Off-Lattice Kinetic Monte Carlo Methods

Reference work entry


Exact modeling of the dynamics of chemical and material systems over experimentally relevant time scales still eludes us even with modern computational resources. Fortunately, many systems can be described as rare event systems where atoms vibrate around equilibrium positions for a long time before a transition is made to a new atomic state. For those systems, the kinetic Monte Carlo (KMC) algorithm provides a powerful solution. In traditional KMC, mechanism and rates are computed beforehand, limiting moves to discretized positions and largely ignoring strain. Many systems of interest, however, are not well-represented by such lattice-based models. Moreover, materials often evolve with complex and concerted mechanisms that cannot be anticipated before the start of a simulation. In this chapter, we describe a class of algorithms, called off-lattice or adaptive KMC, which relaxes both limitations of traditional KMC, with atomic configurations represented in the full configuration space and reaction events are calculated on-the-fly, with the possible use of catalogs to speed up calculations. We discuss a number of implementations of off-lattice KMC developed by different research groups, emphasizing the similarities between the approaches that open modeling to new classes of problems.



This work was supported in part by a grant from the Natural Science and Engineering Research Council of Canada. MT and NM are grateful to Calcul Québec and Compute Canada for providing extensive computer time and computer access. The work in Austin was supported by the National Science Foundation (CHE-0645497, CHE-1152342, and CHE-1534177) and the Welch Foundation (F-1841). Sustained computational resources have been provided by the Texas Advanced Computing Center.


  1. Alexander KC, Schuh CA (2016) Towards the reliable calculation of residence time for off-lattice kinetic Monte Carlo simulations. Model Simul Mater Sci Eng 24(6):65014.,
  2. Althorpe S, Angulo G, Astumian RD, Beniwal V, Bolhuis PG, Brandão J, Ellis J, Fang W, Glowacki DR, Hammes-Schiffer S et al (2016) Application to large systems: general discussion. Faraday Discuss 195:671–698ADSCrossRefGoogle Scholar
  3. Barkema GT, Mousseau N (1996) Event-based relaxation of continuous disordered systems. Phys Rev Lett 77(21):4358–4361ADSCrossRefGoogle Scholar
  4. Béland LK, Mousseau N (2013) Long-time relaxation of ion-bombarded silicon studied with the kinetic activation-relaxation technique: microscopic description of slow aging in a disordered system. Phys Rev B 88(21):214201ADSCrossRefGoogle Scholar
  5. Béland LK, Brommer P, El-Mellouhi F, Joly JF, Mousseau N (2011) Kinetic activation-relaxation technique. Phys Rev E 84(4):046704. ADSCrossRefGoogle Scholar
  6. Béland LK, Anahory Y, Smeets D, Guihard M, Brommer P, Joly JFF, Pothier JcC, Lewis LJ, Mousseau N, Schiettekatte F, Postale C, Centre-ville S (2013) Replenish and relax: explaining logarithmic annealing in ion-implanted c-Si. Phys Rev Lett 111(10):105502–105506.,
  7. Béland LK, Osetsky YN, Stoller RE, Xu H (2015a) Interstitial loop transformations in FeCr. J Alloys Compd 640:219–225CrossRefGoogle Scholar
  8. Béland LK, Osetsky YN, Stoller RE, Xu H (2015b) Slow relaxation of cascade-induced defects in Fe. Phys Rev B 91(5):054108ADSCrossRefGoogle Scholar
  9. Béland LK, Samolyuk GD, Stoller RE (2016) Differences in the accumulation of ion-beam damage in Ni and NiFe explained by atomistic simulations. J Alloys Compd 662:415–420CrossRefGoogle Scholar
  10. Boulougouris GC, Frenkel D (2005) Monte Carlo sampling of a Markov web. J Chem Theory Comput 1:389–393CrossRefGoogle Scholar
  11. Boulougouris GC, Theodorou DN (2007) Dynamical integration of a Markovian web: a first passage time approach. J Chem Phys 127:084903ADSCrossRefGoogle Scholar
  12. Brommer P, Béland LK, Joly JF, Mousseau N (2014) Understanding long-time vacancy aggregation in iron: a kinetic activation-relaxation technique study. Phys Rev B 90(13):134109–134117. ADSCrossRefGoogle Scholar
  13. Chill ST, Henkelman G (2014) Molecular dynamics saddle search adaptive kinetic Monte Carlo. J Chem Phys 140:214110ADSCrossRefGoogle Scholar
  14. Chill ST, Stevenson J, Ruhle V, Shang C, Xiao P, Farrell J, Wales D, Henkelman G (2014a) Benchmarks for characterization of minima, transition states and pathways in atomic systems. J Chem Theory Comput 10:5476–5482CrossRefGoogle Scholar
  15. Chill ST, Welborn M, Terrell R, Zhang L, Berthet JC, Pedersen A, Jónsson H, Henkelman G (2014b) Eon: software for long time scale simulations of atomic scale systems. Model Simul Mater Sci Eng 22:055002ADSCrossRefGoogle Scholar
  16. Duncan J, Harjunmaa A, Terrell R, Drautz R, Henkelman G, Rogal J (2016) Collective atomic displacements during complex phase boundary migration in solid-solid phase transformations. Phys Rev Lett 116(3):035701ADSCrossRefGoogle Scholar
  17. El-Mellouhi F, Mousseau N, Lewis L (2008) Kinetic activation-relaxation technique: an off-lattice self-learning kinetic Monte Carlo algorithm. Phys Rev B 78(15):153202. ADSCrossRefGoogle Scholar
  18. Faken D, Jónsson H (1994) Systematic analysis of local atomic structure combined with 3D computer graphics. Comput Mater Sci 2:279–286CrossRefGoogle Scholar
  19. Fichthorn KA, Lin Y (2013) A local superbasin kinetic Monte Carlo method. J Chem Phys 138(16):164104ADSCrossRefGoogle Scholar
  20. Guteŕrez M, Argaéz C, Jónsson H (2016) Improved minimum mode following method for finding first order saddle points. J Chem Theory Comput 13:125–134CrossRefGoogle Scholar
  21. Henkelman G, Jónsson H (1999) A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives. J Chem Phys 111:7010–7022ADSCrossRefGoogle Scholar
  22. Henkelman G, Jónsson H (2000) Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points. J Chem Phys 113:9978–9985ADSCrossRefGoogle Scholar
  23. Henkelman G, Jónsson H (2001) Long time scale kinetic Monte Carlo simulations without lattice approximation and predefined event table. J Chem Phys 115(21):9657–9666. ADSCrossRefGoogle Scholar
  24. Henkelman G, Uberuaga BP, Jónsson H (2000) A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J Chem Phys 113:9901–9904ADSCrossRefGoogle Scholar
  25. Jay A, Raine M, Richard N, Mousseau N, Goiffon V, Hemeryck A, Magnan P (2017) Simulation of single particle displacement damage in silicon part II: generation and long time relaxation of damage structure. IEEE Trans Nucl Sci 64(1):141–148.,
  26. Joly JF, Béland LK, Brommer P, El-Mellouhi F, Mousseau N (2012) Optimization of the kinetic activation-relaxation technique, an off-lattice and self-learning kinetic Monte-Carlo method. J Phys Conf Ser 341:012007.,
  27. Joly JF, Béland LK, Brommer P, Mousseau N (2013) Contribution of vacancies to relaxation in amorphous materials: a kinetic activation-relaxation technique study. Phys Rev B 87(14):144204. ADSCrossRefGoogle Scholar
  28. Jónsson H, Mills G, Jacobsen KW (1998) Nudged elastic band method for finding minimum energy paths of transitions. In: Berne BJ, Ciccotti G, Coker DF (eds) Classical and quantum dynamics in condensed phase simulations. World Scientific, Singapore, pp 385–404CrossRefGoogle Scholar
  29. Kim WK, Tadmor EB (2014) Entropically stabilized dislocations. Phys Rev Lett 112(10):105501ADSCrossRefGoogle Scholar
  30. Koziatek P, Barrat JL, Derlet P, Rodney D (2013) Inverse Meyer-Neldel behavior for activated processes in model glasses. Phys Rev B 87:224105. ADSCrossRefGoogle Scholar
  31. Lu C, Jin K, Béland LK, Zhang F, Yang T, Qiao L, Zhang Y, Bei H, Christen HM, Stoller RE et al (2016) Direct observation of defect range and evolution in ion-irradiated single crystalline Ni and Ni binary alloys. Sci Rep 6:19994ADSCrossRefGoogle Scholar
  32. Plimpton S (1995) Fast parallel algorithms for short-range molecular dynamics. J Comput Phys 117:1–19ADSzbMATHCrossRefGoogle Scholar
  33. Machado-Charry E, Béland LK, Caliste D, Genovese L, Deutsch T, Mousseau N, Pochet P (2011) Optimized energy landscape exploration using the ab initio based activation-relaxation technique. J Chem Phys 135(3):034102.,
  34. Mahmoud S, Trochet M, Restrepo OA, Mousseau N (2018) Study of point defects diffusion in nickel using kinetic activation-relaxation technique. Acta Mater 144:679–690.,
  35. Malek R, Mousseau N (2000) Dynamics of Lennard-Jones clusters: a characterization of the activation-relaxation technique. Phys Rev E 62(6):7723–7728. ADSCrossRefGoogle Scholar
  36. Marinica MC, Willaime F, Mousseau N (2011) Energy landscape of small clusters of self-interstitial dumbbells in iron. Phys Rev B 83(9):094119. ADSCrossRefGoogle Scholar
  37. Martínez E, Marian J, Kalos MH, Perlado JM (2008) Synchronous parallel kinetic Monte Carlo for continuum diffusion-reaction systems. J Comput Phys 227(8):3804–3823ADSMathSciNetzbMATHCrossRefGoogle Scholar
  38. McKay BD, Piperno A (2014) Practical graph isomorphism, II. J Symb Comput 60:94–112.,
  39. McKay BD et al (1981) Practical graph isomorphism. Congr Numer 30:45–87MathSciNetzbMATHGoogle Scholar
  40. Mousseau N, Barkema GT (1998b) Traveling through potential energy landscapes of disordered materials: the activation-relaxation technique. Phys Rev E 57:2419–2424. ADSCrossRefGoogle Scholar
  41. Munro LJ, Wales DJ (1999) Defect migration in crystalline silicon. Phys Rev B 59:3969ADSCrossRefGoogle Scholar
  42. Nocedal J (1980) Updating quasi-Newton matrices with limited storage. Math Comput 35:773–782MathSciNetzbMATHCrossRefGoogle Scholar
  43. Novotny MA (1995) Monte Carlo algorithms with absorbing Markov chains: fast local algorithms for slow dynamics. Phys Rev Lett 74:1–5ADSCrossRefGoogle Scholar
  44. Novotny MA (2001) A tutorial on advanced dynamic monte carlo methods for systems with discrete state spaces. In: Stauffer D (ed) Annual reviews of computational physics IX. World Scientific, Singapore, pp 153–210CrossRefGoogle Scholar
  45. Ojifinni RA, Froemming NS, Gong J, Pan M, Kim TS, White J, Henkelman G, Mullins CB (2008) Water-enhanced low-temperature CO oxidation and isotope effects on atomic oxygen-covered Au (111). J Am Chem Soc 130(21):6801–6812CrossRefGoogle Scholar
  46. Osetsky YN, Béland LK, Stoller RE (2016) Specific features of defect and mass transport in concentrated FCC alloys. Acta Mater 115:364–371CrossRefGoogle Scholar
  47. Pedersen A, Jónsson H (2009) Simulations of hydrogen diffusion at grain boundaries in aluminum. Acta Mater 57:4036–4045CrossRefGoogle Scholar
  48. Pedersen A, Luiser M (2014) Bowl breakout: escaping the positive region when searching for saddle points. J Chem Phys 141(2):024109ADSCrossRefGoogle Scholar
  49. Pedersen A, Henkelman G, Schiøtz J, Jónsson H (2009) Long time scale simulation of a grain boundary in copper. New J Phys 11:073034CrossRefGoogle Scholar
  50. Pedersen A, Berthet JC, Jónsson H (2012) Simulated annealing with coarse graining and distributed computing. Lect Notes Comput Sci 7134:34–44CrossRefGoogle Scholar
  51. Perez D, Luo SN, Voter AF, Germann TC (2013) Entropic stabilization of nanoscale voids in materials under tension. Phys Rev Lett 110(20):206001ADSCrossRefGoogle Scholar
  52. Puchala B, Falk ML, Garikipati K (2010) An energy basin finding algorithm for kinetic Monte Carlo acceleration. J Chem Phys 132(13):134104.,
  53. Raine M, Jay A, Richard N, Goiffon V, Girard S, Member S, Gaillardin M, Paillet P, Member S (2017) Simulation of single particle displacement damage in silicon part I: global approach and primary interaction simulation. IEEE Trans Nucl Sci 64(1):133–140.,
  54. Restrepo OA, Mousseau N, El-Mellouhi F, Bouhali O, Trochet M, Becquart CS (2016) Diffusion properties of Fe-C systems studied by using kinetic activation-relaxation technique. Comput Mater Sci 112:96–106.,,
  55. Restrepo OA, Becquart CS, El-Mellouhi F, Bouhali O, Mousseau N (2017) Diffusion mechanisms of C in 100, 110 and 111 Fe surfaces studied using kinetic activation-relaxation technique. Acta Mater 136:303–314.,
  56. Shim Y, Amar JG (2005) Semirigorous synchronous sublattice algorithm for parallel kinetic Monte Carlo simulations of thin film growth. Phys Rev B 71:125432ADSCrossRefGoogle Scholar
  57. Shim Y, Amar JG (2006) Hybrid asynchronous algorithm for parallel kinetic Monte Carlo simulations of thin film growth. J Comput Phys 212(1):305–317ADSzbMATHCrossRefGoogle Scholar
  58. Sinha AK (1972) Topologically close-packed structures of transition metal alloys. Prog Mat Sci 15:81CrossRefGoogle Scholar
  59. Sørensen MR, Voter AF (2000) Temperature-accelerated dynamics for simulation of infrequent events. J Chem Phys 112:9599–9606ADSCrossRefGoogle Scholar
  60. Sørensen MR, Jacobsen KW, Jónsson H (1996) Thermal diffusion processes in metal-tip-surface interactions: contact formation and adatom mobility. Phys Rev Lett 77:5067–5070ADSCrossRefGoogle Scholar
  61. Terentyev D, Malerba L, Klaver P, Olsson P (2008) Formation of stable sessile interstitial complexes in reactions between glissile dislocation loops in BCC Fe. J Nucl Mater 382(2):126–133ADSCrossRefGoogle Scholar
  62. Terrell R, Welborn M, Chill ST, Henkelman G (2012) Database of atomistic reaction mechanisms with application to kinetic Monte Carlo. J Chem Phys 137:014105ADSCrossRefGoogle Scholar
  63. Trochet M, Mousseau N (2017) Energy landscape and diffusion kinetics of lithiated silicon: a kinetic activation-relaxation technique study. Phys Rev B 96(13):134118. ADSCrossRefGoogle Scholar
  64. Trochet M, Béland LK, Joly JF, Brommer P, Mousseau N (2015) Diffusion of point defects in crystalline silicon using the kinetic activation-relaxation technique method. Phys Rev B 91(22):224106. ADSCrossRefGoogle Scholar
  65. Trochet M, Sauvé-Lacoursière A, Mousseau N (2017) Algorithmic developments of the kinetic activation-relaxation technique: accessing long-time kinetics of larger and more complex systems. J Chem Phys 147(15):152712. ADSCrossRefGoogle Scholar
  66. Trushin O, Karim A, Kara A, Rahman TS (2005) Self-learning kinetic Monte Carlo method: application to Cu(111). Phys Rev B 72(11):115401. ADSCrossRefGoogle Scholar
  67. Valiquette F, Mousseau N (2003) Energy landscape of relaxed amorphous silicon. Phys Rev B 68:125209. ADSCrossRefGoogle Scholar
  68. Vernon LJ (2010) Modelling the growth of TiO2. Ph.D. thesis, Loughborough UniversityGoogle Scholar
  69. Vernon LJ (2012) PESTO: potential energy surface tools.
  70. Vernon L, Kenny SD, Smith R, Sanville E (2011) Growth mechanisms for TiO2 at its rutile (110) surface. Phys Rev B 83(7):75412. ADSCrossRefGoogle Scholar
  71. Wales DJ (2002) Discrete path sampling. Mol Phys 100:3285–3305ADSCrossRefGoogle Scholar
  72. Xiao P, Wu Q, Henkelman G (2014) Basin constrained κ-dimer method for saddle point finding. J Chem Phys 141:164111ADSCrossRefGoogle Scholar
  73. Xu H, Osetsky YN, Stoller RE (2011) Simulating complex atomistic processes: on-the-fly kinetic Monte Carlo scheme with selective active volumes. Phys Rev B 84(13):132103. ADSCrossRefGoogle Scholar
  74. Xu H, Stoller RE, Osetsky YN, Terentyev D et al (2013) Solving the puzzle of <100> interstitial loop formation in BCC iron. Phys Rev Lett 110(26):265503ADSCrossRefGoogle Scholar
  75. Xu H, Stoller RE, Béland LK, Osetsky YN (2015) Self-evolving atomistic kinetic Monte Carlo simulations of defects in materials. Comput Mater Sci 100:135–143CrossRefGoogle Scholar
  76. Xu L, Mei DH, Henkelman G (2009) Adaptive kinetic Monte Carlo simulation of methanol decomposition on Cu(100). J Chem Phys 131:244520ADSCrossRefGoogle Scholar
  77. Zeng Y, Xiao P, Henkelman G (2014) Unification of algorithms for minimum mode optimization. J Chem Phys 140:044115ADSCrossRefGoogle Scholar
  78. Zhou XW, Wadley HNG, Johnson RA, Larson DJ, Tabat N, Cerezo A, Petford-Long AK, Smith GDW, Clifton PH, Martens RL, Kelly TF (2001) Atomic scale structure of sputtered metal multilayers. Acta Mater 49:4005–4015CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Departement de physique and Regroupement québécois sur les matériaux de pointeUniversité de MontréalMontréalCanada
  2. 2.Departement de physique and Regroupement québécois sur les matériaux de pointeUniversité de MontréalMontréalCanada
  3. 3.Department of PhysicsUniversity de MontréalMontréalCanada
  4. 4.Department of Mechanical and Materials EngineeringQueen’s UniversityKingstonCanada
  5. 5.Department of Chemistry and the Institute for Computational and Engineering SciencesUniversity of Texas at AustinAustinUSA
  6. 6.Department of Chemistry and BiochemistryUniversity of Texas at AustinAustinUSA

Personalised recommendations