1
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Yeh CN, Morales MA. Low-Scaling Algorithms for GW and Constrained Random Phase Approximation Using Symmetry-Adapted Interpolative Separable Density Fitting. J Chem Theory Comput 2024; 20:3184-3198. [PMID: 38597496 DOI: 10.1021/acs.jctc.4c00085] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/11/2024]
Abstract
We present low-scaling algorithms for GW and constrained random phase approximation based on a symmetry-adapted interpolative separable density fitting (ISDF) procedure that incorporates the space-group symmetries of crystalline systems. The resulting formulations scale cubically, with respect to system size, and linearly with the number of k-points, regardless of the choice of single-particle basis and whether a quasiparticle approximation is employed. We validate these methods through comparisons with published literature and demonstrate their efficiency in treating large-scale systems through the construction of downfolded many-body Hamiltonians for carbon dimer defects embedded in hexagonal boron nitride supercells. Our work highlights the efficiency and general applicability of ISDF in the context of large-scale many-body calculations with k-point sampling beyond density functional theory.
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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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2
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Amblard D, Blase X, Duchemin I. Static versus dynamically polarizable environments within the many-body GW formalism. J Chem Phys 2024; 160:154104. [PMID: 38624115 DOI: 10.1063/5.0203637] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/15/2024] [Accepted: 03/28/2024] [Indexed: 04/17/2024] Open
Abstract
Continuum- or discrete-polarizable models for the study of optoelectronic processes in embedded subsystems rely mostly on the restriction of the surrounding electronic dielectric response to its low frequency limit. Such a description hinges on the assumption that the electrons in the surrounding medium react instantaneously to any excitation in the central subsystem, thus treating the environment in the adiabatic limit. Exploiting a recently developed embedded GW formalism with an environment described at the fully ab initio level, we assess the merits of the adiabatic limit with respect to an environment where the full dynamics of the dielectric response are considered. Furthermore, we show how to properly take the static limit of the environment's susceptibility by introducing the so-called Coulomb-hole and screened-exchange contributions to the reaction field. As a first application, we consider a C60 molecule at the surface of a C60 crystal, namely, a case where the dynamics of the embedded and embedding subsystems are similar. The common adiabatic assumption, when properly treated, generates errors below 10% on the polarization energy associated with frontier energy levels and associated energy gaps. Finally, we consider a water molecule inside a metallic nanotube, the worst case for the environment's adiabatic limit. The error on the gap polarization energy remains below 10%, even though the error on the frontier orbital polarization energies can reach a few tenths of an electronvolt.
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Affiliation(s)
- David Amblard
- University Grenoble Alpes, CNRS, Inst NEEL, F-38042 Grenoble, France
| | - Xavier Blase
- University Grenoble Alpes, CNRS, Inst NEEL, F-38042 Grenoble, France
| | - Ivan Duchemin
- University Grenoble Alpes, CEA, IRIG-MEM-L_Sim, 38054 Grenoble, France
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3
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Gao W, Tang Z, Zhao J, Chelikowsky JR. Efficient Full-Frequency GW Calculations Using a Lanczos Method. PHYSICAL REVIEW LETTERS 2024; 132:126402. [PMID: 38579203 DOI: 10.1103/physrevlett.132.126402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/30/2023] [Accepted: 02/21/2024] [Indexed: 04/07/2024]
Abstract
The GW approximation is widely used for reliable and accurate modeling of single-particle excitations. It also serves as a starting point for many theoretical methods, such as its use in the Bethe-Salpeter equation (BSE) and dynamical mean-field theory. However, full-frequency GW calculations for large systems with hundreds of atoms remain computationally challenging, even after years of efforts to reduce the prefactor and improve scaling. We propose a method that reformulates the correlation part of the GW self-energy as a resolvent of a Hermitian matrix, which can be efficiently and accurately computed using the standard Lanczos method. This method enables full-frequency GW calculations of material systems with a few hundred atoms on a single computing workstation. We further demonstrate the efficiency of the method by calculating the defect-state energies of silicon quantum dots with diameters up to 4 nm and nearly 2,000 silicon atoms using only 20 computational nodes.
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Affiliation(s)
- Weiwei Gao
- Key Laboratory of Materials Modification by Laser, Ion and Electron Beams, Ministry of Education, Dalian University of Technology, Dalian 116024, China
| | - Zhao Tang
- Center for Computational Materials, Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas 78712, USA
| | - Jijun Zhao
- Key Laboratory of Atomic and Subatomic Structure and Quantum Control (Ministry of Education), Guangdong Basic Research Center of Excellence for Structure and Fundamental Interactions of Matter, School of Physics, South China Normal University, Guangzhou 510006, China
| | - James R Chelikowsky
- Center for Computational Materials, Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas 78712, USA
- Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA
- McKetta Department of Chemical Engineering, The University of Texas at Austin, Austin, Texas 78712, USA
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4
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Zhang Z, Yin X, Hu W, Yang J. Machine Learning K-Means Clustering of Interpolative Separable Density Fitting Algorithm for Accurate and Efficient Cubic-Scaling Exact Exchange Plus Random Phase Approximation within Plane Waves. J Chem Theory Comput 2024; 20:1944-1961. [PMID: 38361423 DOI: 10.1021/acs.jctc.3c01157] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/17/2024]
Abstract
The exact-exchange plus random-phase approximation (EXX+RPA) method has emerged as a crucial tool for precisely characterizing electronic structures in molecular and solid systems. We present an accurate and efficient implementation of EXX+RPA calculations that scale cubically and are conducted within plane waves. Our approach incorporates the interpolative separable density fitting (ISDF) algorithm, effectively mitigating the computational challenges associated with the plane wave basis set. To overcome the constraints of the conventional ISDF algorithm, characterized by the exceptionally high prefactor in QR factorization for interpolation point selection, we introduce an enhanced machine learning K-means method. This method incorporates a novel empirical weight function called "SSM+" for more precise interpolation point selection, capturing physical information more accurately across diverse systems. Our machine learning approach offers a quasiquadratic scaling alternative, effectively replacing the computationally demanding cubic-scaling QRCP algorithm in plane-wave-based EXX+RPA calculations. Furthermore, we enhance the method's capabilities by optimizing GPU acceleration using MATLAB's integrated GPU toolkit. In particular, our approach reduces the computational scaling of χ0 from 3.80 to 2.13 and the overall computational scaling of EXX from 2.74 to 2.10. We achieve a remarkable GPU acceleration speedup of up to 35×. Regarding CPU computation time, the standard quartic-scaling method requires 22 h to compute Si128, while QRCP completes the calculation in only around 1 h, achieving a speedup up to 20×. However, the utilization of the K-means algorithm reduces the time to 800 s, a substantial improvement of 100× compared to the standard algorithm. By employing the K-means algorithm, the computational time for interpolative point calculation using QRCP decreases from 1 h to 1 min, resulting in a 55× speed increase. With this improved algorithm, we successfully computed the dissociation curve of H2 and the equilibrium polyynic geometry of C18 molecules.
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Affiliation(s)
- Zhenlin Zhang
- 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
| | - Xilin Yin
- 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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5
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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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6
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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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7
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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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8
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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: 2] [Impact Index Per Article: 2.0] [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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9
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Weng G, Pang A, Vlček V. Spatial Decay and Limits of Quantum Solute-Solvent Interactions. J Phys Chem Lett 2023; 14:2473-2480. [PMID: 36867592 DOI: 10.1021/acs.jpclett.3c00208] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/18/2023]
Abstract
Molecular excitations in the liquid-phase environment are renormalized by the surrounding solvent molecules. Herein, we employ the GW approximation to investigate the solvation effects on the ionization energy of phenol in various solvent environments. The electronic effects differ by up to 0.4 eV among the five investigated solvents. This difference depends on both the macroscopic solvent polarizability and the spatial decay of the solvation effects. The latter is probed by separating the electronic subspace and the GW correlation self-energy into fragments. The fragment correlation energy decays with increasing intermolecular distance and vanishes at ∼9 Å, and this pattern is independent of the type of solvent environment. The 9 Å cutoff defines an effective interacting volume within which the ionization energy shift per solvent molecule is proportional to the macroscopic solvent polarizability. Finally, we propose a simple model for computing the ionization energies of molecules in an arbitrary solvent environment.
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Affiliation(s)
- Guorong Weng
- Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106-9510, United States
| | - Amanda Pang
- Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106-9510, United States
| | - Vojtěch Vlček
- Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106-9510, United States
- Materials Department, University of California, Santa Barbara, California 93106-5050, United States
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10
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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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11
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Bruneval F, Dattani N, van Setten MJ. The GW Miracle in Many-Body Perturbation Theory for the Ionization Potential of Molecules. Front Chem 2022; 9:749779. [PMID: 35004607 PMCID: PMC8733722 DOI: 10.3389/fchem.2021.749779] [Citation(s) in RCA: 27] [Impact Index Per Article: 13.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/29/2021] [Accepted: 09/14/2021] [Indexed: 11/30/2022] Open
Abstract
We use the GW100 benchmark set to systematically judge the quality of several perturbation theories against high-level quantum chemistry methods. First of all, we revisit the reference CCSD(T) ionization potentials for this popular benchmark set and establish a revised set of CCSD(T) results. Then, for all of these 100 molecules, we calculate the HOMO energy within second and third-order perturbation theory (PT2 and PT3), and, GW as post-Hartree-Fock methods. We found GW to be the most accurate of these three approximations for the ionization potential, by far. Going beyond GW by adding more diagrams is a tedious and dangerous activity: We tried to complement GW with second-order exchange (SOX), with second-order screened exchange (SOSEX), with interacting electron-hole pairs (WTDHF), and with a GW density-matrix (γGW). Only the γGW result has a positive impact. Finally using an improved hybrid functional for the non-interacting Green’s function, considering it as a cheap way to approximate self-consistency, the accuracy of the simplest GW approximation improves even more. We conclude that GW is a miracle: Its subtle balance makes GW both accurate and fast.
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Affiliation(s)
- Fabien Bruneval
- CEA, Service de Recherches de Métallurgie Physique, Direction des Energies, Université Paris-Saclay, Paris, France
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12
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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: 10] [Impact Index Per Article: 3.3] [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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13
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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: 5] [Impact Index Per Article: 1.7] [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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Duchemin I, Blase X. Cubic-Scaling All-Electron GW Calculations with a Separable Density-Fitting Space-Time Approach. J Chem Theory Comput 2021; 17:2383-2393. [PMID: 33797245 DOI: 10.1021/acs.jctc.1c00101] [Citation(s) in RCA: 45] [Impact Index Per Article: 15.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
We present an implementation of the GW space-time approach that allows cubic-scaling all-electron calculations with standard Gaussian basis sets without exploiting any localization or sparsity considerations. The independent-electron susceptibility is constructed in a time representation over a nonuniform distribution of real-space locations {rk} optimized within a separable resolution-of-the-identity framework to reproduce standard Coulomb-fitting calculations with meV accuracy. The compactness of the obtained {rk} distribution leads to a crossover with the standard Coulomb-fitting scheme for system sizes below a few hundred electrons. The needed analytic continuation follows a recent approach that requires the continuation of the screened Coulomb potential rather than the much more structured self-energy. The present scheme is benchmarked over large molecular sets, and scaling properties are demonstrated on a family of defected hexagonal boron-nitride flakes containing up to 6000 electrons.
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Affiliation(s)
- Ivan Duchemin
- Université Grenoble Alpes, CEA, IRIG-MEM-L_Sim, 38054 Grenoble, France
| | - Xavier Blase
- Université Grenoble Alpes, CNRS, Inst NEEL, F-38042 Grenoble, France
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Wilhelm J, Seewald P, Golze D. Low-Scaling GW with Benchmark Accuracy and Application to Phosphorene Nanosheets. J Chem Theory Comput 2021; 17:1662-1677. [DOI: 10.1021/acs.jctc.0c01282] [Citation(s) in RCA: 19] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/11/2022]
Affiliation(s)
- Jan Wilhelm
- Institute of Theoretical Physics, University of Regensburg, D-93053 Regensburg, Germany
| | - Patrick Seewald
- Department of Chemistry, University of Zurich, CH-8057 Zurich, Switzerland
| | - Dorothea Golze
- Department of Applied Physics, Aalto University, FI-00076 Aalto, Finland
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Zhu T, Chan GKL. All-Electron Gaussian-Based G0W0 for Valence and Core Excitation Energies of Periodic Systems. J Chem Theory Comput 2021; 17:727-741. [DOI: 10.1021/acs.jctc.0c00704] [Citation(s) in RCA: 18] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Tianyu Zhu
- Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, United States
| | - Garnet Kin-Lic Chan
- Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, United States
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17
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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: 18] [Impact Index Per Article: 4.5] [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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18
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Blase X, Duchemin I, Jacquemin D, Loos PF. The Bethe-Salpeter Equation Formalism: From Physics to Chemistry. J Phys Chem Lett 2020; 11:7371-7382. [PMID: 32787315 DOI: 10.1021/acs.jpclett.0c01875] [Citation(s) in RCA: 70] [Impact Index Per Article: 17.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
The Bethe-Salpeter equation (BSE) formalism is steadily asserting itself as a new efficient and accurate tool in the ensemble of computational methods available to chemists in order to predict optical excitations in molecular systems. In particular, the combination of the so-called GW approximation, giving access to reliable ionization energies and electron affinities, and the BSE formalism, able to model UV/vis spectra, has shown to provide accurate singlet excitation energies with a typical error of 0.1-0.3 eV. With a similar computational cost as time-dependent density-functional theory (TD-DFT), BSE is able to provide an accuracy on par with the most accurate global and range-separated hybrid functionals without the unsettling choice of the exchange-correlation functional, resolving further known issues (e.g., charge-transfer excitations). In this Perspective, we provide a historical overview of BSE, with a particular focus on its condensed-matter roots. We also propose a critical review of its strengths and weaknesses in different chemical situations.
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Affiliation(s)
- Xavier Blase
- Université Grenoble Alpes, CNRS, Institut NEEL, F-38042 Grenoble, France
| | - Ivan Duchemin
- Université Grenoble Alpes, CEA, IRIG-MEM-L Sim, 38054 Grenoble, France
| | - Denis Jacquemin
- Université de Nantes, CNRS, CEISAM UMR 6230, F-44000 Nantes, France
| | - Pierre-François Loos
- Laboratoire de Chimie et Physique Quantiques, Université de Toulouse, CNRS, UPS, Toulouse, France
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