  Brillouin-Wigner (BW) perturbation theory is developed for both ground and
excited states of open-shell nuclei. We show that with optimal partitioning of
the many-body Hamiltonian proposed earlier by the authors [Z. Li and N.
Smirnova, arXiv:2306.13629], one can redefine the BW perturbation series for a
given state of the effective Hamiltonian in a small P-space to be converging
under the condition that the energy of this state is below the lowest
eigenvalue of the Hamiltonian matrix block belonging to the complement of the
P-space, characterized by the same good quantum numbers as the state under
consideration. Specifically, the BW perturbative calculations for the lowest
$J^\pi$ states are always converging due to the variational principle. This
property does hold for both soft and hard internucleon interactions in the
harmonic oscillator basis. To illustrate this method and check the convergence
behavior, we present numerical studies of low-energy spectra of $^{5,6,7}$Li
using the Daejeon16 and bare N3LO potentials.
