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Metadynamics: A Unified Framework for Accelerating Rare Events and Sampling Thermodynamics and Kinetics

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Handbook of Materials Modeling

Abstract

Metadynamics is an enhanced sampling algorithm in which the normal evolution of the system is biased by a history-dependent potential constructed as a sum of Gaussians centered along the trajectory followed by a suitably chosen set of collective variables. The sum of Gaussians forces the system to escape from local free energy minima and is used to iteratively build an estimator of the free energy. This original idea has been developed and improved over the years in several variants, which nowadays allow addressing in a unified framework some of the most important tasks of molecular simulations: computing the free energy as a function of the collective variables, accelerating rare events, and estimating unbiased kinetic rate constants. This chapter provides a survey of the many formulations of metadynamics with an emphasis on the underlying theoretical concepts and some hints on the appropriate manner of using this approach for solving complicated real-world problems.

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References

  • Baftizadeh F, Cossio P, Pietrucci F, Laio A (2012) Protein folding and ligand-enzyme binding from bias-exchange meta-dynamics simulations. Curr Phys Chem 2:79–91

    Article  Google Scholar 

  • Barducci A, Bussi G, Parrinello M (2008) Well-tempered metadynamics: a smoothly converging and tunable free-energy method. Phys Rev Lett 1(2):020603. https://doi.org/10.1103/PhysRevLett.100.020603

  • Barducci A, Bonomi M, Parrinello M (2011) Metadynamics. Wiley Interdiscip Rev Comput Mol Sci 1(5):826–843

    Article  Google Scholar 

  • Behler J, Martonak R, Donadio D, Parrinello M (2008) Metadynamics simulations of the high-pressure phases of silicon employing a high-dimensional neural network potential. Phys Rev Lett 100(18):185501. https://doi.org/10.1103/PhysRevLett.100.185501

  • Biarnés X, Pietrucci F, Marinelli F, Laio A (2012) METAGUI. A VMD interface for analyzing metadynamics and molecular dynamics simulations. Comput Phys Commun 183(1):203–211. https://doi.org/10.1016/j.cpc.2011.08.020

    Article  ADS  Google Scholar 

  • Bonomi M, Barducci A, Parrinello M (2009a) Reconstructing the equilibrium Boltzmann distribution from well-tempered metadynamics. J Comput Chem 30(11):1615–1621

    Article  Google Scholar 

  • Bonomi M, Branduardi D, Bussi G, Camilloni C, Provasi D, Raiteri P, Donadio D, Marinelli F, Pietrucci F, Broglia RA et al (2009b) Plumed: a portable plugin for free-energy calculations with molecular dynamics. Comput Phys Commun 180(10):1961–1972

    Article  ADS  Google Scholar 

  • Boyer MJ, Vilčiauskas L, Hwang GS (2016) Structure and li+ ion transport in a mixed carbonate/lipf 6 electrolyte near graphite electrode surfaces: a molecular dynamics study. Phys Chem Chem Phys 18(40):27868–27876

    Article  Google Scholar 

  • Branduardi D, Bussi G, Parrinello M (2012) Metadynamics with adaptive gaussians. J Chem Theory Comput 8(7):2247–2254

    Article  Google Scholar 

  • Bui T, Phan A, Cole DR, Striolo A (2017) Transport mechanism of guest methane in water-filled nanopores. J Phys Chem C 121(29):15675–15686

    Article  Google Scholar 

  • Bussi G, Gervasio FL, Laio A, Parrinello M (2006a) Free-energy landscape for beta hairpin folding from combined parallel tempering and metadynamics. J Am Chem Soc 128(41):13435–13441. https://doi.org/10.1021/ja062463w

    Article  Google Scholar 

  • Bussi G, Laio A, Parrinello M (2006b) Equilibrium free energies from nonequilibrium metadynamics. Phys Rev Lett 96(9):090601. https://doi.org/10.1103/PhysRevLett.96.090601

    Article  ADS  Google Scholar 

  • Bussi G, Branduardi D et al (2015) Free-energy calculations with metadynamics: theory and practice. Rev Comput Chem 28:1–49

    Google Scholar 

  • Camilloni C, Provasi D, Tiana G, Broglia RA (2008) Exploring the protein G helix free-energy surface by solute tempering metadynamics. Proteins Struct Funct Bioinf 71(4):1647–1654. https://doi.org/10.1002/prot.21852

    Article  Google Scholar 

  • Car R, Parrinello M (1985) Unified approach for molecular-dynamics and density-functional theory. Phys Rev Lett 45:2471

    Article  ADS  Google Scholar 

  • Carter EA, Ciccotti G, Hynes JT, Kapral R (1989) Constrained reaction coordinate dynamics for the simulation of rare events. Chem Phys Lett 156:472–477

    Article  ADS  Google Scholar 

  • Ceriani C, Laio A, Fois E, Gamba A, Martonak R, Parrinello M (2004) Molecular dynamics simulation of reconstructive phase transitions on an anhydrous zeolite. Phys Rev B 70(11):113403. https://doi.org/10.1103/PhysRevB.70.113403

    Article  ADS  Google Scholar 

  • Cheng T, Goddard WA, An Q, Xiao H, Merinov B, Morozov S (2017) Mechanism and kinetics of the electrocatalytic reaction responsible for the high cost of hydrogen fuel cells. Phys Chem Chem Phys 19(4):2666–2673

    Article  Google Scholar 

  • Crespo Y, Marinelli F, Pietrucci F, Laio A (2010) Metadynamics convergence law in a multidimensional system. Phys Rev E 81:055701. https://doi.org/10.1103/PhysRevE.81.055701

    Article  ADS  Google Scholar 

  • Cunha RA, Bussi G (2017) Unraveling Mg2+–Rna binding with atomistic molecular dynamics. RNA 23(5):628–638

    Article  Google Scholar 

  • Cvijovic D, Klinowski J (1995) Taboo search – an approach to the multiple minima problem. Science 267:664–666

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Dama JF, Parrinello M, Voth GA (2014) Well-tempered metadynamics converges asymptotically. Phys Rev Lett 112(24):240602

    Article  ADS  Google Scholar 

  • Dellago C, Bolhuis P, Csajka FS, Chandler D (1998) Transition path sampling and the calculation of rate constants. J Chem Phys 108:1964–1977

    Article  ADS  Google Scholar 

  • Dellago C, Bolhuis P, Geissler P (2002) Transition path sampling. Adv Chem Phys 123:1–78

    Google Scholar 

  • Di Pietro E, Pagliai M, Cardini G, Schettino V (2006) Solid-state phase transition induced by pressure in LiOH center dot H2O. J Phys Chem B 110(27):13539–13546. https://doi.org/10.1021/jp061620a

    Article  Google Scholar 

  • Donadio D, Bernasconi M (2005) Ab initio simulation of photoinduced transformation of small rings in amorphous silica. Phys Rev B 71(7):073307. https://doi.org/10.1103/PhysRevB.71.073307

    Article  ADS  Google Scholar 

  • Donadio D, Raiteri P, Parrinello M (2005) Topological defects and bulk melting of hexagonal ice. J Phys Chem B 109:5421–5424

    Article  Google Scholar 

  • Ensing B, De Vivo M, Liu Z, Moore P, Klein ML (2006) Metadynamics as a tool for exploring free energy landscapes of chemical reactions. Acc Chem Res 39(2):73–81

    Article  Google Scholar 

  • Fiorin G, Klein ML, Hénin J (2013) Using collective variables to drive molecular dynamics simulations. Mol Phys 111(22–23):3345–3362

    Article  ADS  Google Scholar 

  • Fitzner M, Sosso GC, Pietrucci F, Pipolo S, Michaelides A (2017) Pre-critical fluctuations and what they disclose about heterogeneous crystal nucleation. Nat Commun 8(1):2257

    Article  ADS  Google Scholar 

  • Fleming KL, Tiwary P, Pfaendtner J (2016) New approach for investigating reaction dynamics and rates with ab initio calculations. J Phys Chem A 120(2):299–305

    Article  Google Scholar 

  • Fu CD, Pfaendtner J (2018) Lifting the curse of dimensionality on enhanced sampling of reaction networks with parallel bias metadynamics. J Chem Theory Comput 14:2516–2525

    Article  Google Scholar 

  • Gil-Ley A, Bussi G (2015) Enhanced conformational sampling using replica exchange with collective-variable tempering. J Chem Theory Comput 11(3):1077–1085

    Article  Google Scholar 

  • Gil-Ley A, Bottaro S, Bussi G (2016) Empirical corrections to the amber RNA force field with target metadynamics. J Chem Theory Comput 12(6):2790–2798

    Article  Google Scholar 

  • Giorgino T, Laio A, Rodriguez A (2017) METAGUI 3: a graphical user interface for choosing the collective variables in molecular dynamics simulations. Comput Phys Commun 217:204–209

    Article  ADS  Google Scholar 

  • Grubmüller H (1995) Predicting slow structural transitions in macromolecular systems: conformational flooding. Phys Rev E 52(3):2893

    Article  ADS  Google Scholar 

  • Hasell T, Miklitz M, Stephenson A, Little MA, Chong SY, Clowes R, Chen L, Holden D, Tribello GA, Jelfs KE et al (2016) Porous organic cages for sulfur hexafluoride separation. J Am Chem Soc 138(5):1653–1659

    Article  Google Scholar 

  • Henin J, Fiorin G, Chipot C, Klein ML (2009) Exploring multidimensional free energy landscapes using time-dependent biases on collective variables. J Chem Theory Comput 6(1):35–47

    Article  Google Scholar 

  • Hosek P, Toulcova D, Bortolato A, Spiwok V (2016) Altruistic metadynamics: multisystem biased simulation. J Phys Chem B 120(9):2209–2215

    Article  Google Scholar 

  • Hu XL, Piccinin S, Laio A, Fabris S (2012) Atomistic structure of cobalt-phosphate nanoparticles for catalytic water oxidationx. ACS Nano 6(12):10497

    Article  Google Scholar 

  • Huber T, Torda A, van Gunsteren W (1994) Local elevation: a method for improving the searching properties of molecular dynamics simulation. J Comput Aided Mol Des 8:695–708

    Article  ADS  Google Scholar 

  • Iannuzzi M, Parrinello M (2004) Proton transfer in heterocycle crystals. Phys Rev Lett 93:025901

    Article  ADS  Google Scholar 

  • Iannuzzi M, Laio A, Parrinello M (2003) Efficient exploration of reactive potential energy surfaces using car-m. parrinelloolecular dynamics. Phys Rev Lett 90:238302

    Google Scholar 

  • Karamertzanis PG, Raiteri P, Parrinello M, Leslie M, Price SL (2008) The thermal stability of lattice-energy minima of 5-fluorouracil: metadynamics as an aid to polymorph prediction. J Phys Chem B 112(14):4298–4308. https://doi.org/10.1021/jp709764e

    Article  Google Scholar 

  • Kevrekidis IG, Gear CW, Hummer G (2004) Equation-free: the computer-aided analysis of comptex multiscale systems. Aiche J 50(7):1346–1355

    Article  Google Scholar 

  • Kumar S, Rosenberg JM, Bouzida D, Swendsen RH, Kollman PA (1995) Multidimensional free-energy calculations using the weighted histogram analysis method. J Comput Chem 16:1339–1350

    Article  Google Scholar 

  • Laino T, Donadio D, Kuo IFW (2007) Migration of positively charged defects in alpha-quartz. Phys Rev B 76(19):195210. https://doi.org/10.1103/PhysRevB.76.195210

    Article  ADS  Google Scholar 

  • Laio A, Gervasio FL (2008) Metadynamics: a method to simulate rare events and reconstruct the free energy in biophysics, chemistry and material science. Rep Prog Phys 71(12):126601

    Article  ADS  Google Scholar 

  • Laio A, Parrinello M (2002) Escaping free energy minima. Proc Natl Acad Sci USA 99:12562–12566

    Article  ADS  Google Scholar 

  • Laio A, Rodriguez-Fortea A, Gervasio FL, Ceccarelli M, Parrinello M (2005) Assessing the accuracy of metadynamics. J Phys Chem B 109:6714–6721

    Article  Google Scholar 

  • Maragliano L, Vanden-Eijnden E (2006) A temperature accelerated method for sampling free energy and determining reaction pathways in rare events simulations. Chem Phys Lett 426(1):168–175

    Article  ADS  Google Scholar 

  • Marinari E, Parisi G (1992) Simulated tempering: a new Monte Carlo scheme. EPL (Europhys Lett) 19(6):451

    Article  ADS  Google Scholar 

  • Marinelli F, Faraldo-Gómez JD (2015) Ensemble-biased metadynamics: a molecular simulation method to sample experimental distributions. Biophys J 108(12):2779–2782

    Article  Google Scholar 

  • Marinelli F, Pietrucci F, Laio A, Piana S (2009) A kinetic model of trp-cage folding from multiple biased molecular dynamics simulations. PLOS Comput Biol 5:1–18. https://doi.org/10.1371/journal.pcbi.1000452

    Article  MathSciNet  Google Scholar 

  • Martoňák R, Laio A, Parrinello M (2003) Predicting crystal structures: the Parrinello-Rahman method revisited. Phys Rev Lett 90:75503

    Article  ADS  Google Scholar 

  • Martoňák R, Laio A, Bernasconi M, Ceriani C, Raiteri P, Parrinello M (2005) Simulation of structural phase transitions by metadynamics. Z Krist 220:489–498

    Google Scholar 

  • Martoňák R, Donadio D, Oganov AR, Parrinello M (2006) Crystal structure transformations in SiO2 from classical and ab initio metadynamics. Nat Mater 5(8):623–626. https://doi.org/10.1038/nmat1696

    Article  ADS  Google Scholar 

  • Martoňák R, Donadio D, Oganov AR, Parrinello M (2007) From four- to six-coordinated silica: transformation pathways from metadynamics. Phys Rev B 76(1):014120. https://doi.org/10.1103/PhysRevB.76.014120

    Article  ADS  Google Scholar 

  • McCarty J, Parrinello M (2017) A variational conformational dynamics approach to the selection of collective variables in metadynamics. J Chem Phys 147(20):204109. https://doi.org/10.1063/1.4998598

    Article  ADS  Google Scholar 

  • Mendels D, McCarty J, Piaggi PM, Parrinello M (2018) Searching for entropically stabilized phases: the case of silver iodide. J Phys Chem C 122(3):1786–1790

    Article  Google Scholar 

  • Merlitz H, Wenzel W (2002) Comparison of stochastic optimization methods for receptor-ligand docking. Chem Phys Lett 362:271–277

    Article  ADS  Google Scholar 

  • Michael M, de Pablo J (2013) A boundary correction algorithm for metadynamics in multiple dimensions. J Chem Phys 139:084102

    Article  ADS  Google Scholar 

  • Micheletti C, Laio A, Parrinello M (2004) Reconstructing the density of states by history-dependent metadynamics. Phys Rev Lett 92:170601

    Article  ADS  Google Scholar 

  • Munro CJ, Hughes ZE, Walsh TR, Knecht MR (2016) Peptide sequence effects control the single pot reduction, nucleation, and growth of au nanoparticles. J Phys Chem C 120(33):18917–18924

    Article  Google Scholar 

  • Oganov A, Martonak R, Laio A, Raiteri P, Parrinello M (2005) Anisotropy of Earth’s D‘’ layer and stacking faults in the MgSiO3 post-perovskite phase. Nature 438(7071):1142–1144. https://doi.org/10.1038/nature04439

    Article  ADS  Google Scholar 

  • Oliveira LF, Fu CD, Pfaendtner J (2018) Density functional tight-binding and infrequent metadynamics can capture entropic effects in intramolecular hydrogen transfer reactions. J Chem Phys 148(15):154101

    Article  ADS  Google Scholar 

  • Palafox-Hernandez JP, Tang Z, Hughes ZE, Li Y, Swihart MT, Prasad PN, Walsh TR, Knecht MR (2014) Comparative study of materials-binding peptide interactions with gold and silver surfaces and nanostructures: a thermodynamic basis for biological selectivity of inorganic materials. Chem Mater 26(17):4960–4969

    Article  Google Scholar 

  • Park S, Pande VS (2007) Choosing weights for simulated tempering. Phys Rev E 76(1):016703

    Article  ADS  Google Scholar 

  • Patey GN, Valleau JP (1975) Monte-carlo method for obtaining interionic potential of mean force in ionic solution. J Chem Phys 63:2334–2339

    Article  ADS  Google Scholar 

  • Pfaendtner J, Bonomi M (2015) Efficient sampling of high-dimensional free-energy landscapes with parallel bias metadynamics. J Chem Theory Comput 11(11):5062–5067

    Article  Google Scholar 

  • Piaggi PM, Valsson O, Parrinello M (2017) Enhancing entropy and enthalpy fluctuations to drive crystallization in atomistic simulations. Phys Rev Lett 119(1):015701

    Article  ADS  Google Scholar 

  • Piana S, Laio A (2007) A bias-exchange approach to protein folding. J Phys Chem B 111(17):4553–4559. https://doi.org/10.1021/jp0678731

    Article  Google Scholar 

  • Pietrucci F, Gerra G, Andreoni W (2010) Cdte surfaces: characterizing dynamical processes with first-principles metadynamics. Appl Phys Lett 97(14):141914

    Article  ADS  Google Scholar 

  • Pitera JW, Chodera JD (2012) On the use of experimental observations to bias simulated ensembles. J Chem Theory Comput 8(10):3445–3451

    Article  Google Scholar 

  • Quigley D, Rodger PM (2008) Metadynamics simulations of ice nucleation and growth. J Comput Phys 128(15):154518. https://doi.org/10.1063/1.2888999

    Google Scholar 

  • Raiteri P, Laio A, Gervasio FL, Micheletti C, Parrinello M (2006) Efficient reconstruction of complex free energy landscapes by multiple walkers metadynamics. J Phys Chem B 110:3533–3539

    Article  Google Scholar 

  • Risken H (1989) The Fokker-Planck equation. Springer

    Book  MATH  Google Scholar 

  • Rohrdanz MA, Zheng W, Clementi C (2013) Discovering mountain passes via torchlight: methods for the definition of reaction coordinates and pathways in complex macromolecular reactions. Ann Rev Phys Chem 64:295–316

    Article  ADS  Google Scholar 

  • Rosso L, Mináry P, Zhu Z, Tuckerman ME (2002) On the use of the adiabatic molecular dynamics technique in the calculation of free energy profiles. J Comput Phys 116(11):4389–4402

    Google Scholar 

  • Salvalaglio M, Tiwary P, Parrinello M (2014) Assessing the reliability of the dynamics reconstructed from metadynamics. J Chem Theor Comput 10(4):1420–1425. https://doi.org/10.1021/ct500040r

    Article  Google Scholar 

  • Salvalaglio M, Perego C, Giberti F, Mazzotti M, Parrinello M (2015) Molecular-dynamics simulations of urea nucleation from aqueous solution. Proc Natl Acad Sci 112(1):E6–E14

    Article  ADS  Google Scholar 

  • Salvalaglio M, Tiwary P, Maggioni GM, Mazzotti M, Parrinello M (2016) Overcoming time scale and finite size limitations to compute nucleation rates from small scale well tempered metadynamics simulations. J Chem Phys 145(21):211925

    Article  ADS  Google Scholar 

  • Sidky H et al, (2018) SSAGES: Software Suite for Advanced General Ensemble Simulations. J Chem Phys 148:044104 https://doi.org/10.1063/1.5008853

    Article  ADS  Google Scholar 

  • Sultan MM, Pande VS (2017) Tica-metadynamics: accelerating metadynamics by using kinetically selected collective variables. J Chem Theory Comput 13(6):2440–2447. https://doi.org/10.1021/acs.jctc.7b00182, pMID:28383914

    Article  Google Scholar 

  • Theodoropoulos C, Qian Y, Kevrekidis IG (2000) Coarse stability and bifurcation analysis using time-steppers: a reaction-diffusion example. Proc Natl Acad Sci USA 97:9840–9843

    Article  ADS  MATH  Google Scholar 

  • Tiwary P (2017) Molecular determinants and bottlenecks in the dissociation dynamics of biotin-streptavidin. J Phys Chem B 121(48):10841–10849. https://doi.org/10.1021/acs.jpcb.7b09510

    Article  Google Scholar 

  • Tiwary P, Berne B (2016a) How wet should be the reaction coordinate for ligand unbinding? J Chem Phys 145(5):054113

    Article  ADS  Google Scholar 

  • Tiwary P, Berne BJ (2016b) Kramers turnover: from energy diffusion to spatial diffusion using metadynamics. J Chem Phys 144(13):134103–134106

    Article  ADS  Google Scholar 

  • Tiwary P, Berne BJ (2016c) Spectral gap optimization of order parameters for sampling complex molecular systems. Proc Natl Acad Sci 113(11):2839–2844. https://doi.org/10.1073/pnas.1600917113

    Article  ADS  Google Scholar 

  • Tiwary P, Berne BJ (2017) Predicting reaction coordinates in energy landscapes with diffusion anisotropy. J Chem Phys 147(15):152701

    Article  ADS  Google Scholar 

  • Tiwary P, Parrinello M (2013) From metadynamics to dynamics. Phys Rev Lett 111:230602–230606. https://doi.org/10.1103/PhysRevLett.111.230602

    Article  ADS  Google Scholar 

  • Tiwary P, Parrinello M (2014) A time-independent free energy estimator for metadynamics. J Phys Chem B 119(3):736–742

    Article  Google Scholar 

  • Tiwary P, van de Walle A (2016) A review of enhanced sampling approaches for accelerated molecular dynamics. In: Multiscale materials modeling for nanomechanics. Springer, New York pp 195–221. https://doi.org/10.1007/978-3-319-33480-6_6

    Google Scholar 

  • Tiwary P, Dama JF, Parrinello M (2015a) A perturbative solution to metadynamics ordinary differential equation. J Chem Phys 143(23):234112

    Article  ADS  Google Scholar 

  • Tiwary P, Mondal J, Morrone JA, Berne B (2015b) Role of water and steric constraints in the kinetics of cavity–ligand unbinding. Proc Natl Acad Sci 112(39):12015–12019

    Article  ADS  Google Scholar 

  • Tribello GA, Ceriotti M, Parrinello M (2012) Using sketch-map coordinates to analyze and bias molecular dynamics simulations. Proc Natl Acad Sci 109(14):5196–5201

    Article  ADS  Google Scholar 

  • Tribello GA, Bonomi M, Branduardi D, Camilloni C, Bussi G (2014) Plumed 2: new feathers for an old bird. Comput Phys Commun 185(2):604–613

    Article  ADS  Google Scholar 

  • Trudu F, Donadio D, Parrinello M (2006) Freezing of a Lennard-Jones fluid: from nucleation to spinodal regime. Phys Rev Lett 97(10):105701. https://doi.org/10.1103/PhysRevLett.97.105701

    Article  ADS  Google Scholar 

  • Valsson O, Tiwary P, Parrinello M (2016) Enhancing important fluctuations: rare events and metadynamics from a conceptual viewpoint. Ann Rev Phys Chem 67(1):159–184

    Article  ADS  Google Scholar 

  • VandeVondele J, Rothlisberger U (2002) Canonical adiabatic free energy sampling (cafes): a novel method for the exploration of free energy surfaces. J Phys Chem B 106(1):203–208

    Article  Google Scholar 

  • Vartak S, Roudgar A, Golovnev A, Eikerling M (2013) Collective proton dynamics at highly charged interfaces studied by ab initio metadynamics. J Phys Chem B 117(2):583–588

    Article  Google Scholar 

  • Voter AF (1997) Hyperdynamics: accelerated molecular dynamics of infrequent events. Phys Rev Lett 78:3908–3911. https://doi.org/10.1103/PhysRevLett.78.3908

    Article  ADS  Google Scholar 

  • Wang F, Landau DP (2001) Efficient, multiple-range random walk algorithm to calculate the density of states. Phys Rev Lett 86:2050

    Article  ADS  Google Scholar 

  • White AD, Dama JF, Voth GA (2015) Designing free energy surfaces that match experimental data with metadynamics. J Chem Theory Comput 11(6):2451–2460

    Article  Google Scholar 

  • Zipoli F, Bernasconi M, Martoňák R (2004) Constant pressure reactive molecular dynamics simulations of phase transitions under pressure: the graphite to diamond conversion revisited. Eur Phys J B 39:41–47

    Article  ADS  Google Scholar 

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Bussi, G., Laio, A., Tiwary, P. (2018). Metadynamics: A Unified Framework for Accelerating Rare Events and Sampling Thermodynamics and Kinetics. In: Andreoni, W., Yip, S. (eds) Handbook of Materials Modeling . Springer, Cham. https://doi.org/10.1007/978-3-319-42913-7_49-1

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