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Review of Feynman’s Path Integral in Quantum Statistics: from the Molecular Schrödinger Equation to Kleinert’s Variational Perturbation Theory

Published online by Cambridge University Press:  03 June 2015

Kin-Yiu Wong
Kin-Yiu Wong*
Affiliation:
Department of Physics, High Performance Cluster Computing Centre, Institute of Computational and Theoretical Studies, Hong Kong Baptist University, 224 Waterloo Road, Kowloon Tong, Hong Kong Institute of Research and Continuing Education, Hong Kong Baptist University (Shenzhen)
*
Corresponding author.Email:wongky@hkbu.edu.hk

Abstract

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Feynman’s path integral reformulates the quantum Schrödinger differential equation to be an integral equation. It has been being widely used to compute internuclear quantum-statistical effects on many-body molecular systems. In this Review, the molecular Schrödinger equation will first be introduced, together with the Born-Oppenheimer approximation that decouples electronic and internuclear motions. Some effective semiclassical potentials, e.g., centroid potential, which are all formulated in terms of Feynman’s path integral, will be discussed and compared. These semiclassical potentials can be used to directly calculate the quantum canonical partition function without individual Schrödinger’s energy eigenvalues. As a result, path integrations are conventionally performed with Monte Carlo and molecular dynamics sampling techniques. To complement these techniques, we will examine how Kleinert’s variational perturbation (KP) theory can provide a complete theoretical foundation for developing non-sampling/non-stochastic methods to systematically calculate centroid potential. To enable the powerful KP theory to be practical for many-body molecular systems, we have proposed a new path-integral method: automated integration-free path-integral (AIF-PI) method. Due to the integration-free and computationally inexpensive characteristics of our AIF-PI method, we have used it to perform ab initio path-integral calculations of kinetic isotope effects on proton-transfer and RNA-related phosphoryl-transfer chemical reactions. The computational procedure of using our AIF-PI method, along with the features of our new centroid path-integral theory at the minimum of the absolute-zero energy (AMAZE), are also highlighted in this review.

Keywords

05.30.-d03.65.-w82.20.-w87.15.A-02.30.RzFeynman’s path integralKleinert’s variational perturbation theorySchrödinger’s equationquantum statisticscentroid potential energyquantum tunnelingzero-point energyisotope effects
Type
Review Article
Information
Communications in Computational Physics , Volume 15 , Issue 4 , April 2014 , pp. 853 - 894
Copyright
Copyright © Global Science Press Limited 2014

References

[1]Kleppner, D., Jackiw, R., Pathways of discovery: One hundred years of quantum physics, Science, 289 (2000) 893–898.CrossRefGoogle Scholar
[2]Gutzwiller, M.C., Resource letter ICQM-1: The interplay between classical and quantum mechanics, Am. J. Phys., 1998, pp. 304–324.Google Scholar
[3]Tanner, G., Richter, K., Rost, J.-M., The theory of two-electron atoms: between ground state and complete fragmentation, Rev. Mod. Phys., 72 (2000) 497–544.Google Scholar
[4]Kohn, W., Nobel lecture: Electronic structure of matter-wave functions and density function-als, Rev. Mod. Phys., 71 (1999) 1253–1266.Google Scholar
[5]Pople, J.A., Nobel lecture: Quantum chemical models, Rev. Mod. Phys., 71 (1999) 1267–1274.Google Scholar
[6]Dirac, P.A.M., Quantum mechanics of many-electron systems, Proc. R. Soc. London A, A123 (1929) 714–733.Google Scholar
[7]Lewis, G.N., The chemical bond, J. Chem. Phys., 1 (1933) 17–28.Google Scholar
[8]Hund, F., The History of Quantum Theory, Harrap, London, 1974.Google Scholar
[9]Weinberger, P., Revisiting Louis de Broglie’s famous 1924 paper in the Philosophical Magazine, Philosophical Magazine Letters, 86 (2006) 405–410.Google Scholar
[10]de Broglie, L., Tentative theory of light quanta, Philosophical Magazine Letters, 2006, pp. 411–423.Google Scholar
[11]de Broglie, L.Recherches sur la théorie des quanta (Researches on the quantum theory); PhD Thesis, Paris, 1924.Google Scholar
[12]Schrödinger, E., Undulatory theory of the mechanics of atoms and molecules, Phys. Rev., 28 (1926) 1049–70.CrossRefGoogle Scholar
[13]Schrödinger, E., Quantisation as a problem of proper values (Part III): Perturbation theory, with application to the Stark effect of the Balmer lines, Annalen der Physik, 1926, pp. 437–476.Google Scholar
[14]Heisenberg, W., The Physical Principles of the Quantum Theory, Dover Publications, New York, 1949. Translated into English by Eckart, C. and Hoyt, F.C.Google Scholar
[15]Heisenberg, W., Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik (The Physical Content of Quantum Kinematics and Mechanics), Zeitschrift für Physik, 1927, pp. 172–198. English translation: Wheeler, J. A. and Zurek, H., Quantum Theory and Measurement, Princeton Univ. Press, 1983, pp. 62–84. Google Scholar
[16]Heisenberg, W., Uber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen (Quantum-Theoretical Re-interpretation of Kinematic and Mechanical Relations), Zeitschrift für Physik, 1925, pp. 879–893.CrossRefGoogle Scholar
[17]Born, M., Jordan, P., Zur Quantenmechanik (On Quantum Mechanics), Zeitschrift für Physik, 1925, pp. 858–888.Google Scholar
[18]Born, M., Heisenberg, W., Jordan, P., Zur Quantenmechanik II (On Quantum Mechanics II), Zeitschrift für Physik, 1925, pp. 557–615.Google Scholar
[19]Born, M., Oppenheimer, J.R., Zur Quantentheorie der Molekeln (On the Quantum Theory of Molecules), Annalen der Physik, 1927, pp. 457–484.Google Scholar
[20]Schrödinger, E., Collected Papers on Wave Mechanics: Together with His Four Lectures on Wave Mechanics, 3rd ed., Chelsea Publishing, New York, 1982.Google Scholar
[21]van der Waerden, B.L., Sourcesof Quantum Mechanics, Dover Publications, New York, 1968, pp. xi, 430 p.Google Scholar
[22]Wheeler, J.A., Zurek, W.H., Quantum Theory and Measurement, Princeton Series in Physics, Princeton University Press, Princeton, N.J., 1983, pp. xxviii, 811 p.Google Scholar
[23]Mehra, J., Rechenberg, H., The historical development of quantum theory, Springer-Verlag, New York, 1982–2001.Google Scholar
[24]Hettema, H., Quantum chemistry: Classic scientific papers, World scientific series in 20th century chemistry; Vol. 8, World Scientific, Singapore; London, 2000, pp. xxxix, 478 p.Google Scholar
[25]Szabo, A., Ostlund, N.S., Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, 2nd ed., Dover Publications, Mineola, N.Y., 1996.Google Scholar
[26]Springborg, M., Methods of Electronic-Structure Calculations: From Molecules to Solids, Wiley, Chichester; New York, 2000.Google Scholar
[27]Hehre, W.J., Radom, L., Schleyer, P.v.R., Pople, J.A., Ab Initio Molecular Orbital Theory, Wiley, New York, 1986.Google Scholar
[28]Dean, D.J., Beyond the nuclear shell model, Physics Today, 60 (2007) 48–53.Google Scholar
[29]Helgaker, T., Jørgensen, P., Olsen, J., Molecular electronic-structure theory, Wiley, Chichester; New York, 2000.Google Scholar
[30]Cohen, E.R., Cvitaš, T., Frey, J.G., Holmström, B., Kuchitsu, K., Marquardt, R., Mills, I., Pavese, F., Quack, M., Stohner, J., Strauss, H., Takami, M., Thor, A.J., Quantities, Units and Symbols in Physical Chemistry, 3rd ed., Royal Society of Chemistry, Cambridge, 2007.Google Scholar
[31]McQuarrie, D.A., Statistical Mechanics, University Science Books,Sausalito, Calif., 2000.Google Scholar
[32]Feynman, R.P., Hibbs, A.R., Styer, D.F., Quantum Mechanics and Path Integrals, Emended ed., Dover Publications, Mineola, N.Y., 2005.Google Scholar
[33]Feynman, R.P., Statistical Mechanics; a Set of Lectures, Benjamin, W. A., Reading, Mass., 1972.Google Scholar
[34]Kleinert, H., Path Integralsin Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 3rd ed., World Scientific, Singapore; River Edge, NJ, 2004.Google Scholar
[35]Ballhausen, C.J., Hansen, A.E., Electronic spectra, Annu. Rev. Phys. Chem., 23 (1972) 15–38.Google Scholar
[36]Kolos, W., Adiabatic approximation and its accuracy, Adv. Quantum Chem., 5 (1970) 99–133.Google Scholar
[37]Hirschfelder, J.O., Meath, W.J., Nature of intermolecular forces, Adv. Chem. Phys., 12 (1967) 3–106.Google Scholar
[38]Mielke, S.L., Peterson, K.A., Schwenke, D.W., Garrett, B.C., Truhlar, D.G., Michael, J.V., Su, M.-C., Sutherland, J.W., H+H2 Thermal Reaction: A Convergence of Theory and Experiment, Phys. Rev. Lett., 91 (2003) 063201.CrossRefGoogle ScholarPubMed
[39]Parr, R.G., Yang, W., Density-Functional Theory of Atoms and Molecules, Oxford University Press; Clarendon Press, New York; Oxford, UK, 1989.Google Scholar
[40]Wong, K.-Y., Lee, T.-S., York, D.M., Active participation of the Mg2+ ion in the reaction coordinate of RNA self-cleavage catalyzed by the hammerhead ribozyme, J. Chem. Theory Comput., 2011, pp. 1–3.Google Scholar
[41]Wong, K.-Y., Gao, J., Insight into the phosphodiesterase mechanism from combined QM/MM free energy simulations, FEBS Journal, 278 (2011) 2579–2595.Google Scholar
[42]Wong, K.-Y., Gao, J., The reaction mechanism of paraoxon hydrolysis by phosphotriesterase from combined QM/MM simulations, Biochemistry, 46 (2007) 13352–13369.Google Scholar
[43]Wu, E.L., Wong, K.-Y., Zhang, X., Han, K., Gao, J., Determination of the structure form of the fourth ligand of zinc in acutolysin A using combined quantum mechanical and molecular mechanical simulation, J. Phys. Chem. B, 113 (2009) 2477–2485.Google Scholar
[44]Wong, K.Y., Smallfield, J.A.O., Fahlman, M., Epstein, A.J., Electronic state of nitrogen containing polypyridine at the interfaces with model sulfonic acid containing polymer and molecule, Synth. Met., 137 (2003) 1031–1032.Google Scholar
[45]Wong, K.Y., Lo, C.F., Sham, W.Y., Fong, H.H., So, S.K., Leung, L.M., Theoretical investigation of a blue hydroxyquinaldine-based aluminum(III) complex, Phys. Lett. A, 321 (2004) 194–198.Google Scholar
[46]Hagler, A.T., Huler, E., Lifson, S., Energy functions for peptides and proteins. I. Derivation of a consistent force field including the hydrogen bond from amide crystals, J. Am. Chem. Soc., 96 (1974) 5319–5327.Google Scholar
[47]Brooks, B.R., Bruccoleri, R.E., Olafson, B.D., States, D.J., Swaminathan, S., Karplus, M., CHARMM: A program for macromolecular energy, minimization, and dynamics calculations, J. Comput. Chem., 4 (1983) 187–217.Google Scholar
[48]Weiner, S.J., Kollman, P.A., Case, D.A., Singh, U.C., Ghio, C., Alagona, G., Profeta, S., Jr., Weiner, P., A new force field for molecular mechanical simulation of nucleic acids and proteins, J. Am. Chem. Soc., 106 (1984) 765–784.Google Scholar
[49]Jorgensen, W.L., Tirado-Rives, J., The OPLS [optimized potentials for liquid simulations] potential functions for proteins, energy minimizations for crystals of cyclic peptides and cram-bin, J. Am. Chem. Soc., 110 (1988) 1657–1666.Google Scholar
[50]Mayo, S.L., Olafson, B.D., Goddard, W.A., III, DREIDING: A generic force field for molecular simulations, J. Phys. Chem., 94 (1990) 8897–8909.Google Scholar
[51]Gao, J., Truhlar, D.G., Quantum mechanical methods for enzyme kinetics, Annu. Rev. Phys. Chem., 53 (2002) 467–505.Google Scholar
[52]Field, M.J., Bash, P.A., Karplus, M., A combined quantum mechanical and molecular mechanical potential for molecular dynamics simulations, J. Comput. Chem., 11 (1990) 700–733.CrossRefGoogle Scholar
[53]Wong, K.-Y., York, D.M., Exact relation between potential of mean force and free-energy profile, J. Chem. Theory Comput., 8 (2012) 3998–4003.Google Scholar
[54]Tanaka, H., Kanoh, H., Yudasaka, M., Iijima, S., Kaneko, K., Quantum effects on hydrogen isotope adsorption on single-wall carbon nanohorns, J. Am. Chem. Soc., 127 (2005) 7511–7516.Google Scholar
[55]Kowalczyk, P., Gauden, P.A., Terzyk, A.P., Bhatia, S.K., Thermodynamics of Hydrogen adsorption in slit-like carbon nanopores at 77K. Classical versus path-integral Monte Carlo simulations, Langmuir, 23 (2007) 3666–3672.Google Scholar
[56]Kowalczyk, P., Gauden, P.A., Terzyk, A.P., Cryogenic Separation of hydrogen isotopes in single-walled carbon and boron-nitride nanotubes: Insight into the mechanism of equilibrium quantum sieving in quasi-one-dimensional pores, J. Phys. Chem. B, 112 (2008) 8275–8284.Google Scholar
[57]Gao, J., Major, D.T., Fan, Y., Lin, Y.-l., Ma, S., Wong, K.-Y., Hybrid quantum and classical methods for computing kinetic isotope effects of chemical reactions in solutions and in enzymes, in: Kukol, A. (Ed.), Molecular Modeling of Proteins, Humana Press, 2008, pp. 37–62.CrossRefGoogle Scholar
[58]Major, D.T., Heroux, A., Orville, A.M., Valley, M.P., Fitzpatrick, P.F., Gao, J., Differential quantum tunneling contributions in nitroalkane oxidase catalyzed and the uncatalyzed proton transfer reaction, Proc. Natl. Acad. Sci. U.S.A., 106 (2009) 20734–20739.CrossRefGoogle ScholarPubMed
[59]Warshel, A., Olsson, M.H.M., Villa-Freixa, J., Computer simulations of isotope effects in enzyme catalysis, in: Kohen, A., Limbach, H.-H. (Eds.), Isotope Effects in Chemistry and Biology, Taylor & Francis, Boca Raton, 2006, pp. 621–644.Google Scholar
[60]Brown, L.M., Feynmanʹs Thesis: A New Approach to Quantum Theory, World Scientific, Singapore; Hackensack, NJ, 2005.Google Scholar
[61]Feynman, R.P., Space-time approachtonon-relativistic quantum mechanics, Rev. Mod. Phys., 20 (1948) 367–387.Google Scholar
[62]Feynman, R.P., The development of the space-time view of quantum electrodynamics, Science, 153 (1966) 699–708.Google Scholar
[63]Kac, M., Probability and related Topics in Physical Sciences, Interscience Publishers, London; New York, 1959 Chapter IV.Google Scholar
[64]Kac, M., On distributions of certain Wiener functionals, Transactions of the American Mathematical Society, 65 (1949) 1–13.Google Scholar
[65]Shankar, R., Principles of Quantum Mechanics, 2nd ed., Plenum Press, New York, 1994.Google Scholar
[66]Chaichian, M., Demichev, A.P., Path Integrals in Physics, Philadelphia, PA, Bristol, UK, 2001.Google Scholar
[67]Schulman, L.S., Techniques and Applications of Path Integration, Wiley, New York, 1981.Google Scholar
[68]Dirac, P.A.M., The Principles of Quantum Mechanics, 4th ed., Clarendon Press, Oxford, England, 1981.Google Scholar
[69]Dirac, P.A.M., The Lagrangian in quantum mechanics, Physikalische Zeitschrift der Sowjetunion, 1933, pp. 64–72.Google Scholar
[70]Putz, M.V., Path integrals for electronic densities, reactivity indices, and localization functions in quantum systems, Int. J. Mol. Sci., 10 (2009) 4816–4940.Google Scholar
[71]Putz, M.V., On Heisenberg Uncertainty Relationship, its extension, and the quantum issue of wave-particle duality, Int. J. Mol. Sci., 11 (2010) 4124–4139.Google ScholarPubMed
[72]Kleinert, H., Pelster, A., Putz, M.V., Variational perturbation theory for Markov processes, Phys. Rev. E, 65 (2002) 066128/1-066128/7.Google Scholar
[73]Putz, M.V., Markovian approach of the electron localization functions, Int. J. Quantum Chem., 105 (2005) 1–11.CrossRefGoogle Scholar
[74]Putz, M.V., Semiclassical electronegativity and chemical hardness, J. Theor. Comput. Chem., 6 (2007) 33–47.Google Scholar
[75]Putz, M.V., Russo, N., Sicilia, E., About the Mulliken electronegativity in DFT, Theor. Chem. Acct., 114 (2005) 38–45.CrossRefGoogle Scholar
[76]Goldstein, H., Poole, C.P., Safko, J.L., Classical Mechanics, 3rd ed., Addison Wesley, San Francisco, 2002.Google Scholar
[77]Wigner, E., On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932) 749–759.Google Scholar
[78]Neumann, M., Zoppi, M., Asymptotic expansions and effective potentials for almost classical N-body systems, Phys. Rev. A, 40 (1989) 4572–84.Google Scholar
[79]Hillery, M., ƠConnell, R.F., Scully, M.O., Wigner, E.P., Distribution functions in physics: fundamentals, Phys. Rep., 106 (1984) 121–167.Google Scholar
[80]Kirkwood, J.G., Quantum statistics of almost classical assemblies, Phys. Rev., 44 (1933) 31–37.CrossRefGoogle Scholar
[81]Fujiwara, Y., Osborn, T.A., Wilk, S.F.J., Wigner-Kirkwood expansions, Phys. Rev. A, 25 (1982) 14.Google Scholar
[82]Hornstein, S.M., Miller, W.H., Quantum corrections (within the classic path approximation) to the Boltzmann density matrix, Chem. Phys. Lett., 13 (1972) 298–300.Google Scholar
[83]Miller, W.H., Improved classical path approximation for the Boltzmann density matrix, J. Chem. Phys., 58 (1973) 1664–1667.Google Scholar
[84]Gillan, M.J., Quantum simulation of hydrogen in metals, Phys. Rev. Lett., 58 (1987) 563–6.Google Scholar
[85]Gillan, M.J., Quantum-classical crossover of the transition rate in the damped double well, Journal of Physics C: Solid State Physics, 20 (1987) 3621–3641.CrossRefGoogle Scholar
[86]Voth, G.A., Feynman path integral formulation of quantum mechanical transition-state theory, J. Phys. Chem., 97 (1993) 8365–8377.Google Scholar
[87]Cao, J., Voth, G.A., The formulation of quantum statistical mechanics based on the Feynman path centroid density. IV. Algorithms for centroid molecular dynamics, J. Chem. Phys., 101 (1994) 6168–83.Google Scholar
[88]Voth, G.A., Path-integral centroid methods in quantum statistical mechanics and dynamics, Adv. Chem. Phys., 93 (1996) 135–218.Google Scholar
[89]Ramírez, R., Loópez-Ciudad, T., Noya, J.C., Feynman effective classical potential in the Schrödinger formulation, Phys. Rev. Lett., 1998, pp. 3303–3306. Comment: Andronico, G.; Branchina, V.; Zappala, Phys, D. Rev. Lett. 2002, 88, 178901; Reply to comment: Ramirez, R.; López-Ciudad, Phys, T. Rev. Lett., 2002, 88, 178902.CrossRefGoogle Scholar
[90]Zwanzig, R.W., High-temperature equation of state by a perturbation method. I. Nonpolar gases, J. Chem. Phys., 22 (1954) 1420–1426.Google Scholar
[91]Kubo, R., Generalized cumulant expansion method, J. Phys. Soc. Jpn., 17 (1962) 1100–1120.Google Scholar
[92]Ramírez, R., Loópez-Ciudad, T., The Schrödinger formulation of the Feynman path centroid density, J. Chem. Phys., 111 (1999) 3339–3348.Google Scholar
[93]Steinfeld, J.I., Francisco, J.S., Hase, W.L., Chemical Kinetics and Dynamics, 2nd ed., Prentice Hall, Upper Saddle River, N.J., 1999.Google Scholar
[94]Kreevoy, M.M., Truhlar, D.G., Transition state theory, in: Bernasconi, C.F. (Ed.), Techniques of Chemistry: Investigation of Rates and Mechanisms of Reactions, Wiley, New York, 1986, pp. 13–95.Google Scholar
[95]Garrett, B.C., Perspective on ˝the transition state method˝, Theor. Chem. Acct., 103 (2000) 200–204.Google Scholar
[96]Petersson, G.A., Perspective on ˝The activated complex in chemical reactions˝, Theor. Chem. Acct., 103 (2000) 190–195.Google Scholar
[97]Truhlar, D.G., Hase, W.L., Hynes, J.T., Current status of transition-state theory, J. Phys. Chem., 87 (1983) 2664–82.Google Scholar
[98]Truhlar, D.G., Garrett, B.C., Klippenstein, S.J., Current status of transition-state theory, J. Phys. Chem., 100 (1996) 12771–12800.Google Scholar
[99]Eyring, H., Activated complex in chemical reactions, J. Chem. Phys., 3 (1935) 107–15.Google Scholar
[100]Kassel, L.S., Statistical mechanical treatment of the activated complex in chemical reactions, J. Chem. Phys., 3 (1935) 399–400.CrossRefGoogle Scholar
[101]Evans, M.G., Polanyi, M., Application of the transition-state method to the calculation of reaction velocities, especially in solution, Transactions of the Faraday Society, 31 (1935) 875–94.Google Scholar
[102]Wigner, E., The transition-state method, Transactions of the Faraday Society, 34 (1938) 29–41.Google Scholar
[103]Eyring, H., Stearn, A.E., The application of the theory of absolute reaction rates to proteins, Chem. Rev., 24 (1939) 253–70.Google Scholar
[104]Glasstone, S., Laidler, K.J.,H. Eyring, , The Theory of Rate Processes: The Kinetics of Chemical Reactions, Viscosity, Diffusion and Electrochemical Phenomena, 1st ed., McGraw-Hill Book Company, Inc., New York; London, 1941.Google Scholar
[105]Griffiths, D.J., Introduction to Quantum Mechanics, Prentice Hall, Englewood Cliffs, N.J., 1995.Google Scholar
[106]Bell, R.P., Tunnel-effect correction for parabolic potential barriers, Transactions of the Faraday Society, 55 (1959) 1–4.Google Scholar
[107]Bell, R.P., The Tunnel Effect in Chemistry, Chapman and Hall, London; New York, 1980.Google Scholar
[108]Johnston, H.S., Gas Phase Reaction Rate Theory, Ronald Press Co., New York, 1966.Google Scholar
[109]Johnston, H.S., Rapp, D., Large tunnelling corrections in chemical reaction rates. II, J. Am. Chem. Soc., 83 (1961) 1–9.Google Scholar
[110]Voth, G.A., Chandler, D., Miller, W.H., Rigorous formulation of quantum transition state theory and its dynamical corrections, J. Chem. Phys., 91 (1989) 7749–60.Google Scholar
[111]Jang, S., Schwieters, C.D., Voth, G.A., A modification of path integral quantum transition state theory for asymmetric and metastable potentials, J. Phys. Chem. A, 103 (1999) 9527–9538.Google Scholar
[112]Jang, S., Voth, G.A., A relationship between centroid dynamics and path integral quantum transition state theory, J. Chem. Phys., 2000, pp. 8747–8757.Google Scholar
[113]Yeon, K.H., Zhang, S., Kim, Y.D., Um, C.I., George, T.F., Quantum solutions for the harmonic-parabola potential system, Phys. Rev. A, 61 (2000) 042103/1-042103/15.Google Scholar
[114]Marx, D., Parrinello, M., Structural quantum effects and three-center two-electron bonding in CH5+, Nature (London), 375 (1995) 216–18.Google Scholar
[115]Tuckerman, M.E., Marx, D., Parrinello, M., The nature and transport mechanism of hydrated hydroxide ions in aqueous solution, Nature (London), 417 (2002) 925-929.Google Scholar
[116]Tuckerman, M.E., Marx, D., Klein, M.L., Parrinello, M., On the quantum nature of the shared proton in hydrogen bonds, Science, 275 (1997) 817–820.Google Scholar
[117]Marx, D., Tuckerman, M.E., Martyna, G.J., Quantum dynamics via adiabaticab initio centroid molecular dynamics, Comput. Phys. Commun., 118 (1999) 166–184.Google Scholar
[118]Paesani, F., Iuchi, S., Voth, G.A., Quantum effects in liquid water from an ab initio-based polarizable force field, J. Chem. Phys., 127 (2007) 074506.Google Scholar
[119]Ohta, Y., Ohta, K., Kinugawa, K., Ab initio centroid path integral molecular dynamics: Application to vibrational dynamics of diatomic molecular systems, J. Chem. Phys., 120 (2004) 312–320.Google Scholar
[120]Hayashi, A., Shiga, M., Tachikawa, M., H/D isotope effect on the dihydrogen bond of NH4+.BeH2 by ab initio path integral molecular dynamics simulation, J. Chem. Phys., 125 (2006) 204310.Google Scholar
[121]Poulsen, J.A., Nyman, G., Rossky, P.J., Practical evaluation of condensed phase quantum correlation functions: A Feynman-Kleinert variational linearized path integral method, J. Chem. Phys., 119 (2003) 12179–12193.Google Scholar
[122]Poulsen, J.A., Nyman, G., Rossky, P.J., Feynman-Kleinert linearized path integral (FK-LPI) algorithms for quantum molecular dynamics, with application to water and He(4), J. Chem. Theory Comput., 2 (2006) 1482–1491.Google Scholar
[123]Poulsen, J.A., Scheers, J., Nyman, G., Rossky, P.J., Quantum density fluctuations in liquid neon from linearized path-integral calculations, Phys. Rev. B, 75 (2007) 224505.Google Scholar
[124]Coker, D.F., Bonella, S., Linearized nonadiabatic dynamics in the adiabatic representation, in: Micha, D.A., Burghardt, I. (Eds.), Quantum Dynamics of Complex Molecular Systems, Springer Series in Chemical Physics, Springer, New York, 2007, pp. 321–342.CrossRefGoogle Scholar
[125]Gao, J., Wong, K.-Y., Major, D.T., Combined QM/MM and path integral simulations of kinetic isotope effects in the proton transfer reaction between nitroethane and acetate ion in water, J. Comput. Chem., 29 (2008) 514–522.Google Scholar
[126]Wang, M., Lu, Z., Yang, W., Nuclear quantum effects on an enzyme-catalyzed reaction with reaction path potential: Proton transfer in triosephosphate isomerase, J. Chem. Phys., 124 (2006) 124516.Google Scholar
[127]Wang, Q., Hammes-Schiffer, S., Hybrid quantum/classical path integral approach for simulation of hydrogen transfer reactions in enzymes, J. Chem. Phys., 125 (2006) 184102.Google Scholar
[128]Major, D.T., Gao, J., A combined quantum mechanical and molecular mechanical study of the reaction mechanism and a-amino acidity in alanine racemase, J. Am. Chem. Soc., 128 (2006) 16345–16357.Google Scholar
[129]Chakrabarti, N., Carrington, T., Jr., , Roux, B., Rate constants in quantum mechanical systems: A rigorous and practical path-integral formulation for computer simulations, Chem. Phys. Lett., 293 (1998) 209–220.CrossRefGoogle Scholar
[130]Field, M.J., Albe, M., Bret, C., MM, F.P.-D., Thomas, A., The dynamo library for molecular simulations using hybrid quantum mechanical and molecular mechanical potentials, J. Comput. Chem., 21 (2000) 1088–1100.Google Scholar
[131]Sauer, T., The Feynman path goes Monte Carlo, in: Janke, W., Pelster, A., Schmidt, H.-J., Bachmann, M. (Eds.), Fluctuating Paths and Fields: Festschrift Dedicated to Hagen Kleinert on the Occasion of His 60th Birthday, World Scientific, River Edge, NJ, 2001, pp. 29–42.Google Scholar
[132]Fosdick, L.D., Numerical estimation of the partition function in quantum statistics, J. Math. Phys., 3 (1962) 1251–1264.CrossRefGoogle Scholar
[133]Fosdick, L.D., The Monte Carlo method in quantum statistics, SIAM Rev., 10 (1968) 315–328.Google Scholar
[134]Morita, T., Solution of the Bloch equation for many-particle systems in terms of the path integral, J. Phys. Soc. Jpn., 35 (1973) 980–4.Google Scholar
[135]Barker, J.A., A quantum-statistical Monte Carlo method; path integrals with boundary conditions, J. Chem. Phys., 70 (1979) 2914–18.Google Scholar
[136]MacKeown, P.K., Evaluation of Feynman path integrals by Monte Carlo methods, Am. J. Phys., 53 (1985) 880–885.Google Scholar
[137]Chandler, D., Wolynes, P.G., Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids, J. Chem. Phys., 74 (1981) 4078–95.Google Scholar
[138]Berne, B.J., Thirumalai, D., On the simulation of quantum systems: path integral methods, Annu. Rev. Phys. Chem., 37 (1986) 401–424.Google Scholar
[139]Ceperley, D.M., Path integrals in the theory of condensed helium, Rev. Mod. Phys., 67 (1995) 279–355.Google Scholar
[140]Mielke, S.L., Truhlar, D.G., A new Fourier path integral method, a more general scheme for extrapolation, and comparison of eight path integral methods for the quantum mechanical calculation of free energies, J. Chem. Phys., 114 (2001) 621–630.Google Scholar
[141]Coalson, R.D., On the connection between Fourier coefficient and Discretized Cartesian path integration, J. Chem. Phys., 85 (1986) 926–936.CrossRefGoogle Scholar
[142]Cao, J., Voth, G.A., The formulation of quantum statistical mechanics based on the Feynman path centroid density. I. Equilibrium properties, J. Chem. Phys., 100 (1994) 5093–105.Google Scholar
[143]Cao, J., Voth, G.A., The formulation of quantum statistical mechanics based on the Feynman path centroid density. II. Dynamic properties, J. Chem. Phys., 100 (1994) 5106–18.Google Scholar
[144]Cao, J., Voth, G.A., The formulation of quantum statistical mechanics based on the Feynman path centroid density. III. Phase space formalism and analysis of centroid molecular dynamics, J. Chem. Phys., 101 (1994) 6157–67.Google Scholar
[145]Cao, J., Voth, G.A., The formulation of quantum statistical mechanics based on the Feynman path centroid density. V. Quantum instantaneous normal mode theory of liquids, J. Chem. Phys., 101 (1994) 6184–92.Google Scholar
[146]Anderson, J.B., Random-walk simulation of the Schrödinger equation. Hydrogen ion (H3+), J. Chem. Phys., 63 (1975) 1499–503.Google Scholar
[147]Lester, W.A., Salomon-Ferrer, R., Some recent developments in quantum Monte Carlo for electronic structure: Methods and application to a bio system, THEOCHEM, 771 (2006) 51–54.Google Scholar
[148]Grossman, J.C., Mitas, L., Efficient quantum Monte Carlo energies for molecular dynamics simulations, Phys. Rev. Lett., 94 (2005) 056403/1-056403/4.Google Scholar
[149]Wagner, L.K., Bajdich, M., Mitas, L., QWalk: A quantum Monte Carlo program for electronic structure, J. Comput. Phys., 228 (2009) 3390–3404.Google Scholar
[150]Wong, K.-Y., Richard, J.P., Gao, J., Theoretical analysis of kinetic isotope effects on proton transfer reactions between substituted alpha-methoxystyrenes and substituted acetic acids, J. Am. Chem. Soc., 131 (2009) 13963–13971.Google Scholar
[151]Wong, K.-Y., Gu, H., Zhang, S., Piccirilli, J.A., Harris, M.E., York, D.M., Characterization of the reaction path and transition states for RNA transphosphorylation models from theory and experiment, Angew. Chem. Int. Ed., 2012, pp. 647–651.Google Scholar
[152]Doll, J.D., Myers, L.E., Semiclassical Monte Carlo methods, J. Chem. Phys., 71 (1979) 2880–3.Google Scholar
[153]Mielke, S.L., Truhlar, D.G., Displaced-points path integral method for including quantum effects in the Monte Carlo evaluation of free energies, J. Chem. Phys., 115 (2001) 652–662.Google Scholar
[154]Giachetti, R., Tognetti, V., Variational approach to quantum statistical mechanics of nonlinear systems with application to sine-Gordon chains, Phys. Rev. Lett., 55 (1985) 912–15.Google Scholar
[155]Feynman, R.P., Kleinert, H., Effective classical partition functions, Phys. Rev. A, 34 (1986) 5080–5084.Google Scholar
[156]Wong, K.-Y., Gao, J., An automated integration-free path-integral method basedon Kleinert’s variational perturbation theory, J. Chem. Phys., 127 (2007) 211103.Google Scholar
[157]Wong, K.-Y., Gao, J., Systematic approach for computing zero-point energy, quantum partition function, and tunneling effect based on Kleinertťs variational perturbation theory, J. Chem. Theory Comput., 4 (2008) 1409–1422.Google Scholar
[158]Wong, K.-Y.Simulating biochemical physics with computers: 1. Enzyme catalysis by phosphotriesterase and phosphodiesterase; 2. Integration-free path-integral method for quantum-statistical calculations; Ph.D. thesis, University of Minnesota (USA), Minneapolis, 2008.Google Scholar
[159]Stratt, R.M., The instantaneous normal modes of liquids, Acc. Chem. Res., 28 (1995) 201–7.Google Scholar
[160]Deng, Y., Ladanyi, B.M., Stratt, R.M., High-frequency vibrational energy relaxation in liquids: The foundations of instantaneous-pair theory and some generalizations, J. Chem. Phys., 117 (2002) 10752–10767.Google Scholar
[161]Wong, K.-Y., Developing a systematic approach for ab initio path-integral simulations, in: L. Wang (Ed.), Molecular Dynamics/Book 1 – Theoretical Developments and Applications in Nanotechnology and Energy, InTech, 2012, pp. 107–132.Google Scholar
[162]Gu, H., Zhang, S., Wong, K.-Y., Radak, B. K., Dissanayake, T., Kellerman, D. L., Dai, Q., Miyagi, M., Anderson, V. E., York, D. M., Piccirilli, J. A., Harris, M. E., Experimental and computational analysis of the transition state for ribonuclease A catalyzed RNA 2′-O-transphosphorylation, Proc. Natl. Acad. Sci. U.S.A., 110, (2013), 13002–13007.Google Scholar