Hartree-Fock Model Research Articles

Convergence acceleration for the multilevel Hartree–Fock model

In the previously introduced multilevel Hartree–Fock (HF) model, the electronic density is optimised in a given region of the molecular system. The approach is based on generating an active occupied and active virtual space by decomposing a start guess density for the entire system. In this work, a diagonalisation based implementation for Roothaan–Hall (RH) with direct inversion in iterative subspace (DIIS) and a quasi-Newton minimisation procedure using the augmented RH (ARH) approach are described for accelerating convergence for the multilevel HF model. The equations are derived to be consistent with convergence acceleration for traditional atomic orbital based HF calculations. The main idea is to formulate all quantities in the molecular orbital basis to exploit that the active molecular orbital basis is significantly smaller than the atomic orbital basis, and thus enable the application of wave function approaches that are well-studied for small molecular systems to large molecular systems. Thus, all equations are formulated such that no atomic orbital density or Fock matrices are needed for the DIIS and ARH algorithms. Results show that the acceleration schemes yield efficient optimisation of the multilevel HF wave function.

Cooling of hypernuclear compact stars: Hartree–Fock models and high-density pairing

ABSTRACT The thermal evolution of hypernuclear compact stars is studied for stellar models constructed on the basis of covariant density functional theory in Hartree and Hartree–Fock approximation. Parametrizations of both types are consistent with the astrophysical mass constraints on compact stars and available hypernuclear data. We discuss the differences of these density functionals and highlight the effects they have on the composition and on the cooling of hypernuclear stars. It is shown that hypernuclear stars computed with density functional models that have a low symmetry energy slope, L, are fairly consistent with the cooling data of observed compact stars. The class of stellar models based on larger L values gives rise to the direct Urca process at low densities, which leads to significantly faster cooling. We conjecture high-density pairing for protons and Λ’s in the P-wave channel and provide simple scaling arguments to obtain these gaps. As a consequence the most massive stellar models with masses 1.8 ≤ M/M⊙ ≤ 2 experience slower cooling by hyperonic dUrca processes which involve Λ’s and protons.

Impact parameter dependence of the yield ratios of light particles as a probe of neutron skin

The yield ratios of neutron/proton (R(n/p)) and $${^3}\hbox {H}/{^3}{\text{He}}$$ ( $$R(t/{^3}{\mathrm{He}})$$ ) with reduced rapidity from 0 to 0.5 are investigated for 50 MeV/u $${^{42,44,46,48,50,52,54,56}}{\text{Ca}} + {^{40}}{\text{Ca}}$$ . This was conducted at whole reduced impact parameters using the isospin-dependent quantum-molecular-dynamics model in which the initial neutron and proton densities are sampled within the Skyrme–Hartree–Fock model, using which the neutron skin thickness ( $$\Delta {R_{\rm{np}}}$$ ) is determined for different neutron-rich Ca isotopes. The results show that both R(n/p) and $$R(t/{^3}{\mathrm{He}})$$ have strong linear correlations with $$\Delta {R_{\rm{np}}}$$ of different Ca isotopic projectiles from five different centralities. It is suggested that R(n/p) and $$R(t/{^3}{\mathrm{He}})$$ , from the same centrality, could be treated as possible experimental observables to extract the neutron skin or halo thickness for neutron-rich isotopic nuclei, including the nuclei near the neutron drip line.

Efficient HF exchange evaluation through Fourier convolution in Cartesian grid for orbital-dependent density functionals.

We present a purely numerical approach in a Cartesian grid, for efficient computation of the Hartree-Fock (HF) exchange contribution in the HF and density functional theory models. This takes inspiration from a recently developed algorithm by Liu et al., in 2017, where the rate-determining step is the accurate evaluation of electrostatic potential. This introduces the Fourier convolution theorem in conjunction with a range-separated Coulomb interaction kernel. The latter is efficiently mapped into a real grid through a simple optimization procedure, giving rise to a constraint in the range-separated parameter. The overall process offers logarithmic scaling with respect to the molecular size. It is then extended toward global hybrid functionals such as B3LYP, PBE0, and BHLYP within pseudopotential Kohn-Sham theory, through an LCAO-MO ansatz in a Cartesian grid, developed earlier in our laboratory. For the sake of comparison, a parallel semi-numerical approach has also been worked out that exploits the familiar Obara-Saika recursion algorithm without any additional techniques. An excellent agreement between these two routes is demonstrated through total energy and orbital energy in a series of atoms and molecules (including 10 π-electron molecules), employing an LANL2DZ-type basis function. A critical analysis of these two algorithms reveals that the proposed numerical scheme could lead to very attractive and competitive scaling. The success of our approach also enables us for further development of optimally tuned range-separated hybrid and hyper functionals.

Investigation of density and form factor of some F isotopes using Hartree-Fock and shell model calculations

Structure of unstable 21,23,25,26F nuclei have been investigatedusing Hartree – Fock (HF) and shell model calculations. The groundstate proton, neutron and matter density distributions, root meansquare (rms) radii and neutron skin thickness of these isotopes arestudied. Shell model calculations are performed using SDBAinteraction. In HF method the selected effective nuclear interactions,namely the Skyrme parameterizations SLy4, Skeσ, SkBsk9 andSkxs25 are used. Also, the elastic electron scattering form factors ofthese isotopes are studied. The calculated form factors in HFcalculations show many diffraction minima in contrary to shellmodel, which predicts less diffraction minima. The long tailbehaviour in nuclear density is noticeable seen in HF more than shellmodel calculations. The deviation occurs between shell model andHF results are attributed to the sensitivity of charge form factors tothe change of the tail part of the charge density. Calculations donefor the rms radii in shell model showed excellent agreement withexperimental values, while HF results showed an overestimation inthe calculated rms radii for 21,23F and good agreement for 25,26F. Ingeneral, it is found that the shell model and HF results have the samebehaviour when the mass number (A) increase.

Evolution of the Position of Single-particle Levels in Neutron-rich Cerium and Neodymium Isotopes

Different Skyrme functional parametrizations were tested for Ce and Nd isotopes for N > 82. The method employed in the study is the axial Skyrme–Hartree–Fock model+BCS. For selected parametrizations, ground-state deformations β2,β3 and β4 have been found and the evolution of single-neutron and single-proton states around the Fermi level along the chain of Ce and Nd isotopes has been investigated. These results were then compared to the existing experimental data.

Spin Symmetry Breaking in the Translation-Invariant Hartree--Fock Electron Gas

We study the breaking of spin symmetry for the nonlinear Hartree-Fock model describing an infinite translation-invariant interacting quantum gas (fluid phase). At zero temperature and for the Coulomb interaction in three space dimensions, we can prove the existence of a unique first order transition between a pure ferromagnetic phase at low density and a paramagnetic phase at high density. Multiple first or second order transitions can happen for other interaction potentials, as we illustrate on some examples. At positive temperature T>0 we compute numerically the phase diagram in the Coulomb case. We find the paramagnetic phase at high temperature or high density and a region where the system is ferromagnetic. We prove that the equilibrium measure is unique and paramagnetic at high temperature or high density.

Differential and Integral Cross Sections of Electron Elastic Scattering by PH3 Molecule in the Energy Ranging from 10 eV up to 20 keV

Differential and integral cross sections for electron elastic scattering by phosphine molecule at the incident energies ranging from 10 eV up to 20 keV are calculated using the well known partial wave expansion formalism. The considered interaction potential is essentially an optical spherical one including a static contribution, exchange and correlation-polarization ones. The first one is numerically calculated considering a single-center Hartree-Fock model, while two expressions of the well-known exchange potential and two others for the correlation-polarization potential are rigorously selected from the literature. The differential and integral cross sections obtained for the four potential combinations investigated in the present work, exhibit a general agreement and show clearly the role played by the exchange and correlation-polarization effects, particularly at lower scattering angles and lower impact energies. In addition, the differential and integral cross sections are compared to the experimental and theoretical results available in the literature and a good agreement is found.

Radiative lifetime measurements and semi-empirical transition parameter calculations for high-lying levels in Ba I

Radiative lifetimes for 18 highly excited levels belonging to 5d4f, 5d6p, and 6snp (n = 10, 12, 14–19) configurations of barium were measured by time-resolved laser-induced fluorescence (TR-LIF) technique. The uncertainties of the results are less than 10%. To our best knowledge, there are 16 lifetimes of Ba I reported for the first time. Two lifetimes were measured to compare with the previous results, and good agreements were achieved. From the combination of these experimental lifetimes and theoretical branching fractions calculated using a pseudo-relativistic Hartree–Fock model including core-polarization effects, a set of semi-empirical transition probabilities and oscillator strengths was derived for 46 Ba I spectral lines in the wavelength region from 240 to 3765 nm. For about half of these transitions, the uncertainties affecting the new decay parameters were found to be equal or better than 50%.

Mean-field approach to reconstructed neutrino energy distributions in accelerator-based experiments

The reconstruction of the neutrino energy is crucial in oscillation experiments that use interactions with nuclei to detect the neutrino. The common reconstruction procedure is based on the kinematics of the final-state lepton. The interpretation of the reconstructed energy in terms of the real neutrino energy must rely on a model for the neutrino-nucleus interaction. The Relativistic Fermi Gas (RFG) model is frequently used in these analyses. In the Hartree-Fock (HF) model for quasielastic nucleon knockout, the bound nucleon wave functions are obtained using an effective nucleon-nucleon force. The final-state wave function is constructed from continuum states in the same potential which have the correct asymptotic behavior. The Continuum Random Phase Approximation (CRPA) model extends the HF approach taking long range correlations into account in a self-consistent way. Considering only single-nucleon processes, the distributions of reconstructed neutrino energies obtained within the HF-CRPA approach are compared with the results of the RFG, an RPWIA calculation, and the RPA+np-nh model of Martini et al. We find that the distributions of reconstructed energies for a fixed incoming energy in the HF-CRPA display additional strength in the low reconstructed energy tails compared to models without elastic distortion of the outgoing nucleon. This asymmetry redistributes strength from higher to lower values of the reconstructed energy. The mean field description of the nuclear dynamics results in a reshaping of the reconstructed energy distribution that cannot be accounted for in a plane wave impulse approximation model, even by modifying ad hoc parameters such as the binding energy. In particular it is shown that in the RFG calculations there is no value of the binding energy which is able to reproduce the entire T2K $\nu_\mu$ oscillated spectrum as calculated in HF-CRPA.

Experimental and Theoretical Radiative Lifetimes, Branching Fractions, Transition Probabilities, and Oscillator Strengths of Some Highly Excited Odd-parity Levels in Ir i

Abstract Radiative lifetimes of 62 odd-parity levels of Ir i in the energy range between 32513.43 and 58625.10 cm−1 were measured using the time-resolved laser-induced fluorescence technique. The lifetime values obtained are in the range from 3.2 to 345 ns. To our best knowledge, 59 results are reported for the first time. These are compared to computed data deduced from a pseudo-relativistic Hartree–Fock model including core-polarization contributions. From the combination of the experimental lifetime measurements and branching fraction calculations, a new set of transition probabilities and oscillator strengths is derived for 134 Ir i spectral lines of astrophysical interest in the wavelength region from 205 to 418 nm.

Occupied-Orbital Fast Multipole Method for Efficient Exact Exchange Evaluation.

We present an efficient algorithm for computing the exact exchange contributions in the Hartree-Fock and hybrid density functional theory models on the basis of the fast multipole method (FMM). Our algorithm is based on the observation that FMM with hierarchical boxes can be efficiently used in the exchange matrix construction, when at least one of the indices of the exchange matrix is constrained to be an occupied orbital. Timing benchmarks are presented for alkane chains (C400H802 and C150H302), a graphene sheet (C150H30), a water cluster [(H2O)100], and a protein Crambin (C202H317O64N55S6). The computational cost of the far-field exchange evaluation for Crambin is roughly 3% that of a self-consistent field iteration when the multipoles up to rank 2 are used.

Density-Based Multilevel Hartree-Fock Model.

We introduce a density-based multilevel Hartree-Fock (HF) method where the electronic density is optimized in a given region of the molecule (the active region). Active molecular orbitals (MOs) are generated by a decomposition of a starting guess atomic orbital (AO) density, whereas the inactive MOs (which constitute the remainder of the density) are never generated or referenced. The MO formulation allows for a significant dimension reduction by transforming from the AO basis to the active MO basis. All interactions between the inactive and active regions of the molecule are retained, and an exponential parametrization of orbital rotations ensures that the active and inactive density matrices separately, and in sum, satisfy the symmetry, trace, and idempotency requirements. Thus, the orbital spaces stay orthogonal, and furthermore, the total density matrix represents a single Slater determinant. In each iteration, the (level-shifted) Newton equations in the active MO basis are solved to obtain the orbital transformation matrix. The approach is equivalent to variationally optimizing only a subset of the MOs of the total system. In this orbital space partitioning, no bonds are broken and no a priori orbital assignments are carried out. In the limit of including all orbitals in the active space, we obtain an MO density-based formulation of full HF.

Lifetime measurements using two-step laser excitation for high-lying even-parity levels and improved theoretical oscillator strengths in Y ii

We report new time-resolved laser-induced fluorescence lifetime measurements for 22 highly excited even-parity levels in singly ionized yttrium (Y II). To populate these levels belonging to the configurations 4d6s, 5s6s 4d5d, 5p2, 4d7s and 4d6d, a two-step laser excitation technique was used. Our previous pseudo-relativistic Hartree-Fock model (Biemont et al. 2011) was improved by extending the configuration interaction up to n = 10 to reproduce the new experimental lifetimes. A set of semi-empirical oscillator strengths extended to transitions falling in the spectral range λλ194-3995 nm, depopulating these 22 even-parity levels in Y II, is presented and compared to the values found in the Kurucz's data base (Kurucz 2011). (Less)

Second spectrum of Chromium (Cr II), Part II: Radiative lifetimes and oscillator strengths of transitions depopulating low lying 3d[formula omitted]4p levels

Radiative lifetime and oscillator strength values were computed by means of a pseudo-relativistic Hartree–Fock model including core-polarization and compared successfully with experimental data given in the literature for respectively numerous low-lying Cr II 3d44p levels, up to 54784 cm−1, and transitions depopulating these levels. Then transition probability and branching fraction data were also deduced in the wavelength range 1825–94400 Å. Furthermore the extracted radial integral values, obtained with the help of the oscillator strength parameterization method, are given for involved transitions in this work, i.e. 3d44s–3d44p, 3d5–3d44p and 3d44p–3d44d. We confirm our observed previous general trends, noting once again a decreasing of transition radial integral values with filling nd shells of the same principal quantum numbers for ndk(n+1)s →ndk(n+1)p transitions.

Quantum Two-Stream Instability with Exchange Interaction

Quantum plasmas can be described in the framework of the self-consistent field approach by a set of time-dependent Schrödinger equations (TDSE) for the electrons in a self-consistent potential. This approach is also well-known as the multistream model of quantum plasmas. The self-consistent field is commonly calculated in the Vlasov approximation from the average charge density of all electrons thereby neglecting the quantum-mechanical exchange interaction. In the present work, we consider a linear Hartree-Fock model with screened Coulomb interactions to examine the role of the exchange interaction. The two-stream instability is analysed and results for the Vlasov, Hartree and Hartree-Fock approximations are compared in detail. It is found that the exchange interaction considerably modifies the stability behavior for weakly screened Coulomb interactions. In particular, crossings of the two branches of the dispersion relation lead to oscillating unstable solutions that are not present in the absence of the exchange interaction.

Correlation effects on the interelectronic distributions of localized electron pairs

It has been shown previously that a wealth of chemical insight may be drawn from studying the interelectronic distribution function for localized electron pairs within a molecular system (Hennessey et al. in Phys Chem Chem Phys 16(46):25548–25556, 2014); however, these accounts have been limited due to the relative inaccuracy of the underlying Hartree–Fock electronic structure model that has been employed. In the current work, we study how the interelectronic distribution function of single electron pairs, represented by localized molecular orbitals, changes between a Hartree–Fock and density functional electronic structure model. We find that localized electron pairs expand, relative to a Hartree–Fock model, in a small but significant way when modelled with density functional theory. More specifically, we generally find that compact electron pairs (i.e. those pairs having narrow distribution functions near small interelectronic distances) result in a smaller change due to correlation than electron pairs more broadly distributed. This counterintuitive effect is attributed to the fact that compact electron pairs are generally held close together by a relatively rigid confining potential (i.e. attraction from local nuclei and repulsion from neighbouring electrons). However, in cases where a series of species all have nearly identical nuclear structure in the vicinity of the LMO of interest, more compact electron pairs did in fact experience the greatest change in interelectronic separation and electron repulsion upon the incorporation of a correlated model. Also, if one allows the geometry of the molecular species to relax, bonds associated with LMOs having compact interelectronic distributions tend to elongate the most with a correlated model (relative to Hartree–Fock) and the resultant deformation of the interelectronic distribution functions then show the most significant changes due to correlation. The results presented herein demonstrate the utility of Kohn–Sham density functional theory for the description of localized chemical space within the scope of the localized pair model.

Supercell Calculations in the Reduced Hartree–Fock Model for Crystals with Local Defects

In this article, we study the speed of convergence of the supercell reduced Hartree-Fock~(rHF) model towards the whole space rHF model in the case where the crystal contains a local defect. We prove that, when the defect is charged, the defect energy in a supercell model converges to the full rHF defect energy with speed $L^{-1}$, where $L^3$ is the volume of the supercell. The convergence constant is identified as the Makov-Payne correction term when the crystal is isotropic cubic. The result is extended to the non-isotropic case.

Nuclear Zemach moments and finite-size corrections to allowedβdecay

The finite-size correction to $\beta$-decay plays an important role in determining the expected antineutrino spectra from reactors at a level that is important for the reactor-neutrino anomaly. Here we express the leading-order finite-size correction to allowed $\beta$-decay in terms of Zemach moments. We calculate the Zemach moments within a Hartree-Fock model using a Skyrme-like energy density functional. We find that the Zemach moments are increased relative to predictions based on the simple assumption of identical uniform nuclear-charge and weak-transition densities. However, for allowed ground-state to ground-state transitions in medium and heavy nuclei, the detailed nuclear structure calculations do not change the finite-size corrections significantly from the simple model predictions, and are only 10-15% larger than the latter even though the densities differ significantly.

Exotic structure of medium-heavy hypernuclei in the Skyrme Hartree-Fock model

We study exotic structure of medium heavy hypernuclei with core nuclei of Ar isotopes and $^{40}\mathrm{Ca}$. Particulary, the superdeformed (SD) states and the corresponding hypernuclei are calculated in the frame of the Skyrme Hartree-Fock (SHF) + BCS model together with a microscopic $\mathrm{\ensuremath{\Lambda}}N$ interaction. We found that the $\mathrm{\ensuremath{\Lambda}}$ separation energy of the ground state is larger than that of the SD state. The result is consistent with the antisymmetrized molecular dynamics (AMD) calculation, but inconsistent with that of relativistic mean field (RMF). The similarity between SHF and AMD and the difference of $\mathrm{\ensuremath{\Lambda}}$ separation energy between SHF and RMF both come from the density distribution in the core nuclei and the corresponding hypernuclei.