Variational time-dependent Hartree-Fock calculations

A pseudopotential framework is used to develop a simplified version of time-dependent Hartree-Fock theory. Variational solutions of the equations are obtained for the alkali metal hydrides LiH, NaH and KH, and excitation energies, oscillator strengths and other properties are calculated. The results are in good agreement with more elaborate theoretical calculations.     

Hartree-Fock calculations of bubble nuclei

Hartree-Fock calculations of ³⁶Ar and of ²⁰⁰Hg in spherical configurations exhibit densities which are very much depleted in the interior region. The depletion agrees well with previous theoretical predictions of bubble nuclei.     

Hartree-Fock calculations of super-heavy nuclei

Hartree-Fock calculations are performed in the region of super heavy nuclei using zero range (Skyrme) forces and the finite range force G-0; the latter predict shell closure for the (α-unstable) nucleus N = 218, Z = 126 while the former do not. Calculations performed using the density matrix expansion (DME or DMEX) approximations give results similar to the Skyrme force. The problem is traced to an inadequacy of the DME, incorrectly placing the nodeless n = 1 high ? states.     

Exact shell-model and projected Hartree-Fock calculations for Mg

The results of a projected Hartree-Fock calculation are compared with those of an exact shell-model calculation for states of J ? 6 in the nucleus ²⁴Mg. The realistic Kuo interaction is used, and gives reasonable agreement with experiment.     

Hartree-Fock and shell model calculations for giant nuclei

The properties of giant nuclear systems like Uranium/Uranium etc. (A?476) have been studied in statical, spherical Hartree-Fock approximation. Various effective forces of Skyrme type were used to calculate binding energies, density distributions and single particle spectra. A shell model of Woods-Saxon type is shown to be more or less equivalent with respect to the shell structure derived from Skyrme models and easy generizable to slightly deformed giant nuclei. Strong magic shells could be found even in this region of the periodic table, but it seems that they cannot contribute significantly to a stabilization of these systems. The total binding energy is in good agreement with liquid drop model extrapolations. Only very briefly we discuss possibilities and problems for calculating deformation energies.     

A different method for solving the Hartree-Fock equations

A method is presented which guarantees the convergence of the solution of HF equations.     

Unrestricted Hartree-Fock calculations for a cluster model of NiO

Self-consistent restricted and unrestricted Hartree-Fock calculations have been performed for a NiO₆^?10 cluster embedded in a point ion field. The unrestricted Hartree-Fock calculations give rise to p-like and d-like states whose energies overlap, but which are essentially non-interacting. The calculated widths of the p-like and d-like states are 4.5 and 2.2 eV respectively, in good agreement with experiment. Furthermore, d-orbital relaxation is found to be much more important for RHF calculation as opposed to UHF calculations.     

Hartree-Fock calculations for finite nuclei with equivalent nonlocal potentials

The method of constructing equivalent regular two-body potentials by a unitary transformation of the two-body Hamiltonian has been generalized to spin-parity dependent nuclear potentials containing tensor- and spin-orbit terms. Starting from the Gammel-Christian-Thaler potential, which includes tensor forces, we obtained a class of equivalent regular, but nonlocal potentials depending on a parameterλ — the range of nonlocality. — These potentials have been used in a Hartree-Fock calculation for the closed-shell nuclei He⁴, C¹², O¹⁶, Si²⁸, S³², Ca⁴⁰. The calculated binding energies show a slowλ-variation with a minimum in the region of 0.7 f. The nuclear radii decrease with increasingλ and are in general too small. The sequence of single particle levels of the nuclei with closedl- shells is in agreement with that obtained with the usual nuclear shell model potential including spin-orbit coupling.     

‘Exact’ Hartree-Fock calculations for atomic-hydrogen crystal

Preliminary ‘exact’ Hartree-Fock calculations are presented for a simple-cubic atomic-hydrogen crystal. Two types of basis orbitals have been considered. Orbital parameters and lattice spacing have been roughly optimized. The Fermi surface is found to be nearly spherical. It is made plausible that the Hartree-Fock limit has been nearly reached.     

Rotationally constrained Hartree-Fock calculations and nuclear fission

Static Hartree-Fock calculations for a model ${}^{144}\text{Nd}$ nucleus are carried out under the assumption of axial symmetry and with constraints on the component of angular momentum perpendicular to the symmetry axis. The resulting energy surfaces exhibit a well defined fission barrier which decreases with increasing angular momentum more rapidly than predicted by liquid drop calculations.     

Exponential convergence and acceleration of Hartree-Fock calculations

We show that we can expect an exponential behaviour for the convergence of the Hartree-Fock solution during the HF iteration procedure. We use this property to extrapolate some collective degrees of freedom, in this case the shape, in order to speed up the self-consistent calculation. For axially deformed nuclei we apply the method to the quadrupole moment which corresponds to a simple scaling transformation on the single particle wave functions. Results are shown for the deformed nuclei ²⁰Ne and ²⁸Si with a Skyrme interaction.     

The minimax technique in relativistic Hartree-Fock calculations

The online version of the original article can be found at     

Hartree-Fock and Hartree-Bogolyubov calculations inp shell nuclei

Hartree-Fock (HF) and Hartree-Fock-Bogolyubov (HFB) calculations have been performed for the 1p shell nuclei. Nuclear deformations are assumed to be at most axially symmetric. The HFB transformation is restricted to allow forp-p andn-n pairing only.Volkov's force, a soft-core, two-body interaction of semi-realistic nature, is used which does not produce any single-particle spin-orbit splittings. Coulomb force and the usual correction for centre-of-mass motion are taken into account. The calculations are carried out in a single-particle basis including all states up to principle oscillator quantum numberN=3 (in some cases,N=4). Binding energies, rms radii, density distributions, and quadrupole moments are calculated and found to be in reasonable agreement with experiment. Large Hartree-Fock energy gaps are obtained. They prevent the pairing correlations considered from becoming effective in an HFB approach and from changing the HF ground-state properties appreciably. In non-selfconjugate nuclei, the Pauli principle, rather than the Coulomb interaction, yields large differences between the charge and mass distributions. A theorem on selfconsistent symmetries is proved. The coefficients of the HFB transformation turn out to be real, if time-reversal and angular momentum projection flip are selfconsistent symmetries.     

The minimax technique in relativistic Hartree-Fock calculations

Using the set of trial spinors and the Dirac-Coulomb Hamiltonian (H _DC) we discuss the role of the minimax theorem in relativistic Hartree-Fock calculations. In principle, the minimax theorem guarantees the occurrence of an upper bound. We also consider a scaling of the functionsu _i and discuss the condition to derive the relativistic hypervirial theorem; the variational procedure represented by the condition serves as an example of the minimax technique. Single zeta calculations onH ₂ ⁺ ,H ₂ and He are analysed. The effect of enlarging the basis is investigated for the He atom. The “upper bound” obtained by usingcoherent basis spinors differs from the result of the (random) linear variation using the kinetically balanced basis set by an amount which is at most of orderc ⁻⁴. Use of thecoherent basis set is advocated. An erratum to this article is available at .     

Time-dependent Hartree-Fock calculations for 16O + 16O reactions

Head-on collisions between ¹⁶O nuclei are treated in the Time-Dependent Hartree-Fock (TDHF) approximation. Reactions at center-of-mass bombarding energies of 2 and 8 MeV per nucleon result in fusion-fission processes with strong internal excitation of the fragments.     

Generalized shells in nuclei: Hartree-Fock calculations for bubble nuclei

Hartree-Fock calculations in the bubble degree of freedom have been performed on a variety of spherical nuclei. Of particular importance are incipient bubble configurations in ³⁶Ar, ⁶⁸Se, ⁸⁴Se, ¹⁰⁰Sn, ¹¹⁶Ce, ¹³⁸Ce and ²⁰⁰Hg, each of which possesses a binding energy which is comparable to that of the normal spherical closed-shell configuration. The densities of the above nuclei display strong deviations from a uniform shape, and give rise to depletions in the nuclear interior. These nonuniformities are due both to the absence of low angular momentum states in otherwise normally occupied spherical shells, and also to strong self-consistency effects. The nonuniformities in the mass density are further enhanced for nuclei whose neutron and proton densities have depressions or peaks at approximately the same distance from the center of the nucleus. A depression of the central density is most pronounced in the nuclei ³⁶Ar, ¹³⁸Ce and ²⁰⁰Hg. Interior depletions of the density are associated with the relatively higher energies of low angular momentum single-particle levels as compared to high angular momentum single-particle levels. This effect can give rise to moderately large gaps at the Fermi surface. Finally, it is shown that in a bubble configuration, the spin-orbit splitting of low lying doublets is sometimes reversed, and that this effect is especially pronounced for levels with low angular momentum.     

Practical angular momentum projection method in multi-shell Hartree-Fock calculations;Na as an example

A pratical method for projecting good angular momentum from non-axial deformed HF intrinsic states involving a large number of particles and j shells, is described. The ²²Na example shows good results for the excitation energies, and E2 and M1 matrix elements.     

A rapidly convergent method for solving the static Hartree-Fock problem

An improved version of the direct density method for solving the static Hartree-Fock equations is presented. A simple relation with the imaginary time method is exhibited.     

Hartree-Fock calculations of surface-symmetry properties with finite-range forces

Complete Hartree-Fock calculations on asymmetric semi-infinite nuclear matter have been performed with a family of finite-range Gogny-type forces. For one of these forces all the computed values of the droplet-model symmetry coefficients J, L, M and Q are in good agreement with recent mass-formula fits.