Session A44: Real-Space Methods for the Electronic Structure Problem I

Exploiting Orbital Locality in Real Space to Enable Large-Scale Condensed-Phase Ab Initio Molecular Dynamics with Hybrid Density Functional Theory

By including a fraction of exact exchange (EXX), hybrid functionals reduce the self-interaction error in semi-local density functional theory (DFT), and thereby furnish a more accurate and reliable description of the underlying electronic structure in systems throughout chemistry, physics, and materials science. However, the high computational cost associated with hybrid DFT has limited its applicability when treating large-scale condensed-phase systems. To overcome this limitation, we have devised a linear-scaling yet formally exact approach that utilizes a local representation of the occupied orbitals to exploit the sparsity in the real-space evaluation of the quantum mechanical exchange interaction in finite-gap systems. Here, we present a detailed description of the theoretical and algorithmic advances required to perform ab initio molecular dynamics (AIMD) simulations of large-scale condensed-phase systems with hybrid DFT. We focus our theoretical discussion on integrating this approach into the framework of Car-Parrinello AIMD, and provide a comprehensive description of our algorithm, which is implemented in the open-source Quantum ESPRESSO program and employs a hybrid MPI/OpenMP parallelization scheme to efficiently utilize the high-performance computing (HPC) resources available on supercomputer architectures. This is followed by a critical assessment of the accuracy and parallel performance (e.g., strong and weak scaling) of this approach when performing AIMD simulations of liquid water in the canonical (NVT) and isobaric-isothermal (NpT) ensembles. With access to HPC resources, we demonstrate that our algorithm enables hybrid DFT based AIMD simulations of condensed-phase systems containing ~1000 atoms with a wall-time cost that is comparable to semi-local DFT. In doing so, this work takes us one step closer to routinely performing AIMD simulations of large-scale condensed-phase systems for sufficiently long timescales at the hybrid DFT level of theory.   

Improving the Scalability of Condensed-Phase Hybrid Density Functional Theory: Computation, Communication, and Load Balancing

Ab initio molecular dynamics (AIMD) simulations at the hybrid density functional theory (DFT) level provide a semi-quantitative description of complex condensed-phase systems such as molecular liquids and crystals. For finite-gap systems, we have developed a linear-scaling and formally exact algorithm for computing the exact exchange interaction in real space based on a localized representation of the occupied orbitals (e.g., maximally localized Wannier functions). Although a massively parallel implementation of this algorithm in Quantum ESPRESSO already enables hybrid DFT based AIMD simulations of condensed-phase systems containing 500-1000 atoms, we have identified three nearly equal contributions to the walltime cost of this approach: computation events, communication overhead, and processor idling due to workload imbalance. In this work, we present a three-pronged strategy that we have employed to attack these contributions and reduce the overall walltime cost by approximately an order of magnitude.   

SPARC-X: Real-space Density Functional Theory for large length and time scales

In this talk, previous and current efforts of the speaker to develop efficient real-space formulations and massively parallel implementations for Density Functional Theory (DFT) will be discussed. These include (i) SPARC: A general purpose framework for performing large-scale electronic structure calculations based on DFT; (ii) Cyclic DFT: A framework for studying systems possessing cyclic symmetry, with application to the bending deformations in nanostructures; (iii) Helical DFT: A framework for studying systems possessing helical symmetry, with application to the torsional deformations in nanostructures; and (iv) SQDFT: A linear-scaling framework for studying materials under extreme conditions. Overall, the speaker will discuss how the above developments enable electronic structure simulations at large length and time scales.   

A Scalable Eigensolver for Real-space Pseudopotential Density Functional Theory: A Polynomial-filtered Spectrum Slicing Method

First-principles electronic structure calculations are a popular avenue for understanding and predicting properties of materials. However, solving the electronic structures of the materials of interest, such as complex biomolecules, nanostructures, and interfacial systems can require descriptions of systems with many atoms, e.g., systems with over 10,000 atoms. Systems of this size pose a challenge to current electronic structure computation software. We will present recent work using a spectrum-slicing algorithm, which is implemented in a real-space pseudopotential density functional theory code, PARSEC. The spectrum slicing method builds an additional layer of parallelization on top of the Chebyshev-filtered subspace iteration. Our approach provides more flexibility to fully utilize the computing power of modern distributed parallel computers. We will demonstrate the scalability of the algorithm and discuss outstanding challenges.   

Treecode-Accelerated Green's Iteration for Kohn-Sham DFT

We present a real-space method for Kohn-Sham Density Functional Theory based on an integral equation formulation of the Kohn-Sham equations, called Treecode-Accelerated Green's Iteration (TAGI). In this approach, the eigenvalue problem for the Kohn-Sham differential operator is converted to a fixed point problem for an integral operator by convolution with a Green's function, then the fixed points are computed using Green's iteration. Essential to this method is the accurate and efficient evaluation of the convolution integrals arising in the iteration. TAGI achieves accuracy and efficiency through the use of adaptive mesh refinement to represent the fields, singularity subtraction schemes to reduce the quadrature error due to the singular Green's functions, and a GPU-accelerated treecode to reduce the computational complexity of the convolution integrals. We have performed all-electron calculations on non-periodic systems and demonstrated systematic convergence to chemical accuracy with respect to tightly converged reference values.   

Accelerating real-space methods by discontinuous projection

By virtue of multiple advances in the past two decades, real-space electronic structure methods have surpassed planewave methods in large-scale calculations of isolated and extended systems alike. Combining advances in both finite-difference and finite-element methods over the decades, we discuss a new approach to accelerate real-space methods further still, while retaining the simplicity, systematic convergence, and parallelizability inherent in the methodology. The key idea is to compress the large, sparse real-space Hamiltonian by projection in a strictly local, systematically improvable, discontinuous basis spanning the occupied subspace. We show how this basis can be constructed and employed to reduce the dimension of the real-space Hamiltonian by up to three orders of magnitude. Molecular dynamics step times of a few minutes for systems containing thousands of atoms demonstrate the scalability of the methodology in a discontinuous Galerkin formulation [1]. Results for 1D, 2D, and 3D systems demonstrate the additional advantages afforded by the new projection formulation [2].
[1] J. Chem. Theory Comput. 14, 2930–2946 (2018).
[2] J. Chem. Phys. 149, 094104 (2018).   

A Space-filling Curve Based Grid Partition to Accelerate Real-space Pseudopotential Density Functional Theory Calculations

Density functional theory (DFT) has become a popular tool to verify, explain, and predict experimental discoveries in materials. In conjunction with pseudopotentials, we can now achieve simulations of systems with tens of thousands of atoms as “routine work.” Real-space DFT has advantages when simulating confined or semi-periodic systems, such as defects, charged systems, and interfaces. Within a finite-difference method, the Hamiltonian matrix is often large and sparse, and requires an efficient implementation of matrix-vector multiplication. We will show through space-filling curves that we can construct a real-space grid whose grid points have excellent locality. Consequently, the communication between compute nodes is reduced. We will also demonstrate that this space-filling curve based grid partition improves the scalability of the matrix-vector multiplications, which is beneficial to polynomial filtering based eigensolvers.   

Reduce Noise in Stochastic Density Functional Theory

Large scale density functional theory (DFT) calculations are necessary for understanding the physics of complex materials. Steep numerical scaling of conventional DFT methods prohibits the routine application to large systems containing tens of thousands of electrons. An alternative to conventional DFT is based on representing the density and density matrix using stochastic sampling of the occupied subspace, allowing for linear or even sub-linear scaling DFT method at the cost of introducing a well-controlled statistical error. It becomes an important task to develop approaches that reduce the stochastic noise in order to improve accuracy and reliability of stochastic DFT. Two different noise reduction techniques have been introduced in stochastic DFT. One is based on decomposing the system into overlapped fragments and the other approach divides the occupied subspace into subspaces according to energy windows. Both noise reduction techniques can significantly reduce the noise level in electronic structures, which leads to orders of magnitude reduction in computational costs. This talk will provide a look into both methods including analysis of noise reduction, scaling, and performance. Illustrations will be given for semiconductor materials with nearly 16,000 electrons.   

A general approach towards continuous translational symmetry in finite difference real-space calculations

We have developed a new scheme to install pseudopotentials on a finite real-space grid that significantly reduces unphysical fluctuations of quantities for fractional grid-point shifts in real space. Instead of interpolating the potential on the grid, this scheme chooses a reference position and use a translation method to represent positions of atoms in real space. This translation is exact for integer grid-point shifts and is designed to minimize the "egg box" effect for fractional grid-point shifts. It provides nonlocal but banded representations for local potentials and is compatible with nonlocal pseudopotential operators. As a demonstration, this scheme is tested in one dimension for three types of potentials: a local ionic potential, a local ionic potential plus a nonlocal operator, and a local ionic potential plus the Hartree and exchange-correlation potential. Fluctuations of examined quantities are reduced by four orders, four orders, and three orders, respectively. This scheme does not require the manipulation of grids and can be easily extended to the three-dimensional case.   

Discrete discontinuous basis projection (DDBP) method for large-scale electronic structure calculations.

The large number of grid points per atom required for accurate real-space Kohn-Sham Density Functional Theory (DFT) calculations restricts their efficiency. In this work, we present an approach to accelerate such calculations several fold, without loss of accuracy, by systematically reducing the cost of the key computational step: the determination of the Kohn-Sham orbitals spanning the occupied subspace. This is achieved by systematically reducing the dimension of the discrete eigenproblem that must be solved, through projection into a highly efficient discrete discontinuous basis. In calculations of quasi-1D, quasi-2D, and bulk metallic systems, we find that accurate energies and forces are obtained with 8–25 basis functions per atom, reducing the dimension of full-matrix eigenproblems by 1-3 orders of magnitude.   

Fast all-electron density functional theory calculations in solids using orthogonalized enriched finite elements

We present a computationally efficient approach to perform real-space all-electron Kohn-Sham DFT calculations for bulk solids using an enriched finite element (FE) basis, wherein the classical FE basis are augmented with atom-centered numerical basis functions constructed from atomic solutions to the Kohn-Sham problem. We term these atom-centered numerical basis functions as enrichment functions. Notably, to improve the conditioning we orthogonalize the enrichment function with respect to the classical FE basis, without compromising on the locality of the resultant basis. In addition to improved conditioning, this orthogonalization procedure also renders the overlap matrix block-diagonal, greatly simplifying its inversion. Subsequently, we use a Chebyshev polynomial based acceleration technique to efficiently compute the occupied eigenspace in each self-consistent iteration. We demonstrate the accuracy and efficiency for periodic unit-cells and supercells, ranging up to 5000 electrons (containing as many as 500 atoms). We observe a staggering 50-100x speedup over the classical FE basis. We also benchmark with (L)APW(+lo) basis for accuracy and performance. Finally, we demonstrate parallel scalability for a system with ~216 Si and C atoms.