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Vu JP, Chen M. Noise reduction of stochastic density functional theory for metals. J Chem Phys 2024; 160:214125. [PMID: 38842491 DOI: 10.1063/5.0207244] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/06/2024] [Accepted: 05/16/2024] [Indexed: 06/07/2024] Open
Abstract
Density Functional Theory (DFT) has become a cornerstone in the modeling of metals. However, accurately simulating metals, particularly under extreme conditions, presents two significant challenges. First, simulating complex metallic systems at low electron temperatures is difficult due to their highly delocalized density matrix. Second, modeling metallic warm-dense materials at very high electron temperatures is challenging because it requires the computation of a large number of partially occupied orbitals. This study demonstrates that both challenges can be effectively addressed using the latest advances in linear-scaling stochastic DFT methodologies. Despite the inherent introduction of noise into all computed properties by stochastic DFT, this research evaluates the efficacy of various noise reduction techniques under different thermal conditions. Our observations indicate that the effectiveness of noise reduction strategies varies significantly with the electron temperature. Furthermore, we provide evidence that the computational cost of stochastic DFT methods scales linearly with system size for metal systems, regardless of the electron temperature regime.
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Affiliation(s)
- Jake P Vu
- Department of Chemistry, Purdue University, West Lafayette, Indiana 47907, USA
| | - Ming Chen
- Department of Chemistry, Purdue University, West Lafayette, Indiana 47907, USA
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2
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Chen M, Baer R, Rabani E. Structure optimization with stochastic density functional theory. J Chem Phys 2023; 158:024111. [PMID: 36641385 DOI: 10.1063/5.0126169] [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
Linear-scaling techniques for Kohn-Sham density functional theory are essential to describe the ground state properties of extended systems. Still, these techniques often rely on the localization of the density matrix or accurate embedding approaches, limiting their applicability. In contrast, stochastic density functional theory (sDFT) achieves linear- and sub-linear scaling by statistically sampling the ground state density without relying on embedding or imposing localization. In return, ground state observables, such as the forces on the nuclei, fluctuate in sDFT, making optimizing the nuclear structure a highly non-trivial problem. In this work, we combine the most recent noise-reduction schemes for sDFT with stochastic optimization algorithms to perform structure optimization within sDFT. We compare the performance of the stochastic gradient descent approach and its variations (stochastic gradient descent with momentum) with stochastic optimization techniques that rely on the Hessian, such as the stochastic Broyden-Fletcher-Goldfarb-Shanno algorithm. We further provide a detailed assessment of the computational efficiency and its dependence on the optimization parameters of each method for determining the ground state structure of bulk silicon with varying supercell dimensions.
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Affiliation(s)
- Ming Chen
- Department of Chemistry, Purdue University, West Lafayette, Indiana 47907, USA
| | - Roi Baer
- Fritz Haber Center of Molecular Dynamics and Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
| | - Eran Rabani
- Department of Chemistry, University of California, Berkeley, California 94720, USA
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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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Baer R, Neuhauser D, Rabani E. Stochastic Vector Techniques in Ground-State Electronic Structure. Annu Rev Phys Chem 2022; 73:255-272. [PMID: 35081326 DOI: 10.1146/annurev-physchem-090519-045916] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Abstract
We review a suite of stochastic vector computational approaches for studying the electronic structure of extended condensed matter systems. These techniques help reduce algorithmic complexity, facilitate efficient parallelization, simplify computational tasks, accelerate calculations, and diminish memory requirements. While their scope is vast, we limit our study to ground-state and finite temperature density functional theory (DFT) and second-order perturbation theory. More advanced topics, such as quasiparticle (charge) and optical (neutral) excitations and higher-order processes, are covered elsewhere. We start by explaining how to use stochastic vectors in computations, characterizing the associated statistical errors. Next, we show how to estimate the electron density in DFT and discuss highly effective techniques to reduce statistical errors. Finally, we review the use of stochastic vector techniques for calculating correlation energies within the second-order Møller-Plesset perturbation theory and its finite temperature variational form. Example calculation results are presented and used to demonstrate the efficacy of the methods. Expected final online publication date for the Annual Review of Physical Chemistry, Volume 73 is April 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
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Affiliation(s)
- Roi Baer
- Fritz Haber Center of Molecular Dynamics and Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem, Israel;
| | - Daniel Neuhauser
- Department of Chemistry and Biochemistry, University of California, Los Angeles, California, USA;
| | - Eran Rabani
- Department of Chemistry, University of California, Berkeley, California, USA; .,Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California, USA.,The Raymond and Beverly Sackler Center of Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv, Israel
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6
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Nguyen M, Li W, Li Y, Rabani E, Baer R, Neuhauser D. Tempering stochastic density functional theory. J Chem Phys 2021; 155:204105. [PMID: 34852484 DOI: 10.1063/5.0063266] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
We introduce a tempering approach with stochastic density functional theory (sDFT), labeled t-sDFT, which reduces the statistical errors in the estimates of observable expectation values. This is achieved by rewriting the electronic density as a sum of a "warm" component complemented by "colder" correction(s). Since the warm component is larger in magnitude but faster to evaluate, we use many more stochastic orbitals for its evaluation than for the smaller-sized colder correction(s). This results in a significant reduction in the statistical fluctuations and systematic deviation compared to sDFT for the same computational effort. We demonstrate the method's performance on large hydrogen-passivated silicon nanocrystals, finding a reduction in the systematic deviation in the energy by more than an order of magnitude, while the systematic deviation in the forces is also quenched. Similarly, the statistical fluctuations are reduced by factors of ≈4-5 for the total energy and ≈1.5-2 for the forces on the atoms. Since the embedding in t-sDFT is fully stochastic, it is possible to combine t-sDFT with other variants of sDFT such as energy-window sDFT and embedded-fragmented sDFT.
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Affiliation(s)
- Minh Nguyen
- Department of Chemistry and Biochemistry, University of California at Los Angeles, Los Angeles, California 90095, USA
| | - Wenfei Li
- Department of Chemistry and Biochemistry, University of California at Los Angeles, Los Angeles, California 90095, USA
| | - Yangtao Li
- Department of Chemistry and Biochemistry, University of California at Los Angeles, 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 and The Raymond and Beverly Sackler Center of Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 69978, Israel
| | - 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 at Los Angeles, and California Nanoscience Institute, Los Angeles, California 90095, USA
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Chen M, Baer R, Neuhauser D, Rabani E. Stochastic density functional theory: Real- and energy-space fragmentation for noise reduction. J Chem Phys 2021; 154:204108. [PMID: 34241170 DOI: 10.1063/5.0044163] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
Stochastic density functional theory (sDFT) is becoming a valuable tool for studying ground-state properties of extended materials. The computational complexity of describing the Kohn-Sham orbitals is replaced by introducing a set of random (stochastic) orbitals leading to linear and often sub-linear scaling of certain ground-state observables at the account of introducing a statistical error. Schemes to reduce the noise are essential, for example, for determining the structure using the forces obtained from sDFT. Recently, we have introduced two embedding schemes to mitigate the statistical fluctuations in the electron density and resultant forces on the nuclei. Both techniques were based on fragmenting the system either in real space or slicing the occupied space into energy windows, allowing for a significant reduction in the statistical fluctuations. For chemical accuracy, further reduction of the noise is required, which could be achieved by increasing the number of stochastic orbitals. However, the convergence is relatively slow as the statistical error scales as 1/Nχ according to the central limit theorem, where Nχ is the number of random orbitals. In this paper, we combined the embedding schemes mentioned above and introduced a new approach that builds on overlapped fragments and energy windows. The new approach significantly lowers the noise for ground-state properties, such as the electron density, total energy, and forces on the nuclei, as demonstrated for a G-center in bulk silicon.
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Affiliation(s)
- Ming Chen
- Department of Chemistry, University of California, 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, Los Angeles, California 90095, USA
| | - Eran Rabani
- Department of Chemistry, University of California, Berkeley, California 94720, USA
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Arnon E, Rabani E, Neuhauser D, Baer R. Efficient Langevin dynamics for “noisy” forces. J Chem Phys 2020; 152:161103. [DOI: 10.1063/5.0004954] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Eitam Arnon
- Fritz Haber Research 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 and The Raymond and Beverly Sackler Center for Computational Molecular and Materials Sciences, Tel Aviv University, Tel Aviv 69978, Israel
| | - Daniel Neuhauser
- Department of Chemistry and Biochemistry, University of California at Los Angeles, Los Angeles, California 90095, USA
| | - Roi Baer
- Fritz Haber Research Center for Molecular Dynamics, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
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9
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Li W, Chen M, Rabani E, Baer R, Neuhauser D. Stochastic embedding DFT: Theory and application to p-nitroaniline in water. J Chem Phys 2019; 151:174115. [PMID: 31703523 DOI: 10.1063/1.5110226] [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
Over this past decade, we combined the idea of stochastic resolution of identity with a variety of electronic structure methods. In our stochastic Kohn-Sham density functional theory (DFT) method, the density is an average over multiple stochastic samples, with stochastic errors that decrease as the inverse square root of the number of sampling orbitals. Here, we develop a stochastic embedding density functional theory method (se-DFT) that selectively reduces the stochastic error (specifically on the forces) for a selected subsystem(s). The motivation, similar to that of other quantum embedding methods, is that for many systems of practical interest, the properties are often determined by only a small subsystem. In stochastic embedding DFT, two sets of orbitals are used: a deterministic one associated with the embedded subspace and the rest, which is described by a stochastic set. The method agrees exactly with deterministic calculations in the limit of a large number of stochastic samples. We apply se-DFT to study a p-nitroaniline molecule in water, where the statistical errors in the forces on the system (the p-nitroaniline molecule) are reduced by an order of magnitude compared with nonembedding stochastic DFT.
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Affiliation(s)
- Wenfei Li
- Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90095, USA
| | - Ming Chen
- Department of Chemistry, University of California and Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
| | - Eran Rabani
- Department of Chemistry, University of California and Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
| | - Roi Baer
- Fritz Haber Center for Molecular Dynamics, 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
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10
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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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11
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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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12
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Chen M, Baer R, Neuhauser D, Rabani E. Overlapped embedded fragment stochastic density functional theory for covalently-bonded materials. J Chem Phys 2019; 150:034106. [DOI: 10.1063/1.5064472] [Citation(s) in RCA: 22] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Ming Chen
- Department of Chemistry, University of California, and Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
| | - Roi Baer
- Fritz Haber Center for 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 Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
- The Raymond and Beverly Sackler Center of Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 69978, Israel
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13
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Wang Z, Chern GW, Batista CD, Barros K. Gradient-based stochastic estimation of the density matrix. J Chem Phys 2018. [DOI: 10.1063/1.5017741] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Zhentao Wang
- Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996, USA
| | - Gia-Wei Chern
- Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA
| | - Cristian D. Batista
- Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996, USA
- Quantum Condensed Matter Division and Shull-Wollan Center, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
| | - Kipton Barros
- Theoretical Division and CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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Neuhauser D, Baer R, Zgid D. Stochastic Self-Consistent Second-Order Green’s Function Method for Correlation Energies of Large Electronic Systems. J Chem Theory Comput 2017; 13:5396-5403. [DOI: 10.1021/acs.jctc.7b00792] [Citation(s) in RCA: 35] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Daniel Neuhauser
- Department
of Chemistry and Biochemistry, University of California at Los Angeles, 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
| | - Dominika Zgid
- Department
of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, United States
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15
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Takeshita TY, de Jong WA, Neuhauser D, Baer R, Rabani E. Stochastic Formulation of the Resolution of Identity: Application to Second Order Møller–Plesset Perturbation Theory. J Chem Theory Comput 2017; 13:4605-4610. [DOI: 10.1021/acs.jctc.7b00343] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Tyler Y. Takeshita
- Department
of Chemistry, University of California Berkeley, Berkeley, California 94720, United States
- Materials
Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States
| | - Wibe A. de Jong
- Computational
Research Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States
| | - Daniel Neuhauser
- Department
of Chemistry and Biochemistry, University of California, Los Angeles, California 90095, United States
| | - Roi Baer
- Fritz
Harber Center for Molecular Dynamics, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
| | - Eran Rabani
- Department
of Chemistry, University of California Berkeley, Berkeley, California 94720, United States
- Materials
Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States
- The Sackler
Center for Computational Molecular Science, Tel Aviv University, Tel Aviv 69978, Israel
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