Single-particle states and the connection between Jastrow and Brueckner theory

Starting from Jastrow theory we derive with the help of variational methods the first orders of the hole-line expansion for the ground-state energy and for the associated wave function of a finite Fermi system. The implications of a variation of the energy expansion with respect to the single-particle basis are discussed both for Jastrow theory and for Brueckner theory. We elucidate the mechanism which leads to the invariance of the complete hole-line expansion with respect to a change of the single-particle basis. As a consequence, variation of the complete expansion in either theory gives a non-trivial result only if that result was built into the expansion from the outset. Nonetheless, variation of the lowest-order terms of the expansion leads to definite results and yields the necessary conditions for natural orbitals.