Method of Correlated Basis Functions

The method of correlated basis functions provides a means of systematically exploiting our intuitive understanding of real physical systems. Here we seek especially to develop a formalism for describing systems of strongly interacting fermions, for example, nuclei and liquid He³, using a complete set of correlated functions Ψₘ=FΦₘ(Φₘ, F²Φₘ)1/2. F is taken as a product of two-body factors, Πi<j^[f(rᵢⱼ)], the f(rᵢⱼ) embodying at least the correlations arising from strong short-range repulsions; and Φₘ, a set of independent-particle-model functions which might serve as a basis in the absence of highly singular interactions. The matrix elements Hₘₙ, Nₘₙ of the Hamiltonian and identity operators with respect to the correlated basis Ψₘ are evaluated by generalization of cluster-expansion techniques developed originally for the calculation of the partition function of a classical imperfect gas and adapted to the quantum-mechanical many-body problem by Iwamoto and Yamada. Contributions to our expansions for the Hₘₙ are classified according to order of magnitude in the quantity ρω, where ω=∫(f²(r)-1)dr is a "correlation parameter" and ρ the average particle density. To the extent that |ρω| is small, it provides a true expansion parameter. A detailed technical discussion of the cluster expansions is given, with special emphasis on their symmetry properties upon truncation. Although the many-body problem is soluble by simultaneous diagonalization of the matrices (Hₘₙ) and (Nₘₙ), it is preferable first to transform to an orthonormal basis of "unperturbed" correlated functions Θₘ with respect to which the Hamiltonian and identity operators have the matrix representations (Hₘₙ) and (Nₘₙ)=(δₘₙ), and then to attempt the separate diagonalization of (Hₘₙ). In general, such a transformation is expedited by the L\"owdin procedure carried out to the desired order in ρω. However, a Schmidt orthogonalization scheme might prove more advantageous in specific cases. Field-theoretic techniques (of both perturbative and nonperturbative nature) may be brought to bear in the diagonalization of (Hₘₙ). The inverse of the unitary transformation which takes the basis Φₘ into the basis Θₘ, applied to the given singular Hamiltonian, leads---for any F in the class considered---to a non-singular "effective Hamiltonian." Following this formal cue, we set out to construct explicitly a second-quantized effective Hamiltonian which generates the same matrix elements, to prescribed order in ρω, when operating between independent-particle kets of the model occupation-number representation, as does the given Hamiltonian in the configuration-space representation, between the corresponding correlated functions of the L\"owdin basis. The quasiparticles described by this effective Hamiltonian interact via well-behaved two-body, three-body, ·{}·{}·{}, Q-body, ·{}·{}·{} potentials, in contrast to the highly singular two-body interactions of the real particles. For small |ρω|, the effective two-body forces predominate. An avenue to deeper understanding of the applicability of the nuclear-shell model and the dressed-quasiparticle theory of liquid He³ has been opened. This article treats only the formalism for a uniform extended medium. Numerical calculations, as well as the extension of the method to finite systems, will follow.