  The Hartree-Fock-Bogoliubov approximation is very useful for treating both
long- and short-range correlations in finite quantum fermion systems, but it
must be extended in order to describe detailed spectroscopic properties. One
problem is the symmetry-breaking character of the HFB approximation. We present
a general and systematic way to restore symmetries and to extend the
configuration space using pfaffian formulas for the many-body matrix elements.
The advantage of those formulas is that the sign of the matrix elements is
unambiguously determined. It is also helpful to extend the space of
configurations by constraining the HFB solutions in some way. A powerful method
for finding these constrained solutions is the gradient method, based on the
generalized Thouless transformation. The gradient method also preserves the
number parity of the Bogoliubov transformation, which facilitates the
application of the theory to systems with odd particle number.
