Kohn-Sham Density Functional Theory Research Articles

Exact decompositions of the total KS-DFT exchange-correlation energy into one- and two-center terms.

In the so-called Interacting Quantum Atoms (IQA) approach, the molecular energy is numerically decomposed as a sum of atomic and diatomic contributions. While proper formulations have been put forward for both Hartree-Fock and post-Hartree-Fock wavefunctions, this is not the case for the Kohn-Sham density functional theory (KS-DFT). In this work, we critically analyze the performance of two fully additive approaches for the IQA decomposition of the KS-DFT energy, namely, the one from Francisco et al., which uses atomic scaling factors, and that from Salvador and Mayer based upon the bond order density (SM-IQA). Atomic and diatomic exchange-correlation (xc) energy components are obtained for a molecular test set comprising different bond types and multiplicities and along the reaction coordinate of a Diels-Alder reaction. Both methodologies behave similarly for all systems considered. In general, the SM-IQA diatomic xc components are less negative than the Hartree-Fock ones, which is in good agreement with the known effect of electron correlation upon (most) covalent bonds. In addition, a new general scheme to minimize the numerical error of the sum of two-electron energy contributions (i.e., Coulomb and exact exchange) in the framework of overlapping atoms is described in detail.

Distributed memory, GPU accelerated Fock construction for hybrid, Gaussian basis density functional theory.

With the growing reliance of modern supercomputers on accelerator-based architecture such a graphics processing units (GPUs), the development and optimization of electronic structure methods to exploit these massively parallel resources has become a recent priority. While significant strides have been made in the development GPU accelerated, distributed memory algorithms for many modern electronic structure methods, the primary focus of GPU development for Gaussian basis atomic orbital methods has been for shared memory systems with only a handful of examples pursing massive parallelism. In the present work, we present a set of distributed memory algorithms for the evaluation of the Coulomb and exact exchange matrices for hybrid Kohn-Sham DFT with Gaussian basis sets via direct density-fitted (DF-J-Engine) and seminumerical (sn-K) methods, respectively. The absolute performance and strong scalability of the developed methods are demonstrated on systems ranging from a few hundred to over one thousand atoms using up to 128 NVIDIA A100 GPUs on the Perlmutter supercomputer.

Hybrid functionals for periodic systems in the density functional tight-binding method

Screened range-separated hybrid (SRSH) functionals within generalized Kohn-Sham density functional theory (GKS-DFT) have been shown to restore a general $1/(r\ensuremath{\varepsilon})$ asymptotic decay of the electrostatic interaction in dielectric environments. Major achievements of SRSH include an improved description of optical properties of solids and correct prediction of polarization-induced fundamental gap renormalization in molecular crystals. The density functional tight-binding method (DFTB) is an approximate DFT that bridges the gap between first-principles methods and empirical electronic structure schemes. While purely long-range corrected RSH are already accessible within DFTB for molecular systems, this work generalizes the theoretical foundation to also include screened range-separated hybrids, with conventional pure hybrid functionals as a special case. The presented formulation and implementation is also valid for periodic boundary conditions (PBC) beyond the $\mathrm{\ensuremath{\Gamma}}$ point. To treat periodic Fock exchange and its integrable singularity in reciprocal space, we resort to techniques successfully employed by DFT, in particular a truncated Coulomb operator and the minimum image convention. Starting from the first-principles Hartree-Fock operator, we derive suitable expressions for the DFTB method, using standard integral approximations and their efficient implementation in the $\mathrm{dftb}+$ software package. Convergence behavior is investigated and demonstrated for the polyacene series as well as two- and three-dimensional materials. Benzene and pentacene molecular and crystalline systems show the correct polarization-induced gap renormalization by SRSH-DFTB at heavily reduced computational cost compared to first-principles methods.

Assessing the source of error in the Thomas-Fermi-von Weizsäcker density functional.

We investigate the source of error in the Thomas-Fermi-von Weizsäcker (TFW) density functional relative to Kohn-Sham density functional theory (DFT). In particular, through numerical studies on a range of materials, for a variety of crystal structures subject to strain and atomic displacements, we find that while the ground state electron density in TFW orbital-free DFT is close to the Kohn-Sham density, the corresponding energy deviates significantly from the Kohn-Sham value. We show that these differences are a consequence of the poor representation of the linear response within the TFW approximation for the electronic kinetic energy, confirming conjectures in the literature. In so doing, we find that the energy computed from a non-self-consistent Kohn-Sham calculation using the TFW electronic ground state density is in very good agreement with that obtained from the fully self-consistent Kohn-Sham solution.

Linear Jacobi-Legendre expansion of the charge density for machine learning-accelerated electronic structure calculations

Kohn–Sham density functional theory (KS-DFT) is a powerful method to obtain key materials’ properties, but the iterative solution of the KS equations is a numerically intensive task, which limits its application to complex systems. To address this issue, machine learning (ML) models can be used as surrogates to find the ground-state charge density and reduce the computational overheads. We develop a grid-centred structural representation, based on Jacobi and Legendre polynomials combined with a linear regression, to accurately learn the converged DFT charge density. This integrates into a ML pipeline that can return any density-dependent observable, including energy and forces, at the quality of a converged DFT calculation, but at a fraction of the computational cost. Fast scanning of energy landscapes and producing starting densities for the DFT self-consistent cycle are among the applications of our scheme.

GPU acceleration of local and semilocal density functional calculations in the SPARC electronic structure code.

We present a Graphics Processing Unit (GPU)-accelerated version of the real-space SPARC electronic structure code for performing Kohn-Sham density functional theory calculations within the local density and generalized gradient approximations. In particular, we develop a modular math-kernel based implementation for NVIDIA architectures wherein the computationally expensive operations are carried out on the GPUs, with the remainder of the workload retained on the central processing units (CPUs). Using representative bulk and slab examples, we show that relative to CPU-only execution, GPUs enable speedups of up to 6× and 60× in node and core hours, respectively, bringing time to solution down to less than 30s for a metallic system with over 14 000 electrons and enabling significant reductions in computational resources required for a given wall time.

TAO-DFT with the Polarizable Continuum Model.

For the ground-state properties of gas-phase nanomolecules with multi-reference character, thermally assisted occupation (TAO) density functional theory (DFT) has recently been found to outperform the widely used Kohn-Sham DFT when traditional exchange-correlation energy functionals are employed. Aiming to explore solvation effects on the ground-state properties of nanomolecules with multi-reference character at a minimal computational cost, we combined TAO-DFT with the PCM (polarizable continuum model). In order to show its usefulness, TAO-DFT-based PCM (TAO-PCM) was used to predict the electronic properties of linear acenes in three different solvents (toluene, chlorobenzene, and water). According to TAO-PCM, in the presence of these solvents, the smaller acenes should have nonradical character, and the larger ones should have increasing polyradical character, revealing striking similarities to the past findings in the gas phase.

Comprehensive Theoretical Study on Four Typical Intramolecular Hydrogen Shift Reactions of Peroxy Radicals: Multireference Character, Recommended Model Chemistry, and Kinetics.

Intramolecular hydrogen shift reactions in peroxy radicals (RO2• → •QOOH) play key roles in the low-temperature combustion and in the atmospheric chemistry. In the present study, we found that a mild-to-moderate multireference character of a potential energy surface (PES) is widely present in four typical hydrogen shift reactions of peroxy radicals (RO2•, R = ethyl, vinyl, formyl methyl, and acetyl) by a systematic assessment based on the T1 diagnostic, %TAE diagnostic, M diagnostic, and contribution of the dominant configuration of the reference CASSCF wavefunction (C02). To assess the effects of these inherent multireference characters on electronic structure calculations, we compared the PESs of the four reactions calculated by the multireference method CASPT2 in the complete basis set (CBS) limit, single-reference method CCSD(T)-F12, and single-reference-based composite method WMS. The results showed that ignoring the multireference character will introduce a mean unsigned deviation (MUD) of 0.46-1.72 kcal/mol from CASPT2/CBS results by using the CCSD(T)-F12 method or a MUD of 0.49-1.37 kcal/mol by WMS for three RO2• reactions (R = vinyl, formyl methyl, and acetyl) with a stronger multireference character. Further tests by single-reference Kohn-Sham (KS) density functional theory methods showed even larger deviations. Therefore, we specifically developed a new hybrid meta-generalized gradient approximation (GGA) functional M06-HS for the four typical H-shift reactions of peroxy radicals based on the WMS results for the ethyl peroxy radical reaction and on the CASPT2/CBS results for the others. The M06-HS method has an averaged MUD of 0.34 kcal/mol over five tested basis sets against the benchmark PESs, performing best in the tested 38 KS functionals. Last, in a temperature range of 200-3000 K, with the new functional, we calculated the high-pressure-limit rate coefficients of these H-shift reactions by the multi-structural variational transition-state theory with the small-curvature tunneling approximation (MS-CVT/SCT) and the thermochemical properties of all of the involved key radicals by the multi-structural torsional (MS-T) anharmonicity approximation method.

Ionization potentials and fundamental gaps in atomic systems from the Ensemble-DFT approach.

Calculations in Kohn-Sham density functional theory crucially rely on high-quality approximations for the exchange-correlation (xc) functional. Standard local and semi-local approximations fail to predict the ionization potential (IP) and the fundamental gap, departing from the Kohn-Sham orbital energies, due to the deviation of the total energy from piecewise-linearity and the absence of the derivative discontinuity. The ensemble generalization procedure introduced in Phys. Rev. Lett. 110, 126403 (2013) restores, to a large extent, these features in any approximate xc functional and improves its ability to predict the IP and the fundamental gap with negligible additional computational effort. In this work we perform an extensive study of atoms and first ions across the Periodic Table, generalizing the local spin-density and the Perdew-Burke-Ernzerhof approximations. By applying the ensemble generalization to a variety of systems, with s-, p-, and d-character, we assess the accuracy of the method and identify important trends. In particular, we find that the accuracy of our approach heavily depends on the character of the frontier orbitals: when d-orbitals are involved, the performance is far less accurate. Possible sources of error are discussed and ways for further improvement are outlined.

Testing Koopmans spectral functionals on the analytically solvable Hooke's atom.

Koopmans spectral functionals are a class of orbital-density-dependent functionals designed to accurately predict spectroscopic properties. They do so markedly better than their Kohn-Sham density-functional theory counterparts, as demonstrated in earlier works on benchmarks of molecules and bulk systems. This work is a complementary study where-instead of comparing against real, many-electron systems-we test Koopmans spectral functionals on Hooke's atom, a toy two-electron system that has analytical solutions for particular strengths of its harmonic confining potential. As these calculations clearly illustrate, Koopmans spectral functionals do an excellent job of describing Hooke's atom across a range of confining potential strengths. This work also provides broader insights into the features and capabilities of Koopmans spectral functionals more generally.

Low-rank approximations for accelerating plane-wave hybrid functional calculations in unrestricted and noncollinear spin density functional theory.

The adaptively compressed exchange (ACE) operator combined with interpolative separable density fitting (ISDF) decomposition has been utilized to accelerate plane-wave hybrid functional calculations for restricted Kohn-Sham density functional theory (DFT), but the neglect of spin degree of freedom has limited its application in the exploration of systems where the spin property of the electron is critical. Herein, we derive the ACE-ISDF formulation for hybrid functional calculations in both unrestricted and noncollinear spin DFT with plane waves and periodic boundary conditions. We proposed an improved ISDF algorithm for the sum of Kohn-Sham orbital pairs to further reduce the computational cost for the spin-noncollinear case. Numerical results demonstrate that these improved ACE-ISDF low-rank approximations can not only significantly reduce the computational time by two orders of magnitude compared with conventional plane-wave hybrid functional calculations but also lead to a good convergence behavior when a moderate rank parameter is set, even for complex periodic magnetic systems. By using these ACE-ISDF approximations, we investigate the electronic and magnetic properties of two-dimensional periodic ferromagnetic semiconductors consisting of triangular zigzag graphene quantum dots and transition metal atoms. Our computational results showcase that hybrid functional calculations in spin DFT can provide not only accurate electronic structures but also accurate magnetic order temperature of ferromagnetic semiconductors compared to local or semilocal functional calculations.

Accelerating self-consistent field iterations in Kohn-Sham density functional theory using a low-rank approximation of the dielectric matrix

We present an efficient preconditioning technique for accelerating the fixed point iteration in real-space Kohn-Sham density functional theory (DFT) calculations. The preconditioner uses a low rank approximation of the dielectric matrix (LRDM) based on G\^ateaux derivatives of the residual of fixed point iteration along appropriately chosen direction functions. We develop a computationally efficient method to evaluate these G\^ateaux derivatives in conjunction with the Chebyshev filtered subspace iteration procedure, an approach widely used in large-scale Kohn-Sham DFT calculations. Further, we propose a variant of LRDM preconditioner based on adaptive accumulation of low-rank approximations from previous SCF iterations, and also extend the LRDM preconditioner to spin-polarized Kohn-Sham DFT calculations. We demonstrate the robustness and efficiency of the LRDM preconditioner against other widely used preconditioners on a range of benchmark systems with sizes ranging from $\sim$ 100-1100 atoms ($\sim$ 500--20,000 electrons). The benchmark systems include various combinations of metal-insulating-semiconducting heterogeneous material systems, nanoparticles with localized $d$ orbitals near the Fermi energy, nanofilm with metal dopants, and magnetic systems. In all benchmark systems, the LRDM preconditioner converges robustly within 20--30 iterations. In contrast, other widely used preconditioners show slow convergence in many cases, as well as divergence of the fixed point iteration in some cases. Finally, we demonstrate the computational efficiency afforded by the LRDM method, with up to 3.4$\times$ reduction in computational cost for the total ground-state calculation compared to other preconditioners.

Target State Optimized Density Functional Theory for Electronic Excited and Diabatic States.

A flexible self-consistent field method, called target state optimization (TSO), is presented for exploring electronic excited configurations and localized diabatic states. The key idea is to partition molecular orbitals into different subspaces according to the excitation or localization pattern for a target state. Because of the orbital-subspace constraint, orbitals belonging to different subspaces do not mix. Furthermore, the determinant wave function for such excited or diabatic configurations can be variationally optimized as a ground state procedure, unlike conventional ΔSCF methods, without the possibility of collapsing back to the ground state or other lower-energy configurations. The TSO method can be applied both in Hartree-Fock theory and in Kohn-Sham density functional theory (DFT). The density projection procedure and the working equations for implementing the TSO method are described along with several illustrative applications. For valence excited states of organic compounds, it was found that the computed excitation energies from TSO-DFT and time-dependent density functional theory (TD-DFT) are of similar quality with average errors of 0.5 and 0.4 eV, respectively. For core excitation, doubly excited states and charge-transfer states, the performance of TSO-DFT is clearly superior to that from conventional TD-DFT calculations. It is shown that variationally optimized charge-localized diabatic states can be defined using TSO-DFT in energy decomposition analysis to gain both qualitative and quantitative insights on intermolecular interactions. Alternatively, the variational diabatic states may be used in molecular dynamics simulation of charge transfer processes. The TSO method can also be used to define basis states in multistate density functional theory for excited states through nonorthogonal state interaction calculations. The software implementing TSO-DFT can be accessed from the authors.

Perturbation Theory Treatment of Spin–Orbit Coupling. III: Coupled Perturbed Method for Solids

A previously proposed noncanonical coupled-perturbed Kohn-Sham density functional theory (KS-DFT)/Hartree-Fock (HF) treatment for spin-orbit coupling is here generalized to infinite periodic systems. The scalar-relativistic periodic KS-DFT/HF solution, obtained with a relativistic effective core potential, is taken as the zeroth-order approximation. Explicit expressions are given for the total energy through third-order, which satisfy the 2N + 1 rule (i.e., requiring only the first-order perturbed wave function for determining the energy through third-order). Expressions for additional second-order corrections to the perturbed wave function (as well as related one-electron properties) are worked out at the uncoupled-perturbed level of theory. The approach is implemented in the Crystal program and validated with calculations of the total energy, electronic band structure, and density variables of spin-current DFT on the tungsten dichalcogenide hexagonal bilayer series (i.e., WSe2, WTe2, WPo2, WLv2), including 6p and 7p elements as a stress test. The computed properties through second- or third-order match well with those from reference two-component self-consistent field (2c-SCF) calculations. For total energies, E(3) was found to consistently improve the agreement against the 2c-SCF reference values. For electronic band structures, visible differences w.r.t. 2c-SCF remained through second-order in only the single-most difficult case of WLv2. As for density variables of spin-current DFT, the perturbed electron density, being vanishing in first-order, is the most challenging for the perturbation theory approach. The visible differences in the electron densities are, however, largest close to the core region of atoms and smaller in the valence region. Perturbed spin-current densities, on the other hand, are well reproduced in all tested cases.

Chemical Bonding in Isoelectronic NdO2and SmO22+

Neodymium dioxide (NdO2) and samarium dioxide cation (SmO22+) are isoelectronic molecules. Here we used calculations of the spin-orbit-free wave functions to study and compare their geometries, spin states, and bonding. We used Kohn-Sham density functional theory with the B97-1 exchange-correlation functional to optimize the geometries and found that the two molecules have different ground spin states and structures. NdO2 favors a linear ONdO triplet structure, and SmO22+ favors a linear SmOO2+ quintet structure. We then used state-averaged complete-active-space self-consistent-field (SA-CASSCF) calculations to investigate the bonding characteristics of NdO2 and SmO22+ in various geometries. We found that in NdOO, one electron is transferred from Nd to O, while in SmO22+, there is no electron transfer between Sm and O. The SA-CASSCF calculation also shows that ONdO has a stronger bonding orbital between a 4f orbital of Nd and a pz orbital of oxygen atoms. We compared three multireference methods, namely, extended multistate complete active space second-order perturbation theory (XMS-CASPT2), extended multistate pair-density functional theory (XMS-PDFT), and compressed multistate pair-density functional theory (CMS-PDFT), for calculating the spin-orbit-free energies of various isomers of both molecules. We found that although XMS-PDFT and CMS-PDFT are at the same cost level as SA-CASSCF, they give results with the same accuracy as given for the much more demanding XMS-CASPT2 calculation. Between the two multistate PDFT methods, CMS-PDFT is better at giving good degeneracies for states that should be degenerate.

Assessing the accuracy of hybrid exchange-correlation functionals for the density response of warm dense electrons.

We assess the accuracy of common hybrid exchange-correlation (XC) functionals (PBE0, PBE0-1/3, HSE06, HSE03, and B3LYP) within the Kohn-Sham density functional theory for the harmonically perturbed electron gas at parameters relevant for the challenging conditions of the warm dense matter. Generated by laser-induced compression and heating in the laboratory, the warm dense matter is a state of matter that also occurs in white dwarfs and planetary interiors. We consider both weak and strong degrees of density inhomogeneity induced by the external field at various wavenumbers. We perform an error analysis by comparing with the exact quantum Monte Carlo results. In the case of a weak perturbation, we report the static linear density response function and the static XC kernel at a metallic density for both the degenerate ground-state limit and for partial degeneracy at the electronic Fermi temperature. Overall, we observe an improvement in the density response when the PBE0, PBE0-1/3, HSE06, and HSE03 functionals are used, compared with the previously reported results for the PBE, PBEsol, local-density approximation, and AM05 functionals; B3LYP, on the other hand, does not perform well for the considered system. Additionally, the PBE0, PBE0-1/3, HSE06, and HSE03 functionals are more accurate for the density response properties than SCAN in the regime of partial degeneracy.

Ab initio study on the electromechanical response of Janus transition metal dihalide nanotubes

We study the electronic response of Janus transition metal dihalide (TMH) nanotubes to mechanical deformations using Kohn–Sham density functional theory. Specifically, considering twelve armchair and zigzag Janus TMH nanotubes that are expected to be stable from the phonon analysis of flat monolayer counterparts, we first compute their equilibrium diameters and then determine the variation in bandgap and effective mass of charge carriers with the application of tensile and torsional deformations. We find that the nanotubes undergo a linear and quadratic decrease in bandgap with tensile and shear strain, respectively. In addition, there is a continual increase and decrease in the effective mass of electrons and holes, respectively. We show that for a given strain, the change in bandgap for the armchair nanotubes can be correlated with the transition metal’s in-plane d orbital’s contribution to the projected density of states at the bottom of the conduction band.

A variational formulation of the Harris functional as a correction to approximate Kohn-Sham density functional theory.

Accurate descriptions of intermolecular interactions are of great importance in simulations of molecular liquids. We present an electronic structure method that combines the accuracy of the Harris functional approach with the computational efficiency of approximately linear-scaling density functional theory (DFT). The non-variational nature of the Harris functional has been addressed by constructing a Lagrangian energy functional, which restores the variational condition by imposing stationarity with respect to the reference density. The associated linear response equations may be treated with linear-scaling efficiency in an atomic orbital based scheme. Key ingredients to describe the structural and dynamical properties of molecular systems are the forces acting on the atoms and the stress tensor. These first-order derivatives of the Harris Lagrangian have been derived and implemented in consistence with the energy correction. The proposed method allows for simulations with accuracies close to the Kohn-Sham DFT reference. Embedded in the CP2K program package, the method is designed to enable ab initio molecular dynamics simulations of molecular solutions for system sizes of several thousand atoms. Available subsystem DFT methods may be used to provide the reference density required for the energy correction at near linear-scaling efficiency. As an example of production applications, we applied the method to molecular dynamics simulations of the binary mixtures cyclohexane-methanol and toluene-methanol, performed within the isobaric-isothermal ensemble, to investigate the hydrogen bonding network in these non-ideal mixtures.

Effects of dispersion corrections on the theoretical description of bulk metals

The addition of a London dispersion correction to standard Kohn-Sham density-functional theory is essential for an accurate description of noncovalent interactions. While several dispersion-corrected density functionals (DC-DFs) have shown excellent performance for hard solids at ambient conditions, their transferability to metallic systems at ambient conditions or under isotropic compression has not been systematically examined. In this study, we assess the ability of selected DC-DFs to describe the equations of state (EOSs) of selected elemental metals and intermetallic compounds up to several gigapascals of pressure. EOS-derived properties, such as the unit-cell volume, the bulk modulus, and its pressure derivative, were then evaluated with and without thermal effects and the results compared with experimental reference data. We also assess the ability of the DC-DFs to predict the phase-transition pressures for a set of intermetallic compounds. The results of this study establish that London dispersion physics, and even dispersion contributions from the core electrons, is important in the description of bulk metals.

Unified Description of Proton NMR Relaxation in Water, Acetonitrile, and Methane from Molecular Dynamics Simulations in the Liquid, Supercritical, and Gas Phases.

A comprehensive calculation of proton NMR relaxation in water, acetonitrile, and methane across a wide range of the phase diagram is provided via ab initio and force-field-based molecular dynamics simulations. The formalism used for the spin-rotation (SR) contribution to relaxation is developed for use with any molecular symmetry and utilizes the full molecular SR tensors, which are calculated from first-principles via Kohn-Sham (KS) DFT. In combination with calculations of the dipolar contribution, near-quantitative agreement with total measured relaxation rates is achieved.