Finite-temperature Hartree-Fock Research Articles

InteraChem: Exploring Excited States in Virtual Reality with Ab Initio Interactive Molecular Dynamics.

InteraChem is an ab initio interactive molecular dynamics (AI-IMD) visualizer that leverages recent advances in virtual reality hardware and software, as well as the graphical processing unit (GPU)-accelerated TeraChem electronic structure package, in order to render quantum chemistry in real time. We introduce the exploration of electronically excited states via AI-IMD using the floating occupation molecular orbital-complete active space configuration interaction method. The optimization tools in InteraChem enable identification of excited state minima as well as minimum energy conical intersections for further characterization of excited state chemistry in small- to medium-sized systems. We demonstrate that finite-temperature Hartree-Fock theory is an efficient method to perform ground state AI-IMD. InteraChem allows users to track electronic properties such as molecular orbitals and bond order in real time, resulting in an interactive visualization tool that aids in the interpretation of excited state chemistry data and makes quantum chemistry more accessible for both research and educational purposes.

Finite-temperature many-body perturbation theory for electrons: Algebraic recursive definitions, second-quantized derivation, linked-diagram theorem, general-order algorithms, and grand canonical and canonical ensembles.

A comprehensive and detailed account is presented for the finite-temperature many-body perturbation theory for electrons that expands in power series all thermodynamic functions on an equal footing. Algebraic recursions in the style of the Rayleigh-Schrödinger perturbation theory are derived for the grand potential, chemical potential, internal energy, and entropy in the grand canonical ensemble and for the Helmholtz energy, internal energy, and entropy in the canonical ensemble, leading to their sum-over-states analytical formulas at any arbitrary order. For the grand canonical ensemble, these sum-over-states formulas are systematically transformed to sum-over-orbitals reduced analytical formulas by the quantum-field-theoretical techniques of normal-ordered second quantization and Feynman diagrams extended to finite temperature. It is found that the perturbation corrections to energies entering the recursions have to be treated as a nondiagonal matrix, whose off-diagonal elements are generally nonzero within a subspace spanned by degenerate Slater determinants. They give rise to a unique set of linked diagrams-renormalization diagrams-whose resolvent lines are displaced upward, which are distinct from the well-known anomalous diagrams of which one or more resolvent lines are erased. A linked-diagram theorem is introduced that proves the size-consistency of the finite-temperature many-body perturbation theory at any order. General-order algorithms implementing the recursions establish the convergence of the perturbation series toward the finite-temperature full-configuration-interaction limit unless the series diverges. The normal-ordered Hamiltonian at finite temperature sheds light on the relationship between the finite-temperature Hartree-Fock and first-order many-body perturbation theories.

Temperature dependence of nuclear properties: A systematic study along the isotopic and isotonic chains of nuclei

In this work, we study the changes in the nuclear properties by performing systematic calculations on the selected isotopic and isotonic chains of nuclei with increasing temperature. The finite temperature Hartree–Fock–Bogoliubov (FT-HFB) calculations are performed using the Skyrme-type SkM* functional and mixed-type pairing interaction. The changes in the pairing properties, internal excitation energies, entropy, two-neutron separation energies, and neutron skin thickness of nuclei are systematically studied. It is shown that both the internal excitation energy and entropy are sensitive to the changes in the pairing properties of nuclei below the critical temperatures. At high temperatures and after T≥1 MeV, both of them increase rapidly. The nuclei near the neutron drip lines are affected more by the temperature effects since the continuum effects start to become dominant around these regions. On the other hand, the internal energy and entropy are not sensitive to the increase in the proton number, and the changes remain almost stable along an isotonic chain with increasing temperature. We also found that some nuclei near the neutron drip lines become bound at finite temperatures, whereas they are unstable against the two-neutron emission (S2n≤0 MeV). Investigation of the neutron skin thickness of nuclei shows that the temperature has a big impact on nuclei close to the neutron drip lines, and it considerably increases the neutron skin thickness of these nuclei at high temperatures.

Benchmarking mean-field approximations to level densities

We assess the accuracy of finite-temperature mean-field theory using as a standard the Hamiltonian and model space of the shell model Monte Carlo calculations. Two examples are considered: the nucleus $^{162}$Dy, representing a heavy deformed nucleus, and $^{148}$Sm, representing a nearby heavy spherical nucleus with strong pairing correlations. The errors inherent in the finite-temperature Hartree-Fock and Hartree-Fock-Bogoliubov approximations are analyzed by comparing the entropies of the grand canonical and canonical ensembles, as well as the level density at the neutron resonance threshold, with shell model Monte Carlo (SMMC) calculations, which are accurate up to well-controlled statistical errors. The main weak points in the mean-field treatments are seen to be: (i) the extraction of number-projected densities from the grand canonical ensembles, and (ii) the symmetry breaking by deformation or by the pairing condensate. In the absence of a pairing condensate, we confirm that the usual saddle-point approximation to extract the number-projected densities is not a significant source of error compared to other errors inherent to the mean-field theory. We also present an alternative formulation of the saddle-point approximation that makes direct use of an approximate particle-number projection and avoids computing the usual three-dimensional Jacobian of the saddle-point integration. We find that the pairing condensate is less amenable to approximate particle-number projection methods due to the explicit violation of particle-number conservation in the pairing condensate. Nevertheless, the Hartree-Fock-Bogoliubov theory is accurate to less than one unit of entropy for $^{148}$Sm at the neutron threshold energy, which is above the pairing phase transition.

Second-order Møller–Plesset perturbation (MP2) theory at finite temperature: relation with Surján’s density matrix MP2 and its application to linear-scaling divide-and-conquer method

In 2005, Surjan showed two explicit formulas for evaluating the second-order Moller–Plesset perturbation (MP2) energy as a functional of the Hartree–Fock density matrix D (Chem Phys Lett 406:318, 2005 ), which are referred to as the Δ E MP2[D] functionals. In this paper, we present the finite-temperature (FT) MP2 energy functionals of the FT Hartree–Fock density matrix. There are also two formulas for the FT-MP2, namely the conventional and renormalized ones; the latter of which has recently been formulated by Hirata and He (J Chem Phys 138:204112, 2013 ). We proved that there exists one-to-one correspondence between the formulas of two FT-MP2 and the Δ E MP2[D] functionals. This fact can explain the different behavior of two Δ E MP2[D] functionals when an approximate Hartree–Fock density matrix is applied, which was previously investigated by Kobayashi and Nakai (Chem Phys Lett 420:250, 2006 ). We also applied the FT-MP2 formalisms to the linear-scaling divide-and-conquer method for improving the accuracy with tiny addition of the computational efforts.

Testing the Monte Carlo–mean field approximation in the one-band Hubbard model

The canonical one-band Hubbard model is studied using a computational method that mixes the Monte Carlo procedure with the mean field approximation. This technique allows us to incorporate thermal fluctuations and the development of short-range magnetic order above ordering temperatures, contrary to the crude finite-temperature Hartree-Fock approximation, which incorrectly predicts a N\'eel temperature $T_N$ that grows linearly with the Hubbard $U/t$. The effective model studied here contains quantum and classical degrees of freedom. It thus belongs to the "spin fermion" model family widely employed in other contexts. Using exact diagonalization, supplemented by the traveling cluster approximation, for the fermionic sector, and classical Monte Carlo for the classical fields, the Hubbard $U/t$ vs. temperature $T/t$ phase diagram is studied employing large three and two dimensional clusters. We demonstrate that the method is capable of capturing the formation of local moments in the normal state without long-range order, the non-monotonicity of $T_N$ with increasing $U/t$, the development of gaps and pseudogaps in the density of states, and the two-peak structure in the specific heat. Extensive comparisons with determinant quantum Monte Carlo results suggest that the present approach is qualitatively, and often quantitatively, accurate, particularly at intermediate and high temperatures. Finally, we study the Hubbard model including plaquette diagonal hopping (i.e. the $t-t^\prime$ Hubbard model) in two dimensions and show that our approach allows us to study low temperature properties where determinant quantum Monte Carlo fails due to the fermion sign problem. Future applications of this method include multi-orbital Hubbard models such as those needed for iron-based superconductors.

Pairing properties of hot nuclei within the finite temperature Hartree Fock+BCS approximation

In this work, finite temperature Hartree Fock BCS (FT+HFBCS) calculations are performed in Sn isotopes with the zero-range Skyrme type interactions. Vanishing of pairing correlations due to the increase in temperature are investigated with various types of Skyrme interactions. In addition, dependence of the critical temperature with respect to the several properties of these energy density functionals are analysed.

Neutral Pion Condensation in Hot and Dense Nuclear Matter

From a viewpoint of the effective interaction and the equivalence between the potential and the field descriptions, a framework to treat the π 0 condensation at finite temperature is presented. This framework is based on the finite-temperature Hartree-Fock method with use of effective two-nucleon potentials. It is successfully applied to the ALS model of π 0 condensate and can be extended to the case including explicitly the isobar Δ(1232) degrees of freedom. In the studies on neutron matter, symmetric nuclear matter and supernova matter, the following points are found; (i) the transition density of π 0 condensation increases remarkably with the temperature, (ii) the cause is a reduction of entropy in the ordered π 0 -condensed phase, (iii) the transition is of first-order and the phase diagram contains a wide mixed phase region, (iv) the equation of state is significantly softened even at finite temperature by the phase transition

Comparison of density functional approximations and the finite-temperature Hartree-Fock approximation in warm dense lithium

We compare the behavior of the finite-temperature Hartree-Fock model with that of thermal density functional theory using both ground-state and temperature-dependent approximate exchange functionals. The test system is bcc Li in the temperature-density regime of warm dense matter (WDM). In this exchange-only case, there are significant qualitative differences in results from the three approaches. Those differences may be important for Born-Oppenheimer molecular dynamics studies of WDM with ground-state approximate density functionals and thermal occupancies. Such calculations require reliable regularized potentials over a demanding range of temperatures and densities. By comparison of pseudopotential and all-electron results at T=0 K for small Li clusters of local bcc symmetry and bond lengths equivalent to high density bulk Li, we determine the density ranges for which standard projector augmented wave (PAW) and norm-conserving pseudopotentials are reliable. Then, we construct and use all-electron PAW data sets with a small cutoff radius that are valid for lithium densities up to at least 80 g/cm{3}.

Solution of the Skyrme–Hartree–Fock–Bogolyubov equations in the Cartesian deformed harmonic-oscillator basis.: (VII) hfodd (v2.49t): A new version of the program

We describe the new version (v2.49t) of the code HFODD which solves the nuclear Skyrme Hartree-Fock (HF) or Skyrme Hartree-Fock-Bogolyubov (HFB) problem by using the Cartesian deformed harmonic-oscillator basis. In the new version, we have implemented the following physics features: (i) the isospin mixing and projection, (ii) the finite temperature formalism for the HFB and HF+BCS methods, (iii) the Lipkin translational energy correction method, (iv) the calculation of the shell correction. A number of specific numerical methods have also been implemented in order to deal with large-scale multi-constraint calculations and hardware limitations: (i) the two-basis method for the HFB method, (ii) the Augmented Lagrangian Method (ALM) for multi-constraint calculations, (iii) the linear constraint method based on the approximation of the RPA matrix for multi-constraint calculations, (iv) an interface with the axial and parity-conserving Skyrme-HFB code HFBTHO, (v) the mixing of the HF or HFB matrix elements instead of the HF fields. Special care has been paid to using the code on massively parallel leadership class computers. For this purpose, the following features are now available with this version: (i) the Message Passing Interface (MPI) framework, (ii) scalable input data routines, (iii) multi-threading via OpenMP pragmas, (iv) parallel diagonalization of the HFB matrix in the simplex breaking case using the ScaLAPACK library. Finally, several little significant errors of the previous published version were corrected.

NUCLEAR EXCITATIONS AND WEAK INTERACTION RATES AT FINITE TEMPERATURE

Self-consistent theory frameworks have been introduced for description of excitations in nuclei at finite temperature, based on energy density functionals formulated using i) relativistic mean field Lagrangian with density dependent meson-nucleon vertex functions, and ii) Skyrme-type functionals. Finite temperature random phase approximation (FTRPA) has been employed in description of multipole excitations and charge-exchange modes. It is shown that in the temperature range T = 1–2 MeV additional transition strength appears at low energies due to thermal unblocking of single-particle orbitals close to the Fermi level. Within the finite temperature Hartree-Fock+RPA based on Skyrme functionals, the electron capture cross sections have been studied for target nuclei at temperatures relevant in modeling the supernova evolution.

Systematic study of fission barriers of excited superheavy nuclei

A systematic study of fission-barrier dependence on excitation energy has been performed using the self-consistent finite-temperature Hartree-Fock+BCS (FT-HF+BCS) formalism with the SkM* Skyrme energy density functional. The calculations have been carried out for even-even superheavy nuclei with Z ranging between 110 and 124. For an accurate description of fission pathways, the effects of triaxial and reflection asymmetric degrees of freedom have been fully incorporated. Our survey demonstrates that the dependence of isentropic fission barriers on excitation energy changes rapidly with particle number, pointing to the importance of shell effects even at large excitation energies characteristic of compound nuclei. The fastest decrease of fission barriers with excitation energy is predicted for deformed nuclei around N = 164 and spherical nuclei around N = 184 that are strongly stabilized by ground-state shell effects. For nuclei ^{240}Pu and ^{256}Fm, which exhibit asymmetric spontaneous fission, our calculations predict a transition to symmetric fission at high excitation energies due to the thermal quenching of static reflection asymmetric deformations.

N/Z DEPENDENCE OF PROJECTILE FRAGMENTATION

The N/Z dependence of projectile fragmentation at relativistic energies has been studied in a recent experiment at the GSI laboratory with the ALADiN forward spectrometer coupled to the LAND neutron detector. Besides a primary beam of 124 Sn , also secondary beams of 124 La and 107 Sn delivered by the FRS fragment separator have been used in order to extend the range of isotopic compositions of the produced spectator sources. With the achieved mass resolution of ΔA/A ≈ 1.5%, lighter isotopes with atomic numbers Z ≤ 10 are individually resolved. The presently ongoing analyses of the measured isotope yields focus on isoscaling and its relation to the properties of hot fragments at freeze-out and on the derivation of chemical freeze-out temperatures which are found to be independent of the isotopic composition of the studied systems. The latter result is at variance with the predictions for limiting temperatures as obtained with finite-temperature Hartree-Fock calculations.

Pairing interactions and vanishing pairing correlations in hot nuclei

Finite temperature Hartree–Fock–Bogoliubov calculations are performed in Sn isotopes using Skyrme and zero-range, density-dependent pairing interactions. For both stable and very neutron-rich nuclei the critical temperature at which pairing correlations vanish is independent of the volume/surface nature of the pairing interaction. The value of the critical temperature follows approximatively the empirical rule T c ≃ 0.5 Δ T = 0 for all the calculated isotopes, showing that the critical temperature could be deduced from the pairing gap at zero temperature. On the other hand, the pairing gap at temperatures just below T c is strongly sensitive to the volume/surface nature of the pairing interaction.

Orbital magnetic properties of quantum dots: The role of electron-electron interactions

We study the magnetic orbital response of a system of N interacting electrons confined in a two-dimensional geometry and subjected to a perpendicular magnetic field in the finite-temperature Hartree-Fock approximation. The electron-electron interaction is modeled by a short-range Yukawa-type potential. We calculate the ground-state energy, magnetization, and magnetic susceptibility as a function of the temperature, potential range, and magnetic field. We show that the amplitude and period of oscillations in the magnetic susceptibility are strongly affected by the electron-electron interaction as evidenced in experimental results. The zero-field susceptibility displays both paramagnetic and diamagnetic phases as a function of temperature and the number of confined electrons.

Analytical expression for the first-order density matrix of ad-dimensional harmonically confined Fermi gas at finite temperature

We present an exact closed form expression for the {\em finite temperature} first-order density matrix of a harmonically trapped ideal Fermi gas in any dimension. This constitutes a much sought after generalization of the recent results in the literature, where exact expressions have been limited to quantities derived from the {\em diagonal} first-order density matrix. We compare our exact results with the Thomas-Fermi approximation (TFA) and demonstrate numerically that the TFA provides an excellent description of the first-order density matrix in the large-N limit. As an interesting application, we derive a closed form expression for the finite temperature Hartree-Fock exchange energy of a two-dimensional parabolically confined quantum dot. We numerically test this exact result against the 2D TF exchange functional, and comment on the applicability of the local-density approximation (LDA) to the exchange energy of an inhomogeneous 2D Fermi gas.

Quantum and thermal phase transitions in the Bechgaard salts and their analogs

In order to investigate quantum and thermal phase transitions in the Bechgaard salts and their sulfur analogs, we perform finite-temperature Hartree-Fock calculations in one dimension with particular emphasis on the interplay between charge ordering and lattice instability. The coexisting charge- and spin-density-wave state as well as its precursor fluctuations in (TMTSF) 2 PF 6 and the lattice tetramerization in (TMTTF) 2 ReO 4 are well interpreted.

Reconstruction of fractional quantum Hall edges.

We study the interplay of electron-electron interaction, confining potential and effects of finite temperature at the edge of a quantum Hall liquid. Our exact diagonalization calculation indicates that edge reconstruction occurs in the fractional quantum Hall regime for a variety of confining potential, including ones that correspond to a "sharp" edge. Our finite temperature Hartree-Fock calculation for integer quantum Hall edges indicates that reconstruction is suppressed above a certain temperature. We discuss the implication of our results on recent edge tunneling and microwave absorption experiments.

Halogen-Bridged Binuclear Metal Complexes Under Static Pressure

Two contrasting observations for halogen ( x )-bridged binuclear platinum complexes R 4 [Pt 2 O 5 H 2 )X] n H 2 O are interpreted by finite-temperature Hartree-Fock calculations. Assuming pressure-induced simultaneous increases of the interdimer electron transfer and Coulomb repulsiion, we reveal that an applied pressure turns the initial charge-polarization state into a charge-density-wave state at high tmperatures, while the charge-density-wave stte oppositely declines with increasing pressure at low tempertures. We predict pressure-induced phase transitions in another family of MMX complexes, Pt 2 (CH 3 CS 2 ) 4 I, as well. MMX chain pressure-induced phase transition Peierls-Hubbard model

Pressure-induced phase transitions of halogen-bridged binuclear metal complexesR4[Pt2(P2O5H2)4X]⋅nH2O

Recent contrasting observations for halogen (X)-bridged binuclear platinum complexes R_4[Pt_2(P_2O_5H_2)_4X]nH_2O, that is, pressure-induced Peierls and reverse Peierls instabilities, are explained by finite-temperature Hartree-Fock calculations. It is demonstrated that increasing pressure transforms the initial charge-polarization state into a charge-density-wave state at high temperatures, whereas the charge-density-wave state oppositely declines with increasing pressure at low temperatures. We further predict that higher-pressure experiments should reveal successive phase transitions around room temperature.