Multiscale Modelling in Computational Heterogeneous Catalysis

Part of the Topics in Current Chemistry book series (TOPCURRCHEM, volume 307)


The goal of multiscale modelling of heterogeneous catalytic reactors is the prediction of all steps, starting from the reaction mechanism at the active centre, the rates of reaction, adsorption and diffusion processes inside the porous system of the catalyst support, based on first principles, quantum chemistry, force field simulations and macroscopic differential equations. The progress in these fields of research will be presented, including linking models between the various levels of description. Alkylation of benzene will be used as an example to demonstrate the various approaches from the active centre to the reactor.


Computational heterogeneous catalysis Molecular dynamics Monte Carlo Multiscale modelling Quantum chemistry 


  1. 1.
    Kohn W, Sham LS (1965) Self-consistent equations including exchange and correlation effects. Phys Rev A 140:1133–1138CrossRefGoogle Scholar
  2. 2.
    Kohanoff J (2006) Electronic structure calculations for solids and molecules: theory and computational methods. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  3. 3.
    Martin RM (2008) Electronic structure: basic theory and practice methods. Cambridge University Press, CambridgeGoogle Scholar
  4. 4.
    Szabo A, Ostlund NS (1996) Modern quantum chemistry: an introduction to advanced electronic structure theory. Dover Publications, Mineola, NYGoogle Scholar
  5. 5.
    Helgaker T, Jørgensen P, Olsen J (2002) Molecular electronic theory. John Wiley & Sons, New YorkGoogle Scholar
  6. 6.
    Shavitt I, Bartlett RJ (2009) Many-body methods in chemistry and physics: MBPT and coupled cluster theory. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  7. 7.
    Truhlar DG, Steckler R, Gordon MS (1987) Potential energy surfaces for polyatomic reaction dynamics. Chem Rev 87:217–236CrossRefGoogle Scholar
  8. 8.
    Pu J, Gao J, Truhlar DG (2006) Multidimensional tunnelling, recrossing, and the transmission coefficient for enzymatic reactions. Chem Rev 106:3140–3169CrossRefGoogle Scholar
  9. 9.
    Gao J, Truhlar DG (2002) Quantum mechanical methods for enzyme kinetics. Ann Rev Phys Chem 53:467–505CrossRefGoogle Scholar
  10. 10.
    Peters B, Heyden A, Bell AT, Chakraborty A (2004) A growing string method for determining transition states: comparison to the nudged elastic band and string methods. J Chem Phys 120:7877–7886CrossRefGoogle Scholar
  11. 11.
    Heyden A, Bell AT, Keil FJ (2005) Efficient methods for finding transition states in chemical reactions: comparison of improved dimer method and partitioned rational function optimization method. J Chem Phys 123:224101-1/14CrossRefGoogle Scholar
  12. 12.
    Kresse G, Hafner J (1993) Ab initio molecular dynamics for liquid metals. Phys Rev B 47:558–561CrossRefGoogle Scholar
  13. 13.
    Shao Y et al (2006) Advances in methods and algorithms in a modern quantum chemistry program package. Phys Chem Chem Phys 8:3172–3191CrossRefGoogle Scholar
  14. 14.
    Banerjee A, Adams N, Simons J, Shepard R (1985) Search for stationary points on surfaces. J Phys Chem 89:52–57CrossRefGoogle Scholar
  15. 15.
    McQuarrie DA (2000) Statistical mechanics. University Science Books, SausalitoGoogle Scholar
  16. 16.
    Hill TL (1987) An introduction to statistical mechanics. Dover Publications, Mineola, NYGoogle Scholar
  17. 17.
    Chandler D (1987) Introduction to modern statistical mechanics. Oxford University Press, OxfordGoogle Scholar
  18. 18.
    Tuckerman ME (2010) Statistical mechanics: theory and molecular simulation. Oxford University Press, OxfordGoogle Scholar
  19. 19.
    Truhlar DG, Garrett BC, Klippenstein SJ (1996) Current status of transition-state theory. J Phys Chem 100:12771–12800CrossRefGoogle Scholar
  20. 20.
    Zener C (1932) Non-adiabatic crossing of energy levels. Proc Roy Soc London Ser A 132:696–702Google Scholar
  21. 21.
    Frenkel D, Smit B (2001) Understanding molecular simulation, 2nd edn. Academic, San DiegoGoogle Scholar
  22. 22.
    Landau DP, Binder K (2009) A guide to Monte Carlo simulations in statistical physics, 3rd edn. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  23. 23.
    Allen MP, Tildesley DJ (1989) Computer simulation of liquids. Oxford University Press, OxfordGoogle Scholar
  24. 24.
    Newman MEJ, Barkema GT (1999) Monte Carlo methods in statistical physics. Oxford University Press, OxfordGoogle Scholar
  25. 25.
    Binder K, Heermann DW (2010) Monte Carlo simulation in statistical physics: an introduction. Springer, HeidelbergCrossRefGoogle Scholar
  26. 26.
    Rapaport DC (2004) The art of molecular dynamics simulation, 2nd edn. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  27. 27.
    Griebel M, Knapek S, Zumbusch G (2009) Numerical simulation in molecular dynamics: numerics, algorithms, parallelization, applications. Springer, HeidelbergGoogle Scholar
  28. 28.
    Tuckerman ME, Martyna GJ (2000) Understanding modern molecular dynamics: techniques and applications. J Phys Chem B 104:159–178CrossRefGoogle Scholar
  29. 29.
    Versteeg HK, Malalasekera W (1995) An introduction to computational fluid dynamics. Prentice Hall, Upper Saddle River, NJGoogle Scholar
  30. 30.
    Lomax H, Pulliam TH, Zingg DW (2001) Fundamentals of computational fluid dynamics. Springer, HeidelbergGoogle Scholar
  31. 31.
    Laney CB (1998) Computational gasdynamics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  32. 32.
    Grotendorst J, Attig N, Blügel S, Marx D (2009) Multiscale simulation methods in molecular sciences. Jülich Supercomputing Centre, JülichGoogle Scholar
  33. 33.
    Sherwood P, Brooks BR, Sansom MSP (2008) Multiscale methods for macromolecular simulations. Curr Opin Struct Biol 18:630–640CrossRefGoogle Scholar
  34. 34.
    Koci P, Novak V, Stepanek F, Marek M, Kubicek M (2010) Multi-scale modelling of reaction and transport in porous catalysts. Chem Eng Sci 65:412–419CrossRefGoogle Scholar
  35. 35.
    Lynbarsev A, Tu YQ, Laaksonen A (2009) Hierarchical multiscale modelling scheme from first principles to mesoscale. J Comput Theor Nanosci 6:951–959CrossRefGoogle Scholar
  36. 36.
    Santiso EE, Gubbins KE (2004) Multi-scale molecular modeling of chemical reactivity. Mol Simul 30:699–748CrossRefGoogle Scholar
  37. 37.
    Starrost F, Carter EA (2002) Modelling the full monty: baring the nature of surfaces across time and space. Surf Sci 500:323–346CrossRefGoogle Scholar
  38. 38.
    Vlachos DG (2005) A review of multiscale analysis: examples from systems biology materials engineering, and other fluid-surface interacting systems. Adv Chem Eng 30:1–61CrossRefGoogle Scholar
  39. 39.
    Baeurle SA (2009) Multiscale modeling of polymer materials using field-theoretic methodologies: a survey about recent developments. J Math Chem 46:363–426CrossRefGoogle Scholar
  40. 40.
    van Santen RA, Neurock M (2006) Molecular heterogeneous catalysis. Wiley-VCH, WeinheimCrossRefGoogle Scholar
  41. 41.
    van Santen RA, Sautet P (2009) Computational methods in catalysis and materials science. Wiley-VCH, WeinheimCrossRefGoogle Scholar
  42. 42.
    Weigend F, Kattannek M, Ahlrichs R (2009) Approximated electron repulsion integrals: Cholesky decomposition versus resolution of the identity methods. J Chem Phys 130:164106CrossRefGoogle Scholar
  43. 43.
    Werner H-J, Knowles PJ (1988) An efficient internally contracted multiconfiguration - reference configuration interaction method. J Chem Phys 89:5803–5814CrossRefGoogle Scholar
  44. 44.
    Andersson K, Roos BO (1995) In: Yarkony DR (ed) Modern electronic structure theory, Advanced Series in Physical Chemistry, Vol 2. World Scientific, SingaporeGoogle Scholar
  45. 45.
    Schüler M, Kovar T, Lischka H, Shepard R, Harrison RJ (1993) A parallel implementation of the COLUMBUS multireference configuration interaction program. Theor Chim Acta 84:489–509CrossRefGoogle Scholar
  46. 46.
    Chwee TS, Szilva AB, Lindh R, Carter EA (2008) Linear scaling multireference singles and doubles configuration interaction. J Chem Phys 128:224106-1/9CrossRefGoogle Scholar
  47. 47.
    Mazziotti DA (2006) Quantum chemistry without wave functions: two-electron reduced density matrices. Acc Chem Res 39:207–215CrossRefGoogle Scholar
  48. 48.
    Chaykin D (2009) Verification of semidefinite optimization problems with application to variational electronic structure calculation. PhD thesis, TU Hamburg-HarburgGoogle Scholar
  49. 49.
    Kutzelnigg W (1985) r 12-dependent terms in the wave function as closed sums of partial wave amplitudes for large I. Theor Chim Acta 68:445–469CrossRefGoogle Scholar
  50. 50.
    Klopper W, Kutzelnigg W (1987) Møller-Plesset calculations taking care of the correlation cusp. Chem Phys Lett 134:17–22CrossRefGoogle Scholar
  51. 51.
    Rychlewski J (2003) Explicitly correlated wave functions in chemistry and physics − theory and applications. Kluwer Academic, DordrechtGoogle Scholar
  52. 52.
    Marchette O, Werner H-J (2008) Accurate calculations of intermolecular interaction energies using explicitly correlated wave functions. Phys Chem Chem Phys 10:3400–3409CrossRefGoogle Scholar
  53. 53.
    Valeev EF (2006) Combining explicitly correlated R12 and Gaussian geminal electronic structure theories. J Chem Phys 125:244106-1/10CrossRefGoogle Scholar
  54. 54.
    Curtiss LA, Redfern PC, Raghavachari K (2007) Gaussian-4 theory. J Chem Phys 126:048108-1/12CrossRefGoogle Scholar
  55. 55.
    Curtiss LA, Redfern PC, Raghavachari K (2005) Assessment of Gaussian-3 and density functional theories on the G3/05 test set of experimental energies. J Chem Phys 123:124107-1/12Google Scholar
  56. 56.
    Montgomery JA, Frisch MJ, Ochterski JW, Petersson GA (1999) A complete basis set model chemistry. J Chem Phys 110:2822–2827CrossRefGoogle Scholar
  57. 57.
    Császár AG, Allen WA, Schaefer HF III (1998) In pursuit of the ab initio limit for conformational energy prototypes. J Chem Phys 108:9751–9765CrossRefGoogle Scholar
  58. 58.
    Dahlke EE, Orthmeyer MA, Truhlar DG (2008) Assessment of multicoefficient correlation methods. J Phys Chem B 112:2372–2381CrossRefGoogle Scholar
  59. 59.
    Bomble YJ, Vazquez J, Kallay M, Michank C, Szalay PG, Császár AG, Gauss J, Stanton JF (2006) High-accuracy extrapolated ab initio thermochemistry. II. Minor improvements to the protocol and a vital simplification. J Chem Phys 125:064108-1/8CrossRefGoogle Scholar
  60. 60.
    Karton A, Taylor PR, Martin JML (2007) Basis set convergence of post-CCSD contributions to molecular atomization energies. J Chem Phys 127:064104-1/11CrossRefGoogle Scholar
  61. 61.
    Hansen N, Kerber T, Sauer J, Bell AT, Keil FJ (2010) Quantum chemical modelling of benzene ethylation over H-ZSM-5 – approaching chemical accuracy: a Hybrid-MP2:DFT study. J Am Chem Soc 132:11525–11538CrossRefGoogle Scholar
  62. 62.
    Sherrill CD (2010) Frontiers in electronic structure theory. J Chem Phys 132:110902CrossRefGoogle Scholar
  63. 63.
    Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev B 136:864–871CrossRefGoogle Scholar
  64. 64.
    Becke AD (1993) Density-functional thermochemistry. III. The role of exact exchange. J Chem Phys 98:5648–5652CrossRefGoogle Scholar
  65. 65.
    Kresse G, Furthmüller J (1996) Efficient iterative schemes for ab-initio total-energy calculations using a plane-wave basis set. Phys Rev B 54:11169–11186CrossRefGoogle Scholar
  66. 66.
    Segall MD, Lindan PLD, Probert MJ, Pickard CJ, Hasnip PJ, Clark SJ, Payne MC (2002) First principles simulation: ideas, illustrations, and the CASTEP code. J Phys Condens Matter 14:2717–2743CrossRefGoogle Scholar
  67. 67.
    Ahlrichs R, Bär M, Häser M, Horn H, Kölmel C (1989) Electronic structure calculations on workstation computers: the program system TURBOMOLE. Chem Phys Lett 162:165–169CrossRefGoogle Scholar
  68. 68.
    Dovesi R, Saunders VR, Roetti R, Orlando R, Zicovick-Wilson CM, Pascale F, Civalleri B, Doll K, Harrison NM, Bush IJ, Llunell M (2009) CRYSTAL09 user’s manual. University of Torino, TorinoGoogle Scholar
  69. 69.
    Jaguar, Version 7.0 (2007) Schrödinger, LLC, New York, NYGoogle Scholar
  70. 70.
    Frisch MJ et al. (2009) Gaussian 09, Revision 1. Gaussian, Inc., Wallingford CTGoogle Scholar
  71. 71.
    Schmidt MW, Baldridge KK, Boatz JA, Elbert ST, Gordon MS, Jensen JH, Koseki S, Matsunaga N, Nguyen SuS, Windus TL, Dupuis M, Montgomery JA (1993) General atomic and molecular electronic structure system. J Comput Chem 14:1347–1363CrossRefGoogle Scholar
  72. 72.
    Dion M, Rydberg H, Schröder E, Langreth DC, Lundqvist BI (2004) Van der Waals density functional for general geometries. Phys Rev Lett 92:246401CrossRefGoogle Scholar
  73. 73.
    Thonhauser T, Cooper VR, Li S, Puzder A, Hyldgaard P, Langreth DC (2007) Van der Waals density functional: self-consistent potential and the nature of the van der Waals bond. Phys Rev B 76:125112CrossRefGoogle Scholar
  74. 74.
    Rudenko A, Keil FJ, Katsnelson MI, Lichtenstein AI (2010) Adsorption of diatomic halogen molecules on graphene: a van der Waals density functional study. Phys Rev B 82:035427CrossRefGoogle Scholar
  75. 75.
    Vydrov OA, Wu Q, Voorhis TV (2008) Self-consistent implementation of a nonlocal van der Waals density functional with a Gaussian basis set. J Chem Phys 129:014106CrossRefGoogle Scholar
  76. 76.
    Grimme S, Antony J, Ehrlich S, Krieg H (2010) A consistent and accurate ab initio parametrisation of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J Chem Phys 132:154104CrossRefGoogle Scholar
  77. 77.
    Silvestri PL (2009) Van der Waals interactions in density functional theory using Wannier functions. J Phys Chem A 113:5224–5234CrossRefGoogle Scholar
  78. 78.
    Chai J-D, Head-Gordon M (2008) Systematic optimization of long range corrected hybrid density functionals. J Chem Phys 128:084106-1/15CrossRefGoogle Scholar
  79. 79.
    Chai J-D, Head-Gordon M (2008) Long-range corrected hybrid density functionals with damped atom-atom dispersion corrections. Phys Chem Chem Phys 10:6615–6620CrossRefGoogle Scholar
  80. 80.
    Pacchioni G (2008) Modeling doped and defective oxides in catalysis with density functional theory methods: room for improvement. J Chem Phys 128:182505CrossRefGoogle Scholar
  81. 81.
    Metropolis NA, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21:1087–1092CrossRefGoogle Scholar
  82. 82.
    Anderson JB (2007) Quantum Monte Carlo. Oxford University Press, OxfordGoogle Scholar
  83. 83.
    Hammond BL, Lester WA, Reynolds PJ (1994) Monte Carlo in ab initio quantum chemistry. World Scientific, SingaporeCrossRefGoogle Scholar
  84. 84.
    Schwartz M (1987) Principles of electrodynamics. Dover Publications, New YorkGoogle Scholar
  85. 85.
    Swart M, van Duijnen PTh, Snijders JG (2001) A charge analysis derived from an atomic multisole expansion. J Comput Chem 22:79–88CrossRefGoogle Scholar
  86. 86.
    Sigfridsson E, Eyde U (1998) Comparison of methods for deriving atomic charges from electrostatic potential and moments. J Comput Chem 19:377–395CrossRefGoogle Scholar
  87. 87.
    Tironi I, Sperb R, Smith PE, van Gunsteren WF (1995) A generalized reaction field method for molecular dynamics simulations. J Chem Phys 102:5451–5459CrossRefGoogle Scholar
  88. 88.
    Hünenberger PH, van Gunsteren WF (1998) Alternative schemes for the inclusion of a reaction field correction into molecular dynamics simulations: influence on the simulated energetic, structural, and dielectric properties of liquid water. J Chem Phys 108:6117–6134CrossRefGoogle Scholar
  89. 89.
    Essmann U, Perera L, Berkowitz M, Darden T, Lee H, Pedersen L (1995) A smooth particle mesh Ewald method. J Chem Phys 103:8577–8593CrossRefGoogle Scholar
  90. 90.
    Wood RH (1995) Continuum electrostatics in a computational universe with finite cutoff radii and periodic boundary conditions. Correction to computed free energies of ionic solvation. J Chem Phys 103:6177–6187CrossRefGoogle Scholar
  91. 91.
    Guillot B (2002) A reappraisal of what we have learnt during three decades of computer simulations on water. J Mol Liquids 101:219–260CrossRefGoogle Scholar
  92. 92.
    Saint-Martin H, Hernández-Cobos J, Ortega-Blake I (2005) Water models based on a single potential energy surface and different molecular degrees of freedom. J Chem Phys 122:224509-1/12CrossRefGoogle Scholar
  93. 93.
    Hernández-Cobos J, Saint-Martin H, Mackie AD, Vega LF, Ortega-Blake I (2005) Water liquid-vapor equilibria predicted by refined ab initio derived potentials. J Chem Phys 123:044506-1/8CrossRefGoogle Scholar
  94. 94.
    Allinger NL, Yuh YH, Lü J-H (1989) Molecular mechanics – the MM3 force-field for hydrocarbons. 1. J Am Chem Soc 111:8551–8566CrossRefGoogle Scholar
  95. 95.
    Allinger NL, Chen KS, Lü J-H (1996) An improved force field (MM4) for saturated hydrocarbons. J Comput Chem 17:642–668CrossRefGoogle Scholar
  96. 96.
    Mayo SL, Olafson BD, Goddard WA III (1990) DREIDING: a generic force field for molecular simulations. J Phys Chem 94:8897–8909CrossRefGoogle Scholar
  97. 97.
    Allured VS, Kelly CM, Landis CR (1991) SHAPES empirical force field: a new treatment of angular potentials and its application to square-planar transition metal complexes. J Am Chem Soc 113:1–13CrossRefGoogle Scholar
  98. 98.
    Root DM, Landis CR, Cleveland T (1993) Valence bond concepts applied to molecular mechanics description of molecular shapes. J Am Chem Soc 115:4201–4209CrossRefGoogle Scholar
  99. 99.
    Rappé AK, Casewit CJ, Colwell KS, Goddard WA III, Skiff WM (1992) UFF, a full periodic table force field for molecular mechanics and molecular dynamics simulations. J Am Chem Soc 114:10024–10035CrossRefGoogle Scholar
  100. 100.
    Maple JR, Hwang M-J, Stockfisch TP, Dinur U, Waldman M, Ewig CS, Hagler AT (1994) Derivation of class II force fields. J Comput Chem 15:161–182CrossRefGoogle Scholar
  101. 101.
    Weiner SJ, Kollman PA, Case DA, Singh UC, Ghio C, Alagona G, Profetta S, Weiner P (1984) A new force-field for molecular mechanical simulation of nucleic acids and proteins. J Am Chem Soc 106:765–784CrossRefGoogle Scholar
  102. 102.
    Brooks BR, Bruccoleri RE, Olafson BD, States DJ, Swaminathan S, Karplus M (1983) CHARMM – a program for macromolecular energy, minimization, and dynamics calculations. J Comput Chem 4:187–217CrossRefGoogle Scholar
  103. 103.
    Jorgensen WL, Maxwell DS, Tirado Rives J (1996) Development and testing of the OPLS all-atom force field on conformational energetics and properties of organic liquids. J Am Chem Soc 118:11225–11236CrossRefGoogle Scholar
  104. 104.
    Halgren TA (1996) Merck molecular force field. 1. Basis form, scope, parametrisation, and performance. J Comput Chem 17:490–519, 520–552, 553–586Google Scholar
  105. 105.
    Daura X, Mark AE, van Gunsteren WF (1998) Parametrization of aliphatic CHn united atoms of GROMOS96 force field. J Comput Chem 19:535–547CrossRefGoogle Scholar
  106. 106.
    Massink SJ, Risselada HJ, Yefimov S, Tieleman DP, de Vries AH (2007) The MARTINI force field: coarse grained model for biomolecular simulations. J Phys Chem B 111:7812–7824CrossRefGoogle Scholar
  107. 107.
    Jalaie M, Lipkowitz KB (2000) Published force field parameters for molecular mechanics, Molecular Dynamics and Monte Carlo simulations. In: Lipkowitz KB, Boyd DB (eds) Reviews in computational chemistry, vol 14. Wiley-VCH, Weinheim, p 441Google Scholar
  108. 108.
    Shelley JC, Shelley MY, Reeder RC, Bandyopadhyay S, Klein ML (2001) A coarse grain model for phospholipid simulations. J Phys Chem B 105:4464–4470CrossRefGoogle Scholar
  109. 109.
    Elezgaray J, Laguerre M (2006) A systematic method to derive force fields for coarse-grained simulations of phospholipids. Comput Phys Commun 145:264–268CrossRefGoogle Scholar
  110. 110.
    Izvekov S, Voth GA (2005) A multiscale coarse-graining method for biomolecular systems. J Phys Chem B 109:2469–2473CrossRefGoogle Scholar
  111. 111.
    Schlick T (2002) Molecular modeling and simulation. Springer, HeidelbergGoogle Scholar
  112. 112.
    Leach AR (2001) Molecular modeling – principles and applications. Prentice Hall, New YorkGoogle Scholar
  113. 113.
    Crespos C, Collins MA, Pijper E, Kroes GJ (2003) Multi-dimensional energy surface determination by modified Shepherd interpolation for a molecule-surface reaction H2 + Pt(111). Chem Phys Lett 376:566–575CrossRefGoogle Scholar
  114. 114.
    Crespos C, Collins MA, Pijper E, Kroes GJ (2004) Application of the modified Shepherd interpolation method to the determination of the potential energy surface for a molecule-surface reaction: H2 + Pt(111). J Chem Phys 120:2392–2406CrossRefGoogle Scholar
  115. 115.
    Zupan J, Gasteiger J (1993) Neural networks for chemists. VCH, WeinheimGoogle Scholar
  116. 116.
    Lorenz S, Scheffler M, Gross A (2006) Descriptions of surface chemical reactions using a neural network representation of the potential energy surface. Phys Rev B 73:115431-1/13CrossRefGoogle Scholar
  117. 117.
    Behler J, Reuter K, Scheffler M (2008) Nonadiabatic effects in the dissociation of oxygen molecules at the Al(111) surface. Phys Rev B 77:115421-1/16CrossRefGoogle Scholar
  118. 118.
    van Duin ACT, Dasgupta S, Lorant F, Goddard WA III (2001) ReaxFF: a reactive force field for hydrocarbons. J Phys Chem A 105:9396–9409CrossRefGoogle Scholar
  119. 119.
    Pauling L (1947) Atomic radii and interatomic distances in metals. J Am Chem Soc 69:542–553CrossRefGoogle Scholar
  120. 120.
    Nielson KD, van Duin ACT, Oxgaard J, Deng W-Q, Goddard WA II (2005) Development of the ReaxFF reactive force field for describing transition metal catalyzed reactions, with application to the initial stages of the catalytic formation of carbon nanotubes. J Phys Chem A 109:493–499CrossRefGoogle Scholar
  121. 121.
    Chenoweth K, van Duin ACT, Persson P, Cheng M-J, Oxgaard J, Goddard WA III (2008) Development and application of a ReaxFF reactive force field for oxidative dehydrogenation on vanadium oxide catalysts. J Phys Chem C 112:14645–14654CrossRefGoogle Scholar
  122. 122.
    Liu B, Lusk MT, Ely JF, van Duin ACT, Goddard WA III (2008) Reactive molecular dynamics force field for the dissociation of light hydrocarbons on Ni(111). Mol Simul 34:967–972CrossRefGoogle Scholar
  123. 123.
    Bortz AB, Kalos MH, Lebowitz JL (1975) New algorithm for Monte Carlo simulations of Ising spin systems. J Comp Phys 17:10–18CrossRefGoogle Scholar
  124. 124.
    Fichthorn KA, Weinberg WH (1991) Theoretical foundations of dynamical Monte Carlo simulations. J Chem Phys 95:1090–1096CrossRefGoogle Scholar
  125. 125.
    Van Kampen NG (2001) Stochastic processes in physics and chemistry. Elsevier, AmsterdamGoogle Scholar
  126. 126.
    Lukkien JJ, Segers JPL, Hilbers PAJ, Gelten RJ, Jansen APJ (1998) Efficient Monte Carlo methods for the simulation of catalytic surface reactions. Phys Rev E 58:2598–2610CrossRefGoogle Scholar
  127. 127.
    Goldstein H, Poole C, Safko J (2002) Classical mechanics. Pearson, Addison Wesley, San FranciscoGoogle Scholar
  128. 128.
    Leimkuhler BJ, Skeel RD (1994) Symplectic numerical integrators in constrained Hamiltonian systems. J Comput Phys 112:117–125CrossRefGoogle Scholar
  129. 129.
    Trotter HF (1959) On the product of semi-groups of operators. Proc Am Math Soc 10:545–551CrossRefGoogle Scholar
  130. 130.
    Reed M, Simon B (1980) Methods of modern mathematical physics. I Functional analysis, 2nd edn. Academic, New YorkGoogle Scholar
  131. 131.
    Hassani S (1999) Mathematical physics. Springer, HeidelbergGoogle Scholar
  132. 132.
    Buneman O (1967) Time-reversible difference procedures. J Comp Phys 1:517–535CrossRefGoogle Scholar
  133. 133.
    Tuckerman ME, Berne BJ, Martyna GJ (1992) Reversible multiple time scales molecular dynamics. J Chem Phys 97:1990–2001CrossRefGoogle Scholar
  134. 134.
    Phillips JC, Braun R, Wang W, Gumbart J, Tajkhorshid E, Villa E, Chipot C, Skeel RD, Kale L, Schulten K (2005) Scalable Molecular Dynamics with NAMD. J Comput Chem 26:1781–1802CrossRefGoogle Scholar
  135. 135.
    Shaw DE (2005) A fast, scalable method for the parallel evaluation of distance-limited pairwise particle interactions. J Comput Chem 26:1318–1328CrossRefGoogle Scholar
  136. 136.
    Ma Q, Izaguirre J, Skeel R (2003) Verlet-I/R-Respa/Impulse is limited by nonlinear instabilities. J Sci Comput 24:1951–1973Google Scholar
  137. 137.
    Morrone JA, Zhou R, Berne BJ (2010) Molecular dynamics with multiple time scales: how to avoid pitfalls. J Chem Theor Comput 6:1798–1804CrossRefGoogle Scholar
  138. 138.
    Skeel RD (1999) Integration schemes for molecular dynamics and related applications. In: Ainsworth M, Levesley J, Marletta M (eds) The graduate student’s guide to numerical analysis. Springer, New York, pp 119–176Google Scholar
  139. 139.
    Schneider T, Stoll E (1978) Molecular dynamics study of a three-dimensional n-component model for distortive phase transitions. Phys Rev B 17:1302–1322CrossRefGoogle Scholar
  140. 140.
    Andersen HC (1980) Molecular dynamics simulations at constant pressure and/or temperature. J Chem Phys 72:2384–2393CrossRefGoogle Scholar
  141. 141.
    Hoover WG, Ladd AJC, Moran B (1982) High strain rate plastic flow studied via non-equilibrium molecular dynamics. Phys Rev Lett 48:1818–1820CrossRefGoogle Scholar
  142. 142.
    Evans DJ, Hoover WG, Failor BH, Moran B, Ladd AJC (1983) Nonequilibrium molecular dynamics via Gauss’s principle of least constraint. Phys Rev A 28:1016–1021CrossRefGoogle Scholar
  143. 143.
    Evans DJ, Morriss GP (1983) The isothermal isobaric molecular dynamics ensemble. Phys Lett A 98:433–436CrossRefGoogle Scholar
  144. 144.
    Evans DJ, Morriss GP (1984) Non-Newtonian molecular dynamics. Phys Rep 1:297–344CrossRefGoogle Scholar
  145. 145.
    Tuckerman ME, Mundy CJ, Martyna GJ (1999) On the classical statistical mechanics of non-Hamiltonian systems. Europhys Lett 45:149–155CrossRefGoogle Scholar
  146. 146.
    Berendsen HJC, Postma JPM, van Gunsteren WF, DiNola A, Haak JR (1984) Molecular dynamics with coupling to an external bath. J Chem Phys 81:3684–3690CrossRefGoogle Scholar
  147. 147.
    Nosé S (1984) A unified formulation of the constant temperature molecular dynamics method. J Chem Phys 81:511–519CrossRefGoogle Scholar
  148. 148.
    Nosé S (1984) A molecular dynamics method for simulations in the canonical ensemble. Mol Phys 52:255–268CrossRefGoogle Scholar
  149. 149.
    Hoover WG (1985) Canonical dynamics: equilibrium phase-space distributions. Phys Rev A 31:1696–1697CrossRefGoogle Scholar
  150. 150.
    Jakobtorweihen S, Verbeeck MG, Lowe CP, Keil FJ, Smit B (2005) Understanding the loading dependence of self-diffusion in carbon nanotubes. Phys Rev Lett 95:044501-1/4CrossRefGoogle Scholar
  151. 151.
    Jakobtorweihen S, Lowe CP, Keil FJ, Smit B (2006) A novel algorithm to model the influence of host lattice flexibility in molecular dynamics simulations: loading dependence of self-diffusion in carbon nanotubes. J Chem Phys 124:154706-1/13CrossRefGoogle Scholar
  152. 152.
    Jakobtorweihen S, Lowe CP, Keil FJ, Smit B (2007) Diffusion of chain molecules and mixtures in carbon nanotubes: the effect of host lattice flexibility and theory of diffusion in the Knudsen regime. J Chem Phys 127:024904-1/11CrossRefGoogle Scholar
  153. 153.
    Swendsen RH, Wang JS (1986) Replica Monte Carlo simulation of spin-glasses. Phys Rev Lett 57:2607–2609CrossRefGoogle Scholar
  154. 154.
    Woods CJ, Esser JW, King MA (2003) The development of replica-exchange based free-energy methods. J Phys Chem B 107:13703–13710CrossRefGoogle Scholar
  155. 155.
    Voter AF (1997) Hyperdynamics: accelerated molecular dynamics of infrequent events. Phys Rev Lett 78:3908–3912CrossRefGoogle Scholar
  156. 156.
    Laio A, Parinello M (2002) Escaping the free energy minima. Proc Natl Acad Sci 99:12562–12566CrossRefGoogle Scholar
  157. 157.
    Carter E, Cicotti G, Hynes JT, Kapral R (1989) Constrained reaction coordinate dynamics for the simulation of rare events. Chem Phys Lett 156:472–477CrossRefGoogle Scholar
  158. 158.
    Sprik M, Cicotti G (1998) Free energy from constrained molecular dynamics. J Phys Chem 109:7737–7744CrossRefGoogle Scholar
  159. 159.
    Torrie GM, Valleau JP (1974) Monte-Carlo free-energy estimates using non-Boltzmann sampling – application to subcritical Lennard-Jones fluid. Chem Phys Lett 28:578–581CrossRefGoogle Scholar
  160. 160.
    Rosso L, Tuckerman ME (2002) An adiabatic molecular dynamics method for the calculation of free energy profiles. Mol Simul 28:91–112CrossRefGoogle Scholar
  161. 161.
    Chipot C, Pohorille A (2007) Free energy calculations: theory and applications in chemistry and biology. Springer, HeidelbergCrossRefGoogle Scholar
  162. 162.
    Washel A, Levitt M (1976) Theoretical studies of enzymatic reactions: dielectric, electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme. J Mol Biol 103:227–249CrossRefGoogle Scholar
  163. 163.
    Sierka M, Sauer J (2005) Hybrid quantum mechanics/molecular mechanics methods and their application. In: Yip S (ed) The handbook of materials modeling, Part A. Methods. Springer, Dordrecht, pp 241–258CrossRefGoogle Scholar
  164. 164.
    Hu H, Yang W (2008) Free energies of chemical reactions in solution and in enzymes with ab initio quantum mechanics/molecular mechanics methods. Annu Rev Phys Chem 59:573–601CrossRefGoogle Scholar
  165. 165.
    Senn HM, Thiel W (2009) QM/MM methods for biomolecular systems. Angew Chem Int Ed 48:1198–1229CrossRefGoogle Scholar
  166. 166.
    York DM, Lee T-S (2009) Multi-scale quantum models for biocatalysis. Springer, DordrechtCrossRefGoogle Scholar
  167. 167.
    Lin H, Truhlar DG (2007) QM/MM: what have we learned, where are we, and where do we go from here? Theor Chem Acc 117:185–199CrossRefGoogle Scholar
  168. 168.
    Park JH, Heyden A (2009) Solving the equations of motion for mixed atomistic and coarse-grained systems. Mol Simul 35:962–973CrossRefGoogle Scholar
  169. 169.
    Mata RA, Werner HJ, Thiel S, Thiel W (2008) Toward accurate barriers for enzymatic reactions: QM/MM case study on p-hydroxybenzoate hydroxylase. J Chem Phys 128:025104-1/8Google Scholar
  170. 170.
    Bakowies D, Thiel W (1996) Hybrid models for combined quantum mechanical and molecular mechanical approaches. J Phys Chem 100:10580–10594CrossRefGoogle Scholar
  171. 171.
    König PH, Hoffmann M, Frauenheim T, Cui Q (2005) A critical evaluation of different QM/MM frontier treatments using SCC-DFTB as the QM method. J Phys Chem B 109:9082–9095CrossRefGoogle Scholar
  172. 172.
    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 113:9978–9985CrossRefGoogle Scholar
  173. 173.
    E W, Ren W, Vanden Eijnden E (2002) String method for the study of rare events. Phys Rev B 66:052301-1/4CrossRefGoogle Scholar
  174. 174.
    Baker J (1986) An algorithm for the location of transition states. J Comp Chem 7:385–395CrossRefGoogle Scholar
  175. 175.
    Henkelman G, Jónsson H (1999) A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives. J Chem Phys 111:7010–7022CrossRefGoogle Scholar
  176. 176.
    Hänggi P, Talkner P, Borkovec M (1990) Reaction-rate theory: fifty years after Kramers. Rev Mod Phys 62:251–341CrossRefGoogle Scholar
  177. 177.
    Bucko T, Hafner J (2010) Entropy effects in hydrocarbon conversion reactions: free energy integrations and transition path sampling. J Phys Condens Matter 22:384201CrossRefGoogle Scholar
  178. 178.
    Pratt LR (1986) A statistical method for identifying transition states in high dimensional problems. J Chem Phys 85:5045–5048CrossRefGoogle Scholar
  179. 179.
    Bolhuis PG, Chandler D, Dellago C, Gassler PL (2002) Transition path sampling: throwing ropes over rough mountain passes, in the dark. Annu Rev Phys Chem 53:291–318CrossRefGoogle Scholar
  180. 180.
    Berendsen HJC (2007) Simulating the physical world. Cambridge University Press, CambridgeGoogle Scholar
  181. 181.
    Broadbelt LJ, Snurr RQ (2000) Applications of molecular modelling in heterogeneous catalysis research. Appl Catal A 200:23–46CrossRefGoogle Scholar
  182. 182.
    Hansen N, Krishna R, van Baten JM, Bell AT, Keil FJ (2009) Analysis of diffusion limitation in the alkylation of benzene over H-ZSM-5 by combining quantum chemical calculations, molecular simulations, and a continuum approach. J Phys Chem C 113:235–246CrossRefGoogle Scholar
  183. 183.
    Hansen N, Krishna R, van Baten JM, Bell AT, Keil FJ (2010) Reactor simulation of benzene ethylation and ethane dehydrogenation catalyzed by ZSM-5: a multiscale approach. Chem Eng Sci 65:2472–2480CrossRefGoogle Scholar
  184. 184.
    Raimondeau S, Vlachos DG (2002) Recent developments on multiscale, hierarchical modelling of chemical reactors. Chem Eng J 90:3–23CrossRefGoogle Scholar
  185. 185.
    Vlachos D (2005) A review of multiscale analysis: examples from systems biology, materials engineering, and other fluid-surface interacting systems. Adv Chem Eng 36:1–61CrossRefGoogle Scholar
  186. 186.
    Christensen CH, Norskov JK (2008) A molecular view of heterogeneous catalysis. J Chem Phys 128:182503-1/8CrossRefGoogle Scholar
  187. 187.
    Hansen N, Kerber T, Sauer J, Bell AT, Keil FJ (2010) Quantum chemical modeling of benzene ethylation over H-ZSM-5 approaching chemical accuracy: a hybrid MP2:DFT study. J Am Chem Soc 132:11525–11538CrossRefGoogle Scholar
  188. 188.
    Heyden A, Peters B, Bell AT, Keil FJ (2005) Comprehensive DFT study of nitrous oxide decomposition over Fe-ZSM-5. J Phys Chem B 109:1857–1873CrossRefGoogle Scholar
  189. 189.
    Heyden A, Hansen N, Bell AT, Keil FJ (2006) Nitrous oxide decomposition over Fe-ZSM-5 in the presence of nitric oxide: a comprehensive DFT study. J Phys Chem B 110:17096–17114CrossRefGoogle Scholar
  190. 190.
    Heyden A, Bell AT, Keil FJ (2005) Kinetic modelling of nitrous oxide decomposition on Fe-ZSM-5 based on parameters obtained from first-principles calculation. J Catal 233:26–35CrossRefGoogle Scholar
  191. 191.
    Brüggemann TC, Przybylski M-D, Balaji SP, Keil FJ (2010) Theoretical investigation of the mechanism of the selective catalytic reduction of nitrogen dioxide with ammonia on H-form zeolites and the role of nitric and nitrous acids as intermediates. J Phys Chem C 114:6567–6587CrossRefGoogle Scholar
  192. 192.
    Liu X, Newsome D, Coppens M-O (2009) Dynamic Monte Carlo simulations of binary self-diffusion in ZSM-5. Microporous Mesoporous Mater 125:149–159CrossRefGoogle Scholar
  193. 193.
    Tuma C, Sauer J (2006) Treating dispersion effects in extended systems by hybrid MP2:DFT calculations – protonation of isobutene in zeolite ferrierite. Phys Chem Chem Phys 8:3955–3965CrossRefGoogle Scholar
  194. 194.
    Myers AL, Prausnitz JM (1965) Thermodynamics of mixed gas adsorption. AIChE J 11:121–127CrossRefGoogle Scholar
  195. 195.
    Huang P, Carter EA (2008) Advances in correlated electronic structure methods for solids, surfaces, and nanostructures. Annu Rev Phys Chem 59:261–290Google Scholar
  196. 196.
    E W, Vanden–Eijnden E (2010) Transition-path theory and path-finding algorithms for the study of rare events. Annu Rev Phys Chem 61:391–420Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Chemical Reaction EngineeringTU Hamburg-HarburgHamburgGermany

Personalised recommendations