Hartree-Fock Results Research Articles

Applicability of noninteger bessel type orbital basis sets: numerical and analytical approaches

Recently, usefulness of the noninteger principal quantum numbers for Bessel type orbitals was discussed by Weniger in (Adv Quant Chem 83:209–237, 2021). In this study, we analyzed the applicability and numerical accuracy of basis sets of noninteger Bessel type orbitals to electronic structure calculations. Both numerical and analytical approaches are applied to two-electron atomic systems. The results of the numerical test demonstrated the potential of the noninteger values of principal quantum number for the improving of Bessel type functions approach in use of LCAO methods. Nevertheless, the analytical approach is still not suitable and in development and needs to be investigated further.The performance of the presented basis functions is also compared to the numerical Hartree–Fock results.

Radial Kohn–Sham problem via integral-equation approach

We present a numerical tool for solving the non-relativistic Kohn–Sham problem for spherically-symmetric atoms. It treats the Schrödinger equation as an integral equation relying heavily on convolutions. The solver supports different types of exchange-correlation functionals including screened and long-range corrected hybrids. We implement a new method for treating range separation based on the complementary error function kernel. The present tool is applied in non-relativistic total energy calculations of atoms. A comparison with ultra-precise reference data[Cinal, JOMC 58, 1571 (2020)] shows a 14-digit agreement for Hartree–Fock results. We provide further benchmark data obtained with 5 different exchange-correlation functionals.

Comparative performance of different hyperbolic cosine functions and generalized B functions basis sets for atomic systems

Recently, we reported a new set of Bessel type functions, which include the radial part of generalized Bessel functions for LCAO calculations of atomic systems. In this study, to achieve further improvement of the performance of generalized Bessel type basis sets in the Hartree–Fock-Roothaan calculations, different hyperbolic cosine functions inserted into the radial part of those generalized Bessel functions. For this purpose, three different generalized hyperbolic cosine functions have been used to construct the generalized Bessel type hyperbolic cosine basis sets. The accuracies of generalized Bessel type hyperbolic cosine functions within the minimal basis sets approach are compared to show their superiority to conventional approaches those in the literature. The performance of the presented basis functions is also compared to the numerical Hartree–Fock results. Our virial ratios are in good agreement to within 8-digits of the −2. It is shown that the results obtained by the new basis sets surpass the quality and accuracy of existing Bessel type basis sets.

Implementation of nuclear time-dependent density-functional theory and its application to the nuclear isovector electric dipole resonance

Following a previous paper [Y. Shi, Phys. Rev. C 98, 014329(2018)], we present an extension of the density-functional theory to allow for dynamic calculations based on the obtained static Hartree-Fock results. We perform extensive benchmark calculations, by comparing the calculated results with that of an existing code Sky3D. To perform linear-response calculations using the TDDFT method, comparisons have been made with the finite-amplitude quasiparticle random-phase approximation (FAM-QRPA) method. We plan to apply the TDDFT method to a systematic description of the IVD resonances in the Zr, Mo, and Ru isotopes. The strengths of IVD resonances are calculated using two complementary methods: TDDFT and FAM-QRPA methods. For the TDDFT results, additional benchmark calculations have been performed using the well-tested code Sky3D. In these three models, the important ingredients which have major influence on the results, such as time-odd potentials, boundary conditions, smoothing procedures, spurious peaks etc., have been carefully examined. The current TDDFT and the Sky3D codes yield almost identical response functions once both codes use the same time-odd mean fields and absorbing boundary conditions. The strengths of the IVD resonances calculated using the TDDFT and FAM-QRPA methods agree reasonably well with the same position of the giant dipole resonance. Upon seeing a reasonable accuracy offered by the implemented code, we perform systematic TDDFT calculations for spherical Zr and Mo isotopes near $N=50$, where experimental data exist. For neutron-rich Zr, Mo, and Ru isotopes where shape evolution exist we predict the photoabsorption cross sections based on oblate and triaxial minima.

Sublinear scaling quantum chemical methods for magnetic shieldings in large molecules.

It is shown that quantum chemical calculations of the nuclear magnetic shieldings of a group of vicinal nuclei in a large molecule can be made sublinear scaling [i.e., O(1)] with respect to the molecular size. This is achieved by making both the diamagnetic and paramagnetic terms translation invariant [following the work of Ochsenfeld et al. J. Chem. Phys. 134, 074102 (2011)] and by making use of a local representation that is independent of the molecular size. Although only Hartree-Fock results are reported here, the idea can readily be extended to correlated wave function methods as well as the relativistic domain.

On approximate projection models

ABSTRACTApproximate projection (AP) schemes are widely used in computational studies of diradicals and transition metal systems to remove spin contamination in unrestricted density functional theory and Hartree–Fock results. Spin contamination results from an inability of single-reference descriptions to correctly model orbital degeneracies. Spin projection methods, including AP, avoid more expensive approaches using multi-reference wave functions to rectify the spin contamination error. Efficient derivations of the AP derivatives have allowed the computation of improved geometries and properties. In this work, we establish a connection between AP and formulations developed using Löwdin's projection operator to obtain an understanding of where the method will work and highlight situations in which caution should be exercised.

A grid-based variational method to the solution of the Schrödinger equation: the q-exponential and the near Hartree-Fock results for the ground state atomic energies.

A grid-based variational method was proposed and applied to the ground state energies of atoms from the first to the third period of the periodic table. The nonuniform grid in the radial coordinate was defined by a q-exponential sequence. Some unusual properties between the optimum q-parameters and the electronic energies or atomic numbers are described. The behavior of the electronic energy, with respect to the q-parameter, yields near Hartree-Fock accuracy with a relatively small number of integration points. A simple relationship between the optimum q-parameters and the atomic numbers was found, which allowed the determination of the optimum q-parameters for atoms of the same period from two results. The remarkable results provide a simple alternative route to reach accurate results. The consistent results also suggest that this is not a random or accidental effect, but some optimum condition achieved by using a q-exponential mesh grid. Graphical abstract The q-exponential and the near Hartree-Fock results for the ground state atomic energies.

Application of the Interacting Quantum Atoms Approach to the S66 and Ionic-Hydrogen-Bond Datasets for Noncovalent Interactions.

The interacting quantum atoms (IQA) method can assess, systematically and in great detail, the strength and physics of both covalent and noncovalent interactions. The lack of a pair density in density functional theory (DFT), which precludes the direct IQA decomposition of the characteristic exchange-correlation energy, has been recently overcome by means of a scaling technique, which can largely expand the applicability of the method. To better assess the utility of the augmented IQA methodology to derive quantum chemical decompositions at the atomic and molecular levels, we report the results of Hartree-Fock (HF) and DFT calculations on the complexes included in the S66 and the ionic H-bond databases of benchmark geometry and binding energies. For all structures, we perform single-point and geometry optimizations using HF and selected DFT methods with triple-ζ basis sets followed by full IQA calculations. Pairwise dispersion energies are accounted for by the D3 method. We analyze the goodness of the HF-D3 and DFT-D3 binding energies, the magnitude of numerical errors, the fragment and atomic distribution of formation energies, etc. It is shown that fragment-based IQA decomposes the formation energies in comparable terms to those of perturbative approaches and that the atomic IQA energies hold the promise of rigorously quantifying atomic and group energy contributions in larger biomolecular systems.

Energy decomposition analysis of single bonds within Kohn–Sham density functional theory

An energy decomposition analysis (EDA) for single chemical bonds is presented within the framework of Kohn-Sham density functional theory based on spin projection equations that are exact within wave function theory. Chemical bond energies can then be understood in terms of stabilization caused by spin-coupling augmented by dispersion, polarization, and charge transfer in competition with destabilizing Pauli repulsions. The EDA reveals distinguishing features of chemical bonds ranging across nonpolar, polar, ionic, and charge-shift bonds. The effect of electron correlation is assessed by comparison with Hartree-Fock results. Substituent effects are illustrated by comparing the C-C bond in ethane against that in bis(diamantane), and dispersion stabilization in the latter is quantified. Finally, three metal-metal bonds in experimentally characterized compounds are examined: a [Formula: see text]-[Formula: see text] dimer, the [Formula: see text]-[Formula: see text] bond in dizincocene, and the Mn-Mn bond in dimanganese decacarbonyl.

On the effectiveness of exponential type orbitals with hyperbolic cosine functions in atomic calculations

Unconventional basis functions, constructed from exponential type orbitals (ETOs) with hyperbolic cosine functions, are applied to Roothaan-Hartree-Fock calculations of atoms within the minimal basis sets framework. The most popular ETOs, Slater type orbitals, B functions and \(\psi ^{(\alpha ^*)}\) functions with \(\alpha ^*=2\), and two types of hyperbolic cosine functions, \(\cosh (\beta r)\) and \(\cosh (\beta r+\gamma )\), are used in this work. The performance of the present basis functions is investigated and compared to the conventional double-zeta Slater-type basis set and numerical Hartree-Fock results. The improvement in the atomic energies clearly demonstrates how the accuracy increases when we move from ETO to ETO with hyperbolic cosine basis functions. The resulting improved minimal basis sets can also be useful in molecular calculations.

Relativistic Brueckner—Hartree—Fock Theory for Finite Nuclei**Supported by the National Basic Research Program of China No 2013CB834400, the National Natural Science Foundation of China under Grants Nos 11175002, 11335002, 11405090, 11375015 and 11621131001, the Research Fund for the Doctoral Program of Higher Education under Grant No 20110001110087, the DFG cluster of excellence “Origin and Structure of the Universe” (www.universe-cluster.de), the CPSC

Starting with a bare nucleon-nucleon interaction, for the first time the full relativistic Brueckner—Hartree—Fock equations are solved for finite nuclei in a Dirac—Woods—Saxon basis. No free parameters are introduced to calculate the ground-state properties of finite nuclei. The nucleus 16O is investigated as an example. The resulting ground-state properties, such as binding energy and charge radius, are considerably improved as compared with the non-relativistic Brueckner—Hartree—Fock results and much closer to the experimental data. This opens the door for ab initio covariant investigations of heavy nuclei.

Communication: Projected Hartree Fock theory as a polynomial similarity transformation theory of single excitations

Spin-projected Hartree-Fock is written as a particle-hole excitation ansatz over a symmetry-adapted reference determinant. Remarkably, this expansion has an analytic expression that we were able to decipher. While the form of the polynomial expansion is universal, the excitation amplitudes need to be optimized. This is equivalent to the optimization of orbitals in the conventional projected Hartree-Fock framework of non-orthogonal determinants. Using the inverse of the particle-hole expansion, we similarity transform the Hamiltonian in a coupled-cluster style theory. The left eigenvector of the non-Hermitian Hamiltonian is constructed in a similar particle-hole expansion fashion, and we show that to numerically reproduce variational projected Hartree-Fock results, one needs as many pair excitations in the bra as the number of strongly correlated entangled pairs in the system. This single-excitation polynomial similarity transformation theory is an alternative to our recently presented double excitation theory, but supports projected Hartree-Fock and coupled cluster simultaneously rather than interpolating between them.

Investigation of Nuclear Ground State Properties of Fuel Materials of 232Th and 238U Using Skyrme-Extended-Thomas-Fermi Approach Method

It has been performed the nuclear ground state properties investigation of 232Th and 238U using Skyrme interaction. The local density, the kinetic energy density, and the spin-orbit density for proton and neutron have been calculated using Extended-Thomas-Fermi approach method. Then the calculation results have been compared with Skyrme-Hartree-Fock results (using HAFOMN and HFBRAD codes). Total energy calculations obtained from this research are -1792.973947 MeV (for 232Th) deviated 0.29244% from experiment energy and -1761.519459 MeV (for 238U) deviated 0.48369% from experiment energy. The distribution profiles of local density and local potential for 232Th and 238U are quite similar with Skyrme- Hartree-Fock results. It is indicated that Skyrme-Extended-Thomas-Fermi method can be used to study the nuclear ground state properties, especially even nuclei.

Fractional Electron Loss in Approximate DFT and Hartree-Fock Theory.

Plots of electronic energy vs electron number, determined using approximate density functional theory (DFT) and Hartree-Fock theory, are typically piecewise convex and piecewise concave, respectively. The curves also commonly exhibit a minimum and maximum, respectively, in the neutral → anion segment, which lead to positive DFT anion HOMO energies and positive Hartree-Fock neutral LUMO energies. These minima/maxima are a consequence of using basis sets that are local to the system, preventing fractional electron loss. Ground-state curves are presented that illustrate the idealized behavior that would occur if the basis set were to be modified to enable fractional electron loss without changing the description in the vicinity of the system. The key feature is that the energy cannot increase when the electron number increases, so the slope cannot be anywhere positive, meaning frontier orbital energies cannot be positive. For the convex (DFT) case, the idealized curve is flat beyond a critical electron number such that any additional fraction of an electron added to the system is unbound. The anion HOMO energy is zero. For the concave (Hartree-Fock) case, the idealized curve is flat up to some critical electron number, beyond which it curves down to the anion energy. A minimum fraction of an electron is required before any binding occurs, but beyond that, the full fraction abruptly binds. The neutral LUMO energy is zero. Approximate DFT and Hartree-Fock results are presented for the F → F(-) segment, and results approaching the idealized behavior are recovered for highly diffuse basis sets. It is noted that if a DFT calculation using a highly diffuse basis set yields a negative LUMO energy then a fraction of an electron must bind and the electron affinity must be positive, irrespective of whether an electron binds experimentally. This is illustrated by calculations on Ne → Ne(-).

Correlation effects in pyrochlore iridate thin films grown along the [111] direction

Over the past few years bulk pyrochlore iridates of the form $A_2$Ir$_2$O$_7$ (where $A$ is a rare earth element, Ir is iridium, and O is oxygen) have been studied as model systems for investigating the interplay of electronic correlations and strong spin-orbit coupling, particularly with the aim of finding correlation-driven topological phases. In this work, we use cellular dynamical mean field theory (CDMFT) to study effects of electronic correlations beyond Hartree-Fock theory in thin films of pyrochlore irradiates grown along the $[111]$ direction. We focus on the bilayer and trilayer systems, and compute the phase diagrams of these systems as a function of electron-electron interaction strength, which is modeled by an on-site Hubbard interaction. By evaluating the $Z_2$ invariant and Chern number using formulas based on the single-particle Green's function and the quasi-particle effective Hamiltonian, we show that on-site correlations can drive an interaction-induced topological phase transition, turning a time-reversal invariant topological insulator and a nearly flat band metal to a correlated Chern insulator (CI) in bilayer and trilayer systems, respectively. By comparing with the Hartree-Fock results, the CDMFT results show that quantum fluctuations enhance the robustness of the interaction-driven CI phase in the thin films. Furthermore, our numerical analysis of the quasiparticle spectrum reveals that the topological phases we find in our many-body calculations are adiabatically connected to those in the single-particle picture.

Exponential type orbitals with generalized hyperbolic cosine functions for atomic systems

Radial basis functions, constructed from Slater type rn∗−1e−ζr and generalized exponential type rn∗−1e−ζrμ functions with the generalized hyperbolic cosine type functions coshpq(βr) and coshpq(βrμ), where p and q are arbitrary parameters, are proposed and applied to Hartree–Fock–Roothaan calculations of atomic systems. The performance of new basis functions within the minimal basis sets framework has been compared to numerical Hartree–Fock results and previous results presented by similar basis functions in the literature. The results obtained by the new basis sets surpass the accuracy of existing basis sets of similar hyperbolic cosine type functions.

Second-order interaction corrections to the Fermi surface and the quasiparticle properties of dipolar fermions in three dimensions

We calculate the renormalized Fermi surface and the quasiparticle properties in the Fermi liquid phase of three-dimensional dipolar fermions to second order in the dipole-dipole interaction. Using parameters relevant to an ultracold gas of erbium atoms, we find that the second-order corrections typically renormalize the Hartree-Fock results by less than one percent. On the other hand, if we use the second-order correction to the compressibility to estimate the regime of stability of the system, the point of instability is already reached for a significantly smaller interaction strength than in the Hartree-Fock approximation.

On the Steady State Correlation Functions of Open Interacting Systems

We address the existence of steady state Green-Keldysh correlation functions of interacting fermions in mesoscopic systems for both the partitioning and partition-free scenarios. Under some spectral assumptions on the non-interacting model and for sufficiently small interaction strength, we show that the system evolves to a NESS which does not depend on the profile of the time-dependent coupling strength/bias. For the partitioned setting we also show that the steady state is independent of the initial state of the inner sample. Closed formulae for the NESS two-point correlation functions (Green-Keldysh functions), in the form of a convergent expansion, are derived. In the partitioning approach, we show that the 0th order term in the interaction strength of the charge current leads to the Landauer-Buettiker formula, while the 1st order correction contains the mean-field (Hartree-Fock) results.

A study of self-consistent Hartree–Fock plus Bardeen–Cooper–Schrieffer calculations with finite-range interactions

In this work we test the validity of a Hartree–Fock plus Bardeen–Cooper–Schrieffer model in which a finite-range interaction is used in the two steps of the calculation by comparing the results obtained to those found in fully self-consistent Hartree–Fock–Bogoliubov calculations using the same interaction. Specifically, we consider the Gogny-type D1S and D1M forces. We study a wide range of spherical nuclei, far from the stability line, in various regions of the nuclear chart, from oxygen to tin isotopes. We calculate various quantities related to the ground state properties of these nuclei, such as binding energies, radii, charge and density distributions, and elastic electron scattering cross sections. The pairing effects are studied by direct comparison with the Hartree–Fock results. Despite its relative simplicity, in most cases, our model provides results very close to those of the Hartree–Fock–Bogoliubov calculations, and it reproduces the empirical evidence of pairing effects rather well in the nuclei investigated.

Analytic cubic and quartic force fields using density-functional theory.

We present the first analytic implementation of cubic and quartic force constants at the level of Kohn-Sham density-functional theory. The implementation is based on an open-ended formalism for the evaluation of energy derivatives in an atomic-orbital basis. The implementation relies on the availability of open-ended codes for evaluation of one- and two-electron integrals differentiated with respect to nuclear displacements as well as automatic differentiation of the exchange-correlation kernels. We use generalized second-order vibrational perturbation theory to calculate the fundamental frequencies of methane, ethane, benzene, and aniline, comparing B3LYP, BLYP, and Hartree-Fock results. The Hartree-Fock anharmonic corrections agree well with the B3LYP corrections when calculated at the B3LYP geometry and from B3LYP normal coordinates, suggesting that the inclusion of electron correlation is not essential for the reliable calculation of cubic and quartic force constants.