Hartree-Fock Approach Research Articles

Charge self-consistent many-body corrections using optimized projected localized orbitals

In order for methods combining ab initio density-functional theory and many-body techniques to become routinely used, a flexible, fast, and easy-to-use implementation is crucial. We present an implementation of a general charge self-consistent scheme based on projected localized orbitals in the projector augmented wave framework in the Vienna Ab Initio Simulation Package. We give a detailed description on how the projectors are optimally chosen and how the total energy is calculated. We benchmark our implementation in combination with dynamical mean-field theory: first we study the charge-transfer insulator NiO using a Hartree–Fock approach to solve the many-body Hamiltonian. We address the advantages of the optimized against non-optimized projectors and furthermore find that charge self-consistency decreases the dependence of the spectral function—especially the gap—on the double counting. Second, using continuous-time quantum Monte Carlo we study a monolayer of SrVO3, where strong orbital polarization occurs due to the reduced dimensionality. Using total-energy calculation for structure determination, we find that electronic correlations have a non-negligible influence on the position of the apical oxygens, and therefore on the thickness of the single SrVO3 layer.

Connecting the nuclear equation of state to the interplay between fusion and quasifission processes in low-energy nuclear reactions

Within the Time Dependent Hartree Fock (TDHF) approach, we investigate the impact of several ingredients of the nuclear effective interaction, such as incompressibility, symmetry energy, effective mass, derivative of the Lane potential and surface terms on the exit channel (fusion vs quasifission) observed in the reaction $^{238}$U+$^{40}$Ca, close to the Coulomb barrier. Our results show that all the ingredients listed above contribute to the competition between fusion and quasifission processes, however the leading role in determining the outcome of the reaction is played by incompressibility, symmetry energy and the isoscalar coefficient of the surface term. This study unravels the complexity of the fusion and quasifission reaction dynamics and helps to understand the microscopic processes responsible for the final outcome of low energy heavy ion collisions in terms of relevant features of the nuclear effective interaction and associated equation of state (EoS).

Production mechanism of neutron-rich nuclei around N = 126 in the multi-nucleon transfer reaction 132Sn + 208Pb**Supported by National Natural Science Foundation of China (11705118, 11475115, 11647026) and Natural Science Foundation of SZU (2016017)

The time-dependent Hartree-Fock approach in three dimensions is employed to study the multi-nucleon transfer reaction 132Sn + 208Pb at various incident energies above the Coulomb barrier. Probabilities for different transfer channels are calculated by using the particle-number projection method. The results indicate that neutron stripping (transfer from the projectile to the target) and proton pick-up (transfer from the target to the projectile) are favored. De-excitation of the primary fragments is treated by using the state-of-art statistical code GEMINI++. Primary and final production cross sections of the target-like fragments (with Z=77 to Z=87) are investigated. The results reveal that fission decay of heavy nuclei plays an important role in the de-excitation process of nuclei with Z > 82. It is also found that the final production cross sections of neutron-rich nuclei depend only slightly on the incident energy, while those of neutron-deficient nuclei depend strongly on the incident energy.

Proton spectral functions in finite nuclei based on the extended Brueckner–Hartree–Fock approach

We have calculated the proton spectral functions in finite nuclei based on the local density approximation where the properties of finite nuclei and nuclear matter are calculated by the Skyrme–Hartree–Fock method and the extended Brueckner–Hartree–Fock approach, respectively. The scaled spectral function from our calculation is in good agreement with experimental results at small momenta while the difference between them becomes apparent at high momenta. Besides, a target dependence of the scaled proton spectral function is also obtained in our calculation as was observed in experiment. A further investigation indicates that the proportion of the high density region of the proton has a significant contribution to this target-dependent behavior since the spectral function in asymmetric nuclear matter increases significantly as a function of density.

Temperature Effects on the Equation of State and Symmetry Energy at Low and High Densities

Thermal properties of symmetric nuclear matter and pure neutron matter are studied in a selfconsistent Green’s function and Brueckner–Hartree–Fock approaches with the inclusion of the contact interaction using CDBONN potential. Also we investigate the temperature dependence of the symmetry energy. The symmetry energy at fixed density is found to generally decrease with temperature. The temperature effects on the nuclear matter symmetry energy are found to be stronger at lower densities while become much weaker at higher densities. The results of several microscopic approaches are compared. Also the results are compared with recent experimental data. There is good agreement between the experimental symmetry energy and those calculated in the Brueckner–Hartree–Fock approach.

Skyrme density functional description of the double magic Ni78 nucleus

We calculate the single particle spectrum of the double magic nucleus $^{78}$Ni in a Hartree-Fock approach using the Skyrme density-dependent effective interaction containing central, spin-orbit and tensor parts. We show that the tensor part has an important effect on the spin-orbit splitting of the proton $1f$ orbit which may explain the survival of magicity so far from the stability valley. We confirm the inversion of the $1f5/2$ and $2p3/2$ levels at the neutron number 48 in the Ni isotopic chain expected from previous Monte Carlo shell model calculations and supported by experimental observation.

Spin Hartree-Fock approach to studying quantum Heisenberg antiferromagnets in low dimensions

We construct a new mean-field theory for quantum (spin-1/2) Heisenberg antiferromagnet in one (1D) and two (2D) dimensions using a Hartree-Fock decoupling of the four-point correlation functions. We show that the solution to the self-consistency equations based on two-point correlation functions does not produce any unphysical finite-temperature phase transition in accord with Mermin-Wagner theorem, unlike the common approach based on the mean-field equation for the order parameter. The next-neighbor spin-spin correlation functions, calculated within this approach, reproduce closely the strong renormalization by quantum fluctuations obtained via Bethe ansatz in 1D and a small renormalization of the classical antiferromagnetic state in 2D. The heat capacity approximates with reasonable accuracy the full Bethe ansatz result at all temperatures in 1D. In 2D, we obtain a reduction of the peak height in the heat capacity at a finite temperature that is accessible by high-order $1/T$ expansions.

Transient and Sharvin resistances of Luttinger liquids

Although the intrinsic conductance of an interacting one-dimensional system is renormalized by the electron-electron correlations, it has been known for some time that this renormalization is washed out by the presence of the (noninteracting) electrodes to which the wire is connected. Here, we study the transient conductance of such a wire: a finite voltage bias is suddenly applied across the wire and we measure the current before it has enough time to reach its stationary value. These calculations allow us to extract the Sharvin (contact) resistance of Luttinger and Fermi liquids. In particular, we find that a perfect junction between a Fermi liquid electrode and a Luttinger liquid electrode is characterized by a contact resistance that consists of half the quantum of conductance in series with half the intrinsic resistance of an infinite Luttinger liquid. These results were obtained using two different methods: a dynamical Hartree-Fock approach and a self-consistent Boltzmann approach. Although these methods are formally approximate, we find a perfect match with the exact results of Luttinger/Fermi liquid theory.

Nuclear matter calculations with chiral interactions

Using two-nucleon and three-nucleon interactions derived in the framework of chiral perturbation theory (ChPT) with and without the explicit Δ isobar contributions, we calculate the energy per particle of symmetric nuclear matter and pure neutron matter employing the microscopic Brueckner–Hartree–Fock approach. Specifically, we present nuclear matter calculations using the new fully local in coordinate-space two-nucleon interaction at the next-to-next-to-next-to-leading-order (N3LO) of ChPT with Δ isobar intermediate states (N3LOΔ) recently developed by Piarulli et al. [1] supplemented with a local N2LO three-nucleon interaction with explicit Δ isobar degrees of freedom. We show that for this combination of two- and three-nucleon interactions it is possible to obtain a good saturation point of symmetric nuclear matter. We also calculate the nuclear symmetry energy and compare our results with the available empirical constraints on this quantity.

The any particle molecular orbital grid-based Hartree-Fock (APMO-GBHF) approach

The any particle molecular orbital grid-based Hartree-Fock approach (APMO-GBHF) is proposed as an initial step to perform multi-component post-Hartree-Fock, explicitly correlated, and density functional theory methods without basis set errors. The method has been applied to a number of electronic and multi-species molecular systems. Results of these calculations show that the APMO-GBHF total energies are comparable with those obtained at the APMO-HF complete basis set limit. In addition, results reveal a considerable improvement in the description of the nuclear cusps of electronic and non-electronic densities.

Nuclear fourth-order symmetry energy and its effects on neutron star properties in the relativistic Hartree-Fock theory

Adopting the density dependent relativistic mean-field (RMF) and relativistic Hartree-Fock (RHF) approaches, the properties of the nuclear fourth-order symmetry energy $S_4$ are studied within the covariant density functional (CDF) theory. It is found that the fourth-order symmetry energies are suppressed in RHF at both saturation and supranuclear densities, where the extra contribution from the Fock terms is demonstrated, specifically via the isoscalar meson-nucleon coupling channels. The reservation of $S_4$ and higher-order symmetry energies in the nuclear equation of state then affects essentially the prediction of neutron star properties, which is illustrated in the quantities such as the proton fraction, the core-crust transition density as well as the fraction of crustal moment of inertia. Since the Fock terms enhance the density dependence of the thermodynamical potential, the RHF calculations predict systematically smaller values of density, proton fraction and pressure at the core-crust transition boundary of neutron stars than density dependent RMF ones. In addition, a linear anti-correlation between the core-crust transition density $\rho_t$ and the density slope of symmetry energy $L$ is found which is then utilized to constrain the core-crust transition density as $\rho_t\thicksim[0.069, 0.098]~\rm{fm}^{-3}$ with the recent empirical information on $L$. The study clarifies the important role of the fourth-order symmetry energy in determining the properties of nuclear matter at extreme isospin or density conditions.

Ab initio simulation of structure and surface energy of low-index surfaces of stoichiometric α-Fe2O3

The structure and surface energy of a series of low-index surfaces of stoichiometric α-Fe2O3 (hematite) are investigated using the periodic Hartree–Fock approach with an a posteriori correction of the correlation energy. The simulations show that, amongst the modeled facets, (011¯2) and (0001) are the most stable surfaces of hematite, which is consistent with the fact that the latter are the dominant growth faces exposed on natural α-Fe2O3. The Fe-terminated (0001) surface is shown to exhibit a large relaxation of the surface atoms. It is argued that this arises mainly due to the fact that the surface cations are located opposite empty cation sites in the filled-filled-unfilled cation sequence along the c-axis. In contrast, the (011¯2) plane cuts the crystal through a plane of empty cation sites, thus giving rise to relatively small relaxations and surface energies. The small relaxations and concomitant exposure of five-coordinate cation sites may be important for the catalytic activity of hematite. The simulations also show that the relative stability of the investigated surfaces changes after a full lattice relaxation with the (0001) and (112¯6) facets relaxing disproportionately large. Wherever possible, the simulations are compared with previous simulation data and experimental results. A Wulff–Gibbs construction is also presented.

Exploring Zeptosecond Quantum Equilibration Dynamics: From Deep-Inelastic to Fusion-Fission Outcomes in ^{58}Ni+^{60}Ni Reactions.

Energy dissipative processes play a key role in how quantum many-body systems dynamically evolve toward equilibrium. In closed quantum systems, such processes are attributed to the transfer of energy from collective motion to single-particle degrees of freedom; however, the quantum many-body dynamics of this evolutionary process is poorly understood. To explore energy dissipative phenomena and equilibration dynamics in one such system, an experimental investigation of deep-inelastic and fusion-fission outcomes in the ^{58}Ni+^{60}Ni reaction has been carried out. Experimental outcomes have been compared to theoretical predictions using time dependent Hartree-Fock and time dependent random phase approximation approaches, which, respectively, incorporate one-body energy dissipation and fluctuations. Excellent quantitative agreement has been found between experiment and calculations, indicating that microscopic models incorporating one-body dissipation and fluctuations provide a potential tool for exploring dissipation in low-energy heavy ion collisions.

Relativistic interpretation of the nature of the nuclear tensor force**Supported by National Natural Science Foundation of China (11375076, 11675065) and the Fundamental Research Funds for the Central Universities (lzujbky-2016-30)

The spin-dependent nature of the nuclear tensor force is studied in detail within the relativistic Hartree-Fock approach. The relativistic formalism for the tensor force is supplemented with an additional Lorentz-invariant tensor formalism in the σ-scalar channel, so as to take into account almost fully the nature of the tensor force brought about by the Fock diagrams in realistic nuclei. Specifically, the tensor sum rules are tested for the spin and pseudo-spin partners with and without nodes, to further understand the nature of the tensor force within the relativistic model. It is shown that the interference between the two components of nucleon spinors causes distinct violations of the tensor sum rules in realistic nuclei, mainly due to the opposite signs on the κ quantities of the upper and lower components, as well as the nodal difference. However, the sum rules can be precisely reproduced if the same radial wave functions are taken for the spin/pseudo-spin partners in addition to neglecting the lower/upper components, revealing clearly the nature of the tensor force.

Analytic model of a multi-electron atom

A fully analytical approximation for the observable characteristics of many-electron atoms is developed via a complete and orthonormal hydrogen-like basis with a single-effective charge parameter for all electrons of a given atom. The basis completeness allows us to employ the secondary-quantized representation for the construction of regular perturbation theory, which includes in a natural way correlation effects, converges fast and enables an effective calculation of the subsequent corrections. The hydrogen-like basis set provides a possibility to perform all summations over intermediate states in closed form, including both the discrete and continuous spectra. This is achieved with the help of the decomposition of the multi-particle Green function in a convolution of single-electronic Coulomb Green functions. We demonstrate that our fully analytical zeroth-order approximation describes the whole spectrum of the system, provides accuracy, which is independent of the number of electrons and is important for applications where the Thomas–Fermi model is still utilized. In addition already in second-order perturbation theory our results become comparable with those via a multi-configuration Hartree–Fock approach.

Hartree-Fock Approach for Rotating Bose-Einstein Condensation in One-dimensional Deep Optical Lattice

In this work, we investigated the thermodynamic behavior of a rotating Bose-Einstein condensation with non-zero interatomic interactions in one-dimensional optical lattice theoretically. Our system is formed by loading three dimensional boson-clouds into 1D optical lattice and subjected to rotate with angular velocity ɷ about the z axis. We employed the semiclassical Hartree-Fock approach to calculate the condensate fraction, critical temperature, entropy and the heat capacity of the system. Thereby, we investigated the effect of the rotation rate, optical potential depth and the interactions parameter on these parameters. The obtained results provide a solid theoretical foundation for the current experiments of rotating interacting Bose-Einstein condensation produced or transferred in optical lattices.

Hartree-Fock treatment of Fermi polarons using the Lee-Low-Pine transformation

We consider the Fermi polaron problem at zero temperature, where a single impurity interacts with non-interacting host fermions. We approach the problem starting with a Frohlich-like Hamiltonian where the impurity is described with canonical position and momentum operators. We apply the Lee-Low-Pine (LLP) transformation to change the fermionic Frohlich Hamiltonian into the fermionic LLP Hamiltonian which describes a many-body system containing host fermions only. We adapt the self-consistent Hartree-Fock (HF) approach, first proposed by Edwards, to the fermionic LLP Hamiltonian in which a pair of host fermions with momenta $\mathbf{k}$ and $\mathbf{k}'$ interact with a potential proportional to $\mathbf{k}\cdot\mathbf{k}'$. We apply the HF theory, which has the advantage of not restricting the number of particle-hole pairs, to repulsive Fermi polarons in one dimension. When the impurity and host fermion masses are equal our variational ansatz, where HF orbitals are expanded in terms of free-particle states, produces results in excellent agreement with McGuire's exact analytical results based on the Bethe ansatz. This work raises the prospect of using the HF ansatz and its time-dependent generalization as building blocks for developing all-coupling theories for both equilibrium and nonequilibrium Fermi polarons in higher dimensions

Triaxial quadrupole dynamics and the inner fission barrier of some heavy even-even nuclei

Background: Inner fission barriers of actinide nuclei have been known for a long time to be unstable with respect to the axial symmetry. On the other hand, taking into account the effect of the relevant adiabatic mass parameter reduces or even may wash out this instability. A proper treatment of the dynamics for both axial and triaxial modes is thus crucial to accurately determine the corresponding fission barriers. This entails in particular an accurate description of pairing correlations.Purpose: We evaluate the potential energies, moments of inertia, and vibrational mass parameters in a two-dimensional relevant deformation space (corresponding to the usual $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ quadrupole deformation parameters) for four actinide nuclei ($^{236}\mathrm{U}, ^{240}\mathrm{Pu}, ^{248}\mathrm{Cm}$, and $^{252}\mathrm{Cf}$). We assess the relevance of our approach to describe the dynamics for a triaxial mode by computing the low energy spectra (exploring thus mainly the equilibrium deformation region). We evaluate the inner fission barrier heights releasing the axial symmetry constraint.Method: Calculations within the Hartree-Fock plus BCS approach are performed using the SkM* Skyrme effective interaction in the particle-hole channel and a seniority force in the particle-particle channel. The intensity of this residual interaction has been fixed to allow a good reproduction of some odd-even mass differences in the actinide region. Adiabatic mass parameters for the rotational and vibrational modes are calculated using the Inglis-Belyaev formula supplemented by a global renormalization factor taking into account the so-called Thouless-Valatin corrections. Spectra are obtained through the diagonalization of the corresponding Bohr collective Hamiltonian.Results: The experimental low energy spectra are qualitatively well reproduced by our calculations for the considered nuclei. Inner fission barrier heights are calculated and compared with available estimates from various experimental data. The reproduction of the data is better for $^{236}\mathrm{U}$ and $^{240}\mathrm{Pu}$ (up to about 300 keV) than for $^{248}\mathrm{Cm}$ and $^{252}\mathrm{Cf}$ (up to about one MeV).Conclusions: While these results are encouraging, they call for, in particular, a better treatment of pairing correlations, especially as far as the particle number conservation is concerned. Besides, these results could provide a basis for the determination of the least action trajectories which would generate better grounds for the evaluation of fission half lives.

Newly identified energy levels and calculated transition rates in astatine atom, the rarest element on Earth

A theoretical investigation of the atomic structure and radiative parameters involving the lowest states within the 6p5, 6p47s and 6p47p configurations of neutral astatine is reported for the first time in the present paper. Using different physical models based on the pseudo-relativistic Hartree–Fock approach, the influence of intravalence, core-valence and core-core electron correlation on the atomic parameters is discussed in detail. This work allowed us to establish the intermediate-coupling composition and to clearly fix the spectroscopic designation of six excited levels within the 6p47s and 6p47p configurations which had been located in previous experimental studies. In addition new tentative identifications of four levels belonging to the 6p49p and 6p410p configurations are reported in this work together with predicted level energies along the 6p4np and 6p4nd Rydberg series up to n = 50.

Radiative Decay Rates for Electric Dipole, Magnetic Dipole and Electric Quadrupole Transitions in Triply Ionized Thulium (Tm IV)

A new set of radiative decay parameters (oscillator strengths, transition probabilities) for spectral lines in triply ionized thulium (Tm IV) has been obtained within the framework of the pseudo-relativistic Hartree-Fock (HFR) approach. The effects of configuration interaction and core-polarization have been investigated in detail and the quality of the results has been assessed through a comparison between different HFR physical models. The spectroscopic data listed in the present paper cover electric dipole as well as magnetic dipole and electric quadrupole transitions in a wide range of wavelengths from extreme ultraviolet to near infrared.