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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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Hadad RE, Roy A, Rabani E, Redmer R, Baer R. Stochastic density functional theory combined with Langevin dynamics for warm dense matter. Phys Rev E 2024; 109:065304. [PMID: 39020867 DOI: 10.1103/physreve.109.065304] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/24/2024] [Accepted: 05/17/2024] [Indexed: 07/19/2024]
Abstract
This study overviews and extends a recently developed stochastic finite-temperature Kohn-Sham density functional theory to study warm dense matter using Langevin dynamics, specifically under periodic boundary conditions. The method's algorithmic complexity exhibits nearly linear scaling with system size and is inversely proportional to the temperature. Additionally, a linear-scaling stochastic approach is introduced to assess the Kubo-Greenwood conductivity, demonstrating exceptional stability for dc conductivity. Utilizing the developed tools, we investigate the equation of state, radial distribution, and electronic conductivity of hydrogen at a temperature of 30 000 K. As for the radial distribution functions, we reveal a transition of hydrogen from gaslike to liquidlike behavior as its density exceeds 4g/cm^{3}. As for the electronic conductivity as a function of the density, we identified a remarkable isosbestic point at frequencies around 7 eV, which may be an additional signature of a gas-liquid transition in hydrogen at 30 000 K.
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Affiliation(s)
| | | | - Eran Rabani
- Department of Chemistry, University of California, Berkeley, California 94720, USA; 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
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3
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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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4
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Shao X, Mi W, Pavanello M. Density Embedding Method for Nanoscale Molecule-Metal Interfaces. J Phys Chem Lett 2022; 13:7147-7154. [PMID: 35901490 DOI: 10.1021/acs.jpclett.2c01424] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/15/2023]
Abstract
In this work, we extend the applicability of standard Kohn-Sham DFT (KS-DFT) to model realistically sized molecule-metal interfaces where the metal slabs venture into the tens of nanometers in size. Employing state-of-the-art noninteracting kinetic energy functionals, we describe metallic subsystems with orbital-free DFT and combine their electronic structure with molecular subsystems computed at the KS-DFT level resulting in a multiscale subsystem DFT method. The method reproduces within a few millielectronvolts the binding energy difference of water and carbon dioxide molecules adsorbed on the top and hollow sites of an Al(111) surface compared to KS-DFT of the combined supersystem. It is also robust for Born-Oppenheimer molecular dynamics simulations. Very large system sizes are approached with standard computing resources thanks to a parallelization scheme that avoids accumulation of memory at the gather-scatter stage. The results as presented are encouraging and open the door to ab initio simulations of realistically sized, mesoscopic molecule-metal interfaces.
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Affiliation(s)
- Xuecheng Shao
- Department of Chemistry, Rutgers University, Newark, New Jersey 07102, United States
| | - Wenhui Mi
- International Center for Computational Method and Software, College of Physics, Jilin University, Changchun 130012, China
| | - Michele Pavanello
- Department of Chemistry, Rutgers University, Newark, New Jersey 07102, United States
- Department of Physics, Rutgers University, Newark, New Jersey 07102, United States
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Weng G, Romanova M, Apelian A, Song H, Vlček V. Reduced Scaling of Optimal Regional Orbital Localization via Sequential Exhaustion of the Single-Particle Space. J Chem Theory Comput 2022; 18:4960-4972. [PMID: 35817013 PMCID: PMC9367006 DOI: 10.1021/acs.jctc.2c00315] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
![]()
Wannier functions have become a powerful tool in the
electronic
structure calculations of extended systems. The generalized Pipek-Mezey
Wannier functions exhibit appealing characteristics (e.g., reaching
an optimal localization and the separation of the σ–π
orbitals) compared with other schemes. However, when applied to giant
nanoscale systems, the orbital localization suffers from a large computational
cost overhead if one is interested in localized states in a small
fragment of the system. Herein, we present a swift, efficient, and
robust approach for obtaining regionally localized orbitals of a subsystem
within the generalized Pipek-Mezey scheme. The proposed algorithm
introduces a reduced work space and sequentially exhausts the entire
orbital space until the convergence of the localization functional.
It tackles systems with ∼10000 electrons within 0.5 h with
no loss in localization quality compared to the traditional approach.
Regionally localized orbitals with a higher extent of localization
are obtained via judiciously extending the subsystem’s size.
Exemplifying on large bulk and a 4 nm wide slab of diamond with an
NV– center, we demonstrate the methodology and discuss
how the choice of the localization region affects the excitation energy
of the defect. Furthermore, we show how the sequential algorithm is
easily extended to stochastic methodologies that do not provide individual
single-particle eigenstates. It is thus a promising tool to obtain
regionally localized states for solving the electronic structure problems
of a subsystem embedded in giant condensed systems.
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Affiliation(s)
- Guorong Weng
- Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106-9510, United States
| | - Mariya Romanova
- Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106-9510, United States
| | - Arsineh Apelian
- Department of Materials, University of California, Santa Barbara, California 93106-9510, United States
| | - Hanbin Song
- 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
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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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White AJ, Collins LA, Nichols K, Hu SX. Mixed stochastic-deterministic time-dependent density functional theory: application to stopping power of warm dense carbon. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2022; 34:174001. [PMID: 35081511 DOI: 10.1088/1361-648x/ac4f1a] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/30/2021] [Accepted: 01/26/2022] [Indexed: 06/14/2023]
Abstract
Warm dense matter (WDM) describes an intermediate phase, between condensed matter and classical plasmas, found in natural and man-made systems. In a laboratory setting, WDM is often created dynamically. It is typically laser or pulse-power generated and can be difficult to characterize experimentally. Measuring the energy loss of high energy ions, caused by a WDM target, is both a promising diagnostic and of fundamental importance to inertial confinement fusion research. However, electron coupling, degeneracy, and quantum effects limit the accuracy of easily calculable kinetic models for stopping power, while high temperatures make the traditional tools of condensed matter, e.g. time-dependent density functional theory (TD-DFT), often intractable. We have developed a mixed stochastic-deterministic approach to TD-DFT which provides more efficient computation while maintaining the required precision for model discrimination. Recently, this approach showed significant improvement compared to models when compared to experimental energy loss measurements in WDM carbon. Here, we describe this approach and demonstrate its application to warm dense carbon stopping across a range of projectile velocities. We compare direct stopping-power calculation to approaches based on combining homogeneous electron gas response with bound electrons, with parameters extracted from our TD-DFT calculations.
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Affiliation(s)
- Alexander J White
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, 87545 NM, United States of America
| | - Lee A Collins
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, 87545 NM, United States of America
| | - Katarina Nichols
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, 87545 NM, United States of America
- Laboratory of Laser Energetics, University of Rochester, Rochester 14623 NY, United States of America
| | - S X Hu
- Laboratory of Laser Energetics, University of Rochester, Rochester 14623 NY, United States of America
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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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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: 6] [Impact Index Per Article: 3.0] [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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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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White AJ, Collins LA. Fast and Universal Kohn-Sham Density Functional Theory Algorithm for Warm Dense Matter to Hot Dense Plasma. PHYSICAL REVIEW LETTERS 2020; 125:055002. [PMID: 32794867 DOI: 10.1103/physrevlett.125.055002] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/06/2020] [Revised: 06/09/2020] [Accepted: 07/15/2020] [Indexed: 06/11/2023]
Abstract
Understanding many processes, e.g., fusion experiments, planetary interiors, and dwarf stars, depends strongly on microscopic physics modeling of warm dense matter and hot dense plasma. This complex state of matter consists of a transient mixture of degenerate and nearly free electrons, molecules, and ions. This regime challenges both experiment and analytical modeling, necessitating predictive ab initio atomistic computation, typically based on quantum mechanical Kohn-Sham density functional theory (KS-DFT). However, cubic computational scaling with temperature and system size prohibits the use of DFT through much of the warm dense matter regime. A recently developed stochastic approach to KS-DFT can be used at high temperatures, with the exact same accuracy as the deterministic approach, but the stochastic error can converge slowly and it remains expensive for intermediate temperatures (<50 eV). We have developed a universal mixed stochastic-deterministic algorithm for DFT at any temperature. This approach leverages the physics of KS-DFT to seamlessly integrate the best aspects of these different approaches. We demonstrate that this method significantly accelerated self-consistent field calculations for temperatures from 3 to 50 eV, while producing stable molecular dynamics and accurate diffusion coefficients.
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Affiliation(s)
- A J White
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | - L A Collins
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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