Variational justification of the dimensional-scaling method in chemical physics: the H-atom

Abstract

The dimensional scaling (D-scaling) method first originated from quantum chromodynamics by using the spatial dimension D as an order parameter. It later has found many useful applications in chemical physics and other fields. It enables, e.g., the calculation of the energies of the Schrodinger equation with Coulomb potentials without having to solve the partial differential equation (PDE). This is done by imbedding the PDE in a D-dimensional space and by letting D tend to infinity. One can avoid the partial derivatives and then solve instead a reduced-order finite dimensional minimization problem. Nevertheless, mathematical proofs for the D-scaling method remain to be rigorously established. In this paper, we will establish this by examining the D-scaling procedures from the variational point of view. We show how the ground state energy of the hydrogen atom model can be calculated by justifying the singular perturbation procedures. In the process, we see in a more clear and mathematical way confirming (Herschbach J Chem Phys 85:838, 1986 Sect. II.A) how the D-dimensional electron wave function “condenses into a particle,” the Dirac delta function, located at the unit Bohr radius.