  A one-dimensional harmonic oscillator in a box is used to introduce the
oblique-basis concept. The method is extended to the nuclear shell model by
combining traditional spherical states, which yield a diagonal representation
of the usual single-particle interaction, with collective configurations that
track deformation. An application to 24Mg, using the realistic two-body
interaction of Wildenthal, is used to explore the validity of this mixed-mode
shell-model scheme. Specifically, the correct binding energy (within 2% of the
full-space result) as well as low-energy configurations that have greater than
90% overlap with full-space results are obtained in a space that spans less
than 10% of the full-space. The theory is also applied to lower pf-shell
nuclei, 44Ti-48Ti and 48Cr, using the Kuo-Brown-3 interaction. These nuclei
show strong SU(3) symmetry breaking due mainly to the single-particle
spin-orbit splitting. Nevertheless, the results also show that yrast band B(E2)
values are insensitive to fragmentation of the SU(3) symmetry. Specifically,
the quadrupole collectivity as measured by B(E2) strengths remains high even
though the SU(3) symmetry is rather badly broken. The IBM and broken-pair
models are considered as alternative basis sets for future oblique-basis
shell-model calculations.
