Approximate Quantum Mechanical Methods for Rate Computation in Complex Systems

  • Steven D. Schwartz


The last 20 years have seen qualitative leaps in the complexity of chemical reactions that have been studied using theoretical methods. While methodologies for small molecule scattering are still of great importance and under active development [1], two important trends have allowed the theoretical study of the rates of reaction in complex molecules, condensed phase systems, and biological systems. First, there has been the explicit recognition that the type of state to state information obtained by rigorous scattering theory is not only not possible for complex systems, but more importantly, not meaningful. Thus, methodologies have been developed that compute averaged rate data directly from a Hamiltonian. Perhaps the most influential of these approaches has been the correlation function formalisms developed by Bill Miller et al. [2]. While these formal expressions for rate theories are certainly not the only correlation function descriptions of quantum rates [3, 4], these expressions of rates directly in terms of evolution operators, and in their coordinate space representations as Feynman Propagators, have lent themselves beautifully to complex systems because many of the approximation methods that have been devised are for Feynman propagator computation. This fact brings us to the second contributor to the blossoming of these approximate methods, the development of a wide variety of approximate mathematical methods to compute the time evolution of quantum systems. Thus the marriage of these mathematical developments has created the necessary powerful tools needed to probe systems of complexity unimagined just a few decades ago.


Proton Transfer Coherent State Evolution Operator Hydride Transfer Transition State Theory 
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  1. [1]
    Y.M. Li and J.Z.H. Zhang, “Theoretical dynamical treatment of chemical reactions”, In: Modern Trends In Chemical Reaction Dynamics Part I: Experiment and Theory by Xueming Yang & Kopin Liu (eds.), 2003.Google Scholar
  2. [2]
    Wm.H. Miller, S.D. Schwartz, and J.W. Tromp, “Quantum mechanical rate constants for bimolecular reactions”, J. Chem. Phys., 79, 4889–4898, 1983.CrossRefADSGoogle Scholar
  3. [3]
    T. Yamamoto, “Quantum statistical mechanical theory of the rate of exchange chemical reactions in the gas phase”, J. Chem. Phys., 33, 281, 1960.CrossRefMathSciNetADSGoogle Scholar
  4. [4]
    D. Chandler, “Statistical mechanics of isomerization dynamics in liquids and the transition state approximation”, J. Chem. Phys., 2959–2970, 1978.Google Scholar
  5. [5]
    For an older but excellent review see: R.B. Bernstein, “Quantum effects in elastic molecular scattering”, Adv. Chem. Phys., 10, 75, 1966.Google Scholar
  6. [6]
    Wm.H. Miller, “Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants”, J. Chem. Phys., 61, 1823–1834, 1974.CrossRefADSGoogle Scholar
  7. [7]
    R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw Hill, New York, 1965.MATHGoogle Scholar
  8. [8]
    J.E. Straub, M. Borkovec, and B.J. Berne, “Molecular dynamics study of an isomerizing diatomic in a Lennard-Jones fluid”, J. Chem. Phys., 89, 4833, 1988.CrossRefADSGoogle Scholar
  9. [9]
    B.J. Gertner, K.R. Wilson, and J.T. Hynes, “Nonequilibrium solvation effects on reaction rates for model SN2 reactions in water”, J. Chem. Phys., 90, 3537, 1988.CrossRefADSGoogle Scholar
  10. [10]
    E. Cortes, B.J. West, and K. Lindenberg, “On the generalized langevin equation: classical and quantum mechanical”, J. Chem. Phys., 82, 2708–2717, 1985.CrossRefMathSciNetADSGoogle Scholar
  11. [11]
    R. Zwanzig, “The nonlinear generalized langevin equation”, J. Stat. Phys., 9, 215, 1973.CrossRefADSGoogle Scholar
  12. [12]
    Wm.H. Miller, “Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants”, J. Chem. Phys., 61, 1823–1834, 1974.CrossRefADSGoogle Scholar
  13. [13]
    P. Pechukas in “Dynamics of molecular collisions”, Part B W.H. Miller, (ed.), Plenum, New York, 1976.Google Scholar
  14. [14]
    S.D. Schwartz, “Accurate quantum mechanics from high order resummed operator expansions”, J. Chem. Phys., 100, 8795–8801, 1994.CrossRefADSGoogle Scholar
  15. [15]
    S.D. Schwartz, “Vibrational energy transfer from resummed evolution operators, J. Chem. Phys., 101, 10436–10441, 1994.CrossRefADSGoogle Scholar
  16. [16]
    D. Antoniou and S.D. Schwartz, “Vibrational energy transfer in linear hydrocarbon chains: new quantum results”, J. Chem. Phys., 103, 7277–7286, 1995.CrossRefADSGoogle Scholar
  17. [17]
    S.D. Schwartz, “The interaction representation and non-adiabatic corrections to adiabatic evolution operators”, J. Chem. Phys., 104, 1394–1398, 1996.CrossRefADSGoogle Scholar
  18. [18]
    D. Antoniou and S.D. Schwartz, “Nonadiabatic effects in a method that combines classical and quantum mechanics”, J. Chem. Phys., 104, 3526–3530, 1996.CrossRefADSGoogle Scholar
  19. [19]
    S.D. Schwartz, “The interaction representation and non-adiabatic corrections to adiabatic evolution operators II: nonlinear quantum systems”, J. Chem. Phys., 104, 7985–7987, 1996.CrossRefADSGoogle Scholar
  20. [20]
    W. Magnus, “On the exponential solution of differential equations for a linear operator”, Comm. Pure and Appl. Math. VII, 649, 1954.CrossRefMathSciNetGoogle Scholar
  21. [21]
    M.D. Feit and J.A. Fleck Jr., “Solution of the schrodinger equation by a spectral method II: vibrational energy levels of triatomic molecules”, J. Chem. Phys., 78, 301, 1983.CrossRefADSGoogle Scholar
  22. [22]
    S.D. Schwartz, “Quantum activated rates — an evolution operator approach”, J. Chem. Phys., 105, 6871–6879, 1996.CrossRefADSGoogle Scholar
  23. [23]
    S.D. Schwartz, “Quantum reaction in a condensed phase — turnover behavior from new adiabatic factorizations and corrections”, J. Chem. Phys., 107, 2424–2429, 1997.CrossRefADSGoogle Scholar
  24. [24]
    D. Antoniou and S.D. Schwartz, “Proton transfer in benzoic acid crystals: another look using quantum operator theory”, J. Chem. Phys., 109, 2287–2293, 1998.CrossRefADSGoogle Scholar
  25. [25]
    D. Antoniou and S.D. Schwartz, “A Molecular dynamics quantum kramers study of proton transfer in solution”, J. Chem. Phys., 110, 465–472, 1999.CrossRefADSGoogle Scholar
  26. [26]
    D. Antoniou and S.D. Schwartz, “Quantum Proton transfer with spatially dependent friction: phenol-amine in methyl chloride”, J. Chem. Phys., 110, 7359–7364, 1999.CrossRefADSGoogle Scholar
  27. [27]
    P. Gross and S.D. Schwartz, “External field control of condensed phase reactions”, J. Chem. Phys., 109, 4843–4851, 1998.CrossRefADSGoogle Scholar
  28. [28]
    R. Karmacharya, P. Gross, and S.D. Schwartz, “The Effect of coupled nonreactive modes on laser control of quantum wavepacket dynamics”, J. Chem. Phys., 111, 6864–6868, 1999.CrossRefADSGoogle Scholar
  29. [29]
    R. Karmacharya, D. Antoniou, and S.D. Schwartz, “Nonequilibrium solvation and the quantum Kramers problem: proton transfer in aqueous glycine”, J. Phys. Chem. (Bill Miller festschrift), B105, 2563–2567, 2001.Google Scholar
  30. [30]
    D. Antoniou, S. Caratzoulas, C. Kalyanaraman, J.S. Mincer, and S.D. Schwartz, “Barrier passage and protein dynamics in enzymatically catalyzed reactions”, European Journal of Biochemistry, 269, 3103–3112, 2002.CrossRefGoogle Scholar
  31. [31]
    D. Antoniou and S.D. Schwartz, “Internal enzyme motions as a source of catalytic activity: rate promoting vibrations and hydrogen tunneling”, J. Phys. Chem., B105, 5553–5558, 2001.Google Scholar
  32. [32]
    R.A. Marcus, “Chemical and electrochemical electron transfer theory”, Ann. Rev. Phys. Chem., 15, 155–181, 1964.CrossRefADSGoogle Scholar
  33. [33]
    V. Babamov and R.A. Marcus, “Dynamics of Hydrogen Atom and Proton Transfer reactions: Symmetric Case”, J. Chem. Phys., 74, 1790, 1981.CrossRefADSGoogle Scholar
  34. [34]
    V.L. Schramm, “Enzymatic transition state analysis and transition-state analogues”, Methods in enzymology 308, 301–354, 1999.CrossRefGoogle Scholar
  35. [35]
    R.L. Schowen, Transition States of Biochemical Processes, Plenum Press, New York, 1978.Google Scholar
  36. [36]
    Wm.H. Miller, “Classical Limit Quantum Mechanics and the Theory of Molecular Collisions”, Adv. Chem. Phys., 25, 69–177, 1974.CrossRefGoogle Scholar
  37. [37]
    P. Pechukas, “Semiclassical scattering theory I”, Phys. Rev, 181, 166–173, 1969.CrossRefADSGoogle Scholar
  38. [38]
    P. Pechukas, “Semiclassical scattering theory II atomic collisions”, Phys. Rev, 181, 174–181, 1969.CrossRefADSGoogle Scholar
  39. [39]
    R.A. Marcus, “Theory of Semiclassical transition probabilities (S matrix) for inelastic and reactive collisions”, J. Chem. Phys., 54, 3965, 1971.CrossRefADSGoogle Scholar
  40. [40]
    M.C. Gutzwiller, “Chaos in classical and quantum mechanics”, Springer New York, 1990.MATHGoogle Scholar
  41. [41]
    Wm.H. Miller, “Classical S Matrix: Numerical application to inelastic collisions”, J. Chem. Phys., 53, 3578–3587, 1970.CrossRefADSGoogle Scholar
  42. [42]
    R.A. Marcus, “Theory of Semiclassical transition probabilities (S matrix) for inelastic and reactive collisions”, J. Chem. Phys., 56, 3548, 1972.CrossRefADSGoogle Scholar
  43. [43]
    M.F. Herman and E. Kluk, “A semiclassical justification for the use of nonspreading wavepackets in dynamics calculations”, Chem. Phys., 91, 27–34, 1984.CrossRefGoogle Scholar
  44. [44]
    E.J. Heller, “Frozen Gaussians: a very simple semiclassical approximation”, J. Chem. Phys., 75, 2923–2931, 1981.CrossRefMathSciNetADSGoogle Scholar
  45. [45]
    Wm.H. Miller, “On the Relation between the semiclassical initial value representation and an exact quantum expansion in time-dependent coherent States”, J. Phys. Chem. B, 106, 8132–8135, 2002.CrossRefGoogle Scholar
  46. [46]
    V.I. Filinov, Nucl. Phys., B271, 717–725, 1986.CrossRefADSGoogle Scholar
  47. [47]
    H. Wang, X. Sun, and Wm.H. Miller, “Semiclassical approximations for the calculation of thermal rate constants for chemical reactions in complex molecular systems”, J. Chem. Phys., 108, 9726–9736, 1998.CrossRefADSGoogle Scholar
  48. [48]
    D.V Shalashin and M.S. Child, “Nine-dimensional quantum molecular dynamics simulation of intramolecular vibrational energy redistribution in the CHD3 molecule with the help of coupled coherent states”, J. Chem. Phys., 119, 1961–1969, 2003.CrossRefADSGoogle Scholar
  49. [49]
    D. Bohm, “A suggested interpretation of the quantum theory in terms of “hidden” variables I”, Phys. Rev., 85, 166, 1952.CrossRefMathSciNetADSGoogle Scholar
  50. [50]
    E.R. Bittner and R.E. Wyatt, “Integrating the quantum Hamilton-Jacobi equations by wavefront expansion and phase space analysis”, J. Chem. Phys., 113, 8888–8897, 2000.CrossRefADSGoogle Scholar
  51. [51]
    J.C. Tully, “Molecular dynamics with electronic transitions”, J. Chem. Phys., 93, 1061–1071 1990.CrossRefADSGoogle Scholar
  52. [52]
    J.C. Tully, In: W.H. Miller (ed.), Dynamics of Molecular Collisions, Part B, Plenum, New York, pp. 217, 1976.Google Scholar
  53. [53]
    G. Wahnstrom and H. Metiu, “The calculation of the thermal rate coefficient by a method combining classical and quantum mechanics”, J. Chem. Phys., 88, 2478–2491, 1988.CrossRefADSGoogle Scholar
  54. [54]
    S. Hammes-Schiffer and J.C. Tully, “Proton transfer in solution: molecular dynamics with quantum transitions”, J. Chem. Phys., 101, 4657–4667, 1994.CrossRefADSGoogle Scholar
  55. [55]
    N. Yu, C.J. Margulis, and D.F. Coker, “Influence of solvation environment on excited state avoided crossings and photo-dissociation dynamics”, J. Phys. Chem. B, 105, 6728–2737, 2001.CrossRefGoogle Scholar
  56. [56]
    C.J. Margulis and D.F. Coker, “Nonadiabatic molecular dynamics simulations of photofragmentation and geminate recombination dynamics in size-selected I2-(CO2)n cluster ions”, J. Chem. Phys., 110, 5677–5690, 1999.CrossRefADSGoogle Scholar
  57. [57]
    D.F. Coker and L. Xiao, “Methods for molecular dynamics with non-adiabatic transitions”, J. Chem. Phys., 102, 496–510, 1995.CrossRefADSGoogle Scholar
  58. [58]
    H.S. Mei and D.F. Coker, “Quantum molecular dynamics studies of H2 transport in water”, J. Chem. Phys., 104, 4755–4767, 1996.CrossRefADSGoogle Scholar
  59. [59]
    S. Nielsen, R. Kapral, and G. Ciccotti, “Mixed quantum-classical surface hopping dynamics”, J. Chem. Phys., 112, 6543–6553, 2000.CrossRefADSGoogle Scholar
  60. [60]
    B.J. Schwartz, E.R. Bittner, O.V. Prezdo, and P.J. Rossky, “Quantum decoherence and the isotope effect in condensed phase nonadiabatic molecular dynamics simulations”, J. Chem. Phys., 104, 5942–5955, 1996.CrossRefADSGoogle Scholar
  61. [61]
    S.R. Billeter, S.P. Webb, P.K. Agarwal, T. Iordanov and S. Hammes-Schiffer, “Hydride transfer in liver alcohol dehydrogenase: quantum dynamics, kinetic isotope effects, and role of enzyme motion”, J.A.C.S., 123, 11262–11272, 2001.CrossRefGoogle Scholar
  62. [62]
    P.K. Agarwal, S.R. Billeter, and S. Hammes Schiffer, “Nuclear quantum effects and enzyme dynamics in dihydrofolate reductase catalysis”, J. Phys. Chem. B, 106, 3238–3293, 2002.CrossRefGoogle Scholar
  63. [63]
    R.M. Nicoll, I. Hillier, D.G. Truhlar, “Quantum mechanical dynamics of hydride transfer in polycyclic hydroxy keytones in the condensed phase”, J.A.C.S., 123, 1459–1463, 2001.CrossRefGoogle Scholar
  64. [64]
    C. Alhambra, J.C. Corchado, M.L. Sanchez, J. Gao, and D.G. Truhlar, “Quantum dynamics of hydride transfer in enzyme catalysis”, J.A.C.S., 122, 8197–8203, 2000.CrossRefGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Steven D. Schwartz
    • 1
  1. 1.Departments of Biophysics and BiochemistryAlbert Einstein College of MedicineNew YorkUSA

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