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Escobedo FA. On the calculation of free energies over Hamiltonian and order parameters via perturbation and thermodynamic integration. J Chem Phys 2021; 155:114112. [PMID: 34551542 DOI: 10.1063/5.0061541] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
In this work, complementary formulas are presented to compute free-energy differences via perturbation (FEP) methods and thermodynamic integration (TI). These formulas are derived by selecting only the most statistically significant data from the information extractable from the simulated points involved. On the one hand, commonly used FEP techniques based on overlap sampling leverage the full information contained in the overlapping macrostate probability distributions. On the other hand, conventional TI methods only use information on the first moments of those distributions, as embodied by the first derivatives of the free energy. Since the accuracy of simulation data degrades considerably for high-order moments (for FEP) or free-energy derivatives (for TI), it is proposed to consider, consistently for both methods, data up to second-order moments/derivatives. This provides a compromise between the limiting strategies embodied by common FEP and TI and leads to simple, optimized expressions to evaluate free-energy differences. The proposed formulas are validated with an analytically solvable harmonic Hamiltonian (for assessing systematic errors), an atomistic system (for computing the potential of mean force with coordinate-dependent order parameters), and a binary-component coarse-grained model (for tracing a solid-liquid phase diagram in an ensemble sampled through alchemical transformations). It is shown that the proposed FEP and TI formulas are straightforward to implement, perform similarly well, and allow robust estimation of free-energy differences even when the spacing of successive points does not guarantee them to have proper overlapping in phase space.
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Affiliation(s)
- Fernando A Escobedo
- Robert Frederick Smith School of Chemistry and Biomolecular Engineering, Cornell University, Ithaca, New York 14853, USA
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2
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Desgranges C, Delhommelle J. Evaluation of the grand-canonical partition function using expanded Wang-Landau simulations. V. Impact of an electric field on the thermodynamic properties and ideality contours of water. J Chem Phys 2016; 145:184504. [DOI: 10.1063/1.4967336] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022] Open
Affiliation(s)
- Caroline Desgranges
- Department of Chemistry, University of North Dakota, 151 Cornell Street Stop 9024, Grand Forks, North Dakota 58202, USA
| | - Jerome Delhommelle
- Department of Chemistry, University of North Dakota, 151 Cornell Street Stop 9024, Grand Forks, North Dakota 58202, USA
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Desgranges C, Delhommelle J. Evaluation of the grand-canonical partition function using expanded Wang-Landau simulations. IV. Performance of many-body force fields and tight-binding schemes for the fluid phases of silicon. J Chem Phys 2016; 144:124510. [PMID: 27036464 DOI: 10.1063/1.4944619] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We extend Expanded Wang-Landau (EWL) simulations beyond classical systems and develop the EWL method for systems modeled with a tight-binding Hamiltonian. We then apply the method to determine the partition function and thus all thermodynamic properties, including the Gibbs free energy and entropy, of the fluid phases of Si. We compare the results from quantum many-body (QMB) tight binding models, which explicitly calculate the overlap between the atomic orbitals of neighboring atoms, to those obtained with classical many-body (CMB) force fields, which allow to recover the tetrahedral organization in condensed phases of Si through, e.g., a repulsive 3-body term that favors the ideal tetrahedral angle. Along the vapor-liquid coexistence, between 3000 K and 6000 K, the densities for the two coexisting phases are found to vary significantly (by 5 orders of magnitude for the vapor and by up to 25% for the liquid) and to provide a stringent test of the models. Transitions from vapor to liquid are predicted to occur for chemical potentials that are 10%-15% higher for CMB models than for QMB models, and a ranking of the force fields is provided by comparing the predictions for the vapor pressure to the experimental data. QMB models also reveal the formation of a gap in the electronic density of states of the coexisting liquid at high temperatures. Subjecting Si to a nanoscopic confinement has a dramatic effect on the phase diagram with, e.g. at 6000 K, a decrease in liquid densities by about 50% for both CMB and QMB models and an increase in vapor densities between 90% (CMB) and 170% (QMB). The results presented here provide a full picture of the impact of the strategy (CMB or QMB) chosen to model many-body effects on the thermodynamic properties of the fluid phases of Si.
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Affiliation(s)
- Caroline Desgranges
- Department of Chemistry, University of North Dakota, 151 Cornell Street Stop 9024, Grand Forks, North Dakota 58202, USA
| | - Jerome Delhommelle
- Department of Chemistry, University of North Dakota, 151 Cornell Street Stop 9024, Grand Forks, North Dakota 58202, USA
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Hemmen A, Panagiotopoulos AZ, Gross J. Grand Canonical Monte Carlo Simulations Guided by an Analytic Equation of State—Transferable Anisotropic Mie Potentials for Ethers. J Phys Chem B 2015; 119:7087-99. [DOI: 10.1021/acs.jpcb.5b01806] [Citation(s) in RCA: 31] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/31/2023]
Affiliation(s)
- Andrea Hemmen
- Institute
of Thermodynamics and Thermal Process Engineering, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany
| | | | - Joachim Gross
- Institute
of Thermodynamics and Thermal Process Engineering, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany
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Desgranges C, Delhommelle J. Evaluation of the grand-canonical partition function using expanded Wang-Landau simulations. III. Impact of combining rules on mixtures properties. J Chem Phys 2014; 140:104109. [DOI: 10.1063/1.4867498] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/16/2022] Open
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Rane KS, Murali S, Errington JR. Monte Carlo Simulation Methods for Computing Liquid–Vapor Saturation Properties of Model Systems. J Chem Theory Comput 2013; 9:2552-66. [DOI: 10.1021/ct400074p] [Citation(s) in RCA: 71] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/08/2023]
Affiliation(s)
- Kaustubh S. Rane
- Department of Chemical
and Biological Engineering, University at Buffalo, The State University of New York, Buffalo, New York 14260-4200,
United States
| | - Sabharish Murali
- Department of Chemical
and Biological Engineering, University at Buffalo, The State University of New York, Buffalo, New York 14260-4200,
United States
| | - Jeffrey R. Errington
- Department of Chemical
and Biological Engineering, University at Buffalo, The State University of New York, Buffalo, New York 14260-4200,
United States
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Meadley SL, Escobedo FA. Thermodynamics and kinetics of bubble nucleation: Simulation methodology. J Chem Phys 2012; 137:074109. [DOI: 10.1063/1.4745082] [Citation(s) in RCA: 52] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Desgranges C, Delhommelle J. Evaluation of the grand-canonical partition function using expanded Wang-Landau simulations. I. Thermodynamic properties in the bulk and at the liquid-vapor phase boundary. J Chem Phys 2012; 136:184107. [DOI: 10.1063/1.4712023] [Citation(s) in RCA: 42] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/25/2023] Open
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Desgranges C, Delhommelle J. Evaluation of the grand-canonical partition function using expanded Wang-Landau simulations. II. Adsorption of atomic and molecular fluids in a porous material. J Chem Phys 2012; 136:184108. [DOI: 10.1063/1.4712025] [Citation(s) in RCA: 38] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/26/2023] Open
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Duncan PD, Masters AJ, Wilson MR. Thermodynamic stability of the cubatic phase of hard cut spheres evaluated by expanded ensemble simulations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:011702. [PMID: 21867191 DOI: 10.1103/physreve.84.011702] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/14/2011] [Revised: 05/23/2011] [Indexed: 05/31/2023]
Abstract
The system of hard cut spheres (disk-shaped particles formed by symmetrically truncating the end caps of a sphere) exhibits an intriguing "cubatic" phase with cubic orientational symmetry. However, it is unclear whether this phase is metastable with respect to the columnar phase. We attempt to provide an answer to this question by carrying out free energy calculations by the expanded ensemble Monte Carlo method. We conclude that there may be a very small region of cubatic stability in the vicinity of the isotropic-cubatic phase transition, but that that transition would need to be determined more accurately to obtain a definitive answer. We also comment on the efficacy of the expanded ensemble method for these kinds of calculations.
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Affiliation(s)
- Peter D Duncan
- Department of Chemistry, University of Durham, South Road, Durham DH1 3LE, UK
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Paluch AS, Shah JK, Maginn EJ. Efficient Solvation Free Energy Calculations of Amino Acid Analogs by Expanded Ensemble Molecular Simulation. J Chem Theory Comput 2011; 7:1394-403. [DOI: 10.1021/ct1006746] [Citation(s) in RCA: 27] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Andrew S. Paluch
- Department of Chemical and Biomolecular Engineering and ‡Center for Research Computing, University of Notre Dame, Notre Dame, Indiana 46556, United States
| | - Jindal K. Shah
- Department of Chemical and Biomolecular Engineering and ‡Center for Research Computing, University of Notre Dame, Notre Dame, Indiana 46556, United States
| | - Edward J. Maginn
- Department of Chemical and Biomolecular Engineering and ‡Center for Research Computing, University of Notre Dame, Notre Dame, Indiana 46556, United States
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Wu D. Understanding free-energy perturbation calculations through a model of harmonic oscillators: theory and implications to improve the sampling efficiency by molecular simulation. J Chem Phys 2011; 133:244116. [PMID: 21197985 DOI: 10.1063/1.3511703] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
Free-energy perturbation calculation is frequently used to calculate free-energy differences because it is easy to implement and the computation is fast. However, the calculation is subject to large inaccuracies in some circumstances due to the insufficient sampling of the relevant tails of the energy-difference distributions. Here we expand this knowledge of insufficient sampling into a two-dimensional (2D) energy space using a model of harmonic oscillators. We show analytically the relation between the energies of the sampling system and those of the desired target energy spaces, which provide the basis to understand the difficulties in free-energy perturbation calculations. We clarify the reasons of the inaccurate calculation in the different harmonic cases that stem from the spatial separations of the reference and the target energy pairs located in the two-dimensional energy space. The potential-energy space introduced into this 2D energy-space model provides additional clues to improve the sampling efficiency. Based on this understanding, we propose two ways to calculate the free-energy differences using the two schemes of the distribution method. We show that the distribution method implemented in the appropriate energy space--the energy-difference space and the potential-energy space, respectively--can improve the calculation of free energies in different circumstances. This analysis implies that the sampling can be improved if it is directed toward the appropriate region in the potential-energy space, which is easily implemented in various types of free-energy calculations. To test this, we calculate the free-energy surface of alanine dipeptide in gas phase and in aqueous phase, respectively. We demonstrate that the free-energy surface calculation is improved when the biased sampling of the potential energy is integrated into the sampling scheme.
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Affiliation(s)
- Di Wu
- Department of Physiology and Biophysics, School of Life Sciences, Fudan University, Shanghai 200433, China.
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Do H, Wheatley RJ, Hirst JD. Molecular simulation of the binary mixture of 1–1–1–2–tetrafluoroethane and carbon dioxide. Phys Chem Chem Phys 2011; 13:15708-13. [DOI: 10.1039/c1cp21419e] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2022]
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Wu D. An efficient umbrella potential for the accurate calculation of free energies by molecular simulation. J Chem Phys 2010; 133:044115. [DOI: 10.1063/1.3464330] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Desgranges C, Hicks J, Magness A, Delhommelle J. Phase equilibria of polyaromatic hydrocarbons by hybrid Monte Carlo Wang–Landau simulations. Mol Phys 2010. [DOI: 10.1080/00268970903544238] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
Affiliation(s)
- C. Desgranges
- a Department of Chemistry , University of North Dakota , 151 Cornell Street Stop 9024, Grand Forks, North Dakota 58202, USA
| | - J.M. Hicks
- a Department of Chemistry , University of North Dakota , 151 Cornell Street Stop 9024, Grand Forks, North Dakota 58202, USA
| | - A. Magness
- b Department of Physics and Astrophysics , University of North Dakota , 101 Cornell Street Stop 7129, Grand Forks, North Dakota 58202, USA
| | - J. Delhommelle
- a Department of Chemistry , University of North Dakota , 151 Cornell Street Stop 9024, Grand Forks, North Dakota 58202, USA
- b Department of Physics and Astrophysics , University of North Dakota , 101 Cornell Street Stop 7129, Grand Forks, North Dakota 58202, USA
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Abreu CRA. Isochronal sampling in non-Boltzmann Monte Carlo methods. J Chem Phys 2009; 131:154113. [DOI: 10.1063/1.3245304] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Charlles R A Abreu
- School of Chemical Engineering, State University of Campinas, Campinas, Sao Paulo 13083-970, Brazil.
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Desgranges C, Delhommelle J. Phase equilibria of molecular fluids via hybrid Monte Carlo Wang–Landau simulations: Applications to benzene and n-alkanes. J Chem Phys 2009; 130:244109. [DOI: 10.1063/1.3158605] [Citation(s) in RCA: 48] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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Escobedo FA, Martinez-Veracoechea FJ. Optimization of expanded ensemble methods. J Chem Phys 2008; 129:154107. [DOI: 10.1063/1.2994717] [Citation(s) in RCA: 47] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/31/2023] Open
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Fenwick MK. A direct multiple histogram reweighting method for optimal computation of the density of states. J Chem Phys 2008; 129:125106. [DOI: 10.1063/1.2981800] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/01/2023] Open
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Wu D. Accurate calculations of free-energy differences by the distribution method. J Chem Phys 2008; 128:224105. [DOI: 10.1063/1.2936987] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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John BS, Juhlin C, Escobedo FA. Phase behavior of colloidal hard perfect tetragonal parallelepipeds. J Chem Phys 2008; 128:044909. [DOI: 10.1063/1.2819091] [Citation(s) in RCA: 85] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Escobedo FA. Optimized expanded ensembles for simulations involving molecular insertions and deletions. II. Open systems. J Chem Phys 2008; 127:174104. [PMID: 17994804 DOI: 10.1063/1.2800321] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
In the Grand Canonical, osmotic, and Gibbs ensembles, chemical potential equilibrium is attained via transfers of molecules between the system and either a reservoir or another subsystem. In this work, the expanded ensemble (EXE) methods described in part I [F. A. Escobedo and F. J. Martinez-Veracoechea, J. Chem. Phys. 127, 174103 (2007)] of this series are extended to these ensembles to overcome the difficulties associated with implementing such whole-molecule transfers. In EXE, such moves occur via a target molecule that undergoes transitions through a number of intermediate coupling states. To minimize the tunneling time between the fully coupled and fully decoupled states, the intermediate states could be either: (i) sampled with an optimal frequency distribution (the sampling problem) or (ii) selected with an optimal spacing distribution (staging problem). The sampling issue is addressed by determining the biasing weights that would allow generating an optimal ensemble; discretized versions of this algorithm (well suited for small number of coupling stages) are also presented. The staging problem is addressed by selecting the intermediate stages in such a way that a flat histogram is the optimized ensemble. The validity of the advocated methods is demonstrated by their application to two model problems, the solvation of large hard spheres into a fluid of small and large spheres, and the vapor-liquid equilibrium of a chain system.
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Affiliation(s)
- Fernando A Escobedo
- School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, New York 14853, USA.
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Escobedo FA, Martínez-Veracoechea FJ. Optimized expanded ensembles for simulations involving molecular insertions and deletions. I. Closed systems. J Chem Phys 2007; 127:174103. [DOI: 10.1063/1.2800320] [Citation(s) in RCA: 39] [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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Ganzenmüller G, Camp PJ. Applications of Wang-Landau sampling to determine phase equilibria in complex fluids. J Chem Phys 2007; 127:154504. [DOI: 10.1063/1.2794042] [Citation(s) in RCA: 79] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Escobedo FA. Simulation of the density of states in isothermal and adiabatic ensembles. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:056701. [PMID: 16803068 DOI: 10.1103/physreve.73.056701] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/04/2006] [Indexed: 05/10/2023]
Abstract
This paper provides a unified treatment of the fundamental methods used to obtain the density of states via molecular simulations with isothermal ensembles (IEs) and adiabatic ensembles (AEs). Our analysis and results show that provides a natural bridge to go back and forth between IE and AE simulation data. They also underline the difference between the density of states of potential energy macrostates and that of total energy macrostates Omega, even though both provide access to the thermodynamic properties of the system. Visited-states approaches and transition matrix methods are described and applied to the Lennard-Jones fluid to target omega and Omega as functions of energy and volume macrostates. It is shown that one can obtain omega via a generalized acceptance-ratio formula that is applicable regardless of the conditions at which the ensemble is simulated. In this way, one can obtain while performing conventional IE or AE simulations, and do it at no extra cost and with a higher accuracy than is achievable with histogram methods.
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Affiliation(s)
- Fernando A Escobedo
- School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, New York 14853, USA
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