Exploring Potential Energy Surfaces with Saddle Point Searches

Reference work entry


The energy surface of an atomic scale representation of a material contains the essential information needed to determine the structure and time evolution of the system at a given temperature. Local minima on the surface represent (meta)stable states of the system, while first-order saddle points characterize the mechanisms of transitions between states. While many well-known methods make it relatively easy to find local minima, the identification of saddle points is more challenging. In this chapter, methods for finding saddle points are discussed as well as applications to materials simulations. Both doubly constrained search methods, where the final and the initial state minima are specified, and singly constrained search methods, where only the initial state is specified, are discussed. The focus is on a classical description of the atom coordinates, but saddle points corresponding to quantum mechanical tunneling are also mentioned. An extension to magnetic systems where the energy surface depends on the orientation of the magnetic vectors is sketched.



This work was supported in part by the Icelandic Research Fund (grant 185405-051) and by the Academy of Finland (grant 278260). V.Á. acknowledges support from a Doctoral Grant of the University of Iceland Research Fund.


  1. Andersen HC (1980) Molecular dynamics simulations at constant pressure and/or temperature. J Chem Phys 72(4):2384–2393ADSCrossRefGoogle Scholar
  2. Andersson S, Nyman G, Arnaldsson A, Manthe U, Jónsson H (2009) Comparison of quantum dynamics and quantum transition state theory estimates of the H + CH4 reaction rate. J Phys Chem A 113:4468CrossRefGoogle Scholar
  3. Antropov VP, Katsnelson MI, Harmon BN, van Schilfgaarde M, Kusnezov D (1996) Spin dynamics in magnets: equation of motion and finite temperature effects. Phys Rev B 54:1019ADSCrossRefGoogle Scholar
  4. Ásgeirsson V, Jónsson H (2018) Efficient evaluation of atom tunneling combined with electronic structure calculations. J Chem Phys 148:102334ADSCrossRefGoogle Scholar
  5. Benderskii VA, Makarov DE, Wight CA (1994) Chemical dynamics at low temperatures. Adv Chem Phys 88:1Google Scholar
  6. Bessarab PF, Uzdin VM, Jónsson H (2012) Harmonic transition state theory of thermal spin transitions. Phys Rev B 85:184409ADSCrossRefGoogle Scholar
  7. Bessarab PF, Uzdin VM, Jónsson H (2013) Potential energy surfaces and rates of spin transitions. Zeitschrift für Physikalische Chemie 227:1543Google Scholar
  8. Bessarab PF, Uzdin VM, Jónsson H (2015) Method for finding mechanism and activation energy of magnetic transitions, applied to skyrmion and antivortex annihilation. Comput Phys Commun 196:335ADSCrossRefGoogle Scholar
  9. Bitzek E, Koskinen P, G’́ahler F, Moseler M, Gumbasch P (2006) Structural relaxation made simple. Phys Rev Lett 97(17):170201Google Scholar
  10. Bligaard T, Jónsson H (2005) Optimization of hyperplanar transition states: application to 2D test problems. Comput Phys Commun 169:284ADSCrossRefGoogle Scholar
  11. Bohner MU, Meisner J, Kastner J (2013) A quadratically-converging nudged elastic band optimizer. J Chem Theory Comput 9(8):3498–3504CrossRefGoogle Scholar
  12. Braun H-B (2012) Topological effects in nanomagnetism: from superparamagnetism to chiral quantum solitons. Adv Phys 61(1):1–116ADSCrossRefGoogle Scholar
  13. Chill ST, Welborn M, Terrell R, Zhang L, Berthet J-C, Pedersen A, Jónsson H, Henkelman G (2014a) EON: software for long time simulations of atomic scale systems. Model Simul Mater Sci Eng 22:055002ADSCrossRefGoogle Scholar
  14. Chill ST, Stevenson J, Ruehle V, Cheng S, Xiao P, Farrel JD, Wales DJ, Henkelman G (2014b) Benchmarks for characterization of minima, transition states, and pathways in atomic, molecular, and condensed matter systems. J Chem Theory Comput 10:5476–5482CrossRefGoogle Scholar
  15. Chu J-W, Trout BL, Brooks BR (2003) A super-linear minimization scheme for the nudged elastic band method. J Chem Phys 119(24):12708–12717ADSCrossRefGoogle Scholar
  16. Ciccotti G, Ferrario M, Laria D, Kapral R (1995) Simulation of classical and quantum activated processes in the condensed phase. In: Reatto L, Manghi F (ed) Progress in computational physics of matter: methods, software and applications. World Scientific, Singapore, p 150CrossRefGoogle Scholar
  17. Einarsdóttir DM, Arnaldsson A, Óskarsson F, Jónsson H (2012) Path optimization with application to tunneling. Lect Notes Comput Sci 7134:45CrossRefGoogle Scholar
  18. Eyring H (1935) The activated complex in chemical reactions. J Chem Phys 3:107ADSCrossRefGoogle Scholar
  19. Feynman RP, Hibbs AR (1965) Quantum mechanics and path integrals. McGraw-Hill, New YorkzbMATHGoogle Scholar
  20. Gillan MJ (1987) Quantum-classical crossover of the transition rate in the damped double well. Phys C Solid State Phys 20(24):3621–3641ADSCrossRefGoogle Scholar
  21. Goumans TPM, Catlow CRA, Brown WA, Kästner J, Sherwood P (2009) An embedded cluster study of the formation of water on interstellar dust grains. Phys Chem Chem Phys 11(26):5431–5436CrossRefGoogle Scholar
  22. Gutiérrez MP, Argáez C, Jónsson H (2017) Improved minimum mode following method for finding first order saddle points. J Chem Theory Comput 13(1):125–134CrossRefGoogle Scholar
  23. Hele TJH, Althorpe SC (2013) Derivation of a true (t → 0+) quantum transition-state theory. I. Uniqueness and equivalence to ring-polymer molecular dynamics transition-state-theory. J Chem Phys 138:084108Google Scholar
  24. Henkelman G, Jónsson H (1999) A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives. J Chem Phys 111:7010ADSCrossRefGoogle Scholar
  25. Henkelman G, Jónsson H (2000) Improved tangent estimate in the NEB method for finding minimum energy paths and saddle points. J Chem Phys 113(22):9978–9985 [Note: There is a typographical error in the Appendix, 2Vi+1 − Vi should be − 2(Vi+1 − Vi)]Google Scholar
  26. Henkelman G, Jónsson H (2001) Theoretical calculations of dissociative adsorption of methane on an Ir(111) surface. Phys Rev Lett 86:664ADSCrossRefGoogle Scholar
  27. Henkelman G, Uberuaga BP, Jónsson H (2000a) A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J Chem Phys 113(22):9901–9904ADSCrossRefGoogle Scholar
  28. Henkelman G, Jóhannesson G, Jónsson H (2000b) Theoretical methods in condensed phase chemistry, methods for finding saddle points and minimum energy paths. In: Schwartz SD (ed) Progress in theoretical chemistry and physics. Kluwer Academic Publishers, Dordrecht, pp 269–302Google Scholar
  29. Henkelman G, Arnaldsson A, Jónsson H (2006) Theoretical calculations of CH4 and H2 associative desorption from Ni(111): could subsurface hydrogen play an important role? J Chem Phys 124:044706ADSCrossRefGoogle Scholar
  30. Jóhannesson GH, Jónsson H (2001) Optimization of hyperplanar transition states. J Chem Phys 115:9644ADSCrossRefGoogle Scholar
  31. Jónsson H, Mills G, Jacobsen KW (1998) Nudged elastic band method for finding minimum energy paths of transitions. In: Berne BJ (ed) Classical and quantum dynamics in condensed phase simulations. World Scientific, Singapore, pp 385–404CrossRefGoogle Scholar
  32. Justo JF, Bazant MZ, Kaxiras E, Bulatov VV, Yip S (1998) Interatomic potential for silicon defects and disordered phases. Phys Rev B 58:2539ADSCrossRefGoogle Scholar
  33. Keck JC (1967) Variational theory of reaction rates. J Chem Phys 13:85Google Scholar
  34. Koistinen O-P, Maras E, Vehtari A, Jónsson H (2016) Minimum energy path calculations with Gaussian process regression. Nanosyst Phys Chem Math 7:925zbMATHCrossRefGoogle Scholar
  35. Koistinen O-P, Dabgjartsdóttir F, Ásgeirsson V, Vehtari A, Jónsson H (2017) Nudged elastic band calculations accelerated with gaussian process regression. J Chem Phys 147(15): 152720ADSCrossRefGoogle Scholar
  36. Kolsbjerg EL, Groves MN, Hammer B (2016) An automated nudged elastic band. J Chem Phys 145(9):094107ADSCrossRefGoogle Scholar
  37. Malek R, Mousseau N (2000) Dynamics of Lennard-Jones clusters: a characterization of the activation-relaxation technique. Phys Rev E 62(6):7723–7728ADSCrossRefGoogle Scholar
  38. Maras E, Trushin O, Stukowski A, Ala-Nissila T, Jónsson H (2016) Global transition path search for dislocation formation in Ge on Si(001). Comput Phys Commun 205:13ADSCrossRefGoogle Scholar
  39. Maras E, Pizzagalli L, Ala-Nissila T, Jónsson H (2017) Atomic scale formation mechanism of edge dislocation relieving lattice strain in a GeSi overlayer on Si(001). Sci Rep 7:11966ADSCrossRefGoogle Scholar
  40. Maronsson JB, Jónsson H, Vegge T (2012) A method for finding the ridge between saddle points applied to rare event rate estimates. Phys Chem Chem Phys 14:2884CrossRefGoogle Scholar
  41. Melander M, Laasonen K, Jónsson H (2015) Removing external degrees of freedom from transition-state search methods using quaternions. J Chem Theory Comput 11(3): 1055–1062CrossRefGoogle Scholar
  42. Mills G, Jónsson H, Schenter GK (1995) Reversible work based transition state theory: application to H2 dissociative adsorption. Surf Sci 324:305–337ADSCrossRefGoogle Scholar
  43. Mills G, Schenter GK, Makarov DE, Jónsson H (1997) Generalized path integral based quantum transition state theory. Chem Phys Lett 278:91ADSCrossRefGoogle Scholar
  44. Mills G, Schenter GK, Makarov DE, Jónsson H (1998) RAW quantum transition state theory. In: Berne BJ, Ciccotti G, Coker DF (eds) Classical and quantum dynamics in condensed phase simulations. World Scientific, Singapore, pp 405–421CrossRefGoogle Scholar
  45. Müller GP, Bessarab PF, Vlasov FM, Lux F, Kiselev NS, Blügel S, Uzdin VM, Jónsson H (2018) Duplication, collapse and escape of magnetic skyrmions revealed using a systematic saddle point search method. Phys. Rev. Lett. 121:197202CrossRefGoogle Scholar
  46. Munro LJ, Wales DJ (1999) Defect migration in crystalline silicon. Phys Rev B 59:3969ADSCrossRefGoogle Scholar
  47. Nocedal J (1980) Updating quasi-Newton matrices with limited storage. Math Comput 35(151):773–782MathSciNetzbMATHCrossRefGoogle Scholar
  48. Olsen RA, Kroes G-J, Henkelman G, Arnaldsson A, Jónsson H (2004) Comparison of methods for finding saddle points without knowledge of the final states. J Chem Phys 121(20):9776–9792ADSCrossRefGoogle Scholar
  49. Pedersen A, Jónsson H (2009) Simulations of hydrogen diffusion at grain boundaries in aluminum. Acta Materialia 57:4036CrossRefGoogle Scholar
  50. Pedersen A, Pizzagalli L, Jónsson H (2009a) Finding mechanism of transitions in complex systems: formation and migration of dislocation kinks in a silicon crystal. J Phys Condens Matter 21:084210ADSCrossRefGoogle Scholar
  51. Pedersen A, Henkelman G, Schioetz J, Jónsson H (2009b) Long timescale simulation of a grain boundary in copper. New J Phys 11:073034CrossRefGoogle Scholar
  52. Pedersen A, Berthet J-C, Jónsson H (2012) Simulated annealing with coarse graining and distributed computing. Lect Notes Comput Sci 7134:34CrossRefGoogle Scholar
  53. Pedersen A, Wikfeldt KT, Karssemeijer LJ, Cuppen HM, Jónsson H (2014) Molecular reordering processes on ice (0001) surfaces from long timescale simulations. J Chem Phys 141:234706ADSCrossRefGoogle Scholar
  54. Pedersen A, Karssemeijer LJ, Cuppen HM, Jónsson H (2015) Long-timescale simulations of H2O admolecule diffusion on Ice Ih(0001) surfaces. J Phys Chem C 119:16528CrossRefGoogle Scholar
  55. Peters B (2017) Reaction rate theory and rare events. Elsevier Science & Technology, AmsterdamGoogle Scholar
  56. Peterson AA (2016) Acceleration of saddle-point searches with machine learning. J Chem Phys 145(7):074106ADSCrossRefGoogle Scholar
  57. Plasencia M, Pedersen A, Arnaldsson A, Berthet J-C, Jónsson H (2014) Geothermal model calibration using a global minimization algorithm based on finding saddle points as well as minima of the objective function. Comput Geosci 65:110ADSCrossRefGoogle Scholar
  58. Richardson JO (2016) J Chem Phys 144:114106ADSCrossRefGoogle Scholar
  59. Richardson JO, Althorpe SC (2009) Ring-polymer molecular dynamics rate-theory in the deep-tunneling regime: connection with semiclassical instanton theory. J Chem Phys 131:214106ADSCrossRefGoogle Scholar
  60. Rommel JB, Kästner J (2011) Adaptive integration grids in instanton theory improve the numerical accuracy at low temperature. J Chem Phys 134:184107ADSCrossRefGoogle Scholar
  61. Sheppard D, Terrell R, Henkelman G (2008) Optimization methods for finding minimum energy paths. J Chem Phys 128(13):134106ADSCrossRefGoogle Scholar
  62. Sheppard D, Xiao P, Chemelewski W, Johnson DD, Henkelman G (2012) A generalized solid-state nudged elastic band method. J Chem Phys 136(7):074103ADSCrossRefGoogle Scholar
  63. Smidstrup S, Pedersen A, Stokbro K, Jónsson H (2014) Improved initial guess for minimum energy path calculations. J Chem Phys 140(21):214106ADSCrossRefGoogle Scholar
  64. Srensen MR, Jacobsen KW, Jónsson H (1996) Thermal diffusion processes in metal tip-surface interactions: contact formation and adatom mobility. Phys Rev Letters 77(25):5067–5070ADSCrossRefGoogle Scholar
  65. Trygubenko SA, Wales DJ (2004) A doubly nudged elastic band method for finding transition states. J Chem Theory Comput 120(5):2082–2094Google Scholar
  66. Uzdin VM, Potkina MN, Lobanov IS, Bessarab PF, Jónsson H (2018) Energy surface and lifetime of magnetic skyrmions. J Magn Magn Mater 459:236–240ADSCrossRefGoogle Scholar
  67. Vineyard GH (1957) Frequency factors and isotope effects in solid state rate processes. J Phys Chem Solids 3:121ADSCrossRefGoogle Scholar
  68. Voter AF, Doll JD (1985) Dynamical corrections to transition state theory for multistate systems: surface self-diffusion in the rare-event regime. J Chem Phys 82(1):80–92ADSCrossRefGoogle Scholar
  69. Wigner E (1938) The transition state method. Trans Faraday Soc 34:29CrossRefGoogle Scholar
  70. Weinan E, Weiqing R, Vanden-Eijnden E (2002) String method for the study of rare events. Phys Rev B 66(4):052301ADSzbMATHGoogle Scholar
  71. Zarkevich NA, Johnson DD (2015) Nudged-elastic band method with two climbing images: finding transition states in complex energy landscapes. J Chem Phys 142(2):024106ADSCrossRefGoogle Scholar
  72. Zhang J, Zhang H, Ye H, Zheng Y (2016) Free-end adaptive nudged elastic band method for locating transition states in minimum energy path calculation. J Chem Phys 145(9):094104ADSCrossRefGoogle Scholar
  73. Zhu T, Li J, Samanta A, Kim HG, Suresh S (2007) Interfacial plasticity governs strain rate sensitivity and ductility in nanostructured metals. Proc Natl Acad Sci USA 104(9):3031–3036ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Science Institute of the University of IcelandReykjavíkIceland
  2. 2.Faculty of Physical SciencesUniversity of IcelandReykjavíkIceland
  3. 3.Department of Energy Conversion and StorageTechnical University of DenmarkLyngbyDenmark

Personalised recommendations