High-accuracy solution of the Dirac equation for three-dimensional hydrogen atom using a variationally optimized diagonalization method

Abstract

High-accuracy solution of the Dirac equation for three-dimensional (3D) hydrogen atom is theoretically presented using a variationally optimized diagonalization method. By imposing the asymptotical conditions, a complete set of orthogonal Laguerre basis functions is constructed explicitly for the solution of the Dirac equation, which eliminates the singularity difficulty at the point nucleus and gives a convergent integral. A fully explicit expression of the matrix elements of the Dirac Hamiltonian has been derived from the method of the generating function of Laguerre polynomials. The relative errors of the relativistic energies are ranged from 11 to 14 digits, showing that the calculated relativistic energies are in excellent agreement with the exact energies. These results show that my proposed orthogonal Laguerre basis set is extremely efficient and stable in calculating high-accuracy solution of the Dirac equation for 3D hydrogen atom.