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.
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.
In this work a unitary correlation operator is presented that explicitly describes the short-ranged central and tensor correlations in the nuclear many-body system. These short-ranged correlations are induced by the repulsive core and the pronounced tensor force of realistic nucleon-nucleon interactions and cannot be described by the simple many-body states of a mean-field or shell model approach. The unitary correlation operator is discussed for the Argonne V18 and the Bonn-A interactions. Applying the correlation operator onto the Hamiltonian a common effective interaction for low momenta is obtained. Calculations for 4He using the one- and two-body part of the correlated Hamiltonian compare favorably with exact many-body methods. Calculations for 16O and 40Ca which are not possible with exact methods are performed using harmonic oscillator shell model states. The observed deviation from the experimental binding energies and radii is attributed to the missing three-body forces. The correlated interaction in a basis-free representation is used in the Fermionic Molecular Dynamics model to calculate the nuclei of the p- and the sd-shell. In this model the uncorrelated many-particle state is given as a Slaterdeterminant of Gaussian wave-packets with spin and isospin degrees of freedom which allows a consistent description of spherically symmetric, intrinsically deformed and clustered nuclei.
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 left-unitary operators that introduce the correlations of the hard core in an uncorrelated bound-state wave function of a finite number A of particles are explicitly determined up to a unitary transformation. Left unitarity is required in order to keep orthonormality. Using a shell-model basis, transformed according to the correlation operator, it is possible to diagonalize the Hamiltonian in a subspace of the full Hilbert space with the technique of Bloch and Horowitz. Ground as well as excited states are treated on the same footing. In practice, the diagonalization is accomplished using the original single-particle basis and transforming back the Hamiltonian. This results in an A-body operator for which a cluster expansion is possible. In this cluster expansion, there are three- and four-body terms easy to evaluate, because they factorize. A particularly interesting application can be made to few-body systems, evaluating the integrals numerically. The unique approximation is then connected with the cut of the basis. The method developed seems comparatively simple and is free of the mathematical problems of the hard core.
A general theory is proposed for the description of short-range correlations of two nucleons in nuclei in the framework of the unitary-model-operator approach. A new effective interaction is derived through the unitary transformation of the original Hamiltonian. The transformed Hamiltonian is given in a cluster expansion form, and the structure of the one- and two-body cluster terms is studied. New aspects of the present approach, compared with the G-matrix theory, are; (i) the new effective interaction is E-independent and Hermitian, (ii) it satisfies the decoupling condition between low- and high-momentum two-particle states, and (iii) it contains both of the contributions of ladder and folded diagrams. It is suggested that the properties (i)∼(iii) of the new effective interaction may offer considerable advantages in accounting manybody correlations and deriving effective interaction among valence nucleons.
A theory is formulated to derive an effective interaction in nuclei in the framework of the unitarymodel-operator approach.A unitary operator is introduced to describe two-body correlations, and it is determined from the condition that the effective interaction should be decoupled between the lowmomentum model space and the high-momentum space.The unitary-transformed Hamiltonian is represented in a cluster expansion form.The one-and two-body cluster terms are mainly discussed.The theory is applied to the calculation of the energies of the ground-and one-body states with the Reid soft core potential.A large gain of the energy has been obtained for the ground state in comparison with the result of the usual G-matrix theory.As for the one-body energies obtained, agreement with the experimental values is fairly good.Although the discrepancies with the experiments still remain at the 1O~ 15 % level, the present approach shows to have some desirable features compared with the nuclear many-body theories proposed so far. § 1. IntroductionA fundamental problem in nuclear physics is to clarify the properties of nuclei in terms of the interaction among their constituent particles.Although there have been many theories or approaches, we have not yet reached the goal of reproducing successfully the experimental data.The investigation of nuclear properties, in the usual nonrelativistic approach, proceeds in two steps: The first one is to determine the interaction among nucleons, and the second is to solve the many-body SchrOdinger equation.Concerning the nucleon-nucleon force, it seems that the phenomenological potential, such as the Reid potential, would be incomplete and one should wait until quark dynamics, probably quantum chromodynamics (QCD), solves finally the problem o~ nuclear force.It is of no doubt that in order to fit the experiments we need more accurate knowledge on the nuclear force including the threebody force.However, if we want to reach the goal, we need also a good many-body theory.Regardless of the accuracy of the nuclear force, we shall not be able to solve the problem unless we have a way of constructing an accurate solution for a many-body system.We believe that many-body theory should share the responsibility for the discrepancy between theory and experiment.It will be sure that nuclear many-body theory is still to be improved.Among many theories for nuclear many-body problem, the most widely applied method is, needless to say, the linked-diagram perturbation theory.This approach may also be referred to as the folded-diagram theory based on the G-matrix interaction especially due 'to Brandow 1l and Kuo et al. 2l An outstanding problem in this approach is the convergence of the perturbation expansion.Since Schucan and Weidenmliller 3 ) pointed out the divergence of the perturbation expansion due to the presence of intruder states, this difficulty has long been a serious problem.Present status of this problem is
A general theory of long range correlations in the ground state of a closed shell nucleus is outlined, using a number of ideas introduced or adumbrated, but not always successfully developed, in previous work of the authors and others. These include the following: (i) Any operator can be formally expressed as a series in particle-hole (p-h) excitation and destruction operators which should converge rapidly for a suitable choice of single-particle basis at the closed shells. By means of these relations and with the help of sum rules all observables can be computed without the explicit use of wave functions. (ii) For a fixed choice of single-particle basis, the ground state energy can be computed from a variational principle in which in lowest nontrivial approximation, in consequence of (i), the parameters are amplitudes of the same type as occur in the random phase approximation (RPA). The precise relation to RPA is fully detailed. (iii) The single particle basis chosen is the natural orbital basis of Lowdin, which presents itself as the obvious choice both for formulation of the dynamics and for the algorithm of solution. (iv) Though the theory deals with pairs of fermion operators, particular attention is paid that all Pauli principle restrictions are satisfied. (v) It is verified that in the perturbation limit, known results are reproduced. The method is generally applicable to any system for which the correlations can be viewed as particle-hole excitations with respect to a reference Slater determinant.