Efficient Computation of the Hartree-Fock Exchange in Real-Space with Projection Operators

Efficient real space formalism for hybrid density functionals

We present an efficient real space formalism for hybrid exchange-correlation functionals in generalized Kohn-Sham density functional theory (DFT). In particular, we develop an efficient representation for any function of the real space finite-difference Laplacian matrix by leveraging its Kronecker product structure, thereby enabling the time to solution of associated linear systems to be highly competitive with the fast Fourier transform scheme while not imposing any restrictions on the boundary conditions. We implement this formalism for both the unscreened and range-separated variants of hybrid functionals. We verify its accuracy and efficiency through comparisons with established planewave codes for isolated as well as bulk systems. In particular, we demonstrate up to an order-of-magnitude speedup in time to solution for the real space method. We also apply the framework to study the structure of liquid water using ab initio molecular dynamics, where we find good agreement with the literature. Overall, the current formalism provides an avenue for efficient real-space DFT calculations with hybrid density functionals.

Sparse-Stochastic Fragmented Exchange for Large-Scale Hybrid Time-Dependent Density Functional Theory Calculations

We extend our recently developed sparse-stochastic fragmented exchange formalism for ground-state near-gap hybrid DFT to calculate absorption spectra within linear-response time-dependent generalized Kohn-Sham DFT (LR-GKS-TDDFT) for systems consisting of thousands of valence electrons within a grid-based/plane-wave representation. A mixed deterministic/fragmented-stochastic compression of the exchange kernel, here using long-range explicit exchange functionals, provides an efficient method for accurate optical spectra. Both real-time propagation as well as frequency-resolved Casida-equation-type approaches for spectra are presented, and the method is applied to large molecular dyes.

High-Throughput Condensed-Phase Hybrid Density Functional Theory for Large-Scale Finite-Gap Systems: The SeA Approach

High-throughput electronic structure calculations (often performed using density functional theory (DFT)) play a central role in screening existing and novel materials, sampling potential energy surfaces, and generating data for machine learning applications. By including a fraction of exact exchange (EXX), hybrid functionals reduce the self-interaction error in semilocal DFT and furnish a more accurate description of the underlying electronic structure, albeit at a computational cost that often prohibits such high-throughput applications. To address this challenge, we have constructed a robust, accurate, and computationally efficient framework for high-throughput condensed-phase hybrid DFT and implemented this approach in the PWSCF module of Quantum ESPRESSO (QE). The resulting SeA approach (SeA = SCDM + exx + ACE) combines and seamlessly integrates: (i) the selected columns of the density matrix method (SCDM, a robust noniterative orbital localization scheme that sidesteps system-dependent optimization protocols), (ii) a recently extended version of exx (a black-box linear-scaling EXX algorithm that exploits sparsity between localized orbitals in real space when evaluating the action of the standard/full-rank V^xx operator), and (iii) adaptively compressed exchange (ACE, a low-rank V^xx approximation). In doing so, SeA harnesses three levels of computational savings: pair selection and domain truncation from SCDM + exx (which only considers spatially overlapping orbitals on orbital-pair-specific and system-size-independent domains) and low-rank V^xx approximation from ACE (which reduces the number of calls to SCDM + exx during the self-consistent field (SCF) procedure). Across a diverse set of 200 nonequilibrium (H₂O)₆₄ configurations (with densities spanning 0.4-1.7 g/cm³), SeA provides a 1-2 order-of-magnitude speedup in the overall time-to-solution, i.e., ≈8-26× compared to the convolution-based PWSCF(ACE) implementation in QE and ≈78-247× compared to the conventional PWSCF(Full) approach, and yields energies, ionic forces, and other properties with high fidelity. As a proof-of-principle high-throughput application, we trained a deep neural network (DNN) potential for ambient liquid water at the hybrid DFT level using SeA via an actively learned data set with ≈8,700 (H₂O)₆₄ configurations. Using an out-of-sample set of (H₂O)₅₁₂ configurations (at nonambient conditions), we confirmed the accuracy of this SeA-trained potential and showcased the capabilities of SeA by computing the ground-truth ionic forces in this challenging system containing >1,500 atoms.

Enabling Large-Scale Condensed-Phase Hybrid Density Functional Theory-Based Ab Initio Molecular Dynamics II: Extensions to the Isobaric-Isoenthalpic and Isobaric-Isothermal Ensembles

In the previous paper of this series [Ko, H.-Y. et al. J. Chem. Theory Comput. 2020, 16, 3757-3785], we presented a theoretical and algorithmic framework based on a localized representation of the occupied space that exploits the inherent sparsity in the real-space evaluation of the exact exchange (EXX) interaction in finite-gap systems. This was accompanied by a detailed description of exx, a massively parallel hybrid message-passing interface MPI/OpenMP implementation of this approach in Quantum ESPRESSO (QE) that enables linear scaling hybrid density functional theory (DFT)-based ab initio molecular dynamics (AIMD) in the microcanonical/canonical (NVE/NVT) ensembles of condensed-phase systems containing 500-1000 atoms (in fixed orthorhombic cells) with a wall time cost comparable to semi-local DFT. In this work, we extend the current capabilities of exx to enable hybrid DFT-based AIMD simulations of large-scale condensed-phase systems with general and fluctuating cells in the isobaric-isoenthalpic/isobaric-isothermal (NpH/NpT) ensembles. The theoretical extensions to this approach include an analytical derivation of the EXX contribution to the stress tensor for systems in general simulation cells with a computational complexity that scales linearly with system size. The corresponding algorithmic extensions to exx include optimized routines that (i) handle both static and fluctuating simulation cells with non-orthogonal lattice symmetries, (ii) solve Poisson's equation in general/non-orthogonal cells via an automated selection of the auxiliary grid directions in the Natan-Kronik representation of the discrete Laplacian operator, and (iii) evaluate the EXX contribution to the stress tensor. Using this approach, we perform a case study on a variety of condensed-phase systems (including liquid water, a benzene molecular crystal polymorph, and semi-conducting crystalline silicon) and demonstrate that the EXX contributions to the energy and stress tensor simultaneously converge with an appropriate choice of exx parameters. This is followed by a critical assessment of the computational performance of the extended exx module across several different high-performance computing architectures via case studies on (i) the computational complexity due to lattice symmetry during NpT simulations of three different ice polymorphs (i.e., ice Ih, II, and III) and (ii) the strong/weak parallel scaling during large-scale NpT simulations of liquid water. We demonstrate that the robust and highly scalable implementation of this approach in the extended exx module is capable of evaluating the EXX contribution to the stress tensor with negligible cost (<1%) as well as all other EXX-related quantities needed during NpT simulations of liquid water (with a very tight 150 Ry planewave cutoff) in ≈5.2 s ((H₂O)₁₂₈) and ≈6.8 s ((H₂O)₂₅₆) per AIMD step. As such, the extended exx module presented in this work brings us another step closer to routinely performing hybrid DFT-based AIMD simulations of sufficient duration for large-scale condensed-phase systems across a wide range of thermodynamic conditions.

Large-Scale Parallel Implementation of Hartree-Fock Exchange Energy on Real-Space Grids Using 3D-Parallel Fast Fourier Transform

Two types of implementation of the Hartree-Fock (HF) exchange energy were developed with the real-space grid approach for the purpose of achieving high efficiency in the parallel execution of the hybrid exchange functional in the density functional theory. First, a parallel implementation of the three-dimensional fast Fourier transform (FFT), referred to as PFFT, was adapted to solve the Poisson equations for the electrostatic potentials of the densities of the orbital pairs. In the other approach, the Poisson equations were solved iteratively through the conjugate gradient (CG) procedures where the operation of Laplacian was parallelized by the domain decomposition scheme. Comparison of the parallel performances for the exchange energy calculation was made between these two approaches, and it was revealed that the calculation with the FFT method is faster than that with CG. The method with FFT is more advantageous than CG because a larger bandwidth can be made available in the collective message passing interface communication associated with the parallel execution of FFT. We also implemented the projection operator to circumvent the laborious calculation of the exchange energy at every self-consistent field step, which made a significant contribution to expedite the convergence. To assess the accuracy of our implementation, the association energies of a hydrated ion were computed, which showed excellent agreement with those given by the Gaussian 09 program employing sophisticated basis sets.

Interpolative Separable Density Fitting Decomposition for Accelerating Hybrid Density Functional Calculations with Applications to Defects in Silicon

We present a new efficient way to perform hybrid density functional theory (DFT)-based electronic structure calculations. The new method uses an interpolative separable density fitting (ISDF) procedure to construct a set of numerical auxiliary basis vectors and a compact approximation of the matrix consisting of products of occupied orbitals represented in a large basis set such as the planewave basis. Such an approximation allows us to reduce the number of Poisson solves from [Formula: see text] to [Formula: see text] when we apply the exchange operator to occupied orbitals in an iterative method for solving the Kohn-Sham equations, where N_e is the number of electrons in the system to be studied. We show that the ISDF procedure can be carried out in [Formula: see text] operations, with a much smaller preconstant compared to methods used in existing approaches. When combined with the recently developed adaptively compressed exchange (ACE) operator formalism, which reduces the number of times the exchange operator needs to be updated, the resulting ACE-ISDF method significantly reduces the computational cost associated with the exchange operator by nearly 2 orders of magnitude compared to existing approaches for a large silicon system with 1000 atoms. We demonstrate that the ACE-ISDF method can produce accurate energies and forces for insulating and metallic systems and that it is possible to obtain converged hybrid functional calculation results for a 1000-atom bulk silicon within 10 min on 2000 computational cores. We also show that ACE-ISDF can scale to 8192 computational cores for a 4096-atom bulk silicon system. We use the ACE-ISDF method to geometrically optimize a 1000-atom silicon system with a vacancy defect using the HSE06 functional and computes its electronic structure. We find that that the computed energy gap from the HSE06 functional is much closer to the experimental value compared to that produced by semilocal functionals in the DFT calculations.

Adaptively Compressed Exchange Operator for Large-Scale Hybrid Density Functional Calculations with Applications to the Adsorption of Water on Silicene

Density functional theory (DFT) calculations using hybrid exchange-correlation functionals have been shown to provide an accurate description of the electronic structures of nanosystems. However, such calculations are often limited to small system sizes due to the high computational cost associated with the construction and application of the Hartree-Fock (HF) exchange operator. In this paper, we demonstrate that the recently developed adaptively compressed exchange (ACE) operator formulation [J. Chem. Theory Comput. 2016, 12, 2242-2249] can enable hybrid functional DFT calculations for nanosystems with thousands of atoms. The cost of constructing the ACE operator is the same as that of applying the exchange operator to the occupied orbitals once, while the cost of applying the Hamiltonian operator with a hybrid functional (after construction of the ACE operator) is only marginally higher than that associated with applying a Hamiltonian constructed from local and semilocal exchange-correlation functionals. Therefore, this new development significantly lowers the computational barrier for using hybrid functionals in large-scale DFT calculations. We demonstrate that a parallel planewave implementation of this method can be used to compute the ground-state electronic structure of a 1000-atom bulk silicon system in less than 30 wall clock minutes and that this method scales beyond 8000 computational cores for a bulk silicon system containing about 4000 atoms. The efficiency of the present methodology in treating large systems enables us to investigate adsorption properties of water molecules on Ag-supported two-dimensional silicene. Our computational results show that water monomer, dimer, and trimer configurations exhibit distinct adsorption behaviors on silicene. In particular, the presence of additional water molecules in the dimer and trimer configurations induces a transition from physisorption to chemisorption, followed by dissociation on Ag-supported silicene. This is caused by the enhanced effect of hydrogen bonds on charge transfer and proton transfer processes. Such a hydrogen bond autocatalytic effect is expected to have broad applications for silicene as an efficient surface catalyst for oxygen reduction reactions and water dissociation.