  The standard Bogoliubov transformation is generalized to enable fermion
number parity breaking. The new transformation can diagonalize fermion
Hamiltonians that are quadratic in fermion and number parity operators. This
new variational freedom cannot lower the energy of Hamiltonians that conserve
number parity. However, it has value as a minimal mechanism for continuously
varying between states of different fermion number, regardless of number
parity.
