1
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Chong S, Grasselli F, Ben Mahmoud C, Morrow JD, Deringer VL, Ceriotti M. Robustness of Local Predictions in Atomistic Machine Learning Models. J Chem Theory Comput 2023; 19:8020-8031. [PMID: 37948446 PMCID: PMC10688186 DOI: 10.1021/acs.jctc.3c00704] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/27/2023] [Revised: 10/12/2023] [Accepted: 10/12/2023] [Indexed: 11/12/2023]
Abstract
Machine learning (ML) models for molecules and materials commonly rely on a decomposition of the global target quantity into local, atom-centered contributions. This approach is convenient from a computational perspective, enabling large-scale ML-driven simulations with a linear-scaling cost and also allows for the identification and posthoc interpretation of contributions from individual chemical environments and motifs to complicated macroscopic properties. However, even though practical justifications exist for the local decomposition, only the global quantity is rigorously defined. Thus, when the atom-centered contributions are used, their sensitivity to the training strategy or the model architecture should be carefully considered. To this end, we introduce a quantitative metric, which we call the local prediction rigidity (LPR), that allows one to assess how robust the locally decomposed predictions of ML models are. We investigate the dependence of the LPR on the aspects of model training, particularly the composition of training data set, for a range of different problems from simple toy models to real chemical systems. We present strategies to systematically enhance the LPR, which can be used to improve the robustness, interpretability, and transferability of atomistic ML models.
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Affiliation(s)
- Sanggyu Chong
- Laboratory
of Computational Science and Modeling, Institute
of Materials, École Polytechnique Fédérale de Lausanne, Lausanne 1015, Switzerland
| | - Federico Grasselli
- Laboratory
of Computational Science and Modeling, Institute
of Materials, École Polytechnique Fédérale de Lausanne, Lausanne 1015, Switzerland
| | - Chiheb Ben Mahmoud
- Laboratory
of Computational Science and Modeling, Institute
of Materials, École Polytechnique Fédérale de Lausanne, Lausanne 1015, Switzerland
| | - Joe D. Morrow
- Department
of Chemistry, Inorganic Chemistry Laboratory, University of Oxford, Oxford OX1 3QR, U.K.
| | - Volker L. Deringer
- Department
of Chemistry, Inorganic Chemistry Laboratory, University of Oxford, Oxford OX1 3QR, U.K.
| | - Michele Ceriotti
- Laboratory
of Computational Science and Modeling, Institute
of Materials, École Polytechnique Fédérale de Lausanne, Lausanne 1015, Switzerland
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2
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Nguyen M, Neuhauser D. Gapped-filtering for efficient Chebyshev expansion of the density projection operator. Chem Phys Lett 2022. [DOI: 10.1016/j.cplett.2022.140036] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
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3
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Fabian MD, Shpiro B, Baer R. Linear Weak Scalability of Density Functional Theory Calculations without Imposing Electron Localization. J Chem Theory Comput 2022; 18:2162-2170. [PMID: 35343234 PMCID: PMC9009081 DOI: 10.1021/acs.jctc.1c00829] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Marcel D. Fabian
- Fritz Haber Research Center for Molecular Dynamics and the Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
| | - Ben Shpiro
- Fritz Haber Research Center for Molecular Dynamics and the Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
| | - Roi Baer
- Fritz Haber Research Center for Molecular Dynamics and the Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
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4
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Shpiro B, Fabian MD, Rabani E, Baer R. Forces from Stochastic Density Functional Theory under Nonorthogonal Atom-Centered Basis Sets. J Chem Theory Comput 2022; 18:1458-1466. [PMID: 35099187 PMCID: PMC8908760 DOI: 10.1021/acs.jctc.1c00794] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
![]()
We
develop a formalism for calculating forces on the nuclei within
the linear-scaling stochastic density functional theory (sDFT) in
a nonorthogonal atom-centered basis set representation (Fabian et al. Wiley Interdiscip. Rev.:
Comput. Mol. Sci.2019, 9, e1412, 10.1002/wcms.1412) and apply it to the Tryptophan Zipper 2 (Trp-zip2) peptide
solvated in water. We use an embedded-fragment approach to reduce
the statistical errors (fluctuation and systematic bias), where the
entire peptide is the main fragment and the remaining 425 water molecules
are grouped into small fragments. We analyze the magnitude of the
statistical errors in the forces and find that the systematic bias
is of the order of 0.065 eV/Å (∼1.2 × 10–3Eh/a0) when 120 stochastic orbitals are used, independently
of system size. This magnitude of bias is sufficiently small to ensure
that the bond lengths estimated by stochastic DFT (within a Langevin
molecular dynamics simulation) will deviate by less than 1% from those
predicted by a deterministic calculation.
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Affiliation(s)
- Ben Shpiro
- Fritz Haber Center for Molecular Dynamics and Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel
| | - Marcel David Fabian
- Fritz Haber Center for Molecular Dynamics and Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel
| | - Eran Rabani
- Department of Chemistry, University of California, Berkeley, California 94720, United States.,Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States.,The Raymond and Beverly Sackler Center of Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 69978, Israel
| | - Roi Baer
- Fritz Haber Center for Molecular Dynamics and Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel
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5
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Vlček V. Stochastic Vertex Corrections: Linear Scaling Methods for Accurate Quasiparticle Energies. J Chem Theory Comput 2019; 15:6254-6266. [DOI: 10.1021/acs.jctc.9b00317] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Vojtěch Vlček
- Department of Chemistry and Biochemistry, University of California, Santa Barbara, Santa Barbara, California 93106, United States
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6
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Chen M, Baer R, Neuhauser D, Rabani E. Energy window stochastic density functional theory. J Chem Phys 2019; 151:114116. [PMID: 31542024 DOI: 10.1063/1.5114984] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Linear scaling density functional theory is important for understanding electronic structure properties of nanometer scale systems. Recently developed stochastic density functional theory can achieve linear or even sublinear scaling for various electronic properties without relying on the sparsity of the density matrix. The basic idea relies on projecting stochastic orbitals onto the occupied space by expanding the Fermi-Dirac operator and repeating this for Nχ stochastic orbitals. Often, a large number of stochastic orbitals are required to reduce the statistical fluctuations (which scale as Nχ -1/2) below a tolerable threshold. In this work, we introduce a new stochastic density functional theory that can efficiently reduce the statistical fluctuations for certain observable and can also be integrated with an embedded fragmentation scheme. The approach is based on dividing the occupied space into energy windows and projecting the stochastic orbitals with a single expansion onto all windows simultaneously. This allows for a significant reduction of the noise as illustrated for bulk silicon with a large supercell. We also provide theoretical analysis to rationalize why the noise can be reduced only for a certain class of ground state properties, such as the forces and electron density.
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Affiliation(s)
- Ming Chen
- Department of Chemistry, University of California and Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
| | - Roi Baer
- Fritz Haber Center of Molecular Dynamics and Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
| | - Daniel Neuhauser
- Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90095, USA
| | - Eran Rabani
- Department of Chemistry, University of California and Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
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7
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Stochastic density functional theory. WILEY INTERDISCIPLINARY REVIEWS-COMPUTATIONAL MOLECULAR SCIENCE 2019. [DOI: 10.1002/wcms.1412] [Citation(s) in RCA: 16] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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8
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Vlček V, Rabani E, Neuhauser D, Baer R. Stochastic GW Calculations for Molecules. J Chem Theory Comput 2017; 13:4997-5003. [DOI: 10.1021/acs.jctc.7b00770] [Citation(s) in RCA: 52] [Impact Index Per Article: 7.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Vojtěch Vlček
- Department
of Chemistry and Biochemistry, University of California, Los Angeles, California 90095, United States
- Fritz
Haber Center for Molecular Dynamics, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
| | - Eran Rabani
- Department
of Chemistry, University of California and Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States
- The
Raymond and Beverly Sackler Center for Computational Molecular and
Materials Science, Tel Aviv University, Tel Aviv, Israel 69978
| | - Daniel Neuhauser
- Department
of Chemistry and Biochemistry, University of California, Los Angeles, California 90095, United States
| | - Roi Baer
- Fritz
Haber Center for Molecular Dynamics, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
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9
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Arnon E, Rabani E, Neuhauser D, Baer R. Equilibrium configurations of large nanostructures using the embedded saturated-fragments stochastic density functional theory. J Chem Phys 2017; 146:224111. [DOI: 10.1063/1.4984931] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Eitam Arnon
- Fritz Haber Center for Molecular Dynamics, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
| | - Eran Rabani
- Department of Chemistry, University of California and Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
- The Raymond and Beverly Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 69978, Israel
| | - Daniel Neuhauser
- Department of Chemistry, University of California, Los Angeles, California 90095, USA
| | - Roi Baer
- Fritz Haber Center for Molecular Dynamics, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
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10
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Aarons J, Sarwar M, Thompsett D, Skylaris CK. Perspective: Methods for large-scale density functional calculations on metallic systems. J Chem Phys 2016; 145:220901. [DOI: 10.1063/1.4972007] [Citation(s) in RCA: 90] [Impact Index Per Article: 11.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Jolyon Aarons
- School of Chemistry, University of Southampton, Southampton SO17 1BJ, United Kingdom
| | - Misbah Sarwar
- Johnson Matthey Technology Centre, Sonning Common, Reading, United Kingdom
| | - David Thompsett
- Johnson Matthey Technology Centre, Sonning Common, Reading, United Kingdom
| | - Chris-Kriton Skylaris
- School of Chemistry, University of Southampton, Southampton SO17 1BJ, United Kingdom
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11
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Truflandier LA, Dianzinga RM, Bowler DR. Communication: Generalized canonical purification for density matrix minimization. J Chem Phys 2016; 144:091102. [PMID: 26957150 DOI: 10.1063/1.4943213] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
A Lagrangian formulation for the constrained search for the N-representable one-particle density matrix based on the McWeeny idempotency error minimization is proposed, which converges systematically to the ground state. A closed form of the canonical purification is derived for which no a posteriori adjustment on the trace of the density matrix is needed. The relationship with comparable methods is discussed, showing their possible generalization through the hole-particle duality. The appealing simplicity of this self-consistent recursion relation along with its low computational complexity could prove useful as an alternative to diagonalization in solving dense and sparse matrix eigenvalue problems.
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Affiliation(s)
- Lionel A Truflandier
- Institut des Sciences Moléculaires, Université Bordeaux, CNRS UMR 5255, 351 cours de la Libération, 33405 Talence cedex, France
| | - Rivo M Dianzinga
- Institut des Sciences Moléculaires, Université Bordeaux, CNRS UMR 5255, 351 cours de la Libération, 33405 Talence cedex, France
| | - David R Bowler
- London Centre for Nanotechnology, UCL, 17-19 Gordon St., London WC1H 0AH, United Kingdom; Department of Physics and Astronomy, UCL, Gower St., London WC1E 6BT, United Kingdom; and International Centre for Materials Nanoarchitechtonics (MANA), National Institute for Materials Science (NIMS), 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan
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12
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Akimov AV, Prezhdo OV. Large-Scale Computations in Chemistry: A Bird’s Eye View of a Vibrant Field. Chem Rev 2015; 115:5797-890. [DOI: 10.1021/cr500524c] [Citation(s) in RCA: 159] [Impact Index Per Article: 17.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/08/2023]
Affiliation(s)
- Alexey V. Akimov
- Department
of Chemistry, University of South California, Los Angeles, California 90089, United States
| | - Oleg V. Prezhdo
- Department
of Chemistry, University of South California, Los Angeles, California 90089, United States
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13
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Richters D, Kühne TD. Self-consistent field theory based molecular dynamics with linear system-size scaling. J Chem Phys 2014; 140:134109. [DOI: 10.1063/1.4869865] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022] Open
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14
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Baer R, Neuhauser D, Rabani E. Self-averaging stochastic Kohn-Sham density-functional theory. PHYSICAL REVIEW LETTERS 2013; 111:106402. [PMID: 25166686 DOI: 10.1103/physrevlett.111.106402] [Citation(s) in RCA: 59] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/15/2013] [Revised: 07/21/2013] [Indexed: 06/03/2023]
Abstract
We formulate the Kohn-Sham density functional theory (KS-DFT) as a statistical theory in which the electron density is determined from an average of correlated stochastic densities in a trace formula. The key idea is that it is sufficient to converge the total energy per electron to within a predefined statistical error in order to obtain reliable estimates of the electronic band structure, the forces on nuclei, the density and its moments, etc. The fluctuations in the total energy per electron are guaranteed to decay to zero as the system size increases. This facilitates "self-averaging" which leads to the first ever report of sublinear scaling KS-DFT electronic structure. The approach sidesteps calculation of the density matrix and thus, is insensitive to its evasive sparseness, as demonstrated here for silicon nanocrystals. The formalism is not only appealing in terms of its promise to far push the limits of application of KS-DFT, but also represents a cognitive change in the way we think of electronic structure calculations as this stochastic theory seamlessly converges to the thermodynamic limit.
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Affiliation(s)
- Roi Baer
- Fritz Haber Center for Molecular Dynamics, Institute of Chemistry, Hebrew University, Jerusalem 91904, Israel
| | - Daniel Neuhauser
- Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90095, USA
| | - Eran Rabani
- School of Chemistry, The Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
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15
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Neuhauser D, Rabani E, Baer R. Expeditious Stochastic Approach for MP2 Energies in Large Electronic Systems. J Chem Theory Comput 2012; 9:24-7. [DOI: 10.1021/ct300946j] [Citation(s) in RCA: 66] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Daniel Neuhauser
- Department of Chemistry and
Biochemistry, University of California, Los Angeles, California, 90095,
United States
| | - Eran Rabani
- School of
Chemistry, The Sackler
Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
| | - Roi Baer
- Fritz Haber Center for Molecular
Dynamics, Institute of Chemistry, Hebrew University, Jerusalem 91904,
Israel
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16
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VandeVondele J, Borštnik U, Hutter J. Linear Scaling Self-Consistent Field Calculations with Millions of Atoms in the Condensed Phase. J Chem Theory Comput 2012; 8:3565-73. [DOI: 10.1021/ct200897x] [Citation(s) in RCA: 116] [Impact Index Per Article: 9.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Joost VandeVondele
- Department of Materials, ETH Zurich, Wolfgang-Pauli-Strasse 27,
8093 Zurich, Switzerland
| | - Urban Borštnik
- Physical Chemistry Institute, University of Zurich, Winterthurerstrasse 190, CH-8057
Zurich, Switzerland
| | - Jürg Hutter
- Physical Chemistry Institute, University of Zurich, Winterthurerstrasse 190, CH-8057
Zurich, Switzerland
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17
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Yam C, Zhang Q, Wang F, Chen G. Linear-scaling quantum mechanical methods for excited states. Chem Soc Rev 2012; 41:3821-38. [DOI: 10.1039/c2cs15259b] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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18
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Rubensson EH. Nonmonotonic Recursive Polynomial Expansions for Linear Scaling Calculation of the Density Matrix. J Chem Theory Comput 2011; 7:1233-6. [DOI: 10.1021/ct2001705] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Emanuel H. Rubensson
- Division of Scientific Computing, Department of Information Technology, Uppsala University, Box 337, SE-751 05 Uppsala, Sweden
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19
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Rudberg E, Rubensson EH. Assessment of density matrix methods for linear scaling electronic structure calculations. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2011; 23:075502. [PMID: 21411885 DOI: 10.1088/0953-8984/23/7/075502] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
Purification and minimization methods for linear scaling computation of the one-particle density matrix for a fixed Hamiltonian matrix are compared. This is done by considering the work needed by each method to achieve a given accuracy in terms of the difference from the exact solution. Numerical tests employing orthogonal as well as non-orthogonal versions of the methods are performed using both element magnitude and cutoff radius based truncation approaches. It is investigated how the convergence speed for the different methods depends on the eigenvalue distribution in the Hamiltonian matrix. An expression for the number of iterations required for the minimization methods studied is derived, taking into account the dependence on both the band gap and the chemical potential. This expression is confirmed by numerical tests. The minimization methods are found to perform at their best when the chemical potential is located near the center of the eigenspectrum. The results indicate that purification is considerably more efficient than the minimization methods studied even when a good starting guess for the minimization is available. In test calculations without a starting guess, purification is more than an order of magnitude more efficient than minimization.
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Affiliation(s)
- Elias Rudberg
- Division of Scientific Computing, Department of Information Technology, Uppsala University, Box 337, SE-751 05 Uppsala, Sweden.
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20
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Rubensson EH, Rudberg E. Bringing about matrix sparsity in linear-scaling electronic structure calculations. J Comput Chem 2011; 32:1411-23. [DOI: 10.1002/jcc.21723] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2010] [Revised: 10/27/2010] [Accepted: 10/28/2010] [Indexed: 11/06/2022]
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21
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Rubensson EH, Rudberg E, Salek P. Methods for Hartree-Fock and Density Functional Theory Electronic Structure Calculations with Linearly Scaling Processor Time and Memory Usage. CHALLENGES AND ADVANCES IN COMPUTATIONAL CHEMISTRY AND PHYSICS 2011. [DOI: 10.1007/978-90-481-2853-2_12] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
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22
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Steele RP, Head-Gordon M. Dual-basis self-consistent field methods: 6-31G* calculations with a minimal 6-4G primary basis. Mol Phys 2010. [DOI: 10.1080/00268970701519754] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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23
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Ceriotti M, Kühne TD, Parrinello M. An efficient and accurate decomposition of the Fermi operator. J Chem Phys 2008; 129:024707. [DOI: 10.1063/1.2949515] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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24
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Rubensson EH, Rudberg E, Sałek P. Density matrix purification with rigorous error control. J Chem Phys 2008; 128:074106. [DOI: 10.1063/1.2826343] [Citation(s) in RCA: 37] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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25
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Nomura S, Iitaka T. Linear scaling calculation of an n-type GaAs quantum dot. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:037701. [PMID: 17930374 DOI: 10.1103/physreve.76.037701] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/21/2007] [Revised: 06/11/2007] [Indexed: 05/25/2023]
Abstract
A linear scale method for calculating electronic properties of large and complex systems is introduced within a local density approximation. The method is based on the Chebyshev polynomial expansion and the time-dependent method, which is tested on the calculation of the electronic structure of a model n-type GaAs quantum dot.
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Affiliation(s)
- Shintaro Nomura
- Institute of Physics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Japan.
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26
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Xiang HJ, Yang J, Hou JG, Zhu Q. Linear scaling calculation of band edge states and doped semiconductors. J Chem Phys 2007; 126:244707. [PMID: 17614577 DOI: 10.1063/1.2746322] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Linear scaling methods provide total energy, but no energy levels and canonical wave functions. From the density matrix computed through the density matrix purification methods, we propose an order-N [O(N)] method for calculating both the energies and wave functions of band edge states, which are important for optical properties and chemical reactions. In addition, we also develop an O(N) algorithm to deal with doped semiconductors based on the O(N) method for band edge states calculation. We illustrate the O(N) behavior of the new method by applying it to boron nitride (BN) nanotubes and BN nanotubes with an adsorbed hydrogen atom. The band gap of various BN nanotubes are investigated systematically and the acceptor levels of BN nanotubes with an isolated adsorbed H atom are computed. Our methods are simple, robust, and especially suited for the application in self-consistent field electronic structure theory.
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Affiliation(s)
- H J Xiang
- Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China
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27
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Steele RP, Shao Y, DiStasio RA, Head-Gordon M. Dual-Basis Analytic Gradients. 1. Self-Consistent Field Theory. J Phys Chem A 2006; 110:13915-22. [PMID: 17181351 DOI: 10.1021/jp065444h] [Citation(s) in RCA: 35] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
Analytic gradients of dual-basis Hartree-Fock and density functional theory energies have been derived and implemented, which provide the opportunity for capturing large basis-set gradient effects at reduced cost. Suggested pairings for gradient calculations are 6-31G/6-31G**, dual[-f,-d]/cc-pVTZ, and 6-311G*/6-311 + +G(3df,3pd). Equilibrium geometries are produced within 0.0005 A of large-basis results for the latter two pairings. Though a single, iterative SCF response equation must be solved (unlike standard SCF gradients), it may be obtained in the smaller basis set, and integral screening further reduces the cost for well-chosen subsets. Total nuclear force calculations exhibit up to 75% savings, relative to large-basis calculations.
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Affiliation(s)
- Ryan P Steele
- Department of Chemistry, University of California-Berkeley, Berkeley, CA 94720, USA.
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28
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Shao Y, Molnar LF, Jung Y, Kussmann J, Ochsenfeld C, Brown ST, Gilbert ATB, Slipchenko LV, Levchenko SV, O'Neill DP, DiStasio RA, Lochan RC, Wang T, Beran GJO, Besley NA, Herbert JM, Lin CY, Van Voorhis T, Chien SH, Sodt A, Steele RP, Rassolov VA, Maslen PE, Korambath PP, Adamson RD, Austin B, Baker J, Byrd EFC, Dachsel H, Doerksen RJ, Dreuw A, Dunietz BD, Dutoi AD, Furlani TR, Gwaltney SR, Heyden A, Hirata S, Hsu CP, Kedziora G, Khalliulin RZ, Klunzinger P, Lee AM, Lee MS, Liang W, Lotan I, Nair N, Peters B, Proynov EI, Pieniazek PA, Rhee YM, Ritchie J, Rosta E, Sherrill CD, Simmonett AC, Subotnik JE, Woodcock HL, Zhang W, Bell AT, Chakraborty AK, Chipman DM, Keil FJ, Warshel A, Hehre WJ, Schaefer HF, Kong J, Krylov AI, Gill PMW, Head-Gordon M. Advances in methods and algorithms in a modern quantum chemistry program package. Phys Chem Chem Phys 2006; 8:3172-91. [PMID: 16902710 DOI: 10.1039/b517914a] [Citation(s) in RCA: 2122] [Impact Index Per Article: 117.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022]
Abstract
Advances in theory and algorithms for electronic structure calculations must be incorporated into program packages to enable them to become routinely used by the broader chemical community. This work reviews advances made over the past five years or so that constitute the major improvements contained in a new release of the Q-Chem quantum chemistry package, together with illustrative timings and applications. Specific developments discussed include fast methods for density functional theory calculations, linear scaling evaluation of energies, NMR chemical shifts and electric properties, fast auxiliary basis function methods for correlated energies and gradients, equation-of-motion coupled cluster methods for ground and excited states, geminal wavefunctions, embedding methods and techniques for exploring potential energy surfaces.
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Affiliation(s)
- Yihan Shao
- Department of Chemistry, University of California, Berkeley, CA 94720, USA
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Xiang HJ, Liang WZ, Yang J, Hou JG, Zhu Q. Spin-unrestricted linear-scaling electronic structure theory and its application to magnetic carbon-doped boron nitride nanotubes. J Chem Phys 2005; 123:124105. [PMID: 16392473 DOI: 10.1063/1.2034448] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We present an extension of the density-matrix-based linear-scaling electronic structure theory to incorporate spin degrees of freedom. When the spin multiplicity of the system can be predetermined, the generalization of the existing linear-scaling methods to spin-unrestricted cases is straightforward. However, without calculations it is hard to determine the spin multiplicity of some complex systems, such as many magnetic nanostuctures and some inorganic or bioinorganic molecules. Here we give a general prescription to obtain the spin-unrestricted ground state of open-shell systems. Our methods are implemented into the linear-scaling trace-correcting density-matrix purification algorithm. The numerical atomic-orbital basis, rather than the commonly adopted Gaussian basis functions, is used. The test systems include O2 molecule and magnetic carbon-doped boron nitride (BN)(5,5) and BN(7,6) nanotubes. Using the newly developed method, we find that the magnetic moments in carbon-doped BN nanotubes couple antiferromagnetically with each other. Our results suggest that the linear-scaling spin-unrestricted trace-correcting purification method is very powerful to treat large magnetic systems.
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Affiliation(s)
- H J Xiang
- Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China (USTC), Hefei, Anhui 230026, People's Republic of China
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Varga K, Zhang Z, Pantelides ST. "Lagrange functions": a family of powerful basis sets for real-space order-N electronic structure calculations. PHYSICAL REVIEW LETTERS 2004; 93:176403. [PMID: 15525095 DOI: 10.1103/physrevlett.93.176403] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/05/2004] [Indexed: 05/24/2023]
Abstract
Plane waves have unparalleled simplicity and systematic convergence by a single monotonic parameter, the energy cutoff, but they are limited to speriodic systems and require Fourier transforms that scale as N(2)logN, where N is the number of atoms. Real-space methods for order-N scaling are computationally complex and convergence depends on several variables. Here we introduce and demonstrate "Lagrange functions" as a family of analytical, complete, and orthonormal basis sets that are suitable for efficient, accurate, real-space, order-N electronic structure calculations. Convergence is controlled by a single monotonic parameter, the dimension of the basis set, and computational complexity is lower than that of plane waves.
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Affiliation(s)
- Kálmán Varga
- Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee, USA
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Affiliation(s)
- Georges Jolicard
- Laboratoire d'Astrophysique de l'Observatoire de Besançon (CNRS UMR 6091), 41 bis Avenue de l'Observatoire, BP1615, 25010 Besançon Cedex, France, and Mathematics Department, University of Hull, Hull HU6 7RX, U.K
| | - David Viennot
- Laboratoire d'Astrophysique de l'Observatoire de Besançon (CNRS UMR 6091), 41 bis Avenue de l'Observatoire, BP1615, 25010 Besançon Cedex, France, and Mathematics Department, University of Hull, Hull HU6 7RX, U.K
| | - John P. Killingbeck
- Laboratoire d'Astrophysique de l'Observatoire de Besançon (CNRS UMR 6091), 41 bis Avenue de l'Observatoire, BP1615, 25010 Besançon Cedex, France, and Mathematics Department, University of Hull, Hull HU6 7RX, U.K
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Liang W, Saravanan C, Shao Y, Baer R, Bell AT, Head-Gordon M. Improved Fermi operator expansion methods for fast electronic structure calculations. J Chem Phys 2003. [DOI: 10.1063/1.1590632] [Citation(s) in RCA: 78] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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Saravanan C, Shao Y, Baer R, Ross PN, Head-Gordon M. Sparse matrix multiplications for linear scaling electronic structure calculations in an atom-centered basis set using multiatom blocks. J Comput Chem 2003; 24:618-22. [PMID: 12632476 DOI: 10.1002/jcc.10224] [Citation(s) in RCA: 50] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
Abstract
A sparse matrix multiplication scheme with multiatom blocks is reported, a tool that can be very useful for developing linear-scaling methods with atom-centered basis functions. Compared to conventional element-by-element sparse matrix multiplication schemes, efficiency is gained by the use of the highly optimized basic linear algebra subroutines (BLAS). However, some sparsity is lost in the multiatom blocking scheme because these matrix blocks will in general contain negligible elements. As a result, an optimal block size that minimizes the CPU time by balancing these two effects is recovered. In calculations on linear alkanes, polyglycines, estane polymers, and water clusters the optimal block size is found to be between 40 and 100 basis functions, where about 55-75% of the machine peak performance was achieved on an IBM RS6000 workstation. In these calculations, the blocked sparse matrix multiplications can be 10 times faster than a standard element-by-element sparse matrix package.
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Affiliation(s)
- Chandra Saravanan
- Department of Chemistry, University of California, Berkeley, California 94720, USA
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Shao Y, Saravanan C, Head-Gordon M, White CA. Curvy steps for density matrix-based energy minimization: Application to large-scale self-consistent-field calculations. J Chem Phys 2003. [DOI: 10.1063/1.1558476] [Citation(s) in RCA: 50] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Santra R, Breidbach J, Zobeley J, Cederbaum LS. Parallel filter diagonalization: A novel method to resolve quantum states in dense spectral regions. J Chem Phys 2000. [DOI: 10.1063/1.481545] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/24/2023] Open
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Wadleigh HH, Ionova IV, Carter EA. Generalized symmetric Rayleigh–Ritz procedure applied to the closed-shell Hartree–Fock problem. J Chem Phys 1999. [DOI: 10.1063/1.478299] [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] Open
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Ayala PY, Scuseria GE. Linear scaling second-order Moller–Plesset theory in the atomic orbital basis for large molecular systems. J Chem Phys 1999. [DOI: 10.1063/1.478256] [Citation(s) in RCA: 353] [Impact Index Per Article: 14.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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Daniels AD, Scuseria GE. What is the best alternative to diagonalization of the Hamiltonian in large scale semiempirical calculations? J Chem Phys 1999. [DOI: 10.1063/1.478008] [Citation(s) in RCA: 105] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Baer R, Head-Gordon M. Electronic structure of large systems: Coping with small gaps using the energy renormalization group method. J Chem Phys 1998. [DOI: 10.1063/1.477709] [Citation(s) in RCA: 21] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Bates KR, Daniels AD, Scuseria GE. Comparison of conjugate gradient density matrix search and Chebyshev expansion methods for avoiding diagonalization in large-scale electronic structure calculations. J Chem Phys 1998. [DOI: 10.1063/1.476927] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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