  Background: Ab initio many-body methods have been developed over the past ten
years to address mid-mass nuclei... As progress in the design of inter-nucleon
interactions is made, further efforts must be made to tailor many-body methods.
  Methods: We formulate a truncated configuration interaction method that
consists of diagonalizing the Hamiltonian in a highly truncated subspace of the
total N-body Hilbert space. The reduced Hilbert space is generated via the
particle-number projected BCS state along with projected seniority-zero two and
four quasi-particle excitations. Furthermore, the extent by which the
underlying BCS state breaks U(1) symmetry is optimized in presence of the
projected two and four quasi-particle excitations... The quality of the newly
designed method is tested against exact solutions of the so-called attractive
pairing Hamiltonian problem.
  Results: By construction, the method reproduce exact results for N=2 and N=4.
For N=(8,16,20) the error on the ground-state correlation energy is less than
(0.006, 0.1, 0.15) % across the entire range of inter-nucleon coupling defining
the pairing Hamiltonian and driving the normal-to-superfluid quantum phase
transition. The presently proposed method offers the advantage to automatically
access the low-lying spectroscopy, which it does with high accuracy.
  Conclusions: The numerical cost of the newly designed variational method is
polynomial (N$^6$) in system size. It achieves an unprecedented accuracy on the
ground-state correlation energy, effective pairing gap and one-body entropy as
well as on the excitation energy of low-lying states of the attractive pairing
Hamiltonian. This constitutes a strong enough motivation to envision its
application to realistic nuclear Hamiltonians in view of providing a
complementary, accurate and versatile ab initio description of mid-mass
open-shell nuclei in the future.
