Use of the Rayleigh-Ritz-Galerkin method with cubic splines for constructing single-particle bound-state radial wavefunctions: The hydrogen atom and its spectrum

The author presents a simple scheme of basis functions piecewise-continuous cubic polynomials, together with a simple quadrature approximation for matrix elements. Applied to the hydrogen atom, the method allows a dozen basis functions to approximate the 14 bound states 1s ... 5f with an average error of 0.03% in energies and 0.2% in the oscillator strengths of the 28 allowed transitions.