1
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Smyser KE, White A, Sharma S. Use of Multigrids to Reduce the Cost of Performing Interpolative Separable Density Fitting. J Phys Chem A 2024; 128:7451-7461. [PMID: 39186251 DOI: 10.1021/acs.jpca.4c02431] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 08/27/2024]
Abstract
In this article, we present an interpolative separable density fitting (ISDF)-based algorithm to calculate the exact exchange in periodic mean field calculations. In the past, decomposing the two-electron integrals into the tensor hypercontraction (THC) form using ISDF was the most expensive step of the entire mean field calculation. Here, we show that by using a multigrid-ISDF algorithm, both the memory and the CPU cost of this step can be reduced. The CPU cost is brought down from cubic scaling to quadratic scaling with a low computational prefactor which reduces the cost by almost 2 orders of magnitude. Thus, in the new algorithm, the cost of performing ISDF is largely negligible compared to other steps. Along with the CPU cost, the memory cost of storing the factorized two-electron integrals is also reduced by a factor of up to 35. With the current algorithm, we can perform Hartree-Fock calculations on a diamond supercell containing more than 17,000 basis functions and more than 1500 electrons on a single node with no disk usage. For this calculation, the cost of constructing the exchange matrix is only a factor of 4 slower than the cost of diagonalizing the Fock matrix. Augmenting our approach with linear scaling algorithms can further speed up the calculations.
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Affiliation(s)
- Kori E Smyser
- Department of Chemistry, University of Colorado, Boulder, Colorado 80302, United States
| | - Alec White
- Quantum Simulation Technologies, Inc., Boston ,Massachusetts02135, United States
| | - Sandeep Sharma
- Department of Chemistry, University of Colorado, Boulder, Colorado 80302, United States
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2
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Sereda M, Allen T, Bradbury NC, Ibrahim KZ, Neuhauser D. Sparse-Stochastic Fragmented Exchange for Large-Scale Hybrid Time-Dependent Density Functional Theory Calculations. J Chem Theory Comput 2024; 20:4196-4204. [PMID: 38713513 DOI: 10.1021/acs.jctc.4c00260] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 05/09/2024]
Abstract
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.
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Affiliation(s)
- Mykola Sereda
- Department of Chemistry and Biochemistry, and California Nanoscience Institute, UCLA, Los Angeles, California 90095-1569, United States
| | - Tucker Allen
- Department of Chemistry and Biochemistry, and California Nanoscience Institute, UCLA, Los Angeles, California 90095-1569, United States
| | - Nadine C Bradbury
- Department of Chemistry and Biochemistry, and California Nanoscience Institute, UCLA, Los Angeles, California 90095-1569, United States
| | - Khaled Z Ibrahim
- Computer Science Department, Lawrence Berkeley National Laboratory, One Cyclotron Road, Berkeley, California 94720, United States
| | - Daniel Neuhauser
- Department of Chemistry and Biochemistry, and California Nanoscience Institute, UCLA, Los Angeles, California 90095-1569, United States
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3
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Li J, Yang L, Wan L, Hu W, Yang J. Machine Learning K-Means Clustering in Interpolative Separable Density Fitting Algorithm: Advancing Accurate and Efficient Cubic-Scaling Density Functional Perturbation Theory Calculations within Plane Waves. J Phys Chem A 2024. [PMID: 38439159 DOI: 10.1021/acs.jpca.3c07159] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/06/2024]
Abstract
Density functional perturbation theory (DFPT) is a crucial tool for accurately describing lattice dynamics. The adaptively compressed polarizability (ACP) method reduces the computational complexity of DFPT calculations from O(N4) to O(N3) by combining the interpolative separable density fitting (ISDF) algorithm. However, the conventional QR factorization with column pivoting (QRCP) algorithm, used for selecting the interpolation points in ISDF, not only incurs a high cubic-scaling computational cost, O(N3), but also leads to suboptimal convergence. This convergence issue is particularly pronounced when considering the complex interplay between the external potential and atomic displacement in ACP-based DFPT calculations. Here, we present a machine learning K-means clustering algorithm to select the interpolation points in ISDF, which offers a more efficient quadratic-scaling O(N2) alternative to the computationally intensive cubic-scaling O(N3) QRCP algorithm. We implement this efficient K-means-based ISDF algorithm to accelerate plane-wave DFPT calculations in KSSOLV, which is a MATLAB toolbox for performing Kohn-Sham density functional theory calculations within plane waves. We demonstrate that this K-means algorithm not only offers comparable accuracy to QRCP in ISDF but also achieves better convergence for ACP-based DFPT calculations. In particular, K-means can remarkably reduce the computational cost of selecting the interpolation points by nearly 2 orders of magnitude compared to QRCP in ISDF.
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Affiliation(s)
- Jielan Li
- Key Laboratory of Precision and Intelligent Chemistry, Department of Chemical Physics and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Liu Yang
- Key Laboratory of Precision and Intelligent Chemistry, Department of Chemical Physics and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Lingyun Wan
- Key Laboratory of Precision and Intelligent Chemistry, Department of Chemical Physics and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Wei Hu
- Key Laboratory of Precision and Intelligent Chemistry, Department of Chemical Physics and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Jinlong Yang
- Key Laboratory of Precision and Intelligent Chemistry, Department of Chemical Physics and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
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4
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Jiao S, Li J, Qin X, Wan L, Hu W, Yang J. Complex-Valued K-Means Clustering of Interpolative Separable Density Fitting Algorithm for Large-Scale Hybrid Functional Enabled Ab Initio Molecular Dynamics Simulations within Plane Waves. J Phys Chem A 2024. [PMID: 38430107 DOI: 10.1021/acs.jpca.3c07172] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/03/2024]
Abstract
K-means clustering, as a classic unsupervised machine learning algorithm, is the key step to select the interpolation sampling points in interpolative separable density fitting (ISDF) decomposition for hybrid functional electronic structure calculations. Real-valued K-means clustering for accelerating the ISDF decomposition has been demonstrated for large-scale hybrid functional enabled ab initio molecular dynamics (hybrid AIMD) simulations within plane-wave basis sets where the Kohn-Sham orbitals are real-valued. However, it is unclear whether such K-means clustering works for complex-valued Kohn-Sham orbitals. Here, we propose an improved weight function defined as the sum of the square modulus of complex-valued Kohn-Sham orbitals in K-means clustering for hybrid AIMD simulations. Numerical results demonstrate that the K-means algorithm with a new weight function yields smoother and more delocalized interpolation sampling points, resulting in smoother energy potential, smaller energy drift, and longer time steps for hybrid AIMD simulations compared to the previous weight function used in the real-valued K-means algorithm. In particular, we find that this improved algorithm can obtain more accurate oxygen-oxygen radial distribution functions in liquid water molecules and a more accurate power spectrum in crystal silicon dioxide compared to the previous K-means algorithm. Finally, we describe a massively parallel implementation of this ISDF decomposition to accelerate large-scale complex-valued hybrid AIMD simulations containing thousands of atoms (2,744 atoms), which can scale up to 5,504 CPU cores on modern supercomputers.
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Affiliation(s)
- Shizhe Jiao
- Hefei National Research Center for Physical Sciences at the Microscale, and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Jielan Li
- Hefei National Research Center for Physical Sciences at the Microscale, and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Xinming Qin
- Hefei National Research Center for Physical Sciences at the Microscale, and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Lingyun Wan
- Hefei National Research Center for Physical Sciences at the Microscale, and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Wei Hu
- Hefei National Research Center for Physical Sciences at the Microscale, and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Jinlong Yang
- Key Laboratory of Precision and Intelligent Chemistry, and Department of Chemical Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
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5
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Delesma FA, Leucke M, Golze D, Rinke P. Benchmarking the accuracy of the separable resolution of the identity approach for correlated methods in the numeric atom-centered orbitals framework. J Chem Phys 2024; 160:024118. [PMID: 38205851 DOI: 10.1063/5.0184406] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/25/2023] [Accepted: 12/19/2023] [Indexed: 01/12/2024] Open
Abstract
Four-center two-electron Coulomb integrals routinely appear in electronic structure algorithms. The resolution-of-the-identity (RI) is a popular technique to reduce the computational cost for the numerical evaluation of these integrals in localized basis-sets codes. Recently, Duchemin and Blase proposed a separable RI scheme [J. Chem. Phys. 150, 174120 (2019)], which preserves the accuracy of the standard global RI method with the Coulomb metric and permits the formulation of cubic-scaling random phase approximation (RPA) and GW approaches. Here, we present the implementation of a separable RI scheme within an all-electron numeric atom-centered orbital framework. We present comprehensive benchmark results using the Thiel and the GW100 test set. Our benchmarks include atomization energies from Hartree-Fock, second-order Møller-Plesset (MP2), coupled-cluster singles and doubles, RPA, and renormalized second-order perturbation theory, as well as quasiparticle energies from GW. We found that the separable RI approach reproduces RI-free HF calculations within 9 meV and MP2 calculations within 1 meV. We have confirmed that the separable RI error is independent of the system size by including disordered carbon clusters up to 116 atoms in our benchmarks.
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Affiliation(s)
| | - Moritz Leucke
- Faculty for Chemistry and Food Chemistry, Technische Universität Dresden, 01062 Dresden, Germany
| | - Dorothea Golze
- Faculty for Chemistry and Food Chemistry, Technische Universität Dresden, 01062 Dresden, Germany
| | - Patrick Rinke
- Department of Applied Physics, Aalto University, FI-02150 Espoo, Finland
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6
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Yeh CN, Morales MA. Low-Scaling Algorithm for the Random Phase Approximation Using Tensor Hypercontraction with k-point Sampling. J Chem Theory Comput 2023; 19:6197-6207. [PMID: 37624575 DOI: 10.1021/acs.jctc.3c00615] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 08/26/2023]
Abstract
We present a low-scaling algorithm for the random phase approximation (RPA) with k-point sampling in the framework of tensor hypercontraction (THC) for electron repulsion integrals (ERIs). The THC factorization is obtained via a revised interpolative separable density fitting (ISDF) procedure with a momentum-dependent auxiliary basis for generic single-particle Bloch orbitals. Our formulation does not require preoptimized interpolating points or auxiliary bases, and the accuracy is systematically controlled by the number of interpolating points. The resulting RPA algorithm scales linearly with the number of k-points and cubically with the system size without any assumption on sparsity or locality of orbitals. The errors of ERIs and RPA energy show rapid convergence with respect to the size of the THC auxiliary basis, suggesting a promising and robust direction to construct efficient algorithms of higher order many-body perturbation theories for large-scale systems.
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Affiliation(s)
- Chia-Nan Yeh
- Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, United States
| | - Miguel A Morales
- Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, United States
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7
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Rettig A, Lee J, Head-Gordon M. Even Faster Exact Exchange for Solids via Tensor Hypercontraction. J Chem Theory Comput 2023; 19:5773-5784. [PMID: 37586065 DOI: 10.1021/acs.jctc.3c00407] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 08/18/2023]
Abstract
Hybrid density functional theory (DFT) remains intractable for large periodic systems due to the demanding computational cost of exact exchange. We apply the tensor hypercontraction (THC) (or interpolative separable density fitting) approximation to periodic hybrid DFT calculations with Gaussian-type orbitals using the Gaussian plane wave approach. This is done to lower the computational scaling with respect to the number of basis functions (N) and k-points (Nk) at a fixed system size. Additionally, we propose an algorithm to fit only occupied orbital products via THC (i.e., a set of points, NISDF) to further reduce computation time and memory usage. This algorithm has linear scaling cost with k-points, no explicit dependence of NISDF on basis set size, and overall cubic scaling with unit cell size. Significant speedups and reduced memory usage may be obtained for moderately sized k-point meshes, with additional gains for large k-point meshes. Adequate accuracy can be obtained using THC-oo-K for self-consistent calculations. We perform illustrative hybrid density function theory calculations on the benzene crystal in the basis set and thermodynamic limits to highlight the utility of this algorithm.
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Affiliation(s)
- Adam Rettig
- Department of Chemistry, University of California, Berkeley, California 94720, United States
- Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States
| | - Joonho Lee
- Department of Chemistry, Columbia University, New York, New York 10027, United States
| | - Martin Head-Gordon
- Department of Chemistry, University of California, Berkeley, California 94720, United States
- Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States
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8
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Qin X, Shang H, Yang J. Efficient implementation of analytical gradients for periodic hybrid functional calculations within fitted numerical atomic orbitals from NAO2GTO. Front Chem 2023; 11:1232425. [PMID: 37577064 PMCID: PMC10413557 DOI: 10.3389/fchem.2023.1232425] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/31/2023] [Accepted: 07/13/2023] [Indexed: 08/15/2023] Open
Abstract
The NAO2GTO scheme provides an efficient way to evaluate the electron repulsion integrals (ERIs) over numerical atomic orbitals (NAOs) with auxiliary Gaussian-type orbitals (GTOs). However, the NAO2GTO fitting will significantly impact the accuracy and convergence of hybrid functional calculations. To address this issue, here we propose to use the fitted orbitals as a new numerical basis to properly handle the mismatch between NAOs and fitted GTOs. We present an efficient and linear-scaling implementation of analytical gradients of Hartree-Fock exchange (HFX) energy for periodic HSE06 calculations with fitted NAOs in the HONPAS package. In our implementation, the ERIs and their derivatives for HFX matrix and forces are evaluated analytically with the auxiliary GTOs, while other terms are calculated using numerically discretized GTOs. Several integral screening techniques are employed to reduce the number of required ERI derivatives. We benchmark the accuracy and efficiency of our implementation and demonstrate that our results of lattice constants, bulk moduli, and band gaps of several typical semiconductors are in good agreement with the experimental values. We also show that the calculation of HFX forces based on a master-worker dynamic parallel scheme has a very high efficiency and scales linearly with respect to system size. Finally, we study the geometry optimization and polaron formation due to an excess electron in rutile TiO2 by means of HSE06 calculations to further validate the applicability of our implementation.
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Affiliation(s)
- Xinming Qin
- Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui, China
| | - Honghui Shang
- Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui, China
| | - Jinlong Yang
- Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui, China
- Key Laboratory of Precision and Intelligent Chemistry, University of Science and Technology of China, Hefei, Anhui, China
- Hefei National Laboratory, University of Science and Technology of China, Hefei, Anhui, China
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9
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Qin X, Hu W, Yang J. Interpolative Separable Density Fitting for Accelerating Two-Electron Integrals: A Theoretical Perspective. J Chem Theory Comput 2023; 19:679-693. [PMID: 36693136 DOI: 10.1021/acs.jctc.2c00927] [Citation(s) in RCA: 5] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/25/2023]
Abstract
Low-rank approximations have long been considered an efficient way to accelerate electronic structure calculations associated with the evaluation of electron repulsion integrals (ERIs). As an accurate and efficient algorithm for compressing the ERI tensor, the interpolative separable density fitting (ISDF) decomposition has recently attracted great attention in this context. In this perspective, we introduce the ISDF decomposition from the theoretical aspects and technique details. The ISDF decomposition can construct a fully separable low-rank approximation (tensor hypercontraction factorization) of ERIs in real space with a cubic cost, offering great flexibility for accelerating high-scaling electronic structure calculations. We review the typical applications of ISDF in hybrid functionals, time-dependent density functional theory, and GW approximation. Finally, we discuss the promising directions for future development of ISDF.
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Affiliation(s)
- Xinming Qin
- Hefei National Research Center for Physical Sciences at the Microscale, Department of Chemical Physics, Synergetic Innovation Center of Quantum Information and Quantum Physics, and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui230026, China
| | - Wei Hu
- Hefei National Research Center for Physical Sciences at the Microscale, Department of Chemical Physics, Synergetic Innovation Center of Quantum Information and Quantum Physics, and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui230026, China
| | - Jinlong Yang
- Hefei National Research Center for Physical Sciences at the Microscale, Department of Chemical Physics, Synergetic Innovation Center of Quantum Information and Quantum Physics, and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui230026, China
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10
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Csóka J, Kállay M. Analytic gradients for local density fitting Hartree-Fock and Kohn-Sham methods. J Chem Phys 2023; 158:024110. [PMID: 36641408 DOI: 10.1063/5.0131683] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/24/2022] Open
Abstract
We present analytic gradients for local density fitting Hartree-Fock (HF) and hybrid Kohn-Sham (KS) density functional methods. Due to the non-variational nature of the local fitting algorithm, the method of Lagrange multipliers is used to avoid the solution of the coupled perturbed HF and KS equations. We propose efficient algorithms for the solution of the arising Z-vector equations and the gradient calculation that preserve the third-order scaling and low memory requirement of the original local fitting algorithm. In order to demonstrate the speed and accuracy of our implementation, gradient calculations and geometry optimizations are presented for various molecular systems. Our results show that significant speedups can be achieved compared to conventional density fitting calculations without sacrificing accuracy.
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Affiliation(s)
- József Csóka
- Department of Physical Chemistry and Materials Science, Faculty of Chemical Technology and Biotechnology, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary
| | - Mihály Kállay
- Department of Physical Chemistry and Materials Science, Faculty of Chemical Technology and Biotechnology, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary
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11
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Ji Y, Lin P, Ren X, He L. Reproducibility of Hybrid Density Functional Calculations for Equation-of-State Properties and Band Gaps. J Phys Chem A 2022; 126:5924-5931. [PMID: 36036969 DOI: 10.1021/acs.jpca.2c05170] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
Hybrid density functional (HDF) approximations usually deliver higher accuracy than local and semilocal approximations to the exchange-correlation functional, but this comes with drastically increased computational cost. Practical implementations of HDFs inevitably involve numerical approximations─even more so than their local and semilocal counterparts due to the additional numerical complexity arising from treating the exact-exchange component. This raises the question regarding the reproducibility of the HDF results yielded by different implementations. In this work, we benchmark the numerical precision of four independent implementations of the popular Heyd-Scuseria-Ernzerhof (HSE) range-separated HDF on describing key materials' properties, including both properties derived from equations of state (EOS) and band gaps of 20 crystalline solids. We find that the energy band gaps obtained by the four codes agree with each other rather satisfactorily. However, for lattice constants and bulk moduli, the deviations between the results computed by different codes are of the same order of magnitude as the deviations between the computational and experimental results. On the one hand, this means that the HSE functional is rather accurate for describing the cohesive properties of simple insulating solids. On the other hand, this also suggests that the numerical precision achieved with current major HSE implementations is not sufficiently high to unambiguously assess the physical accuracy of HDFs. It is found that the pseudopotential treatment of the core electrons is a major factor that contributes to this uncertainty.
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Affiliation(s)
- Yuyang Ji
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, Anhui China
| | - Peize Lin
- Songshan Lake Materials Laboratory, Dongguan 523808, Guangdong, China.,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Xinguo Ren
- Songshan Lake Materials Laboratory, Dongguan 523808, Guangdong, China.,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Lixin He
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, Anhui China
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12
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Computational Characterization of Nanosystems. CHINESE J CHEM PHYS 2022. [DOI: 10.1063/1674-0068/cjcp2111233] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022]
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13
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Wu K, Qin X, Hu W, Yang J. Low-Rank Approximations Accelerated Plane-Wave Hybrid Functional Calculations with k-Point Sampling. J Chem Theory Comput 2021; 18:206-218. [PMID: 34918919 DOI: 10.1021/acs.jctc.1c00874] [Citation(s) in RCA: 12] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
The low-rank approximations of the adaptively compressed exchange (ACE) operator and interpolative separable density fitting (ISDF) algorithms significantly reduce the computational cost and memory usage of hybrid functional calculations in real space, but the lack of k-point sampling hinders their implementation in reciprocal space for periodic systems with the plane-wave basis set. Here, we combine the ACE operator and ISDF decomposition into a new ACE-ISDF algorithm for periodic systems in reciprocal space with k-point sampling. On the basis of the ACE-ISDF algorithm with the improved reciprocal space ACE operator and k-point Fourier convolution, the time complexity of the hybrid functional calculation is reduced from O(Ne4Nk2) to O(Ne3Nklog(Nk)) (Ne and Nk are the number of electrons and k-points, respectively) with a much smaller prefactor and much lower memory consumption compared to the standard method for periodic systems with a plane-wave basis set.
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Affiliation(s)
- Kai Wu
- Hefei National Laboratory for Physical Sciences at the Microscale, Department of Chemical Physics, Synergetic Innovation Center of Quantum Information and Quantum Physics, and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Xinming Qin
- Hefei National Laboratory for Physical Sciences at the Microscale, Department of Chemical Physics, Synergetic Innovation Center of Quantum Information and Quantum Physics, and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Wei Hu
- Hefei National Laboratory for Physical Sciences at the Microscale, Department of Chemical Physics, Synergetic Innovation Center of Quantum Information and Quantum Physics, and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Jinlong Yang
- Hefei National Laboratory for Physical Sciences at the Microscale, Department of Chemical Physics, Synergetic Innovation Center of Quantum Information and Quantum Physics, and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
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14
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Ma H, Wang L, Wan L, Li J, Qin X, Liu J, Hu W, Lin L, Yang C, Yang J. Realizing Effective Cubic-Scaling Coulomb Hole Plus Screened Exchange Approximation in Periodic Systems via Interpolative Separable Density Fitting with a Plane-Wave Basis Set. J Phys Chem A 2021; 125:7545-7557. [PMID: 34428038 DOI: 10.1021/acs.jpca.1c03762] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
The GW approximation is an effective way to accurately describe the single-electron excitations of molecules and the quasiparticle energies of solids. However, a perceived drawback of the GW calculations is their high computational cost and large memory usage, which limit their applications to large systems. Herein, we demonstrate an accurate and effective low-rank approximation to accelerate non-self-consistent GW (G0W0) calculations under the static Coulomb hole plus screened exchange (COHSEX) approximation for periodic systems. Our approach is to adopt the interpolative separable density fitting (ISDF) decomposition and Cauchy's integral to construct low-rank representations of the dielectric matrix ϵ and self-energy matrix Σ. This approach reduces the number of floating point operations from O(Ne4) to O(Ne3) and requires a much smaller memory footprint. Two methods are used to select the interpolation points in ISDF, including the standard QR factorization with column pivoting (QRCP) procedure and the machine learning K-means clustering (K-means) algorithm. We demonstrate that these two methods can yield similar accuracy for both molecules and solids at much lower computational cost. In particular, K-means clustering can significantly reduce the computational cost of selecting the interpolation points by an order of magnitude compared to QRCP, resulting in an overall speedup factor of about ten times ISDF accelerated the static COHSEX calculations compared with conventional COHSEX approximation.
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Affiliation(s)
- Huanhuan Ma
- Hefei National Laboratory for Physical Sciences at the Microscale, Department of Chemical Physics, Synergetic Innovation Center of Quantum Information and Quantum Physics, and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Lei Wang
- Hefei National Laboratory for Physical Sciences at the Microscale, Department of Chemical Physics, Synergetic Innovation Center of Quantum Information and Quantum Physics, and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Lingyun Wan
- Hefei National Laboratory for Physical Sciences at the Microscale, Department of Chemical Physics, Synergetic Innovation Center of Quantum Information and Quantum Physics, and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Jielan Li
- Hefei National Laboratory for Physical Sciences at the Microscale, Department of Chemical Physics, Synergetic Innovation Center of Quantum Information and Quantum Physics, and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Xinming Qin
- Hefei National Laboratory for Physical Sciences at the Microscale, Department of Chemical Physics, Synergetic Innovation Center of Quantum Information and Quantum Physics, and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Jie Liu
- Hefei National Laboratory for Physical Sciences at the Microscale, Department of Chemical Physics, Synergetic Innovation Center of Quantum Information and Quantum Physics, and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Wei Hu
- Hefei National Laboratory for Physical Sciences at the Microscale, Department of Chemical Physics, Synergetic Innovation Center of Quantum Information and Quantum Physics, and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China.,Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States
| | - Lin Lin
- Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States.,Department of Mathematics, University of California, Berkeley, California 94720, United States
| | - Chao Yang
- Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States
| | - Jinlong Yang
- Hefei National Laboratory for Physical Sciences at the Microscale, Department of Chemical Physics, Synergetic Innovation Center of Quantum Information and Quantum Physics, and Anhui Center for Applied Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
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Lin P, Ren X, He L. Efficient Hybrid Density Functional Calculations for Large Periodic Systems Using Numerical Atomic Orbitals. J Chem Theory Comput 2021; 17:222-239. [PMID: 33307678 DOI: 10.1021/acs.jctc.0c00960] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
We present an efficient, linear-scaling implementation for building the (screened) Hartree-Fock exchange (HFX) matrix for periodic systems within the framework of numerical atomic orbital (NAO) basis functions. Our implementation is based on the localized resolution of the identity approximation by which two-electron Coulomb repulsion integrals can be obtained by only computing two-center quantities-a feature that is highly beneficial to NAOs. By exploiting the locality of basis functions and efficient prescreening of the intermediate three- and two-index tensors, one can achieve a linear scaling of the computational cost for building the HFX matrix with respect to the system size. Our implementation is massively parallel, thanks to a MPI/OpenMP hybrid parallelization strategy for distributing the computational load and memory storage. All these factors add together to enable highly efficient hybrid functional calculations for large-scale periodic systems. In this work, we describe the key algorithms and implementation details for the HFX build as implemented in the ABACUS code package. The performance and scalability of our implementation with respect to the system size and the number of CPU cores are demonstrated for selected benchmark systems up to 4096 atoms.
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Affiliation(s)
- Peize Lin
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Xinguo Ren
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Lixin He
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China
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Qin X, Li J, Hu W, Yang J. Machine Learning K-Means Clustering Algorithm for Interpolative Separable Density Fitting to Accelerate Hybrid Functional Calculations with Numerical Atomic Orbitals. J Phys Chem A 2020; 124:10066-10074. [PMID: 33200932 DOI: 10.1021/acs.jpca.0c06019] [Citation(s) in RCA: 20] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
The interpolative separable density fitting (ISDF) is an efficient and accurate low-rank decomposition method to reduce the high computational cost and memory usage of the Hartree-Fock exchange (HFX) calculations with numerical atomic orbitals (NAOs). In this work, we present a machine learning K-means clustering algorithm to select the interpolation points in ISDF, which offers a much cheaper alternative to the expensive QR factorization with column pivoting (QRCP) procedure. We implement this K-means-based ISDF decomposition to accelerate hybrid functional calculations with NAOs in the HONPAS package. We demonstrate that this method can yield a similar accuracy for both molecules and solids at a much lower computational cost. In particular, K-means can remarkably reduce the computational cost of selecting the interpolation points by nearly two orders of magnitude compared to QRCP, resulting in a speedup of ∼10 times for ISDF-based HFX calculations.
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Affiliation(s)
- Xinming Qin
- Hefei National Laboratory for Physical Sciences at the Microscale, Department of Chemical Physics, and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Jielan Li
- Hefei National Laboratory for Physical Sciences at the Microscale, Department of Chemical Physics, and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Wei Hu
- Hefei National Laboratory for Physical Sciences at the Microscale, Department of Chemical Physics, and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Jinlong Yang
- Hefei National Laboratory for Physical Sciences at the Microscale, Department of Chemical Physics, and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
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