  The particle-hole Density Matrix Renormalization Group (p-h DMRG) method is
discussed as a possible new approach to large-scale nuclear shell-model
calculations. Following a general description of the method, we apply it to a
class of problems involving many identical nucleons constrained to move in a
single large j-shell and to interact via a pairing plus quadrupole interaction.
A single-particle term that splits the shell into degenerate doublets is
included so as to accommodate the physics of a Fermi surface in the problem. We
apply the p-h DMRG method to this test problem for two $j$ values, one for
which the shell model can be solved exactly and one for which the size of the
hamiltonian is much too large for exact treatment. In the former case, the
method is able to reproduce the exact results for the ground state energy, the
energies of low-lying excited states, and other observables with extreme
precision. In the latter case, the results exhibit rapid exponential
convergence, suggesting the great promise of this new methodology even for more
realistic nuclear systems. We also compare the results of the test calculation
with those from Hartree-Fock-Bogolyubov approximation and address several other
questions about the p-h DMRG method of relevance to its usefulness when
treating more realistic nuclear systems.
