Hartree-Fock Ground State Research Articles

Weyl laws for interacting particles

We study grand-canonical interacting fermionic systems in the mean-field regime, in a trapping potential. We provide the first order term of integrated and pointwise Weyl laws, but in the case with interaction. More precisely, we prove the convergence of the densities of the grand-canonical Hartree-Fock ground state to the Thomas-Fermi ground state in the semiclassical limit ℏ → 0. For the proof, we write the grand-canonical version of the results of Fournais, Lewin, and Solovej [Calculus Var. Partial Differ. Equations 57, 105 (2018)] and Conlon [Commun. Math. Phys. 88, 133 (1983)].

Utilizing the Hartree-Fock Method to Analyze Carbon Dioxide Molecule

The determination of the ground state of the carbon dioxide molecule is conducted through matrix Hartree-Fock computations for a polyatomic molecule. Utilization of basis sets is implemented to achieve a precision that approaches the sub-μ Hartree of the CO<sub>2</sub> molecule for the energy levels. Utilizing a 28s14p14d14f atom-centered and a 24sl0plld bond-centered Gaussian basis set, the upper limit for the Hartree-Fock ground state energy of CO<sub>2</sub> at its practical picture has been calculated, yielding a value of -187.125408 Hartree. Utilizing our knowledge in diatomic molecules, we approximate the precision of this measurement to fall within 5-6 μ Hartree. The current computations offer a methodology to assess the precision of different basis sets frequently utilized in molecular self-consistent field investigations. These basis sets encompass STOJG, 4-31G, 6-31G, 6-31G(3d), 6-31IG, 6-311+G(3d0, D95, and D95V+(3d)), alongside cc-pVDZ, uug-cc-pVDZ, cc-pVTZ, and aug-cc-pVTZ, which have been proposed for the purpose of investigating electron correlation. We have recently performed calculations on the CO1 ground state at the nuclear geomevy employed in the presenl work using the basis sets proposed by Dunning and co-workers and designated cc-pVQZ. aupcc-pVQZ, cc-pVSZ and nupcc-pVSZ. The estimated basis set truncation emrs for these sets are 0.00371, 0.003 19, 0.00047 and 0.00039 Hartree respectively.

On relative bounds for interacting Fermion operators

We consider a system of interacting fermions whose Hamiltonian is unitarily transformed so that the interaction is a quartic perturbation of the Hartree–Fock effective Hamiltonian. It is shown under natural model assumptions that the interaction does not admit a relative bound with respect to the effective Hamiltonian that is uniform in the system’s size. This bound is exemplified on the Hubbard model with nearest neighbor interaction on a discrete d -dimensional torus of length L around its Hartree–Fock ground state and derive relative bounds of the effective interaction with respect to the effective kinetic energy. It is shown that there are no relative bounds uniform in L .

Revisited relativistic Dirac-Hartree-Fock X-ray scattering factors. I. Neutral atoms with Z = 2-118.

In this first of a series of publications, the X-ray scattering factors for neutral atoms are revisited. Using the recently developed DBSR_HF program [Zatsarinny & Froese Fischer (2016). Comput. Phys. Comm. 202, 287-303] the fully relativistic Dirac-Hartree-Fock ground-state wavefunctions for all atoms with Z = 2-118 (He-Og) have been calculated using the extended average level scheme and including both the Breit interaction correction to the electronic motion due to magnetic and retardation effects, and the Fermi distribution function for the description of the nuclear charge density. The comparison of our wavefunctions with those obtained in several previous studies in terms of the total and orbital (spinor) electronic energies, and a number of local and integrated total and orbital properties, confirmed the quality of the generated wavefunctions. The employed dense radial grid combined with the DBSR_HF's B-spline representation of the relativistic one-electron orbitals allowed for a precise integration of the X-ray scattering factors using a newly developed Fortran program SF. Following the established procedure [Maslen et al. (2006). International Tables for Crystallography, Vol. C, Section 6.1.1, pp. 554-589], the resulting X-ray scattering factors have been interpolated in the 0 ≤ sin θ/λ ≤ 2 Å-1 and 2 ≤ sin θ/λ ≤ 6 Å-1 ranges using the recommended analytical functions with both the four- (which is a current convention) and five-term expansions. An exhaustive comparison of the newly generated X-ray scattering factors with the International Union of Crystallography recommended values and those from a number of previous studies showed an overall good agreement and allowed identification of a number of typos and inconsistencies in the recommended quantities. A detailed analysis of the results suggests that the newly derived values may represent an excellent compromise among all the previous studies. The determined conventional interpolating functions for the two sin θ/λ intervals show, on average, the same accuracy as the recommended parametrizations. However, an extension of each expansion by only a single term provides a significant improvement in the accuracy of the interpolated values for an overwhelming majority of the atoms. As such, an updated set of the fully relativistic X-ray scattering factors and the interpolating functions for neutral atoms with Z = 2-118 can be easily incorporated into the existing X-ray diffraction software with only minor modifications. The outcomes of the undertaken research should be of interest to members of the crystallographic community who push the boundaries of the accuracy and precision of X-ray diffraction studies.

Local magnetic moments due to loop currents in metals

We present Hartree-Fock calculations on a simple model to obtain the conditions of formation of local magnetic moments due to loop currents ${L}_{o}$ and spin-loop currents ${L}_{s}$ and compare them to the conditions of formation of local spin moments $M$ which were given long ago in a similar approximation by Anderson. A model with three degenerate orbitals sitting on an equilateral triangle, with on-site and nearest-neighbor repulsions $U$ and $V$, respectively, and intersite kinetic energy, hybridizing with conduction electrons with a parameter $\mathrm{\ensuremath{\Delta}}$ is investigated. ${L}_{o}$ and ${L}_{s}$ are promoted by large $V/\mathrm{\ensuremath{\Delta}}$ and their magnitude is relatively unaffected by $U/\mathrm{\ensuremath{\Delta}}$. Spin-magnetic moments $M$ promoted by large $U/\mathrm{\ensuremath{\Delta}}$ on the other hand are adversely affected by $V/\mathrm{\ensuremath{\Delta}}$. In this model, ${L}_{o}$ for $V$ multiplied by the number of neighbors is approximately the same as the $M$ promoted by $U$ in Anderson's local model for $M$. ${L}_{o}$ and ${L}_{s}$ are degenerate if exchange interactions and Hund's rule are neglected but ${L}_{o}$ is favored when they are included. Many of the qualitative results are visible in an expression for the Hartree-Fock ground state energy derived as a function of small ${L}_{o}, {L}_{s}$, and $M$. Numerical minimization of the Hartree-Fock energy is presented for larger values. We also briefly discuss the connection and differences of the interaction-generated orbital currents and spin currents discussed here and generalized to a lattice with the topological states in metals and semiconductors.

Multiplicity, localization, and domains in the Hartree–Fock ground state of the two-dimensional Hubbard model

We explore certain properties of the Hartree–Fock approximation to the ground state of the two-dimensional Hubbard model, emphasizing the fact that in the Hartree approach there is an enormous multiplicity of self-consistent solutions which are nearly degenerate in energy, reminiscent of a spin glass, but which may differ substantially in other bulk properties. It is argued that this multiplicity is physically relevant at low temperatures. We study the localization properties of the one-particle wavefunctions comprising the Hartree–Fock states, and find that these are unlocalized at small and moderate values of U/t, in particular in the stripe region, but become highly localized at values corresponding to strong repulsion. We also find rectangular domains as well as stripes in the stripe region of the phase diagram, and study pair correlations in the neighborhood of half-filling.

Cumulant Lanczos tridiagonalization for calculation of the correlation energy in a molecule

The problem of constructing a guaranteed convergent sequence of corrections to the Hartree–Fock ground state energy of a molecule without storing the many-electron wave function is considered. Several methods based on cumulants are considered and it is shown that such sequence is obtained by Lanczos tridiagonalization, in which the elements of the tridiagonal matrix are calculated through cumulants.

Multiplicity, Localization, and Domains in the Hartree-Fock Ground State of the Two-Dimensional Hubbard Model

Multiplicity, Localization, and Domains in the Hartree-Fock Ground State of the Two-Dimensional Hubbard Model

Excited States from State-Specific Orbital-Optimized Pair Coupled Cluster.

The pair coupled cluster doubles (pCCD) method (where the excitation manifold is restricted to electron pairs) has a series of interesting features. Among others, it provides ground-state energies very close to what is obtained with doubly occupied configuration interaction (DOCI), but with a polynomial cost (compared with the exponential cost of the latter). Here, we address whether this similarity holds for excited states by exploring the symmetric dissociation of the linear H4 molecule. When ground-state Hartree–Fock (HF) orbitals are employed, pCCD and DOCI excited-state energies do not match, a feature that is assigned to the poor HF reference. In contrast, by optimizing the orbitals at the pCCD level (oo-pCCD) specifically for each excited state, the discrepancies between pCCD and DOCI decrease by 1 or 2 orders of magnitude. Therefore, the pCCD and DOCI methodologies still provide comparable energies for excited states, but only if suitable, state-specific orbitals are adopted. We also assessed whether a pCCD approach could be used to directly target doubly excited states, without having to resort to the equation-of-motion (EOM) formalism. In our Δoo-pCCD model, excitation energies are extracted from the energy difference between separate oo-pCCD calculations for the ground state and the targeted excited state. For a set comprising the doubly excited states of CH+, BH, nitroxyl, nitrosomethane, and formaldehyde, we found that Δoo-pCCD provides quite accurate excitation energies, with root-mean-square deviations (with respect to full configuration interaction results) lower than those of CC3 and comparable to those of EOM-CCSDT, two methods with a much higher computational cost.

Symmetry breaking in the double moiré superlattices of relaxed twisted bilayer graphene on hexagonal boron nitride

We study the atomic and electronic structures of the commensurate double moir\'{e} superlattices in fully relaxed twisted bilayer graphene (TBG) nearly aligned with the hexagonal boron nitride (BN). The single-particle effective Hamiltonian ($\hat{H}^0$) taking into account the relaxation effect and the full moir\'{e} Hamiltonian introduced by BN has been built for TBG/BN. The mean-field (MF) band structures of the self-consistent Hartree-Fock (SCHF) ground states at different number ($\nu$) of filled flat bands relative to the charge neutrality point (CNP) are obtained based on $\hat{H}^0$ in the plane-wave-like basis. The single-particle flat bands in TBG/BN become separated by the opened gap at CNP due to the symmetry breaking in $\hat{H}^0$. We find that the broken $C_2$ symmetry in $\hat{H}^0$ mainly originates from the intralayer inversion-asymmetric structural deformation in the graphene layer adjacent to BN, which introduces spatially non-uniform modifications of the intralayer Hamiltonian. The gapped flat bands have finite Chern numbers. For TBG/BN with the magic twist angle, the SCHF ground states with $|\nu|$ = 1-3 are all insulating with narrow MF gaps. When the flat conduction bands are filled, the gap at $\nu$ = 1 is smaller than that at $\nu$ = 3, suggesting that the nontrivial topological properties associated with the flat Chern bands are more likely to be observed at $\nu = 3$. This is similar for negative $\nu$ with empty valence bands. The dependence of the electronic structure of TBG/BN on positive $\nu$ is roughly consistent with recent experimental observations.

Performance of the LRESC Model on top of DFT Functionals for Relativistic NMR Shielding Calculations.

The linear response within the elimination of the small component model (LRESC) is an insightful and computationally efficient method for including relativistic effects on molecular properties like the nuclear magnetic shielding constants, spin-rotation constant, g-tensor, and electric field gradient of heavy atom containing molecules with atoms belonging up to the sixth row of the periodic table. One of its main advantages is its capacity to analyze the electronic origin of the different relativistic correcting terms. Until now, it was always applied on top of Hartree-Fock ground-state wave functions (LRESC/HF) to calculate and analyze NMR shieldings. In this work, we show the performance of the LRESC formalism on top of some density functional theory (DFT) functionals to compute tin shielding constants in SnX4 (X = H, F, Cl, Br, I) molecular systems. We analyze the performance of each LRESC/DFT scheme on reproducing the electronic mechanisms of the shieldings, taking as a benchmark the results of relativistic calculations at the RPA level of approach (4c/RPA). As in previous works, we divide the LRESC relativistic correcting terms into two groups: core-dependent and ligand-dependent contributions. It is shown here that core-dependent corrections are well-reproduced for the selected DFT functionals, but some differences arise in the ligand-dependent ones. We focus on the performance of different functionals, including the same electron correlation part but containing different amounts of HF exchange. The best results are obtained for the BHandHLYP functional (50% of HF exchange) and the worst for BLYP (0%). When the percentage of HF exchange increases, ligand-dependent contributions are better described, and the final LRESC/DFT results are closer to those obtained with LRESC/HF and 4c/RPA methods. The spin-orbit correction to the shielding constant is one of the main ligand-dependent contributions (there are two more) with total value depending on the amount of HF exchange included in the functional. When the amount of HF exchange decreases, the spin-orbit contribution becomes larger, overestimating the shielding constant even when nonrelativisitc DFT values are much smaller than the nonrelativistic HF ones, as it happens for the heaviest molecular system studied here (SnI4).

Magnetic stability for the Hartree-Fock ground state in two dimensional Rashba-Gauge electronic systems

The magnetic Hartree Fock ground state stability for a two-dimensional interacting electron system with Rashba-type coupling is studied by implementing the standard many body Green’s function formalism. The externally applied electrical field E enters into the Hamiltonian model through a local gauge-type transformation ∼EjAj(k), with Aj(k) as the spin gauge vector potential. Phase diagrams associated to the average spin polarization, the Fermi energy, electron density and energy band gap at zero temperature are constructed. The magnetic polarization state as a function of E is obtained by minimizing the Helmholtz energy functional F with respect to the average Z-spin: 〈σ̂Z〉. We have found that the electric field might reverse the magnetic ground state, unlike the characteristic decreasing behavior associated to the linear term σ̂·(k×E).

Lower bound on the Hartree-Fock energy of the electron gas

The Hartree-Fock ground state of the Homogeneous Electron Gas is never translation invariant, even at high densities. As proved by Overhauser, the (paramagnetic) free Fermi Gas is always unstable under the formation of spin or charge density waves. We give here the first explicit bound on the energy gain due to the breaking of translational symmetry. Our bound is exponentially small at high density, which justifies posteriori the use of the non-interacting Fermi Gas as a reference state in the large-density expansion of the correlation energy of the Homogeneous Electron Gas. We are also able to discuss the positive temperature phase diagram and prove that the Overhauser instability only occurs at temperatures which are exponentially small at high density. Our work sheds a new light on the Hartree-Fock phase diagram of the Homogeneous Electron Gas.

Constraining symmetry energy at subnormal density by isovector giant dipole resonances of spherical nuclei * * Supported by the National Natural Science Foundation of China (11875328)

In our previous study, the deduced Langevin equation has been applied to investigate the isoscalar giant monopole resonance. In the current study, the framework is extended to study the isovector giant dipole resonance (IVGDR). The potential well in the IVGDR is calculated by separating the neutron and proton densities based on the Hartree-Fock ground state. Subsequently, the Langevin equation is solved self-consistently, resulting in the centroid energy of the IVGDR without width. The symmetry energy around the density of 0.02 fm−3 contributes the most to the potential well in the IVGDR. By comparison with the updated experimental data of IVGDR energies in spherical nuclei, the calculations within 37 sets of Skyrme functionals suggest the symmetry energy to be in the range of 8.13-9.54 MeV at a density of 0.02 fm−3.

Communication: A mean field platform for excited state quantum chemistry

We present a mean field theory for excited states that is broadly analogous to ground state Hartree-Fock theory. Like Hartree-Fock, our approach is deterministic, state-specific, applies a variational principle to a minimally correlated ansatz, produces energy stationary points, relaxes the orbital basis, has a Fock-build cost-scaling, and can serve as the foundation for correlation methods such as perturbation theory and coupled cluster theory. To emphasize this last point, we pair our mean field approach with an excited state analog of second order Møller-Plesset theory and demonstrate that in water, formaldehyde, neon, and stretched lithium fluoride, the resulting accuracy far exceeds that of configuration interaction singles and rivals that of equation of motion coupled cluster.

Anderson localisation for infinitely many interacting particles in Hartree–Fock theory

We prove the occurrence of Anderson localisation for a system of infinitely many particles interacting with a short range potential, within the ground state Hartree–Fock approximation. We assume that the particles hop on a discrete lattice and that they are submitted to an external periodic potential which creates a gap in the non-interacting one particle Hamiltonian. We also assume that the interaction is weak enough to preserve a gap. We prove that the mean-field operator has exponentially localised eigenvectors, either on its whole spectrumor at the edges of its bands, depending on the strength of the disorder.

Quantum correlations in critical XXZ system and LMG model

We investigate the quantum phase transitions for the [Formula: see text] spin-1/2 chains via the quantum correlations between the nearest and next-to-nearest neighbor spins characterized by negativity, information deficit, trace distance discord and local quantum uncertainty. It is shown that all these correlations exhibit the quantum phase transitions at [Formula: see text]. However, only information deficit and local quantum uncertainty can demonstrate quantum phase transitions at [Formula: see text]. The analytical and numerical behaviors of the quantum correlations for the [Formula: see text] system are presented. We also consider quantum correlations in the Hartree–Fock ground state of the Lipkin–Meshkov–Glick (LMG) model.

Temperature dependent structural and vibrational properties of hydroxyapatite: A theoretical and experimental study

Abstract We describe the temperature dependence of the vibrational, rotational and translational partition functions of the activated complex of hydroxyapatite (HAp). Computed data show that the vibrational modes have a larger contribution towards the partition function of HAp compared to the rotational and translational contributions. X-ray diffraction (XRD), and Fourier transform infrared (FTIR) spectroscopy have been applied for HAp in the temperature range from 730 °C to 1030 °C at steps of 100 °C. Temperature dependent density functional theory (DFT), B3LYP, and Ground State Hartree-Fock (HF) with 6–311G basis set calculations were also applied to HAp to calculate FTIR spectra, HOMO, and LUMO energies, and density of states (DOS), and the results have been compared to experimental findings. The present results underline that experimental measurements and theoretical calculations of unit cell parameters, and the intensities of most of the FTIR data for HAp are nearly independent on temperature.

Optimizing phonon space in the phonon-coupling model

We present a new scheme to select the most relevant phonons in the phonon-coupling model, named here time-blocking approximation (TBA). The new criterion, based on the phonon-nucleon coupling strengths rather than on $B(EL)$ values, is more selective and thus produces much smaller phonon spaces in TBA. This is beneficial in two respects: first, it curbs down the computational cost, and second, it reduces the danger of double counting in the expansion basis of TBA. We use here TBA in a form where the coupling strength is regularized to keep the given Hartree-Fock ground state stable. The scheme is implemented in an RPA and TBA code based on the Skyrme energy functional. We first explore carefully the cutoff dependence with the new criterion and can work out a natural (optimal) cutoff parameter. Then we use the freshly developed and tested scheme to a survey of giant resonances and low-lying collective states in six doubly magic nuclei looking also on the dependence of the results when varying the Skyrme parametrization.

A coupled cluster theory with iterative inclusion of triple excitations and associated equation of motion formulation for excitation energy and ionization potential.

A single reference coupled cluster theory that is capable of including the effect of connected triple excitations has been developed and implemented. This is achieved by regrouping the terms appearing in perturbation theory and parametrizing through two different sets of exponential operators: while one of the exponentials, involving general substitution operators, annihilates the ground state but has a non-vanishing effect when it acts on the excited determinant, the other is the regular single and double excitation operator in the sense of conventional coupled cluster theory, which acts on the Hartree-Fock ground state. The two sets of operators are solved as coupled non-linear equations in an iterative manner without significant increase in computational cost than the conventional coupled cluster theory with singles and doubles excitations. A number of physically motivated and computationally advantageous sufficiency conditions are invoked to arrive at the working equations and have been applied to determine the ground state energies of a number of small prototypical systems having weak multi-reference character. With the knowledge of the correlated ground state, we have reconstructed the triple excitation operator and have performed equation of motion with coupled cluster singles, doubles, and triples to obtain the ionization potential and excitation energies of these molecules as well. Our results suggest that this is quite a reasonable scheme to capture the effect of connected triple excitations as long as the ground state remains weakly multi-reference.