Solutions of Atomic and Molecular Schrödinger Equations with One-dimensional Function Approach

Rigorous numerical techniques to solve the Schrödinger equation are both interesting and desirable, particularly with one that can include new features beyond the standard methods. In this article, we review one-dimensional function(1D function) approach developed recently by us to obtain the solutions of the Schrödinger equations of atomic and molecular systems where one-dimensional basis functions have been applied to separate components. A uniform real-space grid representation of the electronic wavefunctions is employed; hence, a refinement technique of residual vector correction can be implemented. The 1D function approach facilitates such convenient numerical integrations that many problems related with the many-electron multi-center potential molecular integrals are circumvented. The converged energy is obtained from a strictly upper bound one, while the obtained two-electron Schrödinger wavefunction exhibits the electron correlation effect on one-electron distribution. Different from density functional theory or Hartree-Fock with the assumed particle-separability, the obtained solution treats more accurately many-body effect of electron correlation found in the electron-electron repulsion energy.