Kohn-Sham Density Functional Theory Research Articles

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.

Reliable Isotropic Electron-Paramagnetic-Resonance Hyperfine Coupling Constants from the Frozen-Density Embedding Quasi-Diabatization Approach.

We present a systematic benchmark of isotropic electron-paramagnetic-resonance hyperfine coupling constants calculated for radical cation and anion complexes of molecules contained in the S22 test set using the frozen-density embedding quasi-diabatization (FDE-diab) approach. The results are compared to those from Kohn-Sham density-functional theory and frozen-density embedding, employing the domain-based local pair natural orbital coupled cluster singles and doubles method as a reference. We demonstrate that our new approach outperforms frozen-density embedding in all cases and provides reliable hyperfine couplings for radical cations using rather simple generalized-gradient approximation-type functionals. By contrast, more sophisticated and computationally less efficient exchange-correlation approximations are required for Kohn-Sham density-functional theory. For the radical anions, FDE-diab can at least provide an accuracy similar to that of Kohn-Sham density-functional theory. Finally, we demonstrate the computational advantages of FDE-diab for a π-stacked benzene octamer radical cation.

Minimal Active Space: NOSCF and NOSI in Multistate Density Functional Theory.

In this Perspective, we introduce a minimal active space (MAS) for the lowest N eigenstates of a molecular system in the framework of multistate density functional theory (MSDFT), consisting of no more than N2 nonorthgonal Slater determinants. In comparison with some methods in wave function theory in which one seeks to expand the ever increasing size of an active space to approximate the wave functions, it is possible to have an upper bound in MSDFT because the auxiliary states in a MAS are used to represent the exact N-dimensional matrix density function D(r). Here, we partition the total Hamiltonian matrix functional into an orbital-dependent part, including multistate kinetic energy Tms and Coulomb-exchange energy EHx plus an external potential energy ∫dr v(r)D(r), and a correlation matrix density functional . The latter accounts for the part of correlation energy not explicitly included in the minimal active space. A major difference from Kohn-Sham DFT is that state interactions are necessary to represent the N-matrix density D(r) in MSDFT, rather than a noninteracting reference state for the scalar ground-state density ρo(r). Two computational approaches are highlighted. We first derive a set of nonorthogonal multistate self-consistent-field (NOSCF) equations for the variational optimization of . We introduce the multistate correlation potential, as the functional derivative of , which includes both correlation effects within the MAS and that from the correlation matrix functional. Alternatively, we describe a nonorthogonal state interaction (NOSI) procedure, in which the determinant functions are optimized separately. Both computational methods are useful for determining the exact eigenstate energies and for constructing variational diabatic states, provided that the universal correlation matrix functional is known. It is hoped that this discussion would stimulate developments of approximate multistate density functionals both for the ground and excited states.

Properties of Oxygen Vacancy and Hydrogen Interstitial Defects in Strontium Titanate: DFT + Ud,p Calculations

This work presents extensive theoretical studies focused on the mixed ion-electron transport in cubic strontium titanate (STO). A new approach to the description of this difficult system was developed within the framework of linear-scaling Kohn–Sham density functional theory, as realized in the ONETEP program. The description we present is free of any empirical parameters and relies on the Hubbard U and Hund’s J corrections applied to both Ti and O atoms. The proposed methodology was validated by considering perfect STO. Its calculated properties were found to be in close agreement with experiments and calculations at higher levels of theory. The validated approach was subsequently employed to study the oxygen vacancy (VO) and the hydrogen interstitial (IH), using very large supercells (625 ± 1 atoms). The relaxed configurations of defects were obtained through fastidious energy minimization and later analyzed from a number of perspectives. The calculated defect formation energies and charge transition levels (CTLs) were found to be in close agreement with the experiment. With the exception of the charge-neutral VO, all considered defects were found to introduce shallow states, located down to 0.2 eV below the conduction band. Our calculations revealed a large 1 eV difference in the thermodynamic and optical CTLs of the neutral VO, explaining the inconsistencies observed─till now─between conduction and spectroscopic measurements. The influence of defects on the bonding characteristics and the crystalline structure of STO was quantified, revealing that both VO and IH defects lead to a significant polarization and strong tilting of the TiO6 octahedra.

Variationally consistent Hellmann–Feynman forces in the finite element formulation of Kohn–Sham density functional theory

Hellmann–Feynman forces are derived within the numerical framework of the finite element method for density functional theory in the Kohn–Sham formalism. The variational consistency of the force expressions in all-electron and pseudopotential settings are carefully examined, with a particular focus on the implications arising from different representations for interaction terms that are associated with electrostatics. Numerical investigations in nonperiodic systems which range from diatomic molecules to carbon allotropes demonstrate the systematic convergence that is offered by the finite element framework, not only for energy and force but also for geometric configuration and molecular statics parameters. A range of higher-order discretizations employing fixed meshes are invoked within these examples based on classical finite elements as well as on isogeometric analysis. Overall, this work contributes to recent advances which demonstrate the viability of the finite element method for carrying out ab initio molecular dynamics.

First-principles equationof state of CHON resin for inertial confinement fusion applications.

A wide-range (0 to 1044.0 g/cm^{3} and 0 to 10^{9}K) equation-of-state (EOS) table for a CH_{1.72}O_{0.37}N_{0.086} quaternary compound has been constructed based on density-functional theory (DFT) molecular-dynamics (MD) calculations using a combination of Kohn-Sham DFT MD, orbital-free DFT MD, and numerical extrapolation. The first-principles EOS data are compared with predictions of simple models, including the fully ionized ideal gas and the Fermi-degenerate electron gas models, to chart their temperature-density conditions of applicability. The shock Hugoniot, thermodynamic properties, and bulk sound velocities are predicted based on the EOS table and compared to those of C-H compounds. The Hugoniot results show the maximum compression ratio of the C-H-O-N resin is larger than that of CH polystyrene due to the existence of oxygen and nitrogen; while the other properties are similar between CHON and CH. Radiation hydrodynamic simulations have been performed using the table for inertial confinement fusion targets with a CHON ablator and compared with a similar design with CH. The simulations show CHON outperforms CH as the ablator for laser-direct-drive target designs.

Accelerating equilibration in first-principles molecular dynamics with orbital-free density functional theory

We introduce a practical hybrid approach that combines orbital-free density functional theory (DFT) with Kohn-Sham DFT for speeding up first-principles molecular dynamics simulations. Equilibrated ionic configurations are generated using orbital-free DFT for subsequent Kohn-Sham DFT molecular dynamics. This leads to a massive reduction of the simulation time without any sacrifice in accuracy. We assess this finding across systems of different sizes and temperature, up to the warm dense matter regime. To that end, we use the cosine distance between the time series of radial distribution functions representing the ionic configurations. Likewise, we show that the equilibrated ionic configurations from this hybrid approach significantly enhance the accuracy of machine-learning models that replace Kohn-Sham DFT. Our hybrid scheme enables systematic first-principles simulations of warm dense matter that are otherwise hampered by the large numbers of atoms and the prevalent high temperatures. Moreover, our finding provides an additional motivation for developing kinetic and noninteracting free-energy functionals for orbital-free DFT.

Subsystem density-functional theory: A reliable tool for spin-density based properties.

Subsystem density-functional theory compiles a set of features that allow for efficiently calculating properties of very large open-shell radical systems such as organic radical crystals, proteins, or deoxyribonucleic acid stacks. It is computationally less costly than correlated ab initio wave function approaches and can pragmatically avoid the overdelocalization problem of Kohn-Sham density-functional theory without employing hard constraints on the electron-density. Additionally, subsystem density-functional theory calculations commonly start from isolated fragment electron densities, pragmatically preserving a priori specified subsystem spin-patterns throughout the calculation. Methods based on subsystem density-functional theory have seen a rapid development over the past years and have become important tools for describing open-shell properties. In this Perspective, we address open questions and possible developments toward challenging future applications in connection with subsystem density-functional theory for spin-dependent properties.

A new framework for frequency-dependent polarizable force fields.

A frequency-dependent extension of the polarizable force field "Atom-Condensed Kohn-Sham density functional theory approximated to the second-order" (ACKS2) [Verstraelen et al., J. Chem. Phys. 141, 194114 (2014)] is proposed, referred to as ACKS2ω. The method enables theoretical predictions of dynamical response properties of finite systems after partitioning of the frequency-dependent molecular response function. Parameters in this model are computed simply as expectation values of an electronic wavefunction, and the hardness matrix is entirely reused from ACKS2 as an adiabatic approximation is used. A numerical validation shows that accurate models can already be obtained with atomic monopoles and dipoles. Absorption spectra of 42 organic and inorganic molecular monomers are evaluated using ACKS2ω, and our results agree well with the time-dependent DFT calculations. Also for the calculation of C6 dispersion coefficients, ACKS2ω closely reproduces its TDDFT reference. When parameters for ACKS2ω are derived from a PBE/aug-cc-pVDZ ground state, it reproduces experimental values for 903 organic and inorganic intermolecular pairs with an MAPE of 3.84%. Our results confirm that ACKS2ω offers a solid connection between the quantum-mechanical description of frequency-dependent response and computationally efficient force-field models.

Accelerating Hybrid Density Functional Theory Molecular Dynamics Simulations by Seminumerical Integration, Resolution-of-the-Identity Approximation, and Graphics Processing Units.

The computationally very demanding evaluation of the 4-center-2-electron (4c2e) integrals and their respective integral derivatives typically represents the major bottleneck within hybrid Kohn-Sham density functional theory molecular dynamics simulations. Building upon our previous works on seminumerical exact-exchange (sn-LinK) [Laqua, H., Thompsons, T. H., Kussmann, J., Ochsenfeld, C., J. Chem. Theory Comput. 2020, 16, 1465] and resolution-of-the-identity Coulomb (RI-J) [Kussmann, J., Laqua, H., Ochsenfeld, C., J. Chem. Theory Comput. 2021, 17, 1512], the expensive 4c2e integral evaluation can be avoided entirely, resulting in a highly efficient electronic structure theory method, allowing for fast ab initio molecular dynamics (AIMD) simulations even with large basis sets. Moreover, we propose to combine the final self-consistent field (SCF) step with the subsequent nuclear forces evaluation, providing the forces at virtually no additional cost after a converged SCF calculation, reducing the total runtime of an AIMD simulation by about another 25%. In addition, multiple independent MD trajectories can be computed concurrently on a single node, leading to a greatly increased utilization of the available hardware─especially when combined with graphics processing unit acceleration─improving the overall throughput by up to another 5 times in this way. With all of those optimizations combined, our proposed method provides nearly 3 orders of magnitude faster execution times than traditional 4c2e integral-based methods. To demonstrate the practical utility of the approach, quantum-mechanical/molecular-mechanical dynamics simulations on double-stranded DNA were performed, investigating the relative hydrogen bond strength between adenine-thymine and guanine-cytosine base pairs. In addition, this illustrative application also contains a general accuracy assessment of the introduced approximations (integration grids, resolution-of-the-identity) within AIMD simulations, serving as a protocol on how to apply these new methods to practical problems.

Vibrational Corrections to NMR Spin-Spin Coupling Constants from Relativistic Four-Component DFT Calculations.

Zero-point vibrational (ZPV) corrections to the nuclear spin–spin coupling constants have been calculated using four-component Dirac–Kohn–Sham DFT for H2X (where X = O, S, Se, Te, Po), XH3 (where X = N, P, As, Sb, Bi), and XH4 (where X = C, Si, Ge, Sn, and Pb) molecules and for HC≡CPbH3. The main goal was to study the influence of relativistic effects on the ZPV corrections and thus results calculated at relativistic and nonrelativistic approaches have been compared. The effects of relativity become notable for the ZPV corrections to the spin–spin coupling constants for compounds with lighter elements (selenium and germanium) than for the spin–spin coupling constants themselves. In the case of molecules containing heavier atoms, for instance BiH3 and PbH4, relativistic effects play a crucial role on the results and approximating ZPV corrections by the nonrelativistic results may lead to larger errors than omitting ZPV corrections altogether.

Plane-wave-based stochastic-deterministic density functional theory for extended systems

Traditional finite-temperature Kohn-Sham density functional theory (KSDFT) has an unfavorable scaling with respect to the electron number or at high temperatures. The evaluation of the ground-state density in KSDFT can be replaced by the Chebyshev trace (CT) method. In addition, the use of stochastic orbitals within the CT method leads to the stochastic density functional theory [Phys. Rev. Lett. 111, 106402 (2013)] (SDFT) and its improved theory, mixed stochastic-deterministic density functional theory [Phys. Rev. Lett. 125, 055002 (2020)] (MDFT). We have implemented the above four methods within the first-principles package ABACUS. All of the four methods are based on the plane-wave basis set with the use of norm-conserving pseudopotentials and the periodic boundary conditions with the use of $k$-point sampling in the Brillouin zone. By using the KSDFT calculation results as benchmarks, we systematically evaluate the accuracy and efficiency of the CT, SDFT, and MDFT methods via examining a series of physical properties, which include the electron density, the free energy, the atomic forces, stress, and density of states for a few condensed phase systems. The results suggest that our implementations of CT, SDFT, and MDFT not only reproduce the KSDFT results with a high accuracy, but also exhibit several advantages over the tradition KSDFT method. We expect these methods can be of great help in studying high-temperature and large-size extended systems such as warm dense matter and dense plasma.