Norm-overlap formula for Hartree-Fock-Bogoliubov states with odd number parity

A formula to calculate the norm overlap between Hartree-Fock-Bogoliubov (HFB) states with odd number parity (one quasi-particle excited states) is derived with the help of Grassmann numbers and Fermion coherent states. The final formula is expressed in terms of a product of the Pfaffian for a neighboring even-even system (the zero quasi-particle state), and an extra factor consisting of the Bogoliubov transformation matrix and the anti-symmetric matrix in Thoulessʼ HFB ansatz for the even-even system.