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 ${\mathrm{He}}^3$, using a complete set of correlated functions ${\Psi }_{\mathrm{m}}=\frac{F{\Phi }_{\mathrm{m}}}{{({\Phi }_{\mathrm{m}}, F^2{\Phi }_{\mathrm{m}})}^{\frac{1}{2}}}$. $F$ is taken as a product of two-body factors, $\Pi {i<j}^{}\left[f\right(r_{\mathrm{ij}}\left)\right]$, the $f\left(r_{\mathrm{ij}}\right)$ embodying at least the correlations arising from strong short-range repulsions; and $\left\{{\Phi }_{\mathrm{m}}\right\}$, a set of independent-particle-model functions which might serve as a basis in the absence of highly singular interactions. The matrix elements ${\mathfrak{H}}_{\mathrm{mn}}$, ${\mathfrak{N}}_{\mathrm{mn}}$ of the Hamiltonian and identity operators with respect to the correlated basis $\left\{{\Psi }_{\mathrm{m}}\right\}$ 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 ${\mathfrak{H}}_{\mathrm{mn}}$ are classified according to order of magnitude in the quantity $\rho \omega$, where $\omega =\int \left(f^2\mathrm{}\right(r)-1)d\mathrm{r}$ is a "correlation parameter" and $\rho$ the average particle density. To the extent that $\left|\rho \omega \right|$ 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 (${\mathfrak{H}}_{\mathrm{mn}}$) and (${\mathfrak{N}}_{\mathrm{mn}}$), it is preferable first to transform to an orthonormal basis of "unperturbed" correlated functions $\left\{{\Theta }_{\mathrm{m}}\right\}$ with respect to which the Hamiltonian and identity operators have the matrix representations ($H_{\mathrm{mn}}$) and $\left(N_{\mathrm{mn}}\right)=\left({\delta }_{\mathrm{mn}}\right)$, and then to attempt the separate diagonalization of ($H_{\mathrm{mn}}$). In general, such a transformation is expedited by the Löwdin procedure carried out to the desired order in $\rho \omega$. 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 ($H_{\mathrm{mn}}$). The inverse of the unitary transformation which takes the basis $\left\{{\Phi }_{\mathrm{m}}\right\}$ into the basis $\left\{{\Theta }_{\mathrm{m}}\right\}$, 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 $\rho \omega$, 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 Löwdin 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 $\left|\rho \omega \right|$, 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 ${\mathrm{He}}^3$ 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.