Fock Exchange Research Articles

Sparse tensor based nuclear gradients for periodic Hartree-Fock and low-scaling correlated wave function methods in the CP2K software package: A massively parallel and GPU accelerated implementation.

The development of novel double-hybrid density functionals offers new levels of accuracy and is leading to fresh insights into the fundamental properties of matter. Hartree-Fock exact exchange and correlated wave function methods, such as second-order Møller-Plesset (MP2) and direct random phase approximation (dRPA), are usually required to build such functionals. Their high computational cost is a concern, and their application to large and periodic systems is, therefore, limited. In this work, low-scaling methods for Hartree-Fock exchange (HFX), SOS-MP2, and direct RPA energy gradients are developed and implemented in the CP2K software package. The use of the resolution-of-the-identity approximation with a short range metric and atom-centered basis functions leads to sparsity, allowing for sparse tensor contractions to take place. These operations are efficiently performed with the newly developed Distributed Block-sparse Tensors (DBT) and Distributed Block-sparse Matrices (DBM) libraries, which scale to hundreds of graphics processing unit (GPU) nodes. The resulting methods, resolution-of-the-identity (RI)-HFX, SOS-MP2, and dRPA, were benchmarked on large supercomputers. They exhibit favorable sub-cubic scaling with system size, good strong scaling performance, and GPU acceleration up to a factor of 3. These developments will allow for double-hybrid level calculations of large and periodic condensed phase systems to take place on a more regular basis.

Constrained density functional theory calculations for estimation of forward and backward intermolecular charge transfer energy

AbstractCharge transfer (CT) in donor–acceptor complexes can occur in two directions: from donor to acceptor (forward CT, CTF) and from acceptor to donor (backward CT, CTB). For an ethylene and tetrafluoroethylene model system, coupled cluster calculations predict that CTF is more stable than CTB. Meanwhile, density functional theory (DFT) and time‐dependent DFT (TDDFT) employing B3LYP functional show a different CT state order depending on the Hartree–Fock exchange (HFX) contribution. The impact of basis sets as a possible cause to induce such unreasonable CT state reverse along with the increase of HFX was investigated. The physical origin of CT state reversal is examined in terms of molecular orbital energy. We analyzed two additional molecular pairs that feature CT state reversal along with the amount of HFX in B3LYP functional. Despite the reduced gap, we suggest constrained DFT using M06‐HF functional as an effective approach to estimate two CT state energies on equal footing.

Incorporating Polarization and Charge Transfer into a Point-Charge Model for Water Using Machine Learning

Rigid nonpolarizable water models with fixed point charges have been widely employed in molecular dynamics simulations due to their efficiency and reasonable accuracy for the potential energy surface. However, the dipole moment surface of water is not necessarily well-described by the same fixed charges, leading to failure in reproducing dipole-related properties. Here, we developed a machine-learning model trained against electronic structure data to assign point charges for water, and the resulting dipole moment surface significantly improved the predictions of the dielectric constant and the low-frequency IR spectrum of liquid water. Our analysis reveals that within our atom-centered point-charge description of the dipole moment surface, the intermolecular charge transfer is the major source of the peak intensity at 200 cm-1, whereas the intramolecular polarization controls the enhancement of the dielectric constant. The effects of exact Hartree-Fock exchange in the hybrid density functional on these properties are also discussed.

Evaluation of DFT+U and HSE Frameworks for Strongly Correlated Iron Oxide

AbstractFor strongly correlated metal oxides, several density functional theories have been used to describe the electronic structure of bulk iron oxide (α‐Fe2O3) and corresponding surfaces, including GGA+U and hybrid functionals (HSE), but the accuracy of these methods is unclear for these metal oxides. In the present study, first‐principles simulations were carried out to examine the effects of a hybrid density functional on α‐Fe2O3 bulk and surface slabs. Further, by using GGA+U, van der Waals corrections and O2 over binding corrections were also examined. It has been found that HSE functionals with 17.5 % Fock exchange predict better properties than those with 12 % or 25 % exchange. Methodological studies indicate that GGA+U predicted zero‐bandgap surface slabs become semiconducting slabs at HSE, and that calculated bandgaps are dependent on the actual exchange addition percentage. Our studies also show that the surface energy and relative stability of different Fe2O3 terminations are less sensitive to the inclusion of dispersion terms. However, when accounting for the GGA‐error in O2 over‐binding, significant changes occur in the computed surface energies. Additionally, HSE06 functional was tested on MnO2(0001) and MnO2(110) surfaces. Both surfaces were identified as metallic by a PBE+U calculation, and an HSE06 analysis revealed a nonzero band gap.

Assessing the Role of the Kohn-Sham Density in the Calculation of the Low-Lying Bethe-Salpeter Excitation Energies.

We adopt the GW many-body perturbation theory in conjunction with the Bethe-Salpeter equation (BSE) to compute 57 excitation energies of a set of 37 molecules. By using the PBEh global hybrid functional and a self-consistent scheme on the eigenvalues in GW, we show a strong dependence of the BSE energy on the starting Kohn-Sham (KS) density functional. This arises from both the quasiparticle energies and the spatial localization of the frozen KS orbitals employed to compute the BSE. In order to address the arbitrariness in the mean field choice, we adopt an orbital-tuning scheme where the amount of Fock exchange, α, is tuned to impose that the KS HOMO matches the GW quasiparticle eigenvalue, thus fulfilling the ionization potential theorem in DFT. The performance of the proposed scheme yields excellent results and it is similar to M06-2X and PBEh with α = 75%, consistent with tuned values of α ranging between 60% and 80%.

SQMBox: Interfacing a semiempirical integral library to modular ab initio electronic structure enables new semiempirical methods.

Ab initio and semiempirical electronic structure methods are usually implemented in separate software packages or use entirely different code paths. As a result, it can be time-consuming to transfer an established ab initio electronic structure scheme to a semiempirical Hamiltonian. We present an approach to unify ab initio and semiempirical electronic structure code paths based on a separation of the wavefunction ansatz and the needed matrix representations of operators. With this separation, the Hamiltonian can refer to either an ab initio or semiempirical treatment of the resulting integrals. We built a semiempirical integral library and interfaced it to the GPU-accelerated electronic structure code TeraChem. Equivalency between ab initio and semiempirical tight-binding Hamiltonian terms is assigned according to their dependence on the one-electron density matrix. The new library provides semiempirical equivalents of the Hamiltonian matrix and gradient intermediates, corresponding to those provided by the ab initio integral library. This enables the straightforward combination of semiempirical Hamiltonians with the full pre-existing ground and excited state functionality of the ab initio electronic structure code. We demonstrate the capability of this approach by combining the extended tight-binding method GFN1-xTB with both spin-restricted ensemble-referenced Kohn-Sham and complete active space methods. We also present a highly efficient GPU implementation of the semiempirical Mulliken-approximated Fock exchange. The additional computational cost for this term becomes negligible even on consumer-grade GPUs, enabling Mulliken-approximated exchange in tight-binding methods for essentially no additional cost.

First-principles evaluation of the second harmonic generation response of reference organic and inorganic crystals.

Using the CRYSTAL17 package at the coupled-perturbed Kohn-Sham (CPKS) level, periodic boundary conditions first-principles calculations are enacted to predict the second harmonic generation second-order nonlinear optical (NLO) susceptibility, χ(2), values of six historical NLO crystals. This selection allowed the comparison between state-of-the-art calculations and experiment. Several computational aspects are tackled to define conditions where the results are converged with respect to the range of lattice summations, to the number of k-points in the first Brillouin zone, to the order of the multipole expansions for evaluating the long-range part of the electrostatic interactions, as well as to the atomic basis set size. A valence triple zeta basis set supplemented with polarization functions has been selected. Then, χ(2) calculations have been performed using a range of exchange-correlation functionals (XCFs). Results show the large impact of the amount of Hartree-Fock (HF) exchange on the amplitude but also on the signon the χ(2) tensor components. To a given extent, these amplitude effects are consistent with results on molecules, but the signreversal effects and the non-monotonic behavior of the χ(2) tensor components as a function of the amount of HF exchange are scarcely found for molecules. Then, using the recommended range-separated hybrid XCFs, the CPKS scheme leads to good agreement with experimental data for potassium dihydrogenophosphate, urea, and χZXX (2) of LiNbO3. The agreement is more questionable for χZZZ (2) of LiNbO3 whereas it remains poor for ammonium dihydrogenophosphate and 2-methyl-4-nitroaniline, with large underestimations by about a factor of 3, opening a path to further fine-tuning of the ranges of inclusion of HF exchange.

How do density functionals affect the Hirshfeld atom refinement?

In this work, the effect of mixing different amounts of Hartree-Fock (HF) exchange with hybrid density functionals applied to the Hirshfeld atom refinement (HAR) of urea and oxalic acid dihydrate is explored. Together, the influence of using different basis sets, methods (including MP2 and HF) and cluster sizes (to model bulk effects) is studied. The results show that changing the amount of HF exchange, no matter the level of theory, has an impact almost exclusively on the H atom refinement parameters. Contrary to pure quantum mechanical calculations where good geometries are obtained with intermediate HF exchange mixtures, in the HAR the best match with neutron diffraction reference values is not necessarily found for these admixtures. While the non-hydrogen covalent bond lengths are insensitive to the combination of method or basis set employed, the X-H bond lengths always increase proportionally to the HF exchange for the analysed systems. This outcome is opposite to what is normally observed from geometry optimisations, i.e., shorter bonds are obtained with greater HF exchange. Additionally, the thermal ellipsoids tend to shrink with larger HF exchange, especially for the H atoms involved in strong hydrogen bonding. Thus, it may be the case that the development of density functionals or basis sets suitable for quantum crystallography should take a different path than those fitted for quantum chemistry calculations.

Photoionization and core resonances from range-separated time-dependent density-functional theory for open-shell states: Example of the lithium atom.

We consider the calculations of photoionization spectra and core resonances of open-shell systems using range-separated time-dependent density-functional theory. Specifically, we use the time-dependent range-separated hybrid (TDRSH) scheme, combining a long-range Hartree-Fock exchange potential and kernel with a short-range potential and kernel from a local density-functional approximation, and the time-dependent locally range-separated hybrid (TDLRSH) scheme, which uses a local range-separation parameter. To efficiently perform the calculations, we formulate a spin-unrestricted linear-response Sternheimer approach in a non-orthogonal B-spline basis set using appropriate frequency-dependent boundary conditions. We illustrate this approach on the Li atom, which suggests that TDRSH and TDLRSH are adequate simple methods for estimating the single-electron photoionization spectra of open-shell systems.

CRYSTAL23: A Program for Computational Solid State Physics and Chemistry.

The Crystal program for quantum-mechanical simulations of materials has been bridging the realm of molecular quantum chemistry to the realm of solid state physics for many years, since its first public version released back in 1988. This peculiarity stems from the use of atom-centered basis functions within a linear combination of atomic orbitals (LCAO) approach and from the corresponding efficiency in the evaluation of the exact Fock exchange series. In particular, this has led to the implementation of a rich variety of hybrid density functional approximations since 1998. Nowadays, it is acknowledged by a broad community of solid state chemists and physicists that the inclusion of a fraction of Fock exchange in the exchange-correlation potential of the density functional theory is key to a better description of many properties of materials (electronic, magnetic, mechanical, spintronic, lattice-dynamical, etc.). Here, the main developments made to the program in the last five years (i.e., since the previous release, Crystal17) are presented and some of their most noteworthy applications reviewed.

Either Accurate Singlet-Triplet Gaps or Excited-State Structures: Testing and Understanding the Performance of TD-DFT for TADF Emitters.

The energy gap between the lowest singlet and triplet excited states (ΔEST) is a key property of thermally activated delayed fluorescence (TADF) emitters, where these states are dominated by charge-transfer (CT) character. Despite its well-known shortcomings concerning CT states, time-dependent density functional theory (TD-DFT) is widely used to predict this gap and study TADF. Moreover, polar CT states exhibit a strong interaction with their molecular environment, which further complicates their computational description. Addressing these two major challenges, this work studies the performance of Tamm-Dancoff-approximated TD-DFT (TDA-DFT) on the recent STGABS27 benchmark set,1 exploring different strategies to include orbital and structural relaxation, as well as dielectric embedding. The results show that the best-performing strategy is to calculate ΔEST at the ground-state structure using functionals with a surprisingly small amount of Fock exchange of ≈10% and without a (complete) solvent model. However, as this approach heavily relies on error cancellation to mimic dielectric relaxation, it is not robust and exhibits large systematic deviations in excited state energies, state characters, and structures. More rigorous approaches, including state-specific solvation, do not share these systematic deviations, but their predicted ΔEST values exhibit larger statistical errors. We thus conclude that for the description of CT states in dielectric environments, none of the tested TDA-DFT methods is competitive with the recently presented ROKS/PCM approach regarding robustness, accuracy, and computational efficiency.

Role of spin currents on electron-electron interaction in the quantum spin Hall phase

The spin current density functional theory (SCDFT) is the generalization of the standard DFT to treat a fermionic system embedded in the effective external field produced by the spin-orbit coupling interaction. Even in the absence of a spin polarization, the SCDFT requires the electron-electron potential to depend on the spin currents ${\mathbf{J}}^{x}, {\mathbf{J}}^{y}$, and ${\mathbf{J}}^{z}$, which only recently was made possible for practical relativistic quantum-mechanical simulations, through the inclusion of a fraction of nonlocal Fock exchange into local-density or generalized-gradient density functional approximations [J. K. Desmarais, J.-P. Flament, and A. Erba, Phys. Rev. B 102, 235118 (2020)]. Here, we apply the SCDFT to the quantum spin Hall phase and show how it improves (even qualitatively) the description of its electronic features relative to the DFT. We study the Bi (001) 2D bilayer and its band insulator to topological insulator phase transition (via $s+{p}_{z}\ensuremath{\leftrightarrow}{p}_{x}+i{p}_{y}$ band inversion) as a function of mechanical strain. We show that the explicit account of spin currents in the electron-electron potential of the SCDFT is key to the appearance of a Dirac cone at the $\mathrm{\ensuremath{\Gamma}}$ point in the valence band structure at the onset of the topological phase transition. Finally, the valence band structure of this system is rationalized using a simple first-order $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ quasidegenerate perturbation theory model.

How good are recent density functionals for ground and excited states of one-electron systems?

Sun et al. [J. Chem. Phys. 144, 191101 (2016)] suggested that common density-functional approximations (DFAs) should exhibit large energy errors for excited states as a necessary consequence of orbital nodality. Motivated by self-interaction corrected density-functional calculations on many-electron systems, we continue their study with the exactly solvable 1s, 2p, and 3d states of 36 hydrogenic one-electron ions (H-Kr35+) and demonstrate with self-consistent calculations that state-of-the-art DFAs indeed exhibit large errors for the 2p and 3d excited states. We consider 56 functionals at the local density approximation (LDA), generalized gradient approximation (GGA) as well as meta-GGA levels, and several hybrid functionals such as the recently proposed machine-learned DM21 local hybrid functional. The best non-hybrid functional for the 1s ground state is revTPSS. As predicted by Sun et al., the 2p and 3d excited states are more difficult for DFAs, and LDA functionals turn out to yield the most systematic accuracy for these states among non-hybrid functionals. The best performance for the three states overall is observed with the BHandH global hybrid GGA functional, which contains 50% Hartree-Fock exchange and 50% LDA exchange. The performance of DM21 is found to be inconsistent, yielding good accuracy for some states and systems and poor accuracy for others. Based on these results, we recommend including a variety of one-electron cations in future training of machine-learned density functionals.

DFT calculations of 1H- and 13C-NMR chemical shifts of 3-methyl-1-phenyl-4-(phenyldiazenyl)-1H-pyrazol-5-amine in solution

Geometries of the 3-methyl-1-phenyl-4-(phenyldiazenyl)-1H-pyrazol-5-amine azo-dye compound and its tautomer were optimized using B3LYP and M06-2X functionals in coupling with TZVP and 6–311 + G(d,p) basis sets. The 1H- and 13C-NMR chemical shifts of all species were predicted using 13 density functional theory (DFT) approaches in coupling with TZVP and 6–311 + G(d,p) basis sets at the different optimized geometries by applying the using GIAO method using the eight geometries. The selected functionals are characterized by having different amount of Hartree–Fock exchange. The selected DFT methods were B3LYP, M06-2X, BP86, B97XD, TPSSTPSS, PBE1PBE, CAM-B3LYP, wB97XD, LSDA, HSEH1PBE, PW91PW91, LC-WPBE, and B3PW91. The results obtained were compared with the available experimental data using different statistical descriptors such as root mean square error (RMSE) and maximum absolute error (MAE). Results revealed that the prediction of the 1H-NMR chemical shifts has more significant dependence on the applied geometry than that of the prediction of the 13C-NMR chemical shifts. Among all the examined functionals, B97D and TPSSTPSS functionals were found to be the most accurate ones, while the M06-2X functional is the least accurate one. Results also revealed that the prediction of NMR chemical shifts using TZVP basis sets results is more accurate results than 6–311 + G(2d,p) basis set.

Nonunique fraction of Fock exchange for defects in two-dimensional materials

By investigating the vacancy and substitutional defects in monolayer ${\mathrm{WS}}_{2}$ with hybrid functionals, we find that there is no unique amount of Fock exchange that concurrently satisfies the generalized Koopmans' condition and reproduces the band-gap and band-edge positions. Fixing the mixing parameter of Fock exchange based on the band gap can lead to qualitatively incorrect defect physics in two-dimensional materials. Instead, excellent agreement is achieved with both experiment and many-body perturbation theory within the $GW$ approximation once the mixing parameters are tuned individually for the defect species and the band edges. We show the departure from a unique optimized mixing parameter is inherent to two-dimensional systems as the band edges experience a reduced screening while the localized defects are subject to bulklike screening.

Modified HSE06 functional applied to anatase TiO2: influence of exchange fraction on the quasiparticle electronic structure and optical response

The influence of non-interacting Kohn–Sham Hamiltonian on the non-self consistent GW(G 0 W 0) quasiparticle gap and Bethe–Salpeter-equation (BSE) optical spectra of anatase TiO2 is systematically evaluated. G 0 W 0 and BSE calculations are carried out starting with HSE06 (Heyd–Scuseria–Ernzerhof) type functionals containing 20%, 25% and 30% exact Hartree–Fock exchange. The results are also compared against G 0 W 0 + BSE calculations starting from semi-local (PBE) functionals. Our results indicate that the G 0 W 0 and BSE calculations of anatase TiO2 depend critically on the mean-field starting point, wherein its dependence is mainly introduced through the dielectric screening evaluated at the intermediate G 0 W 0. We find that the band dispersion, density of states, and consequently the oscillator strengths of optical excitation and spatial localization of excitons are insensitive to the starting points while the quasiparticle gap, optical gap and exciton binding energies are strongly affected. G 0 W 0 quasiparticle gap of anatase TiO2 computed over hybrid functional starting points is typically overestimated compared to measured values. However, by varying the amount of exact exchange, the dielectric screening can be tuned, and thus the quasiparticle gap. Exciton binding energy is shown to increase in proportion to the increase of the amount of exact exchange. A simple extrapolation of the calculated data leads to the exact match with the recently measured value with 13% of the exact exchange. Systematic analysis of G 0 W 0 + BSE calculation starting from screened hybrid functionals provided in this study forms a reference for all such future calculations of pristine anatase TiO2 and its derivatives.

Accurate Nonempirical Range-Separated Hybrid van der Waals Density Functional for Complex Molecular Problems, Solids, and Surfaces

We introduce a new, general-purpose, range-separated hybrid van der Waals density \ph{functional, termed vdW-DF-ahbr,} within the non-empirical vdW-DF method [JPCM 32, 393001 (2020)]. It combines correlation from vdW-DF2 with a screened Fock exchange that is fixed by \ph{a new model of exchange effects} in the density-explicit vdW-DF2-b86r functional [PRB 89, 121103(R) (2014)]. The new vdW-DF2-ahbr prevents spurious exchange binding and has a small-density-gradient form set from many-body perturbation analysis. It is accurate for \ph{bulk as well as layered materials} and it systematically and significantly improves the performance of present vdW-DFs for molecular problems. Importantly, vdW-DF2-ahbr also outperforms present-standard (dispersion-corrected) range-separated hybrids on a broad collection of noncovalent-interaction benchmark sets, while at the same time successfully mitigating the density-driven errors that often affect the description of molecular transition states and isomerization calculations. vdW-DF2-ahbr furthermore improves on state of the art density functional theory approaches by 1) correctly predicting both the substrate structure and the site preference for CO adsorption on Pt(111), 2) outperforming existing non-empirical vdW-DFs for the description of CO$_2$ adsorption in both a functionalized and in a simple metal-organic framework, and 3) being highly accurate \ph{for the} set of base-pair interactions in a model of DNA assembly.

Adaptively compressed exchange in the linearized augmented plane wave formalism

We present an implementation of the adaptively compressed exchange (ACE) operator in the linearized-augmented plane waves formalism. ACE is a low-rank representation of the Fock exchange that avoids any loss of precision for the total energy. Our study shows that this property remains in the all-electron case, as we apply this method in nonrelativistic total-energy calculations with a hybrid exchange-correlation functional PBE0. The obtained data for light atoms and molecules are within a few $\ensuremath{\mu}\mathrm{Ha}$ of the precise multiresolution-analysis calculations. Aside from ACE, another key ingredient to achieve such a high precision with Fock exchange was the use of high-energy local orbitals. Finally, we use this implementation to calculate PBE0 gaps in solids and compare the results to other all-electron results.

THEORETICAL STUDYING SPECTRAL CHARACTERISTICS OF Tm ATOM WITHIN OPTIMIZED RELATIVISTIC MANY-BODY THEORY

Theoretical studying Rydberg spectrum of complex lanthanide atom of Tm has been performed within the relativistic many-body perturbation theory and generalized relativistic energy approach. The zeroth approximation of the relativistic perturbation theory is provided by the optimized Dirac-Kohn-Sham-Breit ones. Optimization has been fulfilled by means of introduction of the special gauge parameter to the Fock and Kohn-Sham exchange potentials and further minimization of the gauge-non-invariant contributions into radiation width of atomic levels with using relativistic orbital sets, generated by the corresponding zeroth approximation Hamiltonian. The calculated energies and widths of autoionization resonant states 4f--1j 6s(J12)nsnp[J] of the Tm atom with n=25-35 are presented and compared with known theoretical results, received within other approaches. Two main types of the Rydberg autoionization resonances decay, namely, the classic Beutler-Fano decay channel and a new Ivanov et al reorientation-type decay channel ar4e studied. It is noted that, for example, for autoionization resonant states with the considered values of n, the decay of resonances occurs along both channels, and it is extremely difficult to understand a priori which of them is dominant.

Correct and Accurate Polymorphic Energy Ordering of Transition-Metal Monoxides Obtained from Semilocal and Onsite-Hybrid Exchange-Correlation Approximations

The relative energetic stability of the structural phases of common antiferromagnetic transition-metal oxides (MnO, FeO, CoO, and NiO) within the semilocal and hybrid density functionals are fraught with difficulties. In particular, MnO is known to be the most difficult case for almost all common semilocal and hybrid density approximations. Here, we show that the meta-generalized gradient approximation (meta-GGA) constructed from the cuspless hydrogen model and Pauli kinetic energy density (MGGAC) can lead to the correct phase as the ground-state of MnO. The relative energy differences of zinc blende (zb) and rock salt (rs) structures as computed using MGGAC are found to be in nice agreement with those obtained from high-level correlation methods like the random phase approximation or quantum Monte Carlo techniques. Besides, we have also applied the onsite hybrid functionals (closely related to DFT+U) based on GGA and meta-GGA functionals, and it is shown that a relatively high amount of Hartree–Fock exchange is necessary to obtain rs as the ground-state phase. Our present investigation suggests the semilocal MGGAC and onsite hybrids, both being computationally cheap, as methods of choice for the calculation of the relative stability of antiferromagnetic transition-metal oxides having potential applications in solid-state physics and structural chemistry.