Kohn-Sham Density Functional Theory Research Articles

Hybrid finite element / multipole expansion method for atomic Kohn-Sham density functional theory calculations

A numerical framework is developed for aspherical atomic Kohn-Sham density functional theory calculations. The framework invokes higher-order finite elements as a radial discretization in combination with a multipole expansion for controlling the spherical resolution. Both all-electron and nonlocal pseudopotential calculations are addressed in a unified setting. The overall approach is validated through a range of numerical examples which demonstrate the systematic convergence of the radial and spherical discretizations as well as the outstanding accuracy that can be efficiently obtained in the presence of strong aspherical external fields. Overall, the presented approach offers a route to adaptive local enrichment for electronic structure calculation in the context of the finite element method.

Version 2.0.0 - M-SPARC: Matlab-Simulation Package for Ab-initio Real-space Calculations

M-SPARC is a Matlab code for performing ab initio Kohn–Sham Density Functional Theory simulations. Version 2.0.0 of the software further extends its capability to include relativistic effects, dispersion interactions, and advanced semilocal/nonlocal exchange–correlation functionals. These features significantly increase the fidelity of first principles calculations that can be performed using M-SPARC.

Development of nonlocal kinetic-energy density functional for the hybrid QM/MM interaction.

Development of the electronic kinetic-energy density functional is a subject of major interest in theoretical physics and chemistry. In this work, the nonlocal kinetic-energy functional is developed in terms of the response function for the molecular system to realize the orbital free density-functional theory (OF-DFT) to be utilized in the hybrid QM/MM (quantum mechanical/molecular mechanical) method. The present approach shows a clear contrast to the previous functionals where the homogeneous electron gas serves as a reference to build the response function. As a benchmark test, we apply the method to a QM water molecule in a dimer system and that embedded in a condensed environment to make comparisons with the results given by the QM/MM calculations employing the Kohn-Sham DFT. It was found that the energetics and the polarization density of the QM solute under the influence of the MM environment can be adequately reproduced with our approach. This work suggests the potential ability of the kinetic-energy functional based on the response functions for the molecular reference systems.

Comparison of long-range corrected kernels and range-separated hybrids for excitons in solids

At the moment, the most accurate theoretical method to describe excitons is the solution of the Bethe-Salpeter equation in the GW approximation (GW-BSE). However, because of its computation cost, time-dependent density functional theory (TDDFT) is becoming the alternative approach to GW-BSE to describe excitons in solids. Nowadays, the most efficient strategy to describe optical spectra of solids in TDDFT is to use long-range corrected exchange-correlation kernels on top of GW or scissor-corrected energies. In recent years, a different strategy based on range-separated hybrid functionals started to be developed in the framework of time-dependent generalized Kohn-Sham density functional theory. Here we compare the performance of long-range corrected kernels with range-separated hybrid functionals for the description of excitons in solids. This comparison has the purpose to weight the pros and cons of using range-separated hybrid functionals, giving new perspectives for theoretical developments of these functionals. We illustrate the comparison for the case of Si and LiF, representative of solid-state excitons.

Supervised learning of a chemistry functional with damped dispersion

Kohn–Sham density functional theory is widely used in chemistry, but no functional can accurately predict the whole range of chemical properties, although recent progress by some doubly hybrid functionals comes close. Here, we optimized a singly hybrid functional called CF22D with higher across-the-board accuracy for chemistry than most of the existing non-doubly hybrid functionals by using a flexible functional form that combines a global hybrid meta-nonseparable gradient approximation that depends on density and occupied orbitals with a damped dispersion term that depends on geometry. We optimized this energy functional by using a large database and performance-triggered iterative supervised training. We combined several databases to create a very large, combined database whose use demonstrated the good performance of CF22D on barrier heights, isomerization energies, thermochemistry, noncovalent interactions, radical and nonradical chemistry, small and large systems, simple and complex systems and transition-metal chemistry.

Performance of Screened-Exchange Functionals for Band Gaps and Lattice Constants of Crystals.

Kohn-Sham density functional theory is the most widely used method for electronic structure calculations of solid-state systems. The screened-exchange functionals developed following the influential work of Scuseria and co-workers in 2003-2006 have significantly improved the accuracy of the predictions of solid-state properties. This work assesses six screened-exchange density functionals for the prediction of 60 band gaps (database BG60) and 68 lattice constants (database LC68). The band gaps are calculated with both consistently calculated lattice constants and experimental lattice constants. Results for the nonlocal screened-exchange functionals are compared with those for six widely used or recently developed local functionals. The results show that all the screened-exchange functionals have smaller mean absolute errors (MAEs) than any of the local functionals. All the functionals except HLE17 overestimate (on average) the lattice constants, and M06-SX gives the best performance among the compared functionals, with a MAE of 0.051 Å. All the functionals underestimate (on average) the band gaps, and M06-SX outperforms all other functionals, with a MAE of 0.47 eV. M06-SX also has the lowest root-mean-squared error for both LC68 and BG60. For the subdatabases of BG60, M06-SX shows better performance for ionic crystals and systems with large band gaps, while HSE12s gives better results for semiconductors and systems with small band gaps. Overall, M06-SX shows the best performance for solid-state systems, followed by N12-SX and HSE12s. The best-performing local functionals are M06-L, revM06-L, and HLE17 for band gaps and M06-L and revM06-L for lattice constants. We found that M06-SX, revM06-L, and N12-SX not only are well optimized for a broad array of chemical properties but also have very good performance for the databases in this paper, making them well-suited for applications involving heterogeneous chemistry.

Thermochemical evaluation of adaptive and fixed density functional theory quadrature schemes.

A systematic study is made of the accuracy and efficiency of a number of existing quadrature schemes for molecular Kohn-Sham Density-Functional Theory (DFT) using 408 molecules and 254 chemical reactions. Included are the fixed SG-x (x = 0-3) grids of Gill et al., Dasgupta, and Herbert, the 3-zone grids of Treutler and Ahlrichs, a fixed five-zone grid implemented in Molpro, and a new adaptive grid scheme. While all methods provide a systematic reduction of errors upon extension of the grid sizes, significant differences are observed in the accuracies for similar grid sizes with various approaches. For the tests in this work, the SG-x fixed grids are less suitable to achieve high accuracies in the DFT integration, while our new adaptive grid performed best among the schemes studied in this work. The extra computational time to generate the adaptive grid scales linearly with molecular size and is negligible compared with the time needed for the self-consistent field iterations for large molecules. A comparison of the grid accuracies using various density functionals shows that meta-GGA functionals need larger integration grids than GGA functionals to reach the same degree of accuracy, confirming previous investigations of the numerical stability of meta-GGA functionals. On the other hand, the grid integration errors are almost independent of the basis set, and the basis set errors are mostly much larger than the errors caused by the numerical integrations, even when using the smallest grids tested in this work.

Consistent Evaluation of Magnetic Exchange Couplings in Multicenter Compounds in KS-DFT: The Recomposition Method.

The use of broken-symmetry calculations in Kohn-Sham density functional theory has offered an affordable route to study magnetic exchange couplings in transition-metal-based compounds. However, computing this property in compounds exhibiting several couplings is still challenging and especially due to the difficulties to overcome the well-known problem of spin contamination. Here, we present a new and general method to compute magnetic exchange couplings in systems featuring several spin sites. To provide a consistent spin decontamination of J values, our strategy exploits the decomposition method of the magnetic exchange coupling proposed by Coulaud et al. and generalizes our previous work on diradical compounds where the overall magnetic exchange coupling is defined as the sum of its three main and properly extracted physical contributions (direct exchange, kinetic exchange, and spin polarization). In this aim, the generalized extraction of all contributions is presented to systems with multiple spin sites bearing one unpaired electron. This is done by proposing a new paradigm to treat the kinetic exchange contribution, which proceeds through monorelaxations of the magnetic orbitals. This method, so-called the recomposition method, is applied to a compound featuring three Cu(II) ions with a linear arrangement and to a recently synthesized complex containing a Cu4O4 cubane unit presenting an unusual magnetic behavior.

On the bending of rectangular atomic monolayers along different directions: an ab initio study

We study the bending of rectangular atomic monolayers along different directions from first principles. Specifically, choosing the phosphorene, GeS, TiS3, and As2S3 monolayers as representative examples, we perform Kohn–Sham density functional theory calculations to determine the variation in transverse flexoelectric coefficient and bending modulus with the direction of bending. We find that while the flexoelectric coefficient is nearly isotropic, there is significant and complex anisotropy in bending modulus that also differs between the monolayers, with extremal values not necessarily occurring along the principal directions. In particular, the commonly adopted orthotropic continuum plate model with uniform thickness fails to describe the observed variations in bending modulus for GeS, TiS3, and As2S3. We determine the direction-dependent effective thickness for use in such continuum models. We also show that the anisotropy in bending modulus is not associated with the rehybridization of atomic orbitals.

Cluster Amplitudes and Their Interplay with Self-Consistency in Density Functional Methods.

Density functional theory (DFT) provides convenient electronic structure methods for the study of molecular systems and materials. Regular Kohn-Sham DFT calculations rely on unitary transformations to determine the ground-state electronic density, ground state energy, and related properties. However, for dissociation of molecular systems into open-shell fragments, due to the self-interaction error present in a large number of density functional approximations, the self-consistent procedure based on the this type of transformation gives rise to the well-known charge delocalization problem. To avoid this issue, we showed previously that the cluster operator of coupled-cluster theory can be utilized within the context of DFT to solve in an alternative and approximate fashion the ground-state self-consistent problem. This work further examines the application of the singles cluster operator to molecular ground state calculations. Two approximations are derived and explored: i) A linearized scheme of the quadratic equation used to determine the cluster amplitudes. ii) The effect of carrying the calculations in a non-self-consistent field fashion. These approaches are found to be capable of improving the energy and density of the system and are quite stable in either case. The theoretical framework discussed in this work could be used to describe, with an added flexibility, quantum systems that display challenging features and require expanded theoretical methods.

Efficient Kohn–Sham density-functional theory implementation of isotropic spectroscopic observables associated with quadratic response functions

For general exchange–correlation functionals with a dependence on the local spin densities and spin-density gradients, we provide computationally tractable expressions for the tensor-averaged quadratic response functions pertinent to the experimental observables in second-harmonic generation (SHG). We demonstrate how the tensor-averaged quantities can be implemented with reference to a derived minimal number of first- and second-order perturbed Fock matrices. Our consideration has the capability of treating a situation of resonance enhancement as it is based on damped response theory and allows for the evaluation of tensor-averaged resonant-convergent quadratic response functions using only ∼25% (one-photon off-resonance regions) and ∼50% (one-photon resonance regions) of the number of auxiliary Fock matrices required when explicitly calculating all the needed individual tensor components. Numerical examples of SHG intensities in the one-photon off-resonance region are provided for a sample of makaluvamine derivatives recognized for their large nonlinear optical responses as well as a benchmark set of small- and medium-sized organic molecules.

Towards in silico mining for superconductors – Cutting the Gordian knot

Utilizing the random forest model, feasibility of training machine learning regressor models to predict critical temperatures of superconductivity from Density Functional Theory (DFT) based electronic band structures is explored. This complementarity between experiment and theory draws inspiration from the merging of Kohn-Sham and Bogoliubov-De Gennes equations [W. Kohn, W, EKU Gross, and LN Oliveira, Int. J. of Quant. Chem.,36(23), 611–615 (1989)]. Features in the Kohn-Sham Density Functional Theory band structure away from EF becoming decisive for the superconducting gap demonstrates this divide-and-conquer physical understanding. Not committing to any microscopic mechanism for the SC at this stage, it implies that in different classes of materials, different electronic features are responsible for the superconductivity. However, training on known members of a class, the performance of new members may be predicted. The method is validated for the A15 materials, including both binary A3X and ternary A6XY intermetallics, A = V, Nb, demonstrating that the two do indeed belong to the same class of superconductors.

Generalization of ETS-NOCV and ALMO-COVP Energy Decomposition Analysis to Connect Any Two Electronic States and Comparative Assessment.

Energy decomposition analysis (EDA) is a useful tool for obtaining chemically meaningful insights into molecular interactions. The extended transition-state method with natural orbitals for chemical valence (ETS-NOCV) and the absolutely localized molecular orbital-based method with complementary occupied-virtual pairs (ALMO-COVP) are two successful EDA schemes. Working within ground-state generalized Kohn-Sham density functional theory (DFT), we extend these methods to perform EDA between any two electronic states that can be connected by a unitary transformation of density matrices. A direct proof that the NOCV eigenvalues are symmetric pairs is given, and we also prove that the charge and energy difference defined by ALMO are invariant under certain orbital rotations, allowing us to define COVPs. We point out that ETS is actually a 1-point quadrature to obtain the effective Fock matrix, and though it is reasonably accurate, it can be systematically further improved by adding more quadrature points. We explain why the calculated amount of transferred charge measured by ALMO-COVP is typically much smaller than that of ETS-NOCV and explain why the ALMO-COVP values should be preferred. While the two schemes are independent, ETS-NOCV and ALMO-COVP in fact give a very similar chemical picture for a variety of chemical interactions, including H-H+, the transition structure for the Diels-Alder reaction between ethene and butadiene, and two hydrogen-bonded complexes, H2O···F- and H2O···HF.

Application of Multiconfiguration Pair-Density Functional Theory to the Diels-Alder Reaction.

Transition states for Diels-Alder reactions are strongly correlated, as evidenced by high-to-very-high M diagnostics, and therefore they require treatment by multireference methods. Multiconfiguration pair-density functional theory (MC-PDFT) combines a multiconfiguration wave function with a functional of the electron density and the on-top pair density to calculate the electronic energy for strongly correlated systems at a much lower cost than wave function methods that do not employ density functionals. Here we apply MC-PDFT to the Diels-Alder cycloaddition reaction of 1,3-butadiene with ethylene, where two kinds of reaction paths have been widely studied: concerted synchronous paths and diradical stepwise paths. The lowest-energy reaction path is now known to be a concerted synchronous one, and a method's ability to predict this is an important test. By comparison to the best available theoretical results in the literature, we test the accuracy of MC-PDFT with several choices of on-top functional for geometries and enthalpies of stable structures along both paths and for the transition state geometries. We also calculate the Arrhenius activation energies for both paths and compare these to experiment. We also compare to Kohn-Sham density functional theory (KS-DFT) with selected exchange-correlation functionals. CAS-PDFT gives consistently good energies and geometries for both the concerted and stepwise mechanisms, but none of the KS-DFT functionals gives accurate activation energies for both. The stepwise transition state is very strongly correlated, and MC-PDFT can treat it, but KS-DFT (which involves a single-configuration treatment) has larger errors. The results confirm that using a multiconfigurational reference function for strongly correlated transition states can significantly improve the reliability and that MC-PDFT can provide good accuracy at a much lower computational cost than competing multireference methods.

Charge-density based convolutional neural networks for stacking fault energy prediction in concentrated alloys

A descriptor-less machine learning (ML) model based only on charge density images extracted from density functional theory (DFT) is developed to predict stacking fault energies (SFE) in concentrated alloys. The model is based on convolutional neural networks (CNNs) as one of the promising ML techniques for dealing with complex images and data. Identification of correct descriptors is a key bottleneck to develop ML models for predicting materials properties. Often, in most ML models, textbook physical descriptors such as atomic radius, valence charge and electronegativity are used as descriptors which have limitations because these properties change in concentrated alloys when multiple elements are mixed to form a solid solution. We illustrate that, within the scope of DFT, the search for descriptors can be circumvented by electronic charge density, which is the backbone of the Kohn-Sham DFT and describes the system completely. The performance of our model is demonstrated by predicting SFE of concentrated alloys with an RMSE and R2 of 6.18 mJ/m2 and 0.87, respectively, validating the accuracy of the proposed approach.

Scale-free-modeling (harmonic) vibrational frequencies: Assessing accuracy and cost-effectiveness by CBS extrapolation.

Empirical scaling of calculated vibrational harmonic frequencies is a popular approach used in the field of molecular sciences. A nonempirical scheme that aims at reducing their basis set error is suggested here. Nearly as cost-effective as the scaled Kohn-Sham density functional theory (KS DFT), it consists of splitting the frequencies into Hartree-Fock and electron correlation contributions, followed by their extrapolation to the complete basis set (CBS) limit. Since the former converges exponentially, the overall cost may actually equal that of CBS extrapolation of the correlation part. Despite shifts in the molecular geometry during vibration, reasons are advanced to justify the approach, with extrapolation from the first two steps of the basis set ladder being effective in accelerating convergence. As benchmark data, a set of harmonic frequencies and zero-point energies for 15 molecules is employed at the second-order Moller-Plesset and coupled-cluster single double triple [CCSD(T)] levels of theory. The results outperform the optimized KS DFT scaled values. As a second test set, equilibrium structures and harmonic frequencies were computed for H2O2, CH2NH, C2H2O, and the trans-isomer of 1,2-C2H2F2. The results are also encouraging, particularly when improved for excess correlation at the CCSD(T)/VDZ level via the focal-point approach. In extreme cases, CBS extrapolation is done from two double-ζ calculations: one canonical and the other using explicit correlation theory. As a further case study, benzene is considered. While the CCSD(T) results show the smallest deviation from the best estimates, the MP2 results also attain good quality: When improved for excess correlation, they show 6-10cm-1 errors relative to the best data, only slightly outperformed at the CCSD(T)/CBS level. Tentative results for the fundamental frequencies are also presented.

Hybrid Functional and Plane Waves based Ab Initio Molecular Dynamics Study of the Aqueous Fe2+ /Fe3+ Redox Reaction.

Kohn-Sham density functional theory and plane wave basis set based ab initio molecular dynamics (AIMD) simulation is a powerful tool for studying complex reactions in solutions, such as electron transfer (ET) reactions involving Fe2+ /Fe3+ ions in water. In most cases, such simulations are performed using density functionals at the level of Generalized Gradient Approximation (GGA). The challenge in modelling ET reactions is the poor quality of GGA functionals in predicting properties of such open-shell systems due to the inevitable self-interaction error (SIE). While hybrid functionals can minimize SIE, standard plane-wave based AIMD at that level of theory is typically 150 times slower than GGA for systems containing ∼100 atoms. Among several approaches reported to speed-up AIMD simulations with hybrid functionals, the noise-stabilized MD (NSMD) procedure, together with the use of localized orbitals to compute the required exchange integrals, is an attractive option. In this work, we demonstrate the application of the NSMD approach for studying the Fe2+ /Fe3+ redox reaction in water. It is shown here that long AIMD trajectories at the level of hybrid density functionals can be obtained using this approach. Redox properties of the aqueous Fe2+ /Fe3+ system computed from these simulations are compared with the available experimental data for validation.

Machine Learning Diffusion Monte Carlo Energies.

We present two machine learning methodologies that are capable of predicting diffusion Monte Carlo (DMC) energies with small data sets (≈60 DMC calculations in total). The first uses voxel deep neural networks (VDNNs) to predict DMC energy densities using Kohn-Sham density functional theory (DFT) electron densities as input. The second uses kernel ridge regression (KRR) to predict atomic contributions to the DMC total energy using atomic environment vectors as input (we used atom-centered symmetry functions, atomic environment vectors from the ANI models, and smooth overlap of atomic positions). We first compare the methodologies on pristine graphene lattices, where we find that the KRR methodology performs best in comparison to gradient boosted decision trees, random forest, Gaussian process regression, and multilayer perceptrons. In addition, KRR outperforms VDNNs by an order of magnitude. Afterward, we study the generalizability of KRR to predict the energy barrier associated with a Stone-Wales defect. Lastly, we move from 2D to 3D materials and use KRR to predict total energies of liquid water. In all cases, we find that the KRR models are more accurate than Kohn-Sham DFT and all mean absolute errors are less than chemical accuracy.

Plutonium Hybrid Materials: A Platform to Explore Assembly and Metal-Ligand Bonding.

We report the synthesis of five new hybrid materials containing the [PuCl6]2- anion and charge-balancing, noncovalent interaction donating 4-X-pyridinium (X = H, Cl, Br, I) cations. Single crystals of the title compounds were grown and harvested from acidic, chloride-rich, aqueous media, and their structures were determined via X-ray diffraction. Compounds 1-4, (4XPyH)2[PuCl6], and 5, (4IPyH)4[PuCl6]·2Cl, exhibit two distinct sheet-like structure types. Structurally relevant noncovalent interactions were tabulated from crystallographic data and verified computationally using electrostatic surface potential maps and the quantum theory of atoms in molecules (QTAIM). The strength of the hydrogen and halogen bonds was quantified using Kohn-Sham density functional theory, and a hierarchy of acceptor-donor pairings was established. The PuIV-Cl bonds were studied using QTAIM and natural localized molecular orbital (NLMO) analyses to delineate the underlying bond mechanism and hybrid atomic orbital contributions therein. The results of the PuIV-Cl bond analyses were compared across compositions via analogous treatments of previously reported [PuO2Cl4]2- and [PuCl3(H2O)5] molecular units. The Pu-Cl bonds are predominately ionic yet exhibit small varying degrees of covalent character that increases from [PuCl3(H2O)5] and [PuO2Cl4]2- to [PuCl6]2-, while the participation of the Pu-based s/d and f orbitals concurrently decreases and increases, respectively.

Training-free hyperparameter optimization of neural networks for electronic structures in matter

A myriad of phenomena in materials science and chemistry rely on quantum-level simulations of the electronic structure in matter. While moving to larger length and time scales has been a pressing issue for decades, such large-scale electronic structure calculations are still challenging despite modern software approaches and advances in high-performance computing. The silver lining in this regard is the use of machine learning to accelerate electronic structure calculations—this line of research has recently gained growing attention. The grand challenge therein is finding a suitable machine-learning model during a process called hyperparameter optimization. This, however, causes a massive computational overhead in addition to that of data generation. We accelerate the construction of neural network models by roughly two orders of magnitude by circumventing excessive training during the hyperparameter optimization phase. We demonstrate our workflow for Kohn–Sham density functional theory, the most popular computational method in materials science and chemistry.