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.
In Fermionic Molecular Dynamics the occurrence of multifragmentation depends strongly on the intrinsic structure of the many-body state. Slater determinants with narrow single-particle states and a cluster substructure show multifragmentation in heavy-ion collisions while those with broad wave functions, which resemble more a shell-model picture, deexcite by particle emission. Which of the two type of states occurs as the ground state minimum or as a local minimum in the energy depends on the effective interaction. Both may equally well reproduce binding energy and radii of nuclei. This ambiguity led us to reinvestigate the derivation of the effective interaction from realistic nucleon-nucleon potentials by means of a unitary correlation operator which is much more suited for dynamical calculations than the G-matrix or the Jastrow method. First results of mixing many Slater determinants are also presented.
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 nucleon momentum distribution
nA(k) for
A=2, 3, 4, 16, and 40 nuclei is systematically analyzed in terms of wave functions resulting from advanced solutions of the nonrelativistic Schr\"odinger equation, obtained within different many-body approaches based upon different realistic bare nucleon-nucleon (
NN) interactions featuring similar short-range repulsion and tensor interactions. Particular attention is paid to the separation of the momentum distributions into the mean-field and short-range correlation (SRC) contributions. It is shown that although at high values of the momentum
k different approaches lead to some quantitative differences, these do not hinder the general conclusion that the high-momentum behavior ($k\ensuremath{\gtrsim}1.5--2$ fm${}^{\ensuremath{-}1}$) of all nuclei considered are very similar, exhibiting the well-known scaling behavior with the mass number
A, independently of the used many-body approach and the details of the bare
NN interaction. To analyze and understand the frequently addressed question concerning the relationships between the nucleus,
nA(k), and the deuteron,
nD(k), momentum distributions, the spin (
S)-isospin (
T) structure of few-nucleon systems and complex nuclei is analyzed in terms of realistic
NN interactions and many-body approaches. To this end, the number of
NN pairs in a given (
ST) state, viz.,
(ST)=(10), (00), (01), and (11), and the contribution of these states to the nucleon momentum distributions are calculated. It is shown that, apart from the (00) state, which has very small effects, all other spin-isospin states contribute to the momentum distribution in a wide range of momenta. It is shown that for all nuclei considered the momentum distributions in the states
T=0 and
T=1 exhibit at $k\ensuremath{\gtrsim}1.5--2$ fm${}^{\ensuremath{-}1}$ very similar behaviors, which represents strong evidence of the
A-independent character of SRCs. The ratio
nA(k)/nD(k) is analyzed in detail, stressing that in the SRC region it always increases with the momentum and the origin of such an increase is discussed and elucidated. The relationships between the one- and two-body momentum distributions, considered in a previous paper, are discussed and clarified, pointing out the relevant role played by the center-of-mass motion of a correlated pair in the (10) state. Eventually, the values of the the probability of high-momentum components in nuclei and the per nucleon probability
a2 of deuteronlike configurations in nuclei are calculated, and the relationship of the present approach with the many-body methods based upon low-momentum effective interactions is briefly discussed.
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.