Ground-state properties of light $4n$ self-conjugate nuclei in $ab$ $initio$ no-core Monte Carlo shell model calculations with nonlocal $NN$ interactions

We report $J^\pi = 0^+$ ground-state energies and point-proton radii of $^4$He, $^8$Be, $^{12}$C, $^{16}$O and $^{20}$Ne nuclei calculated by the $ab$ $initio$ no-core Monte Carlo shell model with the JISP16 and Daejeon16 nonlocal $NN$ interactions. Ground-state energies are obtained in the basis spaces up to seven oscillator shells ($N_{\rm shell} = 7$) with several oscillator energies ($\hbar \omega$) around the optimal oscillator energy for the convergence of ground-state energies. These energy eigenvalues are extrapolated to obtain estimates of converged ground state energies in each basis space using energy variances of computed energy eigenvalues. We further extrapolate these energy-variance-extrapolated energies obtained in the finite basis spaces to infinite basis-space results with an empirical exponential form. This form features a dependence on the basis-space size but is independent of the value of $\hbar\omega$ used for the harmonic-oscillator basis functions. Point-proton radii for these states of atomic nuclei are also calculated following techniques employed for the energies. From these results, it is found that the Daejeon16 $NN$ interaction provides good agreement with experimental data up to approximately $^{16}$O, while the JISP16 $NN$ interaction provides good agreement with experimental data up to approximately $^{12}$C. Beyond these nuclei, the interactions produce overbinding accompanied by radii that are too small. These findings suggest and encourage further revisions of nonlocal $NN$ interactions towards the investigation of nuclear structure in heavier-mass regions.