, as has been demonstrated in this work with hydrogen anion. K E Y W O R D S ground state, hydrogen atom, iteration, one-dimensional basis function, Schrödinger wave equations

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

“…/ -= Unlike the untransformed coordinates, the Table 1. Total energies (in Hartree) of the hydrogenlike atoms calculated with the untransformed method [43], the transformed method, and the exact calculations [9]. The cutoff range in one-dimensional x 0 (in Bohr) for the transformed method is given in the last column.…”

“…¥ 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.…”

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

“…In their study, zeroth-order hydrogenic wave function was used to evaluate the expectation values of the electron-electron repulsion in the two-electron atoms. Finally, Rahman et al [40] used the linear combination of hydrogenic wave functions to obtain accurate energy levels of hydrogenic ions by means of an iteration technique. Their method was also extended to many-body problems and proven to be accurate for the calculation of the helium-like ions, especially the hydrogen anion.…”

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.…”