A new variational approach is developed for studying the properties of systems of particles interacting through singular short-range repulsions that give rise to strong two-particle correlations. The correlated trial function ${\ensuremath{\Psi}}_{\ensuremath{\gamma}}={e}^{S}{\ensuremath{\Phi}}_{\ensuremath{\gamma}}$ (state $\ensuremath{\gamma}$) results, with proper choice of
S, in a simple form for the energy expectation value
〈H〉---as well as for other matrix elements of interest---which is devoid of all reference to the strong repulsions except through
e2S factors and hence is particularly suited to calculation. In many cases an independent-particle type ${\ensuremath{\Phi}}_{\ensuremath{\gamma}}$ seems appropriate. The cluster evaluation of this form for
〈H〉 is discussed, both in the few-particle and many-particle cases. Using the techniques of Iwamoto and Yamada, simplified convergent cluster expansions for the energy expectation value are derived for many-fermion and many-boson systems. A program for application of this method to nuclear problems is being initiated.
Fermionic Molecular Dynamics (FMD) models a system of fermions by means of many-body states which are composed of antisymmetrized products of single-particle states. These consist of one or several Gaussians localized in coordinate and momentum space. The parameters specifying them are the dynamical variables of the model. As the repulsive core of the nucleon-nucleon interaction induces short range correlations which cannot be accommodated by a Slater determinant, a novel approach, the unitary correlation operator method (UCOM), is applied. The unitary correlator moves two particles away from each other whenever their relative distance is within the repulsive core. The time-dependent variational principle yields the equations of motion for the variables. Energies of the stationary ground states are calculated and compared to exact many-body results for nuclei up to Ca 48. Time-dependent solutions are shown for collisions between nuclei.
The short range repulsion between nucleons is treated by a unitary correlation operator which shifts the nucleons away from each other whenever their uncorrelated positions are within the replusive core. By formulating the correlation as a transformation of the relative distance between particle pairs, general analytic expressions for the correlated wave functions and correlated operators are given. The decomposition of correlated operators into irreducible n-body operators is discussed. The one- and two-body-irreducible parts are worked out explicitly and the contribution of three-body correlations is estimated to check convergence. Ground state energies of nuclei up to mass number A=48 are calculated with a spin-isospin-dependent potential and single Slater determinants as uncorrelated states. They show that the deduced energy- and mass-number-independent correlated two-body Hamiltonian reproduces all "exact" many-body calculations surprisingly well.
We study many-body correlations in the ground state of a general quantum system of bosons or fermions by including an additional Jastrow function in our recently proposed variational coupled-cluster method. Our approach combines the advantages of state-dependent correlations in the coupled-cluster theory and of the strong, short-ranged correlations of the Jastrow function. We apply a generalized linked-cluster expansion for the Jastrow wave function and provide a detailed analysis for practical evaluation of the Hamiltonian expectation value as an energy functional of the Jastrow function and the bare density-distribution functions introduced and calculated in our earlier publications; a simple, first-order energy functional is derived and detailed formulas for the higher-order contributions are provided. Our energy functional does not suffer the divergence as most coupled-cluster calculations often do when applying to Hamiltonians with hardcore potentials. We also discuss possible applications of our technique, including applications to strongly correlated fermion systems.
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 ${\ensuremath{\Psi}}_{\mathrm{m}}=\frac{F{\ensuremath{\Phi}}_{\mathrm{m}}}{{({\ensuremath{\Phi}}_{\mathrm{m}}, {F}^{2}{\ensuremath{\Phi}}_{\mathrm{m}})}^{\frac{1}{2}}}$.
F is taken as a product of two-body factors, $\ensuremath{\Pi}{i<j}^{}[f({r}_{\mathrm{ij}})]$, the
f(rij) embodying at least the correlations arising from strong short-range repulsions; and ${{\ensuremath{\Phi}}_{\mathrm{m}}}$, 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 ${{\ensuremath{\Psi}}_{\mathrm{m}}}$ 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 $\ensuremath{\rho}\ensuremath{\omega}$, where $\ensuremath{\omega}=\ensuremath{\int}({f}^{2}(r)\ensuremath{-}1)d\mathrm{r}$ is a "correlation parameter" and $\ensuremath{\rho}$ the average particle density. To the extent that $|\ensuremath{\rho}\ensuremath{\omega}|$ 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 ${{\ensuremath{\Theta}}_{\mathrm{m}}}$ with respect to which the Hamiltonian and identity operators have the matrix representations (
Hmn) and $({N}_{\mathrm{mn}})=({\ensuremath{\delta}}_{\mathrm{mn}})$, 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 $\ensuremath{\rho}\ensuremath{\omega}$. 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 (
Hmn). The inverse of the unitary transformation which takes the basis ${{\ensuremath{\Phi}}_{\mathrm{m}}}$ into the basis ${{\ensuremath{\Theta}}_{\mathrm{m}}}$, 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 $\ensuremath{\rho}\ensuremath{\omega}$, 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 L\"owdin basis. The quasiparticles described by this effective Hamiltonian interact via well-behaved two-body, three-body, \ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{},
Q-body, \ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{} potentials, in contrast to the highly singular two-body interactions of the real particles. For small $|\ensuremath{\rho}\ensuremath{\omega}|$, 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.
Within the framework of fermionic molecular dynamics a method is developed to better account for long-range tensor correlations in nuclei when working with a single Slater determinant. Single-particle states with mixed isospin and broken parity build up an intrinsic Slater determinant, which is then charge and parity projected. By minimizing the energy of this many-body state with respect to the parameters of the single-particle states and projecting afterward on angular momentum, ground-state energies are obtained that are systematically lower than corresponding Hartree-Fock results. The realistic Argonne V18 potential is used and short-range correlations are treated with the unitary correlation operator method. Comparison with exact few-body calculations shows that in
4He about one-fifth of the correlation energy from long-range correlations are accounted for. These correlations, which extend over the whole nucleus, are visualized with the isospin and spin-isospin density of the intrinsic state. The divergence of the spin-isospin density, the source for pion fields, turns out to be of dipole nature.