Large-scale Electronic Structure Calculations Research Articles

Deep learning tight-binding approach for large-scale electronic simulations at finite temperatures with ab initio accuracy

Simulating electronic behavior in materials and devices with realistic large system sizes remains a formidable task within the ab initio framework due to its computational intensity. Here we show DeePTB, an efficient deep learning-based tight-binding approach with ab initio accuracy to address this issue. By training on structural data and corresponding ab initio eigenvalues, the DeePTB model can efficiently predict tight-binding Hamiltonians for unseen structures, enabling efficient simulations of large-size systems under external perturbations such as finite temperatures and strain. This capability is vital for semiconductor band gap engineering and materials design. When combined with molecular dynamics, DeePTB facilitates efficient and accurate finite-temperature simulations of both atomic and electronic behavior simultaneously. This is demonstrated by computing the temperature-dependent electronic properties of a gallium phosphide system with 106 atoms. The availability of DeePTB bridges the gap between accuracy and scalability in electronic simulations, potentially advancing materials science and related fields by enabling large-scale electronic structure calculations.

Random Green’s Function Method for Large-Scale Electronic Structure Calculation

We report a linear-scaling random Green’s function (rGF) method for large-scale electronic structure calculation. In this method, the rGF is defined on a set of random states and is efficiently calculated by projecting onto Krylov subspace. With the rGF method, the Fermi–Dirac operator can be obtained directly, avoiding the polynomial expansion to Fermi–Dirac function. To demonstrate the applicability, we implement the rGF method with the density-functional tight-binding method. It is shown that the Krylov subspace can maintain at small size for materials with different gaps at zero temperature, including H2O and Si clusters. We find with a simple deflation technique that the rGF self-consistent calculation of H2O clusters at T = 0 K can reach an error of ∼ 1 meV per H2O molecule in total energy, compared to deterministic calculations. The rGF method provides an effective stochastic method for large-scale electronic structure simulation.

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.

Speeding up tight binding calculations using zone-folding methods

Tight binding models are widely used in large scale electronic structure calculations of nanostructures. Their atomistic nature makes them flexible, but also means the computational cost increases rapidly with system size. The large number of calculations required to design nanostructures makes computational efficiency desirable. We have developed a method to increase computational speed while retaining most of its accuracy. The method is based on the use of supercells and zone folding combined with a truncation of the Hamiltonians to only include states close to the band-edges. We apply the method to model the band edge energies of a GaAs/AlAs quantum well grown along the [110]-directions with 3D and 2D periodic boundary conditions as well as the density of states and dielectric function of the quantum well. We typically find a speed-up of ten times with only a small loss of accuracy of the calculation result.

All-Atom Nonadiabatic Dynamics Simulation of Hybrid Graphene Nanoribbons Based on Wannier Analysis and Machine Learning.

Trajectory surface hopping combined with ab initio electronic structure calculations is a popular and powerful approach for on-the-fly nonadiabatic dynamics simulations. For large systems, however, this remains a significant challenge because of the unaffordable computational cost of large-scale electronic structure calculations. Here, we present an efficient divide-and-conquer approach to construct the system Hamiltonian based on Wannier analysis and machine learning. In detail, the large system under investigation is first decomposed into small building blocks, and then all possible segments formed by building blocks within a cutoff distance are found out. Ab initio molecular dynamics is carried out to generate a sequence of geometries for each equivalent segment with periodicity. The Hamiltonian matrices in the maximum localized Wannier function (MLWF) basis are obtained for all geometries and utilized to train artificial neural networks (ANNs) for the structure-dependent Hamiltonian elements. Taking advantage of the orthogonality and spatial locality of MLWFs, the one-electron Hamiltonian of a large system at arbitrary geometry can be directly constructed by the trained ANNs. As demonstrations, we study charge transport in a zigzag graphene nanoribbon (GNR), a coved GNR, and a series of hybrid GNRs with a state-of-the-art surface hopping method. The interplay between delocalized and localized states is found to determine the electron dynamics in hybrid GNRs. Our approach has successfully studied GNRs with >10 000 atoms, paving the way for efficient and reliable all-atom nonadiabatic dynamics simulation of general systems.

A parallel orbital-updating based optimization method for electronic structure calculations

In this paper, we propose a parallel orbital-updating based optimization method for electronic structure calculations. With our method, the solution of the minimization problem for the Kohn-Sham energy functional with respect to N orbitals is replaced by the solution of N independent minimization problems for the energy functional with respect to one orbital and the orthogonalization of the N updated orbitals. This new method allows a two-level parallelization. This feature makes our approach has a great advantage in large scale parallel computing. The numerical experiments show that our new method is reliable and efficient. Hence, our new method has a great potential for large scale electronic structure calculations on modern supercomputers.

EQE 2.0: Subsystem DFT beyond GGA functionals

By adopting a divide-and-conquer strategy, subsystem-DFT (sDFT) can dramatically reduce the computational cost of large-scale electronic structure calculations. The key ingredients of sDFT are the nonadditive kinetic energy and exchange-correlation functionals which dominate it's accuracy. Even though, available semilocal nonadditive functionals find a broad range of applications, their accuracy is somewhat limited especially for those systems where achieving balance between exchange-correlation interactions on one side and nonadditive kinetic energy on the other is crucial. In eQE 2.0, we improve dramatically the accuracy of sDFT simulations by (1) implementing nonlocal nonadditive kinetic energy functionals based on the LMGP family of functionals; (2) adapting Quantum ESPRESSO's implementation of rVV10 and vdW-DF nonlocal exchange-correlation functionals to be employed in sDFT simulations; (3) implementing “deorbitalized” meta GGA functionals (e.g., SCAN-L). We carefully assess the performance of the newly implemented tools on the S22-5 test set. eQE 2.0 delivers excellent interaction energies compared to conventional Kohn-Sham DFT and CCSD(T). The improved performance does not come at a loss of computational efficiency. We show that eQE 2.0 with nonlocal nonadditive functionals retains the same linear scaling behavior achieved previously in eQE 1.0 with semilocal nonadditive functionals. Program summaryProgram title: eQECPC Library link to program files:https://doi.org/10.17632/g55n9xmnt2.1Developer's repository link:https://gitlab.com/Pavanello/eqeLicensing provisions: GPLv2Programming language: Fortran 90External routines/libraries: BLACS, MPINature of problem: Solving the electronic structure of molecules and materials with subsystem density functional theorySolution method: This version of eQE uses subsystem Density-Functional Theory (DFT) to compute the electronic structure of molecules and materials. Subsystem DFT is a divide-and-conquer version of DFT that can be massively parallelized and is linear scaling in work and data. However, it requires the use of density functionals for kinetic and exchange-correlation energies. eQE can now employ nonlocal functionals as well as meta-GGA functionals for these energy terms.Additional comments: Url of stable release http://eqe.rutgers.edu

Harnessing the Power of Multi-GPU Acceleration into the Quantum Interaction Computational Kernel Program.

We report a new multi-GPU capable ab initio Hartree-Fock/density functional theory implementation integrated into the open source QUantum Interaction Computational Kernel (QUICK) program. Details on the load balancing algorithms for electron repulsion integrals and exchange correlation quadrature across multiple GPUs are described. Benchmarking studies carried out on up to four GPU nodes, each containing four NVIDIA V100-SXM2 type GPUs demonstrate that our implementation is capable of achieving excellent load balancing and high parallel efficiency. For representative medium to large size protein/organic molecular systems, the observed parallel efficiencies remained above 82% for the Kohn-Sham matrix formation and above 90% for nuclear gradient calculations. The accelerations on NVIDIA A100, P100, and K80 platforms also have realized parallel efficiencies higher than 68% in all tested cases, paving the way for large-scale ab initio electronic structure calculations with QUICK.

Γ valley transition metal dichalcogenide moiré bands

The valence band maxima of most group VI transition metal dichalcogenide thin films remain at the Γ point all of the way from bulk to bilayer. In this paper, we develop a continuum theory of the moiré minibands that are formed in the valence bands of Γ-valley homobilayers by a small relative twist. Our effective theory is benchmarked against large-scale ab initio electronic structure calculations that account for lattice relaxation. As a consequence of an emergent [Formula: see text] symmetry, we find that low-energy Γ-valley moiré holes differ qualitatively from their K-valley counterparts addressed previously; in energetic order, the first three bands realize 1) a single-orbital model on a honeycomb lattice, 2) a two-orbital model on a honeycomb lattice, and 3) a single-orbital model on a kagome lattice.

Pythonic Black-box Electronic Structure Tool (PyBEST). An open-source Python platform for electronic structure calculations at the interface between chemistry and physics

Pythonic Black-box Electronic Structure Tool (PyBEST) represents a fully-fledged modern electronic structure software package developed at Nicolaus Copernicus University in Toruń. The package provides an efficient and reliable platform for electronic structure calculations at the interface between chemistry and physics using unique electronic structure methods, analysis tools, and visualization. Examples are the (orbital-optimized) pCCD-based models for ground- and excited-states electronic structure calculations as well as the quantum entanglement analysis framework based on the single-orbital entropy and orbital-pair mutual information. PyBEST is written primarily in the Python programming language with additional parts written in C＋＋, which are interfaced using Pybind11, a lightweight header-only library. By construction, PyBEST is easy to use, to code, and to interface with other software packages. Moreover, its modularity allows us to conveniently host additional Python packages and software libraries in future releases to enhance its performance. The electronic structure methods available in PyBEST are tested for the half-filled 1-D model Hamiltonian. The capability of PyBEST to perform large-scale electronic structure calculations is demonstrated for the model vitamin B12 compound. The investigated molecule is composed of 190 electrons and 777 orbitals for which an orbital optimization within pCCD and an orbital entanglement and correlation analysis are performed for the first time. Program summaryProgram title: PyBESTCPC Library link to program files:https://doi.org/10.17632/xf9kb7yfwr.1Developer’s repository link:https://zenodo.org/record/3925278#.X5KAZS8Rq6sLicensing provisions: GNU General Public License 3Programming language:Python, C＋＋Nature of problem: Efficient and reliable modeling of electronic structures featuring both weakly- and strongly-correlated electrons. Small- and large-scale quantum-mechanical problems at the interface between chemistry and physics comprising both quantum chemical and model Hamiltonians. Specifically, modeling potential energy surfaces of complex electronic structures including bond breaking/formation, elucidating complex electronic structures through the picture of interacting orbitals, describing noncovalent interactions, ultra-cold trapped quantum gases, and a variety of applications in interdisciplinary quantum mechanical-based problems.Solution method: Modular implementation of a series of unconventional (and conventional) electronic structure models based on the pCCD ansatz to solve the electronic Schrödinger equation. These include the description of both ground- and excited-states, the determination of interaction energies, and the analysis and interpretation of electronic wavefunctions. All modules are implemented in the modern Python programming language, where bottleneck operations are handled by C＋＋ code interfaced by the Pybind11 header-only library. The implemented (wavefunction) modules and modular code structure make PyBEST a very efficient alternative to existing electronic structure packages.Additional comments including restrictions and unusual features:PyBEST features unconventional electronic structure methods (pCCD and post-pCCD methods) that are not available in any other quantum chemistry/physics software package. It also includes a general orbital entanglement and correlation module that supports both pCCD and selected post-pCCD methods. PyBEST is designed to be easy to use and code in. Due to its modularity (for instance of the tensor contraction engine), new Python modules and features can be straightforwardly imported and exploited without changing any wavefunction modules directly.

Stone–Wales defects preserve hyperuniformity in amorphous two-dimensional networks

Disordered hyperuniformity (DHU) is a recently discovered novel state of many-body systems that possesses vanishing normalized infinite-wavelength density fluctuations similar to a perfect crystal and an amorphous structure like a liquid or glass. Here, we discover a hyperuniformity-preserving topological transformation in two-dimensional (2D) network structures that involves continuous introduction of Stone-Wales (SW) defects. Specifically, the static structure factor [Formula: see text] of the resulting defected networks possesses the scaling [Formula: see text] for small wave number k, where [Formula: see text] monotonically decreases as the SW defect concentration p increases, reaches [Formula: see text] at [Formula: see text], and remains almost flat beyond this p. Our findings have important implications for amorphous 2D materials since the SW defects are well known to capture the salient feature of disorder in these materials. Verified by recently synthesized single-layer amorphous graphene, our network models reveal unique electronic transport mechanisms and mechanical behaviors associated with distinct classes of disorder in 2D materials.

Flexibilities of wavelets as a computational basis set for large-scale electronic structure calculations.

The BigDFT project was started in 2005 with the aim of testing the advantages of using a Daubechies wavelet basis set for Kohn-Sham (KS) density functional theory (DFT) with pseudopotentials. This project led to the creation of the BigDFT code, which employs a computational approach with optimal features of flexibility, performance, and precision of the results. In particular, the employed formalism has enabled the implementation of an algorithm able to tackle DFT calculations of large systems, up to many thousands of atoms, with a computational effort that scales linearly with the number of atoms. In this work, we recall some of the features that have been made possible by the peculiar properties of Daubechies wavelets. In particular, we focus our attention on the usage of DFT for large-scale systems. We show how the localized description of the KS problem, emerging from the features of the basis set, is helpful in providing a simplified description of large-scale electronic structure calculations. We provide some examples on how such a simplified description can be employed, and we consider, among the case-studies, the SARS-CoV-2 main protease.

Practical Approach to Large-Scale Electronic Structure Calculations in Electrolyte Solutions via Continuum-Embedded Linear-Scaling Density Functional Theory

We present the implementation of a hybrid continuum-atomistic model for including the effects of a surrounding electrolyte in large-scale density functional theory (DFT) calculations within the Ord...

Resonant and bound states of charged defects in two-dimensional semiconductors

A detailed understanding of charged defects in two-dimensional semiconductors is needed for the development of ultrathin electronic devices. Here, we study negatively charged acceptor impurities in monolayer WS$_2$ using a combination of scanning tunnelling spectroscopy and large-scale atomistic electronic structure calculations. We observe several localized defect states of hydrogenic wave function character in the vicinity of the valence band edge. Some of these defect states are bound, while others are resonant. The resonant states result from the multi-valley valence band structure of WS$_2$, whereby localized states originating from the secondary valence band maximum at $\Gamma$ hybridize with continuum states from the primary valence band maximum at K/K$^{\prime}$. Resonant states have important consequences for electron transport as they can trap mobile carriers for several tens of picoseconds.

Large-Scale Phonon Calculations Using the Real-Space Multigrid Method.

Phonons are fundamental to understanding the dynamical and thermal properties of materials. However, first-principles phonon calculations are usually limited to moderate-size systems due to their high computational requirements. We implemented the finite displacement method (FDM) in the highly parallel real-space multigrid (RMG) suite of codes to study phonon properties. RMG scales from desktops to clusters and supercomputers containing thousands of nodes, fully supports graphics processing units (GPUs), including multiple GPUs per node, and is very suitable for large-scale electronic structure calculations. It is used as the core computational kernel to calculate the force constants matrix with FDM. By comparing with other widely used density functional theory packages and experimental data from inelastic neutron scattering, we demonstrate that RMG is very accurate in calculating forces at small displacements from equilibrium positions. The calculated phonon band structures and vibrational spectra for a variety of different systems are in very good agreement with plane-wave-based density functional theory codes, Quantum ESPRESSO, CASTEP and VASP, and these results have been validated comparing with inelastic neutron scattering experimental data measured at the VISION spectrometer at the Spallation Neutron Source.

Parallelization and scalability analysis of inverse factorization using the chunks and tasks programming model

We present three methods for distributed memory parallel inverse factorization of block-sparse Hermitian positive definite matrices. The three methods are a recursive variant of the AINV inverse Cholesky algorithm, iterative refinement, and localized inverse factorization. All three methods are implemented using the Chunks and Tasks programming model, building on the distributed sparse quad-tree matrix representation and parallel matrix-matrix multiplication in the publicly available Chunks and Tasks Matrix Library (CHTML). Although the algorithms are generally applicable, this work was mainly motivated by the need for efficient and scalable inverse factorization of the basis set overlap matrix in large scale electronic structure calculations. We perform various computational tests on overlap matrices for quasi-linear glutamic acid-alanine molecules and three-dimensional water clusters discretized using the standard Gaussian basis set STO-3G with up to more than 10 million basis functions. We show that for such matrices the computational cost increases only linearly with system size for all the three methods. We show both theoretically and in numerical experiments that the methods based on iterative refinement and localized inverse factorization outperform previous parallel implementations in weak scaling tests where the system size is increased in direct proportion to the number of processes. We show also that, compared to the method based on pure iterative refinement, the localized inverse factorization requires much less communication.

Impact of Disorder on the Optoelectronic Properties of GaNyAs1−x−yBix Alloys and Heterostructures

We perform a systematic theoretical analysis of the nature and importance of alloy disorder effects on the electronic and optical properties of GaN$_{y}$As$_{1-x-y}$Bi$_{x}$ alloys and quantum wells (QWs), using large-scale atomistic supercell electronic structure calculations based on the tight-binding method. Using ordered alloy supercell calculations we also derive and parametrise an extended basis 14-band \textbf{k}$\cdot$\textbf{p} Hamiltonian for GaN$_{y}$As$_{1-x-y}$Bi$_{x}$. Comparison of the results of these models highlights the role played by short-range alloy disorder -- associated with substitutional nitrogen (N) and bismuth (Bi) incorporation -- in determining the details of the electronic and optical properties. Systematic analysis of large alloy supercells reveals that the respective impact of N and Bi on the band structure remain largely independent, a robust conclusion we find to be valid even in the presence of significant alloy disorder where N and Bi atoms share common Ga nearest neighbours. Our calculations reveal that N- (Bi-) related alloy disorder strongly influences the conduction (valence) band edge states, leading in QWs to strong carrier localisation, as well as inhomogeneous broadening and modification of the conventional selection rules for optical transitions. Our analysis provides detailed insight into key properties and trends in this unusual material system, and enables quantitative evaluation of the potential of GaN$_{y}$As$_{1-x-y}$Bi$_{x}$ alloys for applications in photonic and photovoltaic devices.

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

We present an approach to accelerate real-space electronic structure methods several fold, without loss of accuracy, by reducing the dimension of the discrete eigenproblem that must be solved. To accomplish this, we construct an efficient, systematically improvable, discontinuous basis spanning the occupied subspace and project the real-space Hamiltonian onto the span. In calculations on a range of systems, we find that accurate energies and forces are obtained with 8-25 basis functions per atom, reducing the dimension of the associated real-space eigenproblems by 1-3 orders of magnitude.

Solution of the k-th eigenvalue problem in large-scale electronic structure calculations

We consider computing the k-th eigenvalue and its corresponding eigenvector of a generalized Hermitian eigenvalue problem of n×n large sparse matrices. In electronic structure calculations, several properties of materials, such as those of optoelectronic device materials, are governed by the eigenpair with a material-specific index k. We present a three-stage algorithm for computing the k-th eigenpair with validation of its index. In the first stage of the algorithm, we propose an efficient way of finding an interval containing the k-th eigenvalue (1≪k≪n) with a non-standard application of the Lanczos method. In the second stage, spectral bisection for large-scale problems is realized using a sparse direct linear solver to narrow down the interval of the k-th eigenvalue. In the third stage, we switch to a modified shift-and-invert Lanczos method to reduce bisection iterations and compute the k-th eigenpair with validation. Numerical results with problem sizes up to 1.5 million are reported, and the results demonstrate the accuracy and efficiency of the three-stage algorithm.

DL_MG: A Parallel Multigrid Poisson and Poisson-Boltzmann Solver for Electronic Structure Calculations in Vacuum and Solution.

The solution of the Poisson equation is a crucial step in electronic structure calculations, yielding the electrostatic potential-a key component of the quantum mechanical Hamiltonian. In recent decades, theoretical advances and increases in computer performance have made it possible to simulate the electronic structure of extended systems in complex environments. This requires the solution of more complicated variants of the Poisson equation, featuring nonhomogeneous dielectric permittivities, ionic concentrations with nonlinear dependencies, and diverse boundary conditions. The analytic solutions generally used to solve the Poisson equation in vacuum (or with homogeneous permittivity) are not applicable in these circumstances, and numerical methods must be used. In this work, we present DL_MG, a flexible, scalable, and accurate solver library, developed specifically to tackle the challenges of solving the Poisson equation in modern large-scale electronic structure calculations on parallel computers. Our solver is based on the multigrid approach and uses an iterative high-order defect correction method to improve the accuracy of solutions. Using two chemically relevant model systems, we tested the accuracy and computational performance of DL_MG when solving the generalized Poisson and Poisson-Boltzmann equations, demonstrating excellent agreement with analytic solutions and efficient scaling to ∼109 unknowns and 100s of CPU cores. We also applied DL_MG in actual large-scale electronic structure calculations, using the ONETEP linear-scaling electronic structure package to study a 2615 atom protein-ligand complex with routinely available computational resources. In these calculations, the overall execution time with DL_MG was not significantly greater than the time required for calculations using a conventional FFT-based solver.