Constrained Hartree-Fock Research Articles

Electronic transition dipole moments from orthogonality constrained Hartree–Fock wavefunctions

In our recent paper (Glushkov V N and Assfeld X 2016 Theor. Chem. Acc. 135), we have presented some applications of the so-called orthogonality constrained Hartree–Fock (OCHF) theory to calculations of excited state energies of the same symmetry, including highly and core excited states. In this paper we report an attempt to apply this methodology to practical calculations of electronic dipole transition moments focusing on transitions between states having the same spatial symmetry as the ground state. Basic tenets of this method are briefly reviewed and its specific features for ab initio calculations of electronic transitions are discussed. The effect of non-orthogonal HF wave functions on transition dipole moments is studied and analyzed. OCHF electronic transition moments calculated for the HeH molecule are presented as a function of the internuclear separation for five 2Σ+ states (X 2Σ+, A2Σ+, C2Σ+, D2Σ+and F 2Σ+). The results of the calculations are, in general, in good agreement with those obtained by multi-reference configuration interaction methods which take correlation effects into account and show a favorable balance between accuracy and computational efficiency.

Can X-ray constrained Hartree-Fock wavefunctions retrieve electron correlation?

The X-ray constrained wavefunction (XC-WF) method proposed by Jayatilaka [Jayatilaka & Grimwood (2001) ▸, Acta Cryst. A57, 76-86] has attracted much attention because it represents a possible third way of theoretically studying the electronic structure of atoms and molecules, combining features of the more popular wavefunction- and DFT-based approaches. In its original formulation, the XC-WF technique extracts statistically plausible wavefunctions from experimental X-ray diffraction data of molecular crystals. A weight is used to constrain the pure Hartree-Fock solution to the observed X-ray structure factors. Despite the wavefunction being a single Slater determinant, it is generally assumed that its flexibility could guarantee the capture, better than any other experimental model, of electron correlation effects, absent in the Hartree-Fock Hamiltonian but present in the structure factors measured experimentally. However, although the approach has been known for long time, careful testing of this fundamental hypothesis is still missing. Since a formal demonstration is impossible, the validation can only be done heuristically and, to accomplish this task, X-ray constrained Hartree-Fock calculations have been performed using structure factor amplitudes computed at a very high correlation level (coupled cluster) for selected molecules in isolation, in order to avoid the perturbations due to intermolecular interactions. The results show that a single-determinant XC-WF is able to capture the electron correlation effects only partially. The largest amount of electron correlation is extracted when: (i) a large external weight is used (much larger than what has normally been used in XC-WF calculations using experimental data); and (ii) the high-order reflections, which carry less information on the electron correlation, are down-weighted (or even excluded), otherwise they would bias the fitting towards the unconstrained Hartree-Fock wavefunction.

Effects of a realistic tensor force on nuclear quadrupole deformation near the “shore” of the island of inversion

The M3Y-type semirealistic interaction is applied to deformed nuclei for the first time. Constrained Hartree-Fock calculations assuming axial symmetry are implemented for the $N=20$ isotones $^{30}\mathrm{Ne}$, $^{32}\mathrm{Mg}$, $^{34}\mathrm{Si}$ and the $N=28$ isotones $^{40}\mathrm{Mg}$, $^{42}\mathrm{Si}$, $^{44}\mathrm{S}$ with the M3Y-P6 interaction. The results match the experimental data well. Effects of the realistic tensor force on the nuclear quadrupole deformation are investigated in relation to the loss of the $N=20$ and 28 magic numbers. The tensor force is confirmed to favor the deformation for the $N=28$ nuclei owing to the closure of the $jj$ shell (i.e., $n0{f}_{7/2}$), while favoring the sphericity for the $N=20$ nuclei owing to the $\ensuremath{\ell}s$ closure of $N=20$.

Fission dynamics within time-dependent Hartree-Fock: Deformation-induced fission

Background: Nuclear fission is a complex large-amplitude collective decay mode in heavy nuclei. Microscopic density functional studies of fission have previously concentrated on adiabatic approaches based on constrained static calculations ignoring dynamical excitations of the fissioning nucleus, and the daughter products. Purpose: To explore the ability of dynamic mean-field methods to describe fast fission processes beyond the fission barrier, using the nuclide $^{240}$Pu as an example. Methods: Time-dependent Hartree-Fock calculations based on the Skyrme interaction are used to calculate non-adiabatic fission paths, beginning from static constrained Hartree-Fock calculations. The properties of the dynamic states are interpreted in terms of the nature of their collective motion. Fission product properties are compared to data. Results: Parent nuclei constrained to begin dynamic evolution with a deformation less than the fission barrier exhibit giant-resonance-type behaviour. Those beginning just beyond the barrier explore large amplitude motion but do not fission, whereas those beginning beyond the two-fragment pathway crossing fission to final states which differ according to the exact initial deformation. Conclusions: Time-dependent Hartree-Fock is able to give a good qualitative and quantitative description of fast fission, provided one begins from a sufficiently deformed state.

Fission fragment mass yield deduced from density distribution in the pre-scission configuration

Static self-consistent methods usually allow one to determine the most probable fission fragments mass asymmetry. We have applied random neck rupture mechanism to the nuclei in the configuration at the end of fission paths. Fission fragment mass distributions have been deduced from the pre-scission nuclear density distribution obtained from the self-consistent calculations. Potential energy surfaces as well as nuclear shapes have been calculated in the fully microscopic theory, namely the constrained Hartree–Fock–Bogoliubov model with the effective Gogny D1S density-dependent interaction. The method has been applied for analysis of fission of Fm, 252Cf and 180Hg and compared with the experimental data.

Shape of Te isotopes in mean-field formalism

The systematic investigation of ground state shape evolution from gamma unstable O(6) to spherical U(5) for Te isotopes has been presented by using the quadrupole moment constrained Hartree Fock Bogoliubov (HFB) method with the Skyrme force SLy4. 124Te has been pointed out as to be the possible critical point nucleus with E(5) symmetry.

AN INVESTIGATION ON SHAPE EVOLUTION OF Ti ISOTOPES WITH HARTREE–FOCK–BOGOLIUBOV THEORY

Constrained Hartree–Fock–Bogoliubov theory with SLy4 and SLy5 Skyrme forces is used to investigate the shape transition between spherical and γ-unstable nuclei in 38–66 Ti . By examining potential energy curves and neutron single-particle levels of even–even Ti isotopes, 46,52,60 Ti are suggested as possible candidates of the nuclei with E(5) symmetry.

Doubly-Magic 100Sn Nucleus with Delta Excitation under Compression

The energies and matter densities of finite nuclei under radial compression are investigated by using a constrained Hartree–Fock method with the Δ degree of freedom included. The results are presented for the doubly-magic nucleus 100 Sn in a effective baryon–baryon interaction. As the nucleus is compressed, the binding energy increases in a linear form approximately. The increase in binding energy is less in large model space systems. The gap between binding energy of only the nucleon and nucleon + Δ is reduced by increasing the model space. The number of Δ's increases as the model space decreases. It is found that as the nucleus is compressed to about 17 times the ordinary nuclear density, the Δ component is sharply increased to about 17.8% of all baryons in the system. This result is consistent with the values extracted from relativistic heavy-ion collisions. The number of Δ's increases as the radius decreases under compression. The single particle energy levels calculated and their behaviors under compression are examined too. A good agreement between results with effective Hamiltonian and the phenomenological shell model for the low lying single-particle spectra is obtained. The current results suggest that the radial density distribution is reduced at large model space for same radius. The peak of Δ radial density distribution is reduced as the model space increases for the same r rms . A considerable reduction in compressibility for the nucleus, and softening of the equation of state with the inclusion of the Δ's in the nuclear dynamics are suggested by the results.

Delta excitation calculation studies in the ground state of the compressed finite heavy doubly-magic nucleus ^{100}Sn

The energies and matter densities of finite nuclei under radial compression are investigated by using a constrained Hartree-Fock method with the \Delta degree of freedom included. The results are presented for the doubly-magic nucleus ^{100}Sn in an effective baryon-baryon interaction. It is found that as the nucleus is compressed to about three time of the ordinary nuclear density, the \Delta component is sharply increased to about 17% of all baryons in the system. This result is consistent with the values extracted from relativistic heavy-ion collisions. The single particle energy levels calculated and their behaviors under compression examined too. A good agreement between results with effective Hamiltonian and the phenomenological shell model for the low lying single-particle spectra obtained. A considerable reduction in compressibility for the nucleus, and softening of the equation of state with the inclusion of the \Delta 's in the nuclear dynamics are suggested by the results.

Shape mixing dynamics in the low-lying states of proton-rich Kr isotopes

We study the oblate–prolate shape mixing in the low-lying states of proton-rich Kr isotopes using the five-dimensional quadrupole collective Hamiltonian. The collective Hamiltonian is derived microscopically by means of the CHFB (constrained Hartree–Fock–Bogoliubov) + Local QRPA (quasiparticle random phase approximation) method, which we have developed recently on the basis of the adiabatic self-consistent collective coordinate method. The results of the numerical calculation show the importance of large-amplitude collective vibrations in the triaxial shape degree of freedom and rotational effects on the oblate–prolate shape mixing dynamics in the low-lying states of these isotopes.

Microscopic transport theory of nuclear processes

We formulate a microscopic theory of the decay of a compound nucleus through fission which generalizes earlier microscopic approaches of fission dynamics performed in the framework of the adiabatic hypothesis. It is based on the constrained Hartree–Fock–Bogoliubov procedure and the Generator Coordinate Method, and requires an effective nucleon–nucleon interaction as the only input quantity. The basic assumption is that the slow evolution of the nuclear shape must be treated explicitly, whereas the rapidly time-dependent intrinsic excitations can be treated by statistical approximations. More precisely, we introduce a “reference density” ρ P which represents the slow evolution of the nuclear shape by a reduced density matrix and the state of intrinsic excitations by a canonical distribution at each given shape of the nucleus. The shape of the nuclear density distribution is described by parameters (“generator coordinates” q), not by “superabundant” degrees of freedom introduced in addition to the complete set of nucleonic degrees of freedom. We first derive a rigorous equation of motion for the reference density ρ P and, subsequently, simplify this equation on the basis of the “Markov approximation”. The temperature T which appears in the canonical distribution is determined by the requirement that, at each time t, the reference density should correctly reproduce the mean excitation energy at given values of the shape parameters q. The resulting equation for the “local” temperature T ( q , t ) must be solved together with the equations of motion obtained for the reduced density matrix R ( q 1 , q 2 ; t ) .

Role of superfluidity in nuclear incompressibilities

Nuclei are propitious tools to investigate the role of the superfluidity in the compressibility of a Fermionic system. The centroid of the Giant Monopole Resonance (GMR) in Tin isotopes is predicted using a constrained Hartree-Fock Bogoliubov approach, ensuring a full self-consistent treatment. Superfluidity is found to favour the compressibitily of nuclei. Pairing correlations explain why doubly magic nuclei such as {sup 208}Pb are stiffer compared to open-shell nuclei. Fully self-consistent predictions of the GMR on an isotopic chain should be the way to microscopically extract both the incompressibility and the density dependence of a given energy functional. The macroscopic extraction of K{sub sym}, the asymmetry incompressibility, is questioned. Investigations of the GMR in unstable nuclei are called for. Pairing gap dependence of the nuclear matter incompressibility should also be investigated.

Refractive indices for molecular crystals from the response of X-ray constrained Hartree–Fock wavefunctions

Refractive indices for molecular crystals are obtained from Hartree-Fock wavefunctions constrained to reproduce a set of experimental X-ray structure factors. Coupled-perturbed Hartree-Fock theory is used to calculate the in-crystal effective polarizabilities from which the refractive indices are obtained, thus eliminating the need for the calibration procedure used in earlier work by Whitten et al. [J. Chem. Phys. 2006, 125, 174505]. The results clearly demonstrate that these X-ray constrained Hartree-Fock (XCHF) wavefunctions reflect genuine effects of intermolecular interactions in crystals. Molecular dipole moments are consistently in excellent agreement with ab initio MP2 estimates that incorporate the effects of the crystal field. Consistent agreement of the XCHF refractive indices with experimental measurements at optical frequencies confirms that this approach can provide both meaningful results and considerable insight into the relative importance of molecular properties and crystal field effects in determining the detailed nature of the refractivity tensor.

OPTIMIZED DESCRIPTION OF NUCLEAR SHAPES AND SYMMETRIES

We propose a group-theory-based method of analysis of the multipole type (αλμ) deformation spaces for the large scale calculations within nuclear mean-field applications. It allows to find ahead of time which sub-sets of the deformation space, although formally different, in fact represent the same information (the same nuclear shapes expressed by an alternative combination of the deformation parameters). The approach presented allows to save important amounts of the computing time in the large-scale mean-field calculations, both in constrained Hartree-Fock (where αλμ is replaced by the constraint equation [Formula: see text]), and in the Strutinsky type approaches.

Applicability of self-consistent mean-field theory

Within the constrained Hartree-Fock (CHF) theory, an analytic condition is derived to estimate whether a concept of the self-consistent mean field is realized in the level repulsive region. The derived condition states that an iterative calculation of the CHF equation does not converge when the quantum fluctuations coming from two-body residual interaction and quadrupole deformation become larger than a single-particle energy difference between two avoided crossing orbits. By means of numerical calculation, it is shown that the analytic condition works well for a realistic case.

Shape transition and oblate-prolate coexistence inN=Zfpg-shell nuclei

Nuclear shape transition and oblate-prolate coexistence in $N=Z$ nuclei are investigated within the configuration space ($2p_{3/2}$, $1f_{5/2}$, $2p_{1/2}$, and $1g_{9/2}$). We perform shell model calculations for $^{60}$Zn, $^{64}$Ge, and $^{68}$Se and constrained Hartree-Fock (CHF) calculations for $^{60}$Zn, $^{64}$Ge, $^{68}$Se, and $^{72}$Kr, employing an effective pairing plus quadrupole residual interaction with monopole interactions. The shell model calculations reproduce well the experimental energy levels of these nuclei. From the analysis of potential energy surface in the CHF calculations, we found shape transition from prolate to oblate deformation in these $N=Z$ nuclei and oblate-prolate coexistence at $^{68}$Se. The ground state of $^{68}$Se has oblate shape, while the shape of $^{60}$Zn and $^{64}$Ge are prolate. It is shown that the isovector matrix elements between $f_{5/2}$ and $p_{1/2}$ orbits cause the oblate deformation for $^{68}$Se, and four-particle four-hole ($4p-4h$) excitations are important for the oblate configuration.

Fission barriers of heavy nuclei within a microscopic approach

The fission barriers of twenty-six isotopes of Thorium, Uranium, Plutonium, Californium, Fermium and Nobelium have been microscopically calculated up to and beyond the second saddle point within a constrained Hartree-Fock plus pairing approach. The Skyrme density-dependent effective force in its SkM* parametrization --rather well suited to the description of fission barriers-- has been used in the particle-hole channel, whereas the usual HF plus BCS formalism with either a seniority force or a delta force has been implemented to treat pairing correlations. The energy correction due to the rotational zero-point motion has been approximately taken into account and the effects of triaxial and reflection asymmetric deformations have been investigated. When known, the experimental fission barrier heights are reproduced within about 1-2MeV.

Characteristic feature of self-consistent mean-field in level crossing region

A shape change of the self-consistent mean-field induced by a configuration change is discussed within the conventional constrained Hartree–Fock (CHF) theory. It is stressed that a single-particle level crossing dynamics should be treated carefully, because the shape of the mean-field in such a finite many-body system as the nucleus strongly changes depending on its configuration. This situation is clearly shown by applying an adiabatic assumption, where the most energetically favorable single-particle states are assumed to be occupied. The excited HF states and the continuously-connected potential energy curves are given by applying the configuration dictated CHF method. The effect of pairing correlation is discussed in the level crossing region. Triaxial deformed results in our Hartree–Fock–Bogoliubov (HFB) calculation with Gogny force nicely reproduce the available experimental data of Ge isotopes. From our numerical calculation, it is concluded that the CHFB state is more fragile than the CHF state in the level crossing region.

Characteristic feature of self-consistent mean-field in level crossing region

A shape change of the self-consistent mean-field induced by a configuration change is discussed within the conventional constrained Hartree–Fock (CHF) theory. It is stressed that a single-particle level crossing dynamics should be treated carefully, because the shape of the mean-field in such a finite many-body system as the nucleus strongly changes depending on its configuration. This situation is clearly shown by applying an adiabatic assumption, where the most energetically favorable single-particle states are assumed to be occupied. The excited HF states and the continuously-connected potential energy curves are given by applying the configuration dictated CHF method. The effect of pairing correlation is discussed in the level crossing region. Triaxial deformed results in our Hartree–Fock–Bogoliubov (HFB) calculation with Gogny force nicely reproduce the available experimental data of Ge isotopes. From our numerical calculation, it is concluded that the CHFB state is more fragile than the CHF state in the level crossing region.

Electron localization functions obtained from X-ray constrained Hartree-Fock wavefunctions for molecular crystals of ammonia, urea and alloxan.

Constrained Hartree-Fock wavefunctions in the superposition of isolated molecules model have been calculated from precise X-ray diffraction data on crystals of ammonia (NH(3)), urea [CO(NH(2))(2)] and alloxan [(CO)(4)(NH)(2)]. The X-ray constrained wavefunctions have been used to derive and examine 'electron localization' information, quantified by the true and approximate electron localization function (ELF), and the true and approximate Fermi hole mobility function (FHMF). The plots of the Fermi hole mobility function are the first to appear in the literature. The results are compared with corresponding isolated-molecule Hartree-Fock calculations to gauge the effect of the crystal environment on the isolated molecules. An error analysis is performed to indicate the features in the plots which are well determined from the experimental data. The results from all plots are broadly consistent, but the approximate ELF shows some artifacts relative to the true ELF.