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

We present a transformed-coordinates method to solve the Schrödinger equation for H-like, He-like, and Li-like systems. Each Cartesian axes of the original Schrödinger equation is transformed to another coordinate system with the square root transformation x′= x1/2. The resulting Hamiltonian contains the first and the second derivative for the kinetic energy part and with the potential proportional to the power of four, decaying faster than the original Coulomb potential. The total energies, their components, and the virial ratio are superior to those of the untransformed coordinates due to the considerably many data-points obtained and long-range sampling. Furthermore, a five-times or better computational efficiency is demonstrated in comparison to the standard method with much-improved accuracy. In agreement with the accurate method, the obtained wavefunction includes not only the radial but also the angular electron correlation of many-electron ions or atoms.