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Starting from realistic nuclear forces, the chiral N$^3$LO and JISP16, wehave applied many-body perturbation theory (MBPT) to the structure ofclosed-shell nuclei, $^4$He and $^{16}$O. The two-body N$^3$LO interaction issoftened by a similarity renormalization group transformation while JISP16 isadopted without renormalization. The MBPT calculations are performed within theHartree-Fock (HF) bases. The angular momentum coupled scheme is used, which canreduce the computational task. Corrections up to the third order in energy andup to the second order in radius are evaluated. Higher-order corrections in theHF basis are small relative to the leading-order perturbative result. Using theanti-symmetrized Goldstone diagram expansions of the wave function, we directlycorrect the one-body density for the calculation of the radius, rather thancalculate corrections to the occupation propabilities of single-particle orbitsas found in other treatments. We compare our results with other methods whereavailable and find good agreement. This supports the conclusion that ourmethods produce reasonably converged results with these interactions. We alsocompare our results with experimental data.

$\it{Ab}$ $\it{initio}$ nuclear many-body perturbation calculations in the Hartree-Fock basis