Matrix Elements of a Fermion System in a Representation of Correlated Basis Functions

The ground state and low excited states of liquid ${\mathrm{He}}^3$ (and other fermion systems) can be constructed from a set of basis functions $\Psi \mathrm{}\left(\right|\mathrm{}\mathrm{n})={{\psi }_0}^B\Phi \mathrm{}(\left|\mathrm{}\mathrm{n}\right)$ in which ${{\psi }_0}^B$ is the ground-state boson-type solution of the Schrödinger equation and the model functions $\Phi \mathrm{}\left(\right|\mathrm{}\mathrm{n})$ are Slater determinants suitable for describing states of the noninteracting Fermion system. Diagonal and nondiagonal matrix elements of the identity and the Hamiltonian operator are evaluated by a cluster-expansion technique. An orthonormal basis system is constructed from $\Psi \mathrm{}\left(\right|\mathrm{}\mathrm{n})$ and used to express the Hamiltonian operator in quasiparticle form: a large diagonal component containing constant, linear, quadratic, and cubic terms in free-quasiparticle occupation-number operators and a nondiagonal component representing the residual interactions involved in collisions of two and three free quasiparticles.