Bibliography tool, search by references, institution filtering. Learn more. Brink Oxford U. Published in: Phys. C 5 DOI: Citations per year 0 10 20 30 40 50 Abstract: APS.

References Figures 0. Davies Oak Ridge. Baranger Carnegie Mellon U. C 1Phys. C 2 erratum. McCarthy Carnegie Mellon U. C 1 C 14 Theory of Finite Nuclei K. Brueckner Pennsylvania U. Gammel Los Alamos.

Weitzner Harvard U. Properties of Finite Nuclei K. Brueckner UC, San Diego. Lockett Los Alamos. Rotenberg Illinois U. Study of finite nuclei in the local density approximation J. B 32 In computational physics and chemistrythe Hartreeâ€”Fock HF method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state.

The Hartreeâ€”Fock method often assumes that the exact N -body wave function of the system can be approximated by a single Slater determinant in the case where the particles are fermions or by a single permanent in the case of bosons of N spin-orbitals. By invoking the variational methodone can derive a set of N -coupled equations for the N spin orbitals. A solution of these equations yields the Hartreeâ€”Fock wave function and energy of the system.

Especially in the older literature, the Hartreeâ€”Fock method is also called the self-consistent field method SCF. Thus, self-consistency was a requirement of the solution. The solutions to the non-linear Hartreeâ€”Fock equations also behave as if each particle is subjected to the mean field created by all other particles see the Fock operator belowand hence the terminology continued.

The equations are almost universally solved by means of an iterative method, although the fixed-point iteration algorithm does not always converge. See Hartreeâ€”Fockâ€”Bogoliubov method for a discussion of its application in nuclear structure theory. In atomic structure theory, calculations may be for a spectrum with many excited energy levels and consequently the Hartreeâ€”Fock method for atoms assumes the wave function is a single configuration state function with well-defined quantum numbers and that the energy level is not necessarily the ground state.

For both atoms and molecules, the Hartreeâ€”Fock solution is the central starting point for most methods that describe the many-electron system more accurately. The rest of this article will focus on applications in electronic structure theory suitable for molecules with the atom as a special case.

The discussion here is only for the Restricted Hartreeâ€”Fock method, where the atom or molecule is a closed-shell system with all orbitals atomic or molecular doubly occupied. Open-shell systems, where some of the electrons are not paired, can be dealt with by either the restricted open-shell or the unrestricted Hartreeâ€”Fock methods. Douglas Hartree's methods were guided by some earlier, semi-empirical methods of the early s by E.

Fues, R. Lindsayand himself set in the old quantum theory of Bohr. It was observed from atomic spectra that the energy levels of many-electron atoms are well described by applying a modified version of Bohr's formula.

### Nuclear structure

The existence of a non-zero quantum defect was attributed to electronâ€”electron repulsion, which clearly does not exist in the isolated hydrogen atom. This repulsion resulted in partial screening of the bare nuclear charge. These early researchers later introduced other potentials containing additional empirical parameters with the hope of better reproducing the experimental data. InD. Hartree introduced a procedure, which he called the self-consistent field method, to calculate approximate wave functions and energies for atoms and ions.

His first proposed method of solution became known as the Hartree methodor Hartree product. However, in J. Slater and J.

## Nonrelativistic mean-field description of the deformation of Î› hypernuclei

Gaunt independently showed that the Hartree method could be couched on a sounder theoretical basis by applying the variational principle to an ansatz trial wave function as a product of single-particle functions.The time-dependent Hartree-Fock calculation with a full Skyrme energy functional has been carried out on the three-dimensional Cartesian lattice space to study E1 giant dipole resonances GDR in light nuclei.

The outgoing boundary condition for the continuum states is treated by the absorbing complex potential. The calculation for GDR in 16 O suggests a significant influence of the residual interaction associated with time-odd densities in the Skyrme functional. We also predict a large damping for superdeformed 14 Be at the neutron drip line. Time-dependent approach to nuclear response in the continuum The quantum-mechanical problems are usually solved in the energy timeindependent representation.

Namely, we either solve an energy eigenvalue problem for bound states or, for scattering states, we calculate a wave function with a proper boundary condition at a given energy. However, if one wishes to calculate physical quantities in a wide energy region, the time-dependent approach is very useful because a single time propagation provides information for a certain range of energy.

Another advantage may be an intuitive picture provided by the time evolution of the wave function. In Ref. The results indicate the capability and efficiency. Documents: Advanced Search Include Citations.

Abstract The time-dependent Hartree-Fock calculation with a full Skyrme energy functional has been carried out on the three-dimensional Cartesian lattice space to study E1 giant dipole resonances GDR in light nuclei.

Powered by:.Li, X. Zhou, H. Much interest has recently been devoted to the shape evolution of neutron-rich Si isotopes both experimentally and theoretically. We use an empirical pairing parameter deduced from the experimental binding energy data for each nucleus and each Skyrme parametrization. The later claim has actually been theoretically pointed out in several studies using the shell-model and mean-field approaches [ 4â€”11 ].

In Ref. The reason for this was attributed recently to the tensor effect on the change of shell structure in neutron-rich nuclei [ 10â€”14 ]. This feature was later analyzed in detail by Tarpanov et al. Also, the magnitude of the change is consistent with the empirical observations. It is necessary to extend the study of tensor correlations not only for spin-orbit splitting but also for the evolution of deformed nuclei, as will be done in the present work.

In addition, the inclusion of tensor terms in the calculations have achieved considerable success in explaining several nuclear structure problems, not only in the ground states [ 16â€”18 ] but also in the excited states [ 19â€”21 ].

In the present study we focus on the tensor effect on the shape evolution of Si isotopes; e. As is well known, the standard BCS treatment of pairing was criticized as having an unphysical particleâ€”gas problem [ 25 ] when applied to neutron-rich isotopes, namely, there is an unrealistic pairing of highly excited states. To cure this defect to some extent, a smooth energy-dependent cut-off weight was introduced [ 2627 ] in the evaluation of the local pair density, to confine the region of the influence of the pairing potential to the vicinity of the Fermi surface.

Some work also tried a variant calculation with weak pairing [ 428 ]. In the present study, the above-mentioned cut-off weight is incorporated into the BCS approximation, and for each nucleus the pairing parameter is obtained with full respect of the empirical data of pairing gaps, extracted by using the experimental binding energies of Ref.

We employ several recent Skyrme parametrizations with various tensor couplings to explore the role of the tensor force in the shape of neutron-rich Si isotopes, and the theoretical results will be compared directly with recent experiments.

The paper is organized as follows.

In Sect. The numerical results and discussions are given in Sect. Finally, Sect.We present a systematic analysis of the description of odd nuclei by the Skyrme-Hartree-Fock approach augmented with pairing in BCS approximation and blocking of the odd nucleon.

Current and spin densities in the Skyrme functional produce time-odd mean fields TOMF for odd nuclei. Their effect on basic properties binding energies, odd-even staggering, separation energies and spectra is investigated for the three Skyrme parameterizations SkI3, SLy6, and SV-bas.

The calculations demonstrate that the TOMF effect is generally small, although not fully negligible. The influence of the Skyrme parameterization and the consistency of the calculations are much more important. With a proper choice of the parameterization, a good description of binding energies and their differences is obtained, comparable to that for even nuclei. The description of low-energy excitation spectra of odd nuclei is of varying quality depending on the nucleus.

This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Bender, P. Heenen, P. Reinhard, Rev. Vretenar, A. Afanasjev, G. Lalazissis, P. Ring, Phys. Stone, P.

### Hartreeâ€“Fock method

Reinhard, Prog. Engel, D. Brink, K. Goeke, S. Krieger, D. Vauterin, Nucl. A Dobaczewski, J. Dudek, Phys. C 52 Afanasjev, P. C 62R Satula, R.Understanding the structure of the atomic nucleus is one of the central challenges in nuclear physics. The quantum mechanical nature of these particles appears via the Pauli exclusion principlewhich states that no two nucleons of the same kind can be at the same state. Thus the fluid is actually what is known as a Fermi liquid. This simple model reproduces the main features of the binding energy of nuclei.

The assumption of nucleus as a drop of Fermi liquid is still widely used in the form of Finite Range Droplet Model FRDMdue to the possible good reproduction of nuclear binding energy on the whole chart, with the necessary accuracy for predictions of unknown nuclei. The expression "shell model" is ambiguous in that it refers to two different eras in the state of the art.

It was previously used to describe the existence of nucleon shells in the nucleus according to an approach closer to what is now called mean field theory. Nowadays, it refers to a formalism analogous to the configuration interaction formalism used in quantum chemistry. We shall introduce the latter here. Systematic measurements of the binding energy of atomic nuclei show systematic deviations with respect to those estimated from the liquid drop model.

This observation led scientists to assume the existence of a shell structure of nucleons protons and neutrons within the nucleus, like that of electrons within atoms. Indeed, nucleons are quantum objects. Strictly speaking, one should not speak of energies of individual nucleons, because they are all correlated with each other. However, as an approximation one may envision an average nucleus, within which nucleons propagate individually.

Owing to their quantum character, they may only occupy discrete energy levels. These levels are by no means uniformly distributed; some intervals of energy are crowded, and some are empty, generating a gap in possible energies. A shell is such a set of levels separated from the other ones by a wide empty gap. Each level may be occupied by a nucleon, or empty. Some levels accommodate several different quantum states with the same energy; they are said to be degenerate.

This occurs in particular if the average nucleus has some symmetry. The concept of shells allows one to understand why some nuclei are bound more tightly than others. This is because two nucleons of the same kind cannot be in the same state Pauli exclusion principle.

So the lowest-energy state of the nucleus is one where nucleons fill all energy levels from the bottom up to some level. A nucleus with full shells is exceptionally stable, as will be explained. As with electrons in the electron shell model, protons in the outermost shell are relatively loosely bound to the nucleus if there are only few protons in that shell, because they are farthest from the center of the nucleus.

Therefore, nuclei which have a full outer proton shell will be more tightly bound and have a higher binding energy than other nuclei with a similar total number of protons. All this is also true for neutrons. Furthermore, the energy needed to excite the nucleus i.

Whenever this unoccupied level is the next after a full shell, the only way to excite the nucleus is to raise one nucleon across the gapthus spending a large amount of energy. Otherwise, if the highest occupied energy level lies in a partly filled shell, much less energy is required to raise a nucleon to a higher state in the same shell. Some evolution of the shell structure observed in stable nuclei is expected away from the valley of stability.

For example, observations of unstable isotopes have shown shifting and even a reordering of the single particle levels of which the shell structure is composed. Some basic hypotheses are made in order to give a precise conceptual framework to the shell model:. The general process used in the shell model calculations is the following.

First a Hamiltonian for the nucleus is defined. Usually, for computational practicality, only one- and two-body terms are taken into account in this definition. The interaction is an effective theory : it contains free parameters which have to be fitted with experimental data. The next step consists in defining a basis of single-particle states, i.PwC UK, Harnessing the fourth industrial revolution for sustainable emerging cities. Fourth industrial revolution for the earth series.

Meitner, L. Disintegration of uranium by neutrons: A new type of nuclear reaction. Goriely, S. Towards a prediction of fission cross sections on the basis of microscopic nuclear inputs. Capote, R. Data Sheets. Bouland, O.

**Quantum Chemistry 9.7 - Hartree-Fock Atomic Energy**

R-matrix analysis and prediction of low-energy neutron-induced fission cross sections for a range of Pu isotopes. Vautherin, D.

Hartree-Fock calculations with Skyrmes interaction. Spherical nuclei. Axially deformed nuclei. Flocard, H. Self-consistent calculation of the fission barrier of Pu. Bonneau, L. Effect of core polarization on magnetic dipole moments in deformed odd-mass nuclei. Koh, M. Band-head spectra of low-energy single-particle excitations in some well-deformed, odd-mass heavy nuclei within a microscopic approach.

Fission barriers of two odd-neutron actinide nuclei taking into account the time-reversal symmetry breaking at the mean-field level. Bender, M. Self-consistent mean-field models for nuclear structure. Nobre, G. Toward a microscopic reaction description based on energy-density-functional structure models. Bardeen, J. Theory of superconductivity. Pototzky, K. Properties of odd nuclei and the impact of time-odd mean fields: A systematic Skyrme-Hartree-Fock analysis.

Beiner, M. Nuclear ground-state properties and self-consistent calculations with the Skyrme interaction. Spherical Description Nucl. Fission-barriers and energy spectra of odd-mass actinide nuclei in self-consistent mean-eld calculations.

University of Bordeaux â€” Universiti Teknologi Malaysia.

## comments so far

## Zulkilkis Posted on 10:12 pm - Oct 2, 2012

Ganz richtig! Die Idee gut, ist mit Ihnen einverstanden.