A full numerical solution of the electronic structure of atoms, usually through diagonalization of the many-body Hamiltonian using configuration-interaction, is not always computationally tractable. Therefore we benchmark three standard approximations, Hartree-Fock (HF), projected Hartree-Fock (PHF), and the random phase approximation (RPA), against 'exact' configuration-interaction (CI) using the same input parameters for all four calculations. These inputs, the atomic one- and two-body interactions, are computed analytically using Slater-type orbitals (STO). This gives a direct comparison between methods for the ground state energies, ionization potentials and electron affinities, for the atoms lithium through neon. These methods could be useful when optimizing basis sets. Obtaining exact results at the beginning of the optimization process is not necessary, therefore, a less accurate method can allow for a faster exploration of basis sets. With this motivation in mind, we perform a sample basis optimization on carbon and boron to determine which method, particularly PHF and RPA, can be used as a proxy for CI.