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Fractional-spin errors are inherent in all current approximate density functionals, including Hartree-Fock theory, and their origin has been related to strong static correlation effects. The conventional way to encode fractional-spin calculations is to construct an ensemble density that scales between the high-spin and low-spin densities. In this article, we explore the variation of the Hartree-Fock fractional-spin (or ghost-interaction) error in one-electron systems using restricted and unrestricted ensemble densities, and the exact generalized Hartree-Fock representation. By considering the hydrogen atom and H$_2^+$ cation, we analyze how the unrestricted and generalized Hartree-Fock schemes minimize this error by localizing the electrons or rotating the spin coordinates. We also reveal a clear similarity between the Coulomb hole of He-like ions and the density depletion near the nucleus induced by the fractional-spin error in the unpolarized hydrogen atom. Finally, we analyze the effect of the fractional-spin error on the M{\o}ller-Plesset adiabatic connection, excited states, and functional- and density-driven errors., Comment: 12 pages, 9 figures

In the present work, the bulk nuclear properties and features of some of the excited states in the neutron-rich even–even 46−54Ca isotopes have been investigated. In order to identify the magicity and the existence of new magic numbers in these isotopes, the shell model and Hartree–Fock–Bogoliubov method were implemented based on Skyrme parameterizations. In particular, root-mean-square charge radius, binding energies, one- and two-neutron separation energies, pairing gaps, reduced transition probabilities, excitation energies, energy levels, and quadrupole deformation parameters have been investigated. The calculated results were compared with available experimental data.

This paper is intended to serve as a bridge between introductory textbooks on quantum mechanics, which typically do not cover the Hartree-Fock self-consistent-field method, and more advanced ones which treat this important computational method for fermionic many-body systems in an abstract and formal way. We derive the Hartree-Fock equation for the 1s orbital of a realistic two-electron atom. By employing a two-dimensional basis-set representation, we avoid the use of variational calculus and are able to visualize key aspects of the method. We explain the basic self-consistent-field algorithm and provide a python script to illustrate how the algorithm works in practice. Utilizing perturbation theory, we perform an analysis of the convergence behavior of the self-consistent-field algorithm, thereby facilitating a deeper understanding of the numerical examples presented. We expect that this work will be useful for teaching computational techniques to physics students.

Predictions on the alpha decay and spontaneous fission of the superheavy nuclei in the range of Z = 106–118 have been systematically done within the Hartree-Fock-Bogoliubov model and different Skyrme type interactions. A comparison of the results of the current study with experimental values showed good agreement. This study was extended to unknown isotopes in the range of Z = 116–118. The calculated alpha decay half-lives are in the range of 10 − 5 to 102 s which lie within measurable values. To identify the long-lived isotopes in the mass region, spontaneous fission half-lives were performed using a semi-empirical formula. The results revealed that the isotopes 276,275,274,272,268110, 276,275,273111, 282,280,279112, 281,280,279113, 284,283114, 286115, 289116, 296,295,292,291117, 297,296118, and 295,293,292118 survived fission and had alpha decay mode as the predominant mode of decay which could be synthesized in the laboratory.

We extend our linear-scaling approach for the calculation of Hartree–Fock exchange energy using localized in situ optimized orbitals [Dziedzic et al., J. Chem. Phys. 139, 214103 (2013)] to leverage massive parallelism. Our approach has been implemented in the onetep (Order-N Electronic Total Energy Package) density functional theory framework, which employs a basis of non-orthogonal generalized Wannier functions (NGWFs) to achieve linear scaling with system size while retaining controllable near-complete-basis-set accuracy. For the calculation of Hartree–Fock exchange, we use a resolution-of-identity approach, where an auxiliary basis set of truncated spherical waves is used to fit products of NGWFs. The fact that the electrostatic potential of spherical waves (SWs) is known analytically, combined with the use of a distance-based cutoff for exchange interactions, leads to a calculation cost that scales linearly with the system size. Our new implementation, which we describe in detail, combines distributed memory parallelism (using the message passing interface) with shared memory parallelism (OpenMP threads) to efficiently utilize numbers of central processing unit cores comparable to, or exceeding, the number of atoms in the system. We show how the use of multiple time-memory trade-offs substantially increases performance, enabling our approach to achieve superlinear strong parallel scaling in many cases and excellent, although sublinear, parallel scaling otherwise. We demonstrate that in scenarios with low available memory, which preclude or limit the use of time-memory trade-offs, the performance degradation of our algorithm is graceful. We show that, crucially, linear scaling with system size is maintained in all cases. We demonstrate the practicability of our approach by performing a set of fully converged production calculations with a hybrid functional on large imogolite nanotubes up to over 1400 atoms. We finish with a brief study of how the employed approximations (exchange cutoff and the quality of the SW basis) affect the calculation walltime and the accuracy of the obtained results.

Within the KaKB, KaLa, and KBLa shells in the position space, the properties of a series of three-electron systems, for instance, B+2, C+3, and N+4 ions, have been studied. This required the partitioning of the two-particle space-spin density and was explicit for the Hartree–Fock description which have been proposed by considering a basis set based on single-zeta B-type orbitals (BTOs). The one- and two-body radial electronic densities R(r1), R(r1, r2), moments ⟨rn1⟩, X-ray form factor F(s), nucleus density R(0), nuclear magnetic shielding constant qd, and the diamagnetic susceptibility бs in the position space are reported. Our results are realized via the Mathematica program and compared with previous theoretical values in the literature., Вивчено властивостi низки триелектронних систем в межах просторових оболонок KaKB, KaLa i KBLa, напри-клад, B+2, C+3 i N+4 йонiв. Це вимагало проведення розбиття двочастинкових функцiй просторової i спiнової густин, що можливо в рамках моделi Хартрi–Фока з базисами, побудованими на 1z орбiталях B-типу. Знайдено одно- i дво-частинковi радiальнi електроннi розподiли густини R(r1) I R(r1, r2), моменти ⟨rn1⟩, формфактори рентгенiвського випромiнювання F(s), густину ядра R(0), константу ядерного магнiтного екранування qd i дiамагнiтну сприйнятливiсть бs. Результати отримано з використанням програми Mathematica i порiвняно з вiдомими теоретичними даними.

The frequently-used Z k and Y k functions in the framework of numerical Hartree–Fock method are formulated for atomic systems embedded in weakly-coupled plasmas modeled by Debye–Huckel potential. The Gegenbauer expansion is used to decompose the screened interelectronic Coulomb potential . Two numerical methods , the integration and differentiation ones, are developed to accurately and efficiently calculate the Z k and Y k integrals with each one showing special advantages in the implementation of numerical Hartree–Fock programs. Comparing to the analytical results specialized for Slater-type radial functions, it is generally concluded that the integration method possesses more accuracy and robustness than the differentiation one. Calculations on the two-electron C 4 + ion in three different screening models are performed to demonstrate the applicability and stability of present methods in a wider range of screening parameters than existing Hartree–Fock calculations. With benchmark Hartree–Fock energies obtained in this work and accurate non-relativistic energies calculated based on Hylleraas-CI basis functions, it is found that the correlation energies are smallest in the fully screening model.

The multi-configuration Dirac–Hartree–Fock (MCDHF) method and the active space approach have been employed to investigate the energy levels, wavelengths, transition probabilities, and line strengths of Si-like Se ions. The contributions of Breit interaction (BI) and quantum electrodynamic (QED) correction are taken into consideration. The wavelengths, transition probabilities, and line strengths of Si-like Se ions have also been calculated. Compared with other theoretical and experimental values of these parameters, our values are in good agreement with others, and the number of energy levels we considered is larger than that of any other theoretical calculations.

Several approaches are presented to improve the efficiency of Hartree–Fock and Kohn–Sham self-consistent field (SCF) calculations relying on a simple first-order energy correction reminiscent of the scheme used in dual-basis SCF methods. The basic idea is to perform an initial SCF calculation computing approximate Fock-matrices and, in the final iteration step, to use a more complete Fock-matrix builder together with the energy correction to diminish the error. The approximation is tested for conventional and local density fitting (DF) SCF approaches combining various auxiliary basis sets, fitting metrics, and Fock-matrix construction algorithms in the initial and final iterations as well as for seminumerical SCF methods combining integration grids of different qualities. We also report the implementation of the occupied orbital resolution of identity exchange construction algorithm with local DF approximations. Benchmark calculations are presented for total energies, reaction energies, and molecular geometries. Our results show that speedups of up to 80% can be expected utilizing the new approaches without significant loss of accuracy.

Energy levels, weighted oscillator strengths and transition probabilities, lifetimes, hyperfine interaction constants, Lande gJ factors and isotope shifts have been calculated for all levels of 1 s 2 and 1 snl ( n = 2 - 8 , l ⩽ 7 ) configurations of He-like oxygen ion (O VII). The calculations were performed using the Multiconfigurational Dirac-Hartree–Fock (MCDHF) and the Relativistic Configuration Interaction (RCI) methods. A second set of calculations was obtained with the Many Body Perturbation Theory (MBPT) method. Transition probabilities are reported for all E 1 , E 2 , M 1 and M 2 transitions. Breit interactions and quantum electrodynamics effects were included in the RCI calculations. Comparisons have been made with the compiled data from the NIST ASD and other calculations and a good agreement was found which confirms the reliability of our results. The accuracy of the present calculations is high enough to facilitate identification of many observed spectral lines. For the first time, we present some missing data for the He-like oxygen ion.

Electron correlation effects play a key role in stabilising two-electron atoms near the critical nuclear charge, representing the smallest charge required to bind two electrons. However, deciphering the importance of these effects relies on fully understanding the uncorrelated Hartree-Fock description. Here, we investigate the properties of the ground state wave function in the small nuclear charge limit using various symmetry-restricted Hartree-Fock formalisms. We identify the nuclear charge where spin-symmetry breaking occurs to give an unrestricted wave function that predicts the ionisation of an electron. We also discover another critical nuclear charge where the closed-shell electron density detaches from the nucleus, and identify the importance of fractional spin errors and static correlation in this limit., 7 pages, 4 figures

We investigate the coupling of different quantum-embedding approaches with a third molecular-mechanics layer, which can be either polarizable or non-polarizable. In particular, such a coupling is discussed for the multilevel families of methods, in which the system is divided into an active and an inactive orbital space. The computational cost of the resulting three-layer approaches is reduced by treating the long-range interactions at the classical level. The developed methods are tested by the calculation of excitation energies of molecular systems in aqueous solution, for which an atomistic description of the environment is crucial to correctly reproduce the specific solute-solvent interactions, such as hydrogen bonding. In particular, we present the results obtained for three different moieties - acrolein, pyridine and para-nitroaniline - showing that an almost perfect agreement with experimental data can be achieved when the relevant physico-chemical interactions are included in the modeling of the condensed phase.

We explore dynamics of disordered and quasi-periodic interacting lattice models using a self-consistent time-dependent Hartree-Fock (TDHF) approximation, accessing both large systems (up to $L = 400$ sites) and very long times (up to $t = 10^5$). We find that, in the $t \to \infty$ limit, the many-body localization (MBL) is always destroyed within the TDHF approximation. At the same time, this approximation provides important information on the long-time character of dynamics in the ergodic side of the MBL transition. Specifically, for one-dimensional (1D) disordered chains, we find slow power-law transport up to the longest times, supporting the rare-region (Griffiths) picture. The information on this subdiffusive dynamics is obtained by the analysis of three different observables - temporal decay $\sim t^{-\beta}$ of real-space and energy-space imbalances as well as domain wall melting - which all yield consistent results. For two-dimensional (2D) systems, the decay is faster than a power law, in consistency with theoretical predictions that $\beta$ grows as $\log t$ for the decay governed by rare regions. At longest times and moderately strong disorder, $\beta$ approaches the limiting value $\beta=1$ corresponding to 2D diffusion. In quasi-periodic (Aubry-Andr\'e) 1D systems, where rare regions are absent, we find considerably faster decay that reaches the ballistic value $\beta=1$, which provides further support to the Griffiths picture of the slow transport in random systems., Comment: 35 pages including appendix, 14 figures. Comments welcome

The electronic properties of azobenzene (AB) in interaction with gold clusters and adsorbed on the Au(111) surface are investigated by adopting a near-Hartree–Fock–Kohn–Sham (HFKS) scheme. This scheme relies on a hybrid Perdew–Burke–Ernzerhof functional, in which the exact non-local HF exchange contribution to the energy is taken as 3/4. Ionization energies and electron affinities for gas phase AB are in very good agreement with experimental data and outer valence Green’s function) calculations. The presence of C—H⋯Au interactions in AB–Aun complexes illustrates the role played by weak interactions between molecular systems and Au nanoparticles, which is in line with recent works on Au–H bonding. In AB–Aun complexes, the frontier orbitals are mainly localized on the gold platform when n ≥ 10, which indicates the transition from a molecular to a semiconducting regime. In the latter regime, the electronic density reorganization in AB–Aun clusters is characterized by significant polarization effects on the Au platform. The accuracy of the near-HFKS scheme for predicting adsorption energies of AB on Au(111) and the interest of combining exact non-local HF exchange with a non-local representation of the dispersion energy are discussed. Taking into account the significant computational cost of the exact non-local HF exchange contribution, calculations for the adsorption energies and density of states for AB adsorbed on Au(111) were carried out by using a quantum mechanics/molecular mechanics approach. The results strongly support near-HFKS as a promising methodology for predicting the electronic properties of hybrid organic–metal systems.

The amines are major source of environment pollutants emitted in atmosphere from various anthropogenic sources. The non-thermal plasma (NTP)-based technology has proved successful in controlling the emitted amines reaching the atmosphere. The efficient NTP reactors rely on accurate electron–molecule collision data. The electron impact cross-sections are thus obtained for a few amines from ionisation threshold to 5000 eV using the single centre expansion (SCE) formalism. The molecular wave function of each target is obtained from the multicentre expansion of the Gaussian-type orbitals within a single determinant Hartree–Fock self-consistent field scheme. The expansion of wave function, density and potential is carried out at the centre of mass of the molecules. The interaction potential included to model the electron interaction in the target comprises static, correlation-polarisation and exchange types of potentials. The elastic cross-sections are obtained after solving the coupled scattering equations using Volterra integral form. The inelastic effects contributing to electron–molecule scattering are approximated by the ionisation cross-sections. The total cross-sections obtained after summing the elastic and ionisation cross-sections are in good agreement with the available data. We have also tried to explain the effect of polarisation potential on scattering cross-sections. A semiempirical formula based on the spatial extent of the molecule is proposed to estimate the cross-sections for the homologous series of amine molecules.

Description of static nuclear properties is one of the main aims of theoretical nuclear physics studies. This work presents the uncertainties due to pairing fit procedure for three static nuclear quantities of rare-earth nuclei, namely the nuclear charge radius, electric quadrupole moment, and the moment of inertia. The Hartree-Fock (HF)-plus-pairing approach was employed with pairing correlations treated within the Bardeen-Cooper-Schrieffer (BCS) framework. The Skyrme SIII parametrization were chosen to approximate the effective nucleon-nucleon interaction while seniority force is used for the pairing interaction. In this work, two sets of pairing strengths obtained from different fit procedures were chosen. Calculated result shows that the moment of inertia is hugely dependent on variation of pairing strengths.

In the present research, the nuclear deformation of the Ne, Mg, Si, S, Ar, and Kr even–even isotopes has been investigated within the framework of Hartree–Fock–Bogoliubov method and SLy4 Skyrme parameterization. In particular, the deform shapes of the effect of nucleons collective motion by coupling between the single-particle motion and the potential surface have been studied. Furthermore, binding energy, the single-particle nuclear density distributions, the corresponding nuclear radii, and quadrupole deformation parameter have been also calculated and compared with the available experimental data. From the outcome of our investigation, it is possible to conclude that the deforming effects cannot be neglected in a characterization of the structure of the neutron-rich nuclei. The relation between the single-particle motion and the potential surface leads to note that the change in the interactions between the nucleons causes the evolution of nuclear surface and leads to variation in the potential shape.

Background: In the previous paper, we predicted reaction cross sections σ R for 40 − 60 , 62 , 64 Ca+12C scattering at 280 MeV/nucleon, using the chiral g -matrix folding model with the densities calculated with the Gogny-D1S Hartree–Fock-Bogoliubov (GHFB) with and without the angular momentum projection (AMP), since Tanaka et al. measured interaction cross sections σ I ( ≈ σ R ) for 42 − 51 Ca in RIKEN and determined neutron skin r skin ( RIKEN ) using the optical limit of the Glauber model with the Woos-Saxon densities. Purpose: Our purpose is to reanalyze the r skin from the σ I using the chiral g -matrix folding model. Our analysis is superior to theirs, since the chiral g -matrix folding model (the GHFB and GHFB+AMP densities) is much better than the optical limit of the Glauber model (the Woos-Saxon densities). Methods: Our model is the chiral g -matrix folding model with the densities scaled from the GHFB and GHFB+AMP densities. Results: We scale the GHFB and GHFB+AMP densities so that the σ R of the scaled densities can agree with the central values of σ I under the condition that the proton radius of the scaled proton density equals the data determined from the isotope shift based on the electron scattering. The r skin thus determined are close to their results r skin 42 − 51 ( RIKEN ) , except for 48Ca. For 48Ca, our value r skin 48 is 0.105 ± 0.06 fm, while their value is r m 48 ( RIKEN ) = 0 . 146 ± 0 . 06 fm. We then take the weighted mean and its error of our result r skin 48 ( σ I ) = 0 . 105 ± 0 . 06 fm and the result r skin 48 ( E 1 pE ) = 0 . 17 ± 0 . 03 fm of the high-resolution E 1 polarizability experiment (E1pE). Our final result is r skin 48 = 0 . 157 ± 0 . 027 fm. Conclusion: Our conclusion is r skin 48 = 0 . 157 ± 0 . 027 fm for 48Ca. For 42 − 47 , 49 − 51 Ca, our results on r skin are similar to theirs. Our result for 48Ca is related to CREX.

An effective set of the Hartree-Fock (HF) equations are derived for electrons of the muonic systems, i.e., molecules containing a positively charged muon, conceiving the muon as a quantum oscillator, which are completely equivalent to the usual two-component HF equations used to derive stationary states of the muonic molecules. In these effective equations, a non-Coulombic potential is added to the orthodox coulomb and exchange potential energy terms, which describes the interaction of the muon and the electrons effectively and is optimized during the self-consistent field cycles. While in the two-component HF equations muon is treated as a quantum wave, in the effective HF equations it is absorbed into the effective potential and practically transformed into an effective potential field experienced by electrons. The explicit form of the effective potential depends on the nature of muon vibrations and is derivable from the basis set used to expand the muonic spatial orbital. The resulting effective Hartree-Fock equations are implemented computationally and used successfully, as a proof of concept, in a series of the muonic molecules containing all atoms from the second and third rows of the Periodic Table. To solve the algebraic version of the equations muon-specific gaussian basis sets are designed for both muon and surrounding electrons and it is demonstrated that the optimized exponents are quite distinct from those derived for the hydrogen isotopes. The developed effective HF theory is quite general and in principle can be used for any muonic system while it is the starting point for a general effective electronic structure theory that incorporates various types of quantum correlations into the muonic systems beyond the HF equations.

Using the Skyrme functional with SIII, SKM*, SLy4, and UNE0 sets of parameters and the Hartree–Fock–Bogoliubov mean-field method; the ground-state properties of even-even and even-odd neutron-rich Mg isotopes have been investigated. The results of calculations of the binding energy per nucleon (B/A), the one- and two-neutron separation energies (Sn and S2n), proton and neutron rms radii, neutron pairing gap, and quadrupole deformation parameter (B2) have been compared with the available experimental data, the results of Hartree–Fock–Bogoliubov calculations based on the D1S Gogny force, and predictions of some nuclear models such as the Finite Range Droplet Model (FRDM) and Relativistic Mean-Field (RMF) model. Our results show good agreements in comparison with the experimental data and the results of the mentioned models.

We introduce hybrid gausslet/Gaussian basis sets, where a standard Gaussian basis is added to a gausslet basis in order to increase accuracy near the nuclei while keeping the spacing of the grid of gausslets relatively large. The Gaussians are orthogonalized to the gausslets, which are already orthonormal, and approximations are introduced to maintain the diagonal property of the two electron part of the Hamiltonian, so that it continues to scale as the second power of the number of basis functions, rather than the fourth. We introduce several corrections to the Hamiltonian designed to enforce certain exact properties, such as the values of certain two-electron integrals. We also introduce a simple universal energy correction which compensates for the incompleteness of the basis stemming from the electron-electron cusps, based on the measured double occupancy of each basis function. We perform a number of Hartree Fock and full configuration interaction (full-CI) test calculations on two electron systems, and Hartree Fock on a ten-atom hydrogen chain, to benchmark these techniques. The inclusion of the cusp correction allows us to obtain complete basis set full-CI results, for the two electron cases, at the level of several microHartrees, and we see similar apparent accuracy for Hartree Fock on the ten-atom hydrogen chain.

Shell model and Hartree–Fock (HF) calculations have been adopted to study the nuclear structure for some odd-A nuclei; 17O, 27Al, 39K, and 51V. The single-particle nuclear density distributions, the associated nuclear radii, and the transverse magnetic form factors were investigated. The wave functions have been utilized from the shell model using USDA and SAS two-body effective interactions for these nuclei with the sd and d3f7 shell model spaces. On the other hand, different Skyrme parameterizations have been used within the HF method to get the single-particle potential which is used to calculate the single-particle matrix elements. The calculated form factors have been compared with available experimental data.

The results of Hartree-Fock simulations of the (0 0 0 1) surface of hematite including an a posteriori calculation of the correlation energy are presented. Structural as well as electronic and magnetic properties of the surface atoms are considered. Infrared and Raman bands of the hematite slab are also presented. The calculated surface relaxation is shown to be in good agreement with experimental data originating from a recent LEED analysis. The iron-oxygen bond at the single-iron-terminated surface is shown to be less ionic than in the bulk. Also, the magnetic dipole moment of the surface iron atom is smaller than in the bulk. Finally, the surface energy of the (0 0 0 1) facet is presented for hematite, and for comparison also for α-alumina and α-chromia for various hybrid functionals. Here it is shown that the effects of electron correlation play an important role for hematite and chromia due to their localized 3d electrons.

The nuclear structure of 208Pb is studied in the framework of the self-consistent random phase approximation (SCRPA). The Hartree–Fock mean field and single particle states are used to implement a completely SCRPA with Skyrme-type interactions. The Hamiltonian is diagonalised within a model space using five Skyrme parameter sets, namely LNS, SkI3, SkO, SkP and SLy4. In view of the huge number of the existing Skyrme-force parameterizations, the question remains which of them provide the best description of data. The approach attempts to accurately describe the structure of the spherical even–even nucleus 208Pb. To illustrate our approach, we compared the binding energy, charge density distribution, excitation energy levels scheme with the available experimental data. Moreover, we calculated isoscalar and isovector monopole, dipole, and quadrupole transition densities and strength functions.

International audience; Nuclear structure data plays an important role in nuclear physics studies and applications such as nuclear power generation. This article presents evaluations of a mean-field approach to describe two nuclear structure quantities namely the electric and magnetic moments. The Hartree-Fock-plus-pairing approach was employed with pairing correlations treated within the Bardeen-Cooper-Schrieffer (BCS) framework. The Skyrme SIII parametrization and seniority force are chosen to approximate the effective nucleon-nucleon and pairing interactions, respectively. Calculated results show that the self-consistent blocking procedure which takes into account time-reversal symmetry breaking is important to reproduce experimental magnetic moment

We show that the lowest-energy solution of the Hartree-Fock-Bogoliubov (HFB) equation has the even particle-number parity as long as the time-reversal symmetry is conserved in the HFB Hamiltonian without null eigenvalues. Based on this finding, we give a rigorous foundation of a method for solving the HFB equation to describe the ground state of odd-mass nuclei by employing a time-reversal anti-symmetric constraint operator to the Hamiltonian, where one obtains directly the ground state as a self-consistent solution of the cranked-HFB-type equation. Numerical analysis is done for the neutron-rich Mg isotopes with a reasonable choice for the operator, and it is demonstrated that the anomalous increase in the matter radius of $^{37}$Mg is well described when the last neutron occupies a low angular-momentum orbital in the framework of the nuclear energy-density-functional method, revealing the deformed halo structure., Comment: 34 pages, 4 figures, to appear in PTEP

The Hartree-Fock problem provides the conceptual and mathematical underpinning of a large portion of quantum chemistry. As efforts in quantum technology aim to enhance computational chemistry algorithms, the Hartree-Fock method, central to many other numerical approaches, is a natural target for quantum enhanced algorithms. While quantum computers and quantum simulation offer many prospects for the future of modern chemistry, the non-deterministic polynomial-complete Hartree-Fock problem is not a likely candidate. We highlight this fact from a number of perspectives including computational complexity, practical examples, and the full characterization of energy landscapes for simple systems.

We show that, as in Hartree Fock theory, the orbitals for excited state mean field theory can be optimized via a self-consistent one-electron equation in which electron-electron repulsion is accounted for through mean field operators. In addition to showing that this excited state ansatz is sufficiently close to a mean field product state to admit a one-electron formulation, this approach brings the orbital optimization speed to within roughly a factor of two of ground state mean field theory. The approach parallels Hartree Fock theory in multiple ways, including the presence of a commutator condition, a one-electron mean-field working equation, and acceleration via direct inversion in the iterative subspace. When combined with a configuration interaction singles Davidson solver for the excitation coefficients, the self consistent field formulation dramatically reduces the cost of the theory compared to previous approaches based on quasi-Newton descent., Comment: 6 pages, 3 tables, 1 figure, plus supplementary material

We calculate new equations of state (EOSs) for astrophysical simulations in the framework of the extended nuclear statistical equilibrium, in which we minimize the free energy density for the full ensemble of nuclei in a hot and dense stellar environment. To evaluate bulk and surface energies of heavy nuclei and free energies of uniformly distributed nucleons, we use fitting formulae for the interaction energies and single-nucleon potentials at zero temperature of a Dirac–Brückner Hartree–Fock (DBHF) theory, one of the modern approaches to describe homogeneous nuclear matter. We find that the DBHF EOS exhibits larger mass fractions for medium-mass nuclei and smaller mass fractions for the other nuclei than the EOS obtained using the variational method (VM), another modern model for homogeneous nuclear matter. This effect is due to the more deeply bound energy for symmetric nuclear matter and the larger symmetry energy encoded in the DBHF EOS. At supra-nuclear densities, the DBHF EOS exhibits characteristics of a larger free energy, a higher pressure, and a larger neutron chemical potential of neutron-rich matter, which lead to a larger radius of cold neutron stars than that obtained by the VM EOS.

The nuclear mean-field model based on Skyrme forces or related density functionals has found widespread application to the description of nuclear ground states, collective vibrational excitations, and heavy-ion collisions. The code Sky3D solves the static or dynamic equations on a three-dimensional Cartesian mesh with isolated or periodic boundary conditions and no further symmetry assumptions. Pairing can be included in the BCS approximation for the static case. The code is implemented with a view to allow easy modifications for including additional physics or special analysis of the results.

The properties of kaonic nuclei are studied using a two-dimensional Skyrme–Hartree–Fock model with a $KN$ Skyrme force. We focus in particular on the instability of the solutions for a too strong $KN$ interaction, which determines a maximum value of the kaon binding in this approach. We then analyze the change of the deformation properties of several core-deformed nuclei caused by the added kaon, and find a shrinking of the core and in some cases a complete loss of deformation.

In this study, the fully self-consistent Hartree–Fock (HF)-based random phase approximation (RPA) calculations were done for the $$^{40}\hbox {Ca}$$ and $$^{48}\hbox {Ca}$$ nuclei using 20 Skyrme-type interactions: KDE0, KDE0v1, SLy4, SLy5, SLy6, SK255, SKI2, SKI3, SKI5, SKM, SKMP, SKP, LNS, SGII, RAPT, SV-bas, SV-m56-O, SV-m64-O, SV-min and T6. Having a large number of Skyrme-force parameterisations requires a continuous search for the best for describing the experimental data. To examine our results, we compared the strength functions S(E), the charge density distribution and centroid energies $$E_{\mathrm{CEN}}$$ of the isoscalar giant monopole resonance (ISGMR), $$J^{\pi } = 0^{+}$$ , $$T = 0$$ , the isovector giant dipole resonance (IVGDR), $$J^{\pi } = 1^{-}$$ , $$T = 1$$ , and isoscalar giant quadrupole resonance (ISGQR), $$J^{\pi } = 2^{+}$$ , $$T = 0$$ with the available experimental data. Moreover, we discussed the sensitivities of the centroid energy $$m_{1}/m_{0}$$ and moment $$m_{1}$$ of the S(E) to the bulk properties of nuclear matter (NM), such as $$K_\mathrm{NM}$$ , the effective mass m* / m and the enhancement factor $$\kappa $$ .

We report detailed investigation of the existence and stability of mixed and demixed modes in binary atomic Bose-Einstein condensates with repulsive interactions in a ring-trap geometry. The stability of such states is examined through eigenvalue spectra for small perturbations, produced by the Bogoliubov-de Gennes equations, and directly verified by simulations based on the coupled Gross-Pitaevskii equations, varying inter- and intra-species scattering lengths so as to probe the entire range of miscibility-immiscibility transitions. In the limit of the one-dimensional (1D) ring, i.e., a very narrow one, stability of mixed states is studied analytically, including hidden-vorticity (HV) modes, i.e., those with opposite vorticities of the two components and zero total angular momentum. The consideration of demixed 1D states reveals, in addition to stable composite single-peak structures, double- and triple-peak ones, above a certain particle-number threshold. In the 2D annular geometry, stable demixed states exist both in radial and azimuthal configurations. We find that stable radially-demixed states can carry arbitrary vorticity and, counter-intuitively, the increase of the vorticity enhances stability of such states, while unstable ones evolve into randomly oscillating angular demixed modes. The consideration of HV states in the 2D geometry expands the stability range of radially-demixed states., to be published in New J. Phys

Differentiable programming is an emerging programming paradigm that allows people to take derivative of an output of arbitrary code snippet with respect to its input. It is the workhorse behind several well known deep learning frameworks, and has attracted significant attention in scientific machine learning community. In this paper, we introduce and implement a density matrix based Hartree–Fock method that naturally fits into the demands of this paradigm, and demonstrate it by performing fully variational ground state calculation on several representative chemical molecules.

The Tsallis entropies are reported, as a function of the Tsallis index q and the screening parameter μ, in position space, for the ground state of the helium atom. Three different potentials, namely Coulomb, Yukawa, and exponential cosine screened Coulomb potentials and their combinations are used in the calculations. The simple Hylleraas correlated wave function, with a three-parameter, is applied to focus on getting the most characteristic physical features of the electron-electron repulsion results. The values are calculated in one- and two-electron charge densities. Moreover, for a comparative study, the Hartree-Fock approximation, which implements a numerical grid method, is employed. Additionally, the modified Tsallis index, Q q , is proposed and calculated to predict the best value of q in the combined potentials, and it was found that 0.5 ≤ q 1.0 .

The codes HFgradZ.py and HFgradT.py find axially symmetric minima of a Hartree-Fock energy functional for a Hamiltonian supplied in a shell-model basis. The functional to be minimized is the Hartree-Fock energy for zero-temperature properties or the Hartree-Fock grand potential for finite-temperature properties. Various constraints may be imposed on the minima., 17 pages, 1 figure, Submitted to Computer Physics Communications

The potential application of carbonaceous C 24 nanocluster framework as an anode in Li-ion batteries (LIBs) is investigated using density functional theory calculations. We find that this fullerene unexpectedly gives an imaginary cell voltage and cannot be used as an anode in LIBs. Here, we explain the origin of this unusual behavior and introduce a strategy to make it suitable for anode materials. We show that there is no energy barrier for Li + diffusion through two neighboring hydrogenated C 24 fullerenes. The percentage of Hartree Fock (HF) exchange of density functionals reversely affects the adsorption energies of Li and Li + , so that it is decreased and increased by increasing %HF exchange, respectively. Also, a linear relationship between %HF and HOMO or LUMO level of the studied systems is predicted.

We report a qualitative difference in molecular band structures and frontier orbital nodal structures in DFT calculations using GGA vs. hybrid functionals and Hartree Fock in molecules used in electrochemical power sources. This can have a significant effect in applications sensitive to redox potentials and to orbital overlaps (excitations, electron transfer rates) but for which the use of hybrid functionals is impractical, such as solids or interfaces used in electrochemical energy conversion and storage technologies. We show that correct band structures and nodal structures (ordering) of frontier orbitals can be obtained by applying a Hubbard correction to selected atomic states.

A B-spline version of a general Dirac–Hartree–Fock program is described. The usual differential equations are replaced by a set of generalized eigenvalue problems of the form ( H a − e a B ) P a = 0 , where H a and B are the Hamiltonian and overlap matrices, respectively, and P a is the two-component relativistic orbit in the B-spline basis. A default universal grid allows for flexible adjustment to different nuclear models. When two orthogonal orbitals are both varied, the energy must also be stationary with respect to orthonormal transformations. At such a stationary point the off-diagonal Lagrange multipliers may be eliminated through projection operators. The self-consistent field procedure exhibits excellent convergence. Several atomic states can be considered simultaneously, including some configuration-interaction calculations. The program provides several options for the treatment of Breit interaction and QED corrections. The information about atoms up to Z = 104 is stored by the program. Along with a simple interface through command-line arguments, this information allows the user to run the program with minimal initial preparations. Program summary Program title: DBSR_HF Catalogue identifier: AEZK_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEZK_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 22643 No. of bytes in distributed program, including test data, etc.: 354629 Distribution format: tar.gz Programming language: Fortran 95. Computer: No specific requirements to the computer. Operating system: Any system with a Fortran 95 compiler. Classification: 2.1. External routines: LAPACK ( http://www.netlib.org/lapack/ ) Nature of problem: Relativistic Dirac–Hartree–Fock wavefunctions are determined for atoms in a bound state. These wavefunctions may be used to predict a variety of atomic properties. Solution method: The radial functions for large and small components of the one-electron spinor are expanded in B-spline bases. The variational principle applied to an energy functional that includes Lagrange multipliers for orthonormal constraints defines the Dirac–Hartree–Fock matrix for each orbital. Orthonormal transformations for a stationary solution were applied and Lagrange multipliers eliminated through projection operators. Restrictions: There is no restriction on calculations for the average or specific term energy of any atomic configuration with shells whose angular momenta are less than or equal to 9/2. Unusual features: The program allows the consideration of a few atomic states simultaneously. A simple interface through the command-line arguments allows the user to run the program with minimal initial preparations. Running time: From a few seconds to a few minutes depending on the atom under consideration.

The spectroscopic properties of poly (styrene–co–p–vinylbenzyl chloride) (poly (St-co-VBC)) were investigated by Fourier transform infrared spectroscopy and 1H nuclear magnetic resonance spectroscopic techniques. The molecular geometry and vibrational frequencies of macroinitiator, poly (St-co-VBC), were calculated by using density functional theory (DFT) and Hartree–Fock (HF) methods with 6–31 G+ (d, p) as a basis set. Calculated theoretical values are shown to be in good agreement with that of experimental values. An excellent harmony between the two data sets was verified. Besides, the experimental data of macroinitiator were compared with experimental data of its corresponding monomers such as St and VBC. The dimer and trimer forms of macroinitiator are used as significant contributions for getting an accurate interpretation of the experimental frequencies of poly (St-co-VBC). The results revealed that the change from St and VBC to poly (St-co-VBC) should be characterized by the disappearance of the CH2=CH bonds of the vinyl group and the appearance of the aliphatic C–H and CH2 bonds. The geometrical parameters, Mulliken atomic charges and frontier molecular orbitals energies were also calculated using the same theoretical methods. The chemical shifts were calculated by using the gauge–including atomic orbital method and all the theoretically predicted values were shown to be in good agreement with experimental values. Molecular orbital properties, molecular electrostatic potential, and the potential energy surface for the atom transfer radical polymerization (ATRP) of the macroinitiator were studied with DFT and HF calculations. The potential energy surface of the ATRP initiator is decided by their electronic effect and steric hindrance effect simultaneously.

A Gaussian-based range-separation approach is applied to the Hartree–Fock exchange interaction and the second-order perturbation theory, comparing the results with those provided through the erfc short-range potential approach. The Gaussian short-range potential's behavior resembles more that of the bare 1/r potential than that of the erfc short-range potential around the origin. Thus, the Gaussian-based range-separation of the exchange term agrees better with the ordinary HF method and the Moller–Plesset perturbation theory.

For each atom of a molecule, we introduce the structure unit group (SUG) consisted of the atom (called as core-atom) and the neighbor atoms. The kinds of SUG are sampled from small molecules, and the Gaussian type orbitals (GTOs) only for core- and hydrogen atoms of each SUG are scaled in the SUG environment. The SUG-scale factors might be used in Hartree-Fock calculations of organic molecules and can make lower a few milli-hartree in molecular energy and cause the change of several percent in functional properties such as Mulliken charge and multi-pole moment.

The nuclear mean-field model based on Skyrme forces can predict a variety of properties of nuclear ground states. We present the Code SkyAx solving the Hartree–Fock equations in two spatial dimensions assuming axial symmetry. Pairing can be included in the BCS approximation. The code is implemented with a view on computational speed. Program summary Program title: SkyAx CPC Library link to program files: http://dx.doi.org/10.17632/fd453hc4jb.1 Licensing provisions: GPLv3 Programming language: Fortran 90 and parallel version with OpenMP. External routines/libraries: BLAS, LAPACK. Nature of problem: The Hartree–Fock equations can be used to determine static properties of nuclei all over the nuclear chart, e.g., nuclear masses, charge radii and deformations. This code implements the widely used Skyrme forces as interaction model and offers in addition to include the pairing interaction through the BCS theory. Due to its two-dimensional nature, the code allows only for axial deformations, which is suitable for most nuclei in the chart of nuclides. Solution method: The nucleonic wave functions are represented on a two-dimensional mesh assuming axial symmetry. The Coulomb potential is calculated for an isolated charge distribution by splitting the problem into a short-range and a long-range part. All spatial derivatives are evaluated using the finite Fourier transform method. The code solves the static Hartree–Fock equations with a damped gradient iteration method. It also allows for constraint iterations, where the monopole, quadrupole, octupole and hexadecapole moments can be fixed. Additional comments including restrictions and unusual features: The current implementation is restricted to even–even nuclei. Furthermore the Hartree–Fock + BCS model is valid only for well-bound nuclei. For nuclei near the neutron and proton drip lines, a full Hartree–Fock–Bogolyubov treatment would be more suitable. The code allows for multipole constraints up to l=4. Furthermore the code can be used to calculate fission paths or landscapes of deformations (potential energy surfaces) in one single run.

It is well known that Brillouin’s theorem (BT) holds in the restricted open-shell Hartree–Fock (ROHF) method for three kinds of single excitations, c → o, c → v, and o → v, where c, o, and v are the orbitals of the closed, open, and virtual shells, respectively. For these excitations, the conditions imposed by BT on the orbitals of a system under study are physically equivalent to the conditions imposed by the variational principle, and this provides a fundamental meaning of BT. Together with this, BT is not satisfied for some excitations of the kind o → o, in which both orbitals participating in excitation belong to the open shell. This limitation of BT is known, for example, for the helium atom, where BT is satisfied for excitation from the ground state S01 (1s2) to the state S11 of the configuration 1s12s1 and is not satisfied for excitations S11 → S01 and S11 → S21 (2s2). In this work, we prove that Brillouin’s conditions for two latter excitations cannot be related to the fundamental conditions imposed by the variational principle due to specific symmetry restrictions. Based on this finding, we give a rigorous proof of fulfillment of BT for the alternative o → o excitation, which takes in the helium atom the form S11 → S31, where both the initial and excited states are treated as arising from the same open-shell configuration 1s12s1, and the state S31 is described by the symmetry-adapted ROHF wave function Ψ(S31) = [Ψ(S21) − Ψ(S01)]/2. The new formulation of BT obeys all the necessary variational and symmetrical conditions, and its validity is illustrated by the results of computations of atom He and molecule LiH in their singlet states arising from different closed-shell and open-shell configurations performed using both ROHF and limited configuration interaction methods.

We report on a quadratically convergent self-consistent field (QC-SCF) algorithm for the spin-projected unrestricted Hartree-Fock (SUHF) to mitigate the slow convergence of SUHF due to the presence of small eigenvalues in the orbital Hessian matrix. The new QC-SCF is robust and stable, allowing us to obtain the SUHF solutions very quickly. To demonstrate the applicability of the method, we present results for test systems with abundant non-dynamic correlation in comparison with the Roothaan repeated diagonalization, Pople extrapolation, and direct inversion of iterative subspace.

Nonrelativistic Hartree–Fock limit calculations are performed for the group 18 atoms using Lambda functions, which are Laguerre-type basis functions (LTFs). Since Lambda functions form a complete orthonormal set for bound states, the total energy approaches the Hartree–Fock limit monotonically upon increasing the number of expansion terms, as determined by the maximum number (N) of the principal quantum number (n) in a set. The convergence behavior of the total energy in relation to the number of expansion terms is investigated. For Rn, N = 116 is required to satisfy the convergence criterion ∣ Δ E / E ∣ < 10 − 15 . Here E is the total energy and ΔE is (E(N−1) − E(N)). Total energies are obtained with 30 significant digits for He, with 14 digits for Ne, Ar, Kr, Xe, and Rn, and with 13 digits for Og. This approach gives −2.86167999561223887877554374002 au as the Hartree–Fock energy of He.

BSHF solves the Hartree–Fock equations in a B-spline basis for atoms, negatively charged ions, and systems of N electrons in arbitrary central potentials. In the B-spline basis the Hartree–Fock integro-differential equations are reduced to a computationally simpler eigenvalue problem. As well as solving this for the ground-state electronic structure self-consistently, the program can calculate discrete and/or continuum excited states of an additional electron or positron in the field of the frozen-target N -electron ground state. It thus provides an effectively complete orthonormal basis that can be used for higher-order many-body theory calculations. Robust and efficient convergence in the self-consistent iterations is achieved by a number of strategies, including by gradually increasing the strength of the electron–electron interaction by scaling the electron charge from a reduced value to its true value. The functionality and operation of the program is described in a tutorial style example. Program summary Program Title: BSHF Program Files doi: http://dx.doi.org/10.17632/fj3y6c58dy.1 Code Ocean Capsule: https://doi.org/10.24433/CO.1226817.v2 Licensing provisions: GPLv3 Programming language : Fortran 90. External routines/libraries: LAPACK. Nature of problem: Self-consistent solution of electronic structure for atoms and electrons in arbitrary central potentials in the Hartree–Fock approximation. Solution method : A B-spline basis is employed that transforms the Hartree–Fock integro-differential equations to a computationally simpler eigenvalue problem. The eigenvalue problem is solved iteratively until self-consistency is achieved. Unusual or notable features: 1. Robust and efficient convergence in the self-consistent iterations is achieved by gradually increasing the value of the electron charge from a reduced value to its true value, i.e., increasing the strength of the electron–electron interaction. In this way all orbitals are calculated simultaneously at every iteration of the self-consistency loop, i.e., without the need to successively fill the occupied shells from the core to valence (as most other Hartree–Fock codes require for convergence). 2. In addition to atoms and negative ions , the program solves the Hartree–Fock equations for systems of N electrons confined in an arbitrary central potential specified by the user: thus the program can be used to e.g., calculate the structure of electrons confined in harmonic potentials , which are known to approximate the electron gas . 3. In addition to calculating the ground state of the N -electron system, the program calculates the discrete and/or continuum excited states for an additional electron or positron in the field of the ‘frozen’ N -electron target. It thus provides an orthonormal basis that can be used as input for many-body theory calculations. Restrictions: The program solves non-relativistic Hartree–Fock equations for spherically symmetric systems only (in modelling systems other than atoms, e.g., harmonically confined electron gas or jellium-type models of clusters, relativistic effects can often be negligible and a non-relativistic treatment is perfectly suitable). For open-shell atoms the central-field approximation is used, i.e., the potential is angularly averaged.

We show that the leading term in the strong-interaction limit of the adiabatic connection that has as weak-interaction expansion the Møller-Plesset perturbation theory can be fully determined from a functional of the Hartree-Fock density. We analyze this functional and highlight similarities and differences with the strong-interaction limit of the density-fixed adiabatic connection case of Kohn-Sham density functional theory.

We perform hybrid functional and quasi-particle band structure calculations with spin-orbit interaction to investigate the band structures of Mg2Si, Mg2Ge, and Mg2Sn. For all Mg2X materials, where X = Si, Ge, and Sn, the characteristics of band edge states, i.e., band and valley degeneracies, and orbital characters, are found to be conserved, independent of the computational schemes such as density functional generalized gradient approximation, hybrid functionals, or quasi-particle calculations. However, the magnitude of the calculated band gap varies significantly with the computational schemes. Within density-functional calculations, the one-particle band gaps of Mg2Si, Mg2Ge, and Mg2Sn are 0.191, 0.090, and -0.346 eV, respectively, and thus severely underestimated compared to the experimental gaps, due to the band gap error in the density functional theory and the significant relativistic effect on the low-energy band structures. By employing hybrid-functional calculations with a 35% fraction of the exact Hartree-Fock exchange energy (HSE-35%), we overcame the negative band gap issue in Mg2Sn. Finally, in quasi-particle calculations on top of the HSE-35% Hamiltonians, we obtained band gaps of 0.835, 0.759, and 0.244 eV for Mg2Si, Mg2Ge, and Mg2Sn, respectively, consistent with the experimental band gaps of 0.77, 0.74, and 0.36 eV, respectively., Comment: 23 pages, including 84 references, 5 tables, 3 figures

We assess the utility of Hartree-Fock (HF) trial wavefunctions in performing phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) on the uniform electron gas (UEG) model. The combination of ph-AFQMC with spin-restricted HF (RHF+ph-AFQMC), was found to be highly accurate and efficient for systems containing up to 114 electrons in 2109 orbitals, particularly for $r_s$ $\le$ 2.0. Compared to spin-restricted coupled-cluster (RCC) methods, we found that RHF+ph-AFQMC performs better than CC with singles, doubles, and triples (RCCSDT) and similarly to or slightly worse than CC with singles, doubles, triples, and quadruples (RCCSDTQ) for $r_s$ $\le$ 3.0 in the 14-electron UEG model. With the 54-electron, we found RHF+ph-AFQMC to be nearly exact for $r_s$ $\le$ 2.0 and pointed out potential biases in existing benchmarks. Encouraged by these, we performed RHF+ph-AFQMC on the 114-electron UEG model for $r_s$ $\le$ 2.0 and provided new benchmark data for future method development. We found that the UEG models with $r_s$ = 5.0 remain to be challenging for RHF+ph-AFQMC. Employing non-orthogonal configuration expansions or unrestricted HF states as trial wavefunctions was also found to be ineffective in the case of the 14-electron UEG model with $r_s$ = 5.0. We emphasize the need for a better trial wavefunction for ph-AFQMC in simulating strongly correlated systems. With the 54-electron and 114-electron UEG models, we stress the potential utility of RHF+ph-AFQMC for simulating dense solids.