We calculate the finite nucleus Dirac mean field spectrum in a Galerkin approach using finite basis splines. We review the method and present results for the relativistic σ-ω model for the closed-shell nuclei ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ and ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$. We study the convergence of the method as a function of the size of the basis and the closure properties of the spectrum using an energy-weighted dipole sum rule. We apply the method to the Dirac random-phase-approximation response and present results for the isoscalar $1^{\mathrm{-}}$ and $3^{\mathrm{-}}$ longitudinal form factors of ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ and ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$. We also use a B-spline spectral representation of the positive-energy projector to evaluate partial energy-weighted sum rules and compare with nonrelativistic sum rule results.

• Received 22 August 1988

DOI:https://doi.org/10.1103/PhysRevC.40.399