Simulating the physical world; hierarchical modeling from quantum mechanics to fluid dynamics

This page intentionally left blank www.pdfgrip.com SIMULATING THE PHYSICAL WORLD The simulation of physical systems requires a simplified, hierarchical approach, which models each level from the atomistic to the macroscopic scale From quantum mechanics to fluid dynamics, this book systematically treats the broad scope of computer modeling and simulations, describing the fundamental theory behind each level of approximation Berendsen evaluates each stage in relation to their applications giving the reader insight into the possibilities and limitations of the models Practical guidance for applications and sample programs in Python are provided With a strong emphasis on molecular models in chemistry and biochemistry, this book will be suitable for advanced undergraduate and graduate courses on molecular modeling and simulation within physics, biophysics, physical chemistry and materials science It will also be a useful reference to all those working in the field Additional resources for this title including solutions for instructors and programs are available online at www.cambridge.org/9780521835275 H e r m a n J C B e r e n d s e n is Emeritus Professor of Physical Chemistry at the University of Groningen His research focuses on biomolecular modeling and computer simulations of complex systems He has taught hierarchical modeling worldwide and is highly regarded in this field www.pdfgrip.com www.pdfgrip.com SIMULATING THE PHYSICAL WORLD Hierarchical Modeling from Quantum Mechanics to Fluid Dynamics HERMAN J C BERENDSEN Emeritus Professor of Physical Chemistry, University of Groningen, the Netherlands www.pdfgrip.com CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521835275 © H J C Berendsen 2007 This publication is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2007 eBook (EBL) ISBN-13 978-0-511-29491-4 ISBN-10 0-511-29491-3 eBook (EBL) hardback ISBN-13 978-0-521-83527-5 hardback ISBN-10 0-521-83527-5 paperback ISBN-13 978-0-521-54294-4 paperback ISBN-10 0-521-54294-4 Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate www.pdfgrip.com Contents Preface Symbols, units and constants Part I page xi xv A Modeling Hierarchy for Simulations 1 Introduction 1.1 What is this book about? 1.2 A modeling hierarchy 1.3 Trajectories and distributions 1.4 Further reading 3 13 14 Quantum mechanics: principles and relativistic effects 2.1 The wave character of particles 2.2 Non-relativistic single free particle 2.3 Relativistic energy relations for a free particle 2.4 Electrodynamic interactions 2.5 Fermions, bosons and the parity rule 19 19 23 25 31 36 From quantum to classical mechanics: when and how 3.1 Introduction 3.2 From quantum to classical dynamics 3.3 Path integral quantum mechanics 3.4 Quantum hydrodynamics 3.5 Quantum corrections to classical behavior 39 39 42 44 64 70 Quantum chemistry: solving the time-independent Schră odinger equation 77 4.1 Introduction 77 4.2 Stationary solutions of the TDSE 78 4.3 The few-particle problem 79 4.4 The Born–Oppenheimer approximation 97 v www.pdfgrip.com vi Contents 4.5 4.6 4.7 4.8 4.9 4.10 The many-electron problem of quantum chemistry Hartree–Fock methods Density functional theory Excited-state quantum mechanics Approximate quantum methods Nuclear quantum states 98 99 102 105 106 107 Dynamics of mixed quantum/classical systems 5.1 Introduction 5.2 Quantum dynamics in a non-stationary potential 5.3 Embedding in a classical environment 109 109 114 129 Molecular dynamics 6.1 Introduction 6.2 Boundary conditions of the system 6.3 Force field descriptions 6.4 Solving the equations of motion 6.5 Controlling the system 6.6 Replica exchange method 6.7 Applications of molecular dynamics 139 139 140 149 189 194 204 207 Free 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 211 211 213 218 221 227 231 234 239 Stochastic dynamics: reducing degrees of freedom 8.1 Distinguishing relevant degrees of freedom 8.2 The generalized Langevin equation 8.3 The potential of mean force 8.4 Superatom approach 8.5 The fluctuation–dissipation theorem 8.6 Langevin dynamics 8.7 Brownian dynamics 8.8 Probability distributions and Fokker–Planck equations 8.9 Smart Monte Carlo methods 8.10 How to obtain the friction tensor energy, entropy and potential of mean force Introduction Free energy determination by spatial integration Thermodynamic potentials and particle insertion Free energy by perturbation and integration Free energy and potentials of mean force Reconstruction of free energy from PMF Methods to derive the potential of mean force Free energy from non-equilibrium processes www.pdfgrip.com 249 249 251 255 256 257 263 268 269 272 274 Contents vii Coarse graining from particles to fluid dynamics 9.1 Introduction 9.2 The macroscopic equations of fluid dynamics 9.3 Coarse graining in space 9.4 Conclusion 279 279 281 288 295 10 Mesoscopic continuum dynamics 10.1 Introduction 10.2 Connection to irreversible thermodynamics 10.3 The mean field approach to the chemical potential 297 297 298 301 11 Dissipative particle dynamics 11.1 Representing continuum equations by particles 11.2 Prescribing fluid parameters 11.3 Numerical solutions 11.4 Applications 305 307 308 309 309 Part II 313 Physical and Theoretical Concepts 12 Fourier transforms 12.1 Definitions and properties 12.2 Convolution and autocorrelation 12.3 Operators 12.4 Uncertainty relations 12.5 Examples of functions and transforms 12.6 Discrete Fourier transforms 12.7 Fast Fourier transforms 12.8 Autocorrelation and spectral density from FFT 12.9 Multidimensional Fourier transforms 315 315 316 317 318 320 323 324 325 331 13 Electromagnetism 13.1 Maxwell’s equation for vacuum 13.2 Maxwell’s equation for polarizable matter 13.3 Integrated form of Maxwell’s equations 13.4 Potentials 13.5 Waves 13.6 Energies 13.7 Quasi-stationary electrostatics 13.8 Multipole expansion 13.9 Potentials and fields in non-periodic systems 13.10 Potentials and fields in periodic systems of charges 335 335 336 337 337 338 339 340 353 362 362 www.pdfgrip.com viii Contents 14 Vectors, operators and vector spaces 14.1 Introduction 14.2 Definitions 14.3 Hilbert spaces of wave functions 14.4 Operators in Hilbert space 14.5 Transformations of the basis set 14.6 Exponential operators and matrices 14.7 Equations of motion 14.8 The density matrix 379 379 380 381 382 384 385 390 392 15 Lagrangian and Hamiltonian mechanics 15.1 Introduction 15.2 Lagrangian mechanics 15.3 Hamiltonian mechanics 15.4 Cyclic coordinates 15.5 Coordinate transformations 15.6 Translation and rotation 15.7 Rigid body motion 15.8 Holonomic constraints 397 397 398 399 400 401 403 405 417 16 Review of thermodynamics 16.1 Introduction and history 16.2 Definitions 16.3 Thermodynamic equilibrium relations 16.4 The second law 16.5 Phase behavior 16.6 Activities and standard states 16.7 Reaction equilibria 16.8 Colligative properties 16.9 Tabulated thermodynamic quantities 16.10 Thermodynamics of irreversible processes 423 423 425 429 432 433 435 437 441 443 444 17 Review of statistical mechanics 17.1 Introduction 17.2 Ensembles and the postulates of statistical mechanics 17.3 Identification of thermodynamical variables 17.4 Other ensembles 17.5 Fermi–Dirac, Bose–Einstein and Boltzmann statistics 17.6 The classical approximation 17.7 Pressure and virial 17.8 Liouville equations in phase space 17.9 Canonical distribution functions 453 453 454 457 459 463 472 479 492 497 www.pdfgrip.com 582 References Trotter, H F (1959) On the product of semigroup of operators Proc Amer math Soc., 10, 545–51 Tuckerman, M., Berne, B J and Martyna, G J (1992) Reversible multiple time scales molecular dynamics J Chem Phys., 97, 1990–2001 Tuckerman, M E., Mundy, C J and Martyna, G J (1999) On the classical statistical mechanics of non-Hamiltonian systems Europhys Lett., 45, 149–55 Tully, J C (1990) Molecular dynamics with electronic transitions J Chem Phys., 93, 1061–71 Tully, J C and Preston, R K (1971) Trajectory surface hopping approach to nonadiabatic molecular collisions: the reaction of H with D2 J Chem Phys., 55, 562–72 Uhlenbeck, G E and Gropper, L (1932) The equation of state of a non-ideal Einstein-Bose or Fermi–Dirac gas Phys Rev., 41, 79–90 Umrigar, C J., Nightingale, M P and Runge, K (1993) A diffusion Monte Carlo algorithm with very small time-step errors J Chem 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Vink, J C (1993) Quantum mechanics in terms of discrete beables Phys Rev., A 48, 1808–18 Wajnryb, E., Altenberger, A R and Dahler, J S (1995) Uniqueness of the microscopic stress tensor J Chem Phys., 103, 9782–7 Wallqvist A and Berne, B J (1985) Path-integral simulation of pure water Chem Phys Lett., 117, 214–9 Warshel, A and Levitt, M (1976) Theoretical studies of enzymic reactions: Dielectric, electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme J Mol Biol., 103, 227–49 Wassenaar, T A (2006) Molecular Dynamics of Sense and Sensibility in Processing and Analysis of Data Ph D thesis, University of Groningen, the Netherlands (electronic version available from http://dissertations.rug.nl/ Wassenaar, T A and Mark, A E (2006) The effect of box shape on the dynamic properties of proteins simulated under periodic boundary conditions J Comput Chem., 27, 316–25 Wax, N., ed (1954) Noise and Stochastic Processes New York, Dover Publ Webster, F., Rossky, P J and Friesner, R.A (1991) Nonadiabatic processes in condensed matter: semi-classical theory and implementation Comp Phys Comm., 63, 494–522 Weisstein, E W (2005) Spherical Harmonic Addition Theorem MathWorld–A Wolfram Web Resource, available at http://mathworld.wolfram.com Weizel, W (1954) Ableitung der quantenmechanischen Wellengleichung des Mehrteilchensystems aus einem klassischen Modell Z Physik, 136, 582–604 Wesselingh, J A and Krishna, R (1990) Mass Transfer New York and London, Ellis Horwood Widom, B (1963) Some topics in the theory of fluids J Chem Phys., 39, 2808–12 Widom, B (2002) Statistical Mechanics A concise introduction for chemists Cambridge, UK, Cambridge University Press Wigner, E (1932) On the quantum correction for thermodynamic equilibrium Phys Rev., 40, 749–59 Wittenburg, J (1977) Dynamics of Systems of Rigid Bodies Stuttgart, Teubner Wolniewicz, L (1993) Relativistic energies of the ground state of the hydrogen molecule J Chem Phys., 99, 1851–68 Wood, R H (1991) Estimation of errors in free energy calculations due to the lag between the Hamiltonian and the system configuration J Phys Chem., 95, 4838– 42 Wood, R H (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, 6177–87 www.pdfgrip.com 584 References Woolfson, M M and Pert, G J (1999) An Introduction on Computer Simulation Oxford, Oxford University Press Wormer, P E S and van der Avoird, A (2000) Intermolecular potentials, internal motions, and spectra of van der Waals and hydrogen-bonded complexes Chem Rev., 100, 4109–43 Wu, Y -S M., Kuppermann, A and Anderson, J B (1999) A very high accuracy potential energy surface for H3 Phys Chem Chem Phys., 1, 929–37 Wyatt, R E (2005) Quantum Dynamics with Trajectories Introduction to Quantum Hydrodynamics New York, Springer–Verlag Yang, W (1991a) Direct calculation of electron density in density-functional theory Phys Rev Lett., 66, 1438–41 Yang, W (1991b) Direct calculation of electron density in density-functional theory: Implementation for benzene and a tetrapeptide Phys Rev., A 66, 7823–6 Yang, W and Lee, T S (1995) A density-matrix divide-and-conquer approach for electronic structure calculations in large molecules J Chem Phys., 103, 5674–8 Yoneya, M., Berendsen, H J C and Hirasawa, K (1994) A non-iterative matrix method for constraint molecular dynamics simulation Mol Simul.,13, 395–405 Ytreberg, F M and Zuckerman, D M (2004) Efficient use of non-equilibrium measurement to estimate free energy differences for molecular systems J Comput Chem., 25, 1749–59 Yu, H., Hansson, T and van Gunsteren, W F (2003) Development of a simple, self-consistent polarizable model for liquid water J Chem Phys., 118, 221–34 Zeiss, G D and Meath, W J (1975) The H2 O–H2 O dispersion energy constant and the dispersion of the specific refractivity of diluted water vapour Mol Phys., 30, 161–9 Zhang, Y and Yang, W (1999) A pseudobond approach to combining quantum mechanical and molecular mechanical methods J Chem Phys., 110, 46–54 Zhou, J., Reich, S and Brooks, B R (2000) Elastic molecular dynamics with selfconsistent flexible constraints J Chem Phys., 112, 7919–29 Zhu, S -B., Singh, S and Robinson, G W (1991) A new flexible/polarizable water model J Chem Phys., 95, 2791–9 Zimmerman J A., Web III, E B., Hoyt, J J et al (2004) Calculation of stress in atomistic simulation Modelling Simul Mater Sci Eng., 12, 319–32 Zvelindovsky, A V., Sevink, G J A., van Vlimmeren, B A C., Maurits, N M and Fraaije, J G E M (1998a) Three-dimensional mesoscale dynamics of block copolymers under shear: The dynamic density-functional approach Phys Rev E, 57, R4879-82 Zvelindovsky, A V., van Vlimmeren, B A C., Sevink, G J A., Maurits, N M and Fraaije, J G E M (1998b) Three-dimensional simulation of hexagonal phase of a specific polymer system under shear: The dynamic density functional approach J Chem Phys., 109, 8751-4 Zwanzig, R W (1954) High-temperature equation of state by a perturbation method I Nonpolar gases J Chem Phys., 22 1420–6 www.pdfgrip.com References 585 Zwanzig, R (1960) Ensemble method in the theory of irreversibility J Chem Phys., 33, 1338–41 Zwanzig, R (1961) Memory effects in irreversible thermodynamics Phys Rev., 124, 983–92 Zwanzig, R (1965) Time-correlation functions and transport coefficients in statistical mechanics Ann Rev Phys Chem., 16, 67–102 Zwanzig, R (1973) Nonlinear generalized Langevin equations J Stat Phys., 9, 215–20 www.pdfgrip.com www.pdfgrip.com Index a.u., see atomic units ab initio molecular dynamics, 151 Abelian group, 380 absolute activity, 462 acid dissociation constant, 439 action, in mechanics, 398 activity coefficient, 435 activity, in thermodynamic potential, 436 adiabatic limit, 132 advancement, reaction, 448 affinity, 448 AFM, see atomic force microscope alchemy, computational, 225 alias, 324 AM1, see Austin model Ampere’s law, 337 Andersen thermostat, 198 Angstrom (unit), xv angular acceleration, 410 angular momentum definition, 405 equation of motion, 405 angular velocity, 409 annealing, 205 associated Legendre functions, 361 atmosphere (unit), xv atomic force microscopy, 237 atomic units, xvii, xvii au, see atomic units Austin model 1, 106 autocorrelation function, 317, 325 axial vectors, 409 Axilrod–Teller effect, 187 azurin, 329 B-splines, 548 back reaction mean field, 131 surface hopping, 134 Banach space, 381 bar (unit), xv BE, see Bose-Einstein beables, 64 Beeman algorithm, 192 beta identification with temperature, 458 binary rotations, 414 Bohmian dynamics, 64 Bohr magneton, 34 Bohr, Copenhagen interpretation, 65 boiling point elevation, 442 Boltzmann statistics, 470 Born energy charge in cavity, 343 in dielectric medium, 344 with ionic strength, 345 Born reaction potential, 345 Born–Oppenheimer approximation, 97 Bose–Einstein statistics, 467 boson, 37 boundary conditions, dielectric, 342 boundary conditions artifacts PBC, 145 conducting, 166 continuum, 148 periodic, 141 restrained shell, 149 tin-foil, 166 triclinic unit cell, 143 boundary element method, 342 Brownian dynamics with acceptance criterion, 273 Brownian dynamics, 268 Bspline, Python program, 551 Caley–Hamilton relation, 389 Caley–Klein parameters, 407 candela (unit), xxiv canonical distribution cartesian coordinates, 497 generalized coordinates, 498 quantum, 457 canonical partition function, 457 cap atom, 189 Car–Parrinello method, 151 cardinal B-splines, 548 587 www.pdfgrip.com 588 Index CASSCF, see complete active space SCF center of mass, 404 CGPM, xv charge spread function cubic, 369 Gaussian, 368 general, 363 chemical potential, see thermodynamic potential chi-square distribution, 532 chi-square table, 532 cintegral, Python program, 544 cinterpol, Python program, 543 Clausius’ virial, 485 Clausius–Clapeyron equation, 435 coalescing drops, DPD, 309 coarse graining, 249 coarse-graining weight function, 289 coexistence region, 434 colligative properties, 441 commutator, 384 complete active space SCF, 105, 136 compressibility adiabatic, 286, 426 in fluid dynamics, 286 in phase space, 496 isothermal, 308, 426 phase space, 198 computational alchemy, 224 conducting boundary conditions, 166 conformation, 204 conical intersection, 136 constraint matrix, 421 constraints coordinate resetting, 419 flexible, 158 force, 418 generalized coordinates, 418 holonomic, 417 hypersurface, 418 in MD algorithm, 194 in thermodynamic integration, 226 Lagrange multiplier, 418 LINCS, 422 linear, 422 metric tensor effect, 499 projection method, 418, 421 rotational, 145 SHAKE, 420 continuity equation, 282 continuum corrections, 168 convolution, 316 coordinate four-vector, 26 coordinate transformations, 384 Copenhagen interpretation, 65 corrections continuum environment, 168 correlation function, classical, 511 Coulomb energy, 340 coupling parameter, 221 Crank–Nicholson algorithm, 116 cspline, Python program, 545 cubic splines, 526 cyclic coordinates, 400 DAC, see divide-and-conquer, de Broglie wavelength, 469 Debye (unit), xv Debye length, 342 DebyeHă uckel approximation, 341 degree of advancement, 448 delta-response, 507 density functional theory description, 102 time-dependent, 105 density matrix and partition function, 394 coordinate representation, 394 definition, 392 ensemble-averaged, 394 time evolution, 392 transformation, 393 detailed balance in Monte Carlo, 13, 273 in replica exchange, 206 in smart Monte Carlo, 273 DFT, see density functional theory dielectric boundary conditions, 342 dielectric displacement, 336 dielectric properties, from fluctuations, 514 differential relations, thermodynamics, 431 differential exact, 426 total, 426 diffusion constant from Einstein relation, 519 from NEMD, 520 diffusive systems, 214 dimerization constant, 438 dipole correction, lattice sum, 372 Dirac equation, 27 displacement current density, 337 dissipative particle dynamics, 305 divide-and-conquer, 106 double well, 82 DPD, see dissipative particle dynamics dummy particles, 157 dynamics Brownian, 268 dissipative particle, 305 fluid continuum, 281 Langevin, 252 mesoscopic, 249, 297 molecular, 139 non-inertial, 268 replica-exchange, 205 scaled particle, 306 dyne (unit), xv effective Born radius, 353 effective pair potentials, 174 Ehrenfest www.pdfgrip.com Index classical limit, 43 Einstein relation, 519 electric constant, xvi electric dipole density, 336 electric factor, xviii electric field definition, 335 fast multipole methods, 362 hierarchical methods, 362 in conducting medium, 348 long-range contribution, 365 long-range Fourier solution, 366 of charge in cavity, 343 periodic systems, 362 short-range contribution, 364 electron acceptor, 439 electron donor, 439 electron transfer reaction, 439 electrostatic unit, 377 electrostatics, 340 energy density, electromagnetic, 339 ensemble canonical, 460 grand-canonical, 461 isobaric-isothermal, 460 microcanonical, 460 size effects, 462 enthalpy, definition, 426 entropy production in reaction, 448 irreversible, 444 entropy finite-size effects, 463 in statistical mechanics, 458 thermodynamic definition, 426 EOS, see equation of state equation of state, 433 equilibrium constant, 438 equipartition theorem, 502 equipartition generalized, 503 kinetic energy, 503 erg (unit), xv ergodic postulate, 454 essential dynamics, 216 esu, see electrostatic unit Euler angles, 412 Euler equations, 407 Euler exponential splines, 553 Euler parameters, 413 Euler’s theorem, 414 Euler–Rodrigues parameters, 413 Eulerian derivative, 281 Ewald summation, 368 exact differential, 426 excess pressure, coarse-graining, 294 excess thermodynamic potential, 219 exchange potential, 479 exchange-correlation potential, 104 excited-state quantum mechanics, 105 expectation, observable, 391 exponential matrices, 385 exponential operators, 385 exponential splines, 553 Faraday (unit), 440 Faraday’s induction law, 337 fast Fourier transform, 323 fast multipole method, 168 FD, see Fermi-Dirac FENE model, 256 Fermi–Dirac statistics, 467 fermion, 37 Feynman path integral, 44 Feynman–Hibbs potential, 71 FFT, see fast Fourier transform fine-structure constant, xviii fitspline, Python program, 546 fitting splines, 530 Fixman’s theorem, 501 flexible constraint, 158 fluctuation–dissipation theorem first, 259, 511 in Langevin dynamics, 257 restricted form, 258 second, 260 fluid dynamics coarse-graining, 288 continuity equation, 282 energy balance, 288 equation of motion, 282 equation of state, 285 equations, 281 Eulerian derivative, 281 heat conduction, 287 incompressible fluids, 285 Lagrangian derivative, 281 momentum conservation, 283 momentum flux, 283 stress tensor, 283 viscosity, 285 viscous stress tensor, 284 FMM, see fast multipole method Fock operator, 101 Fokker–Planck equation, 268 for Brownian dynamics, 272 for diagonal friction, 272 for generalized Langevin, 271 for velocity, 268 general equation, 270 force field bond angle, 154 bonded interactions, 153 Buckingham potential, 156 constraints, 155 Coulomb interaction, 156 dihedrals, 154 effective potentials, 174 empirical adjustments, 150 flexible constraint, 158 ideal, 183 www.pdfgrip.com 589 590 Index improper dihedral, 155 Lennard–Jones, 155 long-range Coulomb, 164 long-range dispersion, 159 Morse potential, 154 multipoles, 158 parent model, 187 polarizability, 157 polarizable, 171 pressure correction, 161 QM/MM, 188 quartic bond potential, 154 reaction field, 164 Ryckaert–Bellemans potential, 155 shifted potential, 160 switching function, 160 truncated potential, 160 united atom, 153 force fields, 149 four-dimensional dynamics, 225 four-vectors, 26 Fourier transform convolution, 316 definitions, 315 discrete, 323 FFT, 323 Gaussian function, 322 multidimensional, 331 square pulse, 320 triangular pulse, 321 free energy, see also potential of mean force free energy by integration, 222 by perturbation, 222 by slow growth, 223 by spatial integration, 213 non-diffusive systems, 213 quantum correction, 477 freezing point depression, 442 frequency response, 508 friction kernel, 252 friction tensor, 274 friction and noise, 259 from diffusion, 275 from Stokes’ law, 275 hydrodynamic interaction, 276 solute particle, 275 frictional force in Langevin dynamics, 251 FT, see Fourier transform fugacity, 435 gauge invariance, 338 Gauss equation, 337 Gauss thermostat, 198 GB, see generalized Born solvation model Gear algorithm, 191 generalized Born solvation model, 352 generalized equipartition theorem, 503 generalized Langevin equation, 253 generalized momentum, 399 GENERIC formalism, 306 ghost particle, 219 Gibbs free energy, definition, 426 Gibbs function, see Gibbs free energy Gibbs’ paradox, 453, 470 Gibbs’ phase rule, 433 Gibbs–Duhem relation, 429, 447 Gibbs–Helmholtz relation, 430 grand ensemble, 461 grand-canonical ensemble, 460 Green’s function QMC, 96 Green’s function, 45 GROMACS, xii group velocity, 24 half-reaction, electrochemical, 439 haloalkane dehydrogenase, 172 Hamiltonian classical, 399 dipole in field, 35 generalized, 402 quantum, 43 relativistic, 28 with vector potential, 32 Zeeman, 34 hartree (unit), xviii Hartree-Fock equation, 101 limit, 101 methods, 99 heat capacity isobaric, 426 isochoric, 426 heat conduction coefficient, 287 Heisenberg interpretation QM, 65 uncertainty relation, 22, 319 helium quantum effects, 41 Quantum Monte Carlo, 92 Hellmann–Feynman force, 131 Helmholtz free energy, definition, 426 Helmholtz free energy, 459 Helmholtz function, see Helmholtz free energy Henry’s constant, 438 hermitian conjugate, 382 hermitian operator, 383 hidden variables, 65 Hilbert space, 381 Hohenberg–Kohn first theorem, 102 second theorem, 103 hydrodynamic interaction, 276 hydrodynamics classical, 281 quantum, 64 hydrogen fluoride classical vibration, quantum vibration, 87 hydrogen www.pdfgrip.com Index quantum effects, 41 hypercomplex numbers, 413 hyperpolarizability, 172 ideal solution, 436 image charge in sphere, 350 imaginary time path integral, 51 inch (unit), xv inertia tensor, 408 information and Boltzmann H-function, 455 in statistical mechanics, 455 Shannon, 455 integrating factor, 426 ion hydration correction, 170 ionic strength, 342 irrelevant degrees of freedom, 249 irreversible processes entropy production, 444 Onsager relations, 450 reactions, 448 stationary state, 450 thermodynamic forces, 444 thermodynamics, 444 isobaric-isothermal ensemble, 460 isokinetic thermostat, 198 Itˆ o-Stratonovich dilemma, 254 Itˆ o, 254 Jacobian, 495 Jarzynski equation, generalized, 241 Jarzynski equation, 239 Jaynes, information, 455 Joule heat, 339 Joule–Thomson coefficient, 426 kcal (unit), xv Kelvin (unit), xxiv kilogram (unit), xxiv kilogramforce (unit), xvii Kirkwood quantum correction, 476 Kohn-Sham orbitals, 103 Kramers equation, 271 Kramers–Kronig relations, 509 Lagrange equations, 398 Lagrange multipliers, 456 Lagrangian derivative, 281 Lagrangian, 398 Lande g-factor, 34 Langevin equation generalized, 253 Markovian, 261 pure, 258 simple pure, 267 simple, 254, 267 Laplace equation, 346 leap-frog algorithm, 192 Legendre polynomials, 360 length, SI unit of, xxiv 591 Lie–Trotter–Suzuki expansion, 386 light intensity, SI unit of, xxiv linear constraint solver, 422 linear response, 506 linear scaling methods, 106 Liouville equation classical, 492 generalized, 494 proper, 494 Liouville operator classical, 493 generalized, 493 Liouville–von Neumann equation, 392 Lorentz convention, 338, 33 Lorentz force, 335 Lorentz transformation, 25 Lyapunov exponent, 204 Madelung fluid, 64 magnetic dipole density, 336 magnetic field definition, 335 magnetic intensity, 336 many-electron quantum methods, 98 Markovian Langevin equation, 261 mass flux density, definition, 282 mass tensor, 401, 498 mass, SI unit of, xxiv mass-metric tensor, 401 material derivative, 281 maximum entropy, 215 Maxwell relations, 431 Maxwell’s equations integrated form, 337 polarizable media, 336 vacuum, 335 MC, see Monte Carlo simulations MCDHO model, see water MD, see molecular dynamics mdstep, Python program, mean-field back reaction, 131 mercury relativistic effects, 30 mesh Ewald methods, 168 mesoscopic dynamics, 249, 297 meter (unit), xxiv metric tensor effect of constraints, 502 metric tensor, 143 Metropolis Monte Carlo, 272 microcanonical ensemble, 459 Minkovsky space, 25 MNDO, see modified neglect of differential overlap mobility matrix, 276 modified neglect of differential overlap, 106 mol (unit), xxiv mole, xxiv molecular dynamics, 139 molecular units, xviii, xvii moment of inertia, 408 www.pdfgrip.com 592 Index wave vector, 317 Oseen tensor, 276 osmolality, 441 osmotic coefficient, 437 oversampling, 324 oxidant, 439 momentum density, 283 momentum flux tensor, 481 momentum four vector, 26 Monte Carlo simulations detailed balance, 13 diffusional quantum, 89 Green’s function quantum, 96 Metropolis scheme, 272 multicanonical, 205 path integral, 61 references, 13 replica exchange, 205 smart MC, 273 variational quantum, 84 MOPAC, 106 Morse oscillator classical, quantum, 87 Mulliken analysis, 156 multi-level system, 128 multicanonical Monte Carlo, 205 multiple time-step algorithms, 193 multipole expansion, 353 multipoles definition, 356 in force field, 158 in spherical harmonics, 359 traceless, 358 Navier–Stokes equation general, 285 incompressible fluids, 285 NDDO, see neglect of diatomic differential overlap neglect of diatomic differential overlap, 106 NEMD, see non-equilibrium molecular dynamics non-diffusive systems, 214 non-equilibrium molecular dynamics, 519 non-inertial dynamics, 268 Nos´ e–Hoover thermostat, 201 nuclear quantum states, 107 Numerov’s method, 80 numerov, Python program, 80 Nyquist’s theorem, 324 octupole moment, 356 octupole, traceless, 359 Onsager, reciprocal relations, 450 open systems, 433 operator commuting, 384 eigenvalue equation, 383 energy, 22 exponential, 385 hermitian, 383 in Fourier space, 317 in Hilbert space, 382 momentum, 22 projection, 388 quantum mechanical, 22 parameterized model 3, 106 parity, 36 Parseval’s theorem, 317 partial differential relations, thermodynamics, 431 partial molar quantities, 427 particle insertion, 219, 237 particle mesh Ewald method, 168, 374 particle–particle particle–mesh, 374 partition function classical, 476 quantum correction, 476 quantum, 457 trace of matrix, 465 path integral classical limit, 48 concept, 44 equivalence Schră odinger eqn, 47 free particle, 55 harmonic potential, 57 imaginary time, 51 minimum action, 48 molecular dynamics, 61 Monte Carlo, 61 normalization, 49 string of beads, 62 Pauli exclusion principle, 37 parity rule, 37 PBC, see periodic boundary conditions PCA, see principal component analysis periodic boundary conditions, 141 periodic cubic splines, 526 periodic disturbance, 508 perturbation time-dependent, 120 time-independent, 119 pH, definition, 439 phase diagram, 434 phase rule, 433 phase space compressibility, 496 definition, 492 incompressible, 494 metric tensor, 495 phase velocity, 20 phenomenological coefficients, 449 phenomenological relations, 449 photoactive yellow protein, 136 pK, definition, 439 PM3, see parameterized model PME, see particle mesh Ewald method PMF, see potential of mean force Poisson bracket, 496 www.pdfgrip.com Index Poisson equation, 341 Poisson–Boltzmann equation full, 341 linearized, 342 polarizability anisotropic, 176 choices, 176 energies and forces, 181 fluctuating charges, 176 in force field, 157 induced dipoles, 176 shell model, 176 Thole model, 177 units, xvi positrons, 28 potential of mean force and free energy, 227, 231 and metric, 231 definition, 230 derivation, 233 harmonic well, 231 in Langevin equations, 255 nonequilibrium, 239 particle insertion, 237 quantum corrections, 232 thermodynamic integration, 234 thermodynamic perturbation, 235 umbrella sampling, 235 potential-derived charges, 156 potential scalar, 337 vector, 337 poundforce (unit), xvii Poynting vector, 338 PPPM, see particle-particle particle-mesh pressure and virial, 485 average, 484 diatomic example, 491 from c.o.m., 489 internal, 432 kinetic, 432 localization, 483 relation to stress, 481 tensor, 481 thermodynamic, 485 principal component analysis, 216 proton acceptor, 439 proton donor, 438 proton transfer reaction, 438 pseudovectors, 409 psi, Python program, 81 psips, 89 Pulay force, 131, 133 pulling simulations, 237 pure Langevin equation, 258 Python, essentials, xii Python program Bspline, 551 cintegral, 544 cinterpol, 543 593 cspline, 545 fitspline, 546 mdstep, numerov, 80 psi, 81 spectrum from time series, 329 SRstep, 87 walkers, 94 QM/MM, 188 QMC, see quantum Monte Carlo quadrupole moment, 356 quadrupole, traceless, 358 quantity, SI unit of, xxiv quantum chemistry, 98 quantum correction, as potential, 478 quantum correction exchange, 479 free energy, 477 harmonic oscillator, 58, 74 partition function, 476 quantum dynamics, back reaction, 109 quantum dynamics adiabatic, 110 crossing states, 111 grid integration, 115 mean field, 112 non-stationary potential, 114 solvated electron, 118 surface hopping, 113 quantum force, 65 quantum hydrodynamics, 64 quantum Monte Carlo diffusional, 89 Green’s function, 96 helium, 92 variational, 84 quantum potential, 68 quantum stochastic dynamics, 64 quantum stress tensor, 69 quantum width, 40 quaternions and rotation matrices, 416 definition, 413 equation of motion, 416 normalized, 413 random force in Brownian dynamics, 269 in Langevin dynamics, 251 random phase approximation, 394 Raoult’s law, 442 reaction coordinate, 230 reaction equilibria, 437 reaction field gradient, 349 reaction field dipole in cavity, 347 from charges in cavity, 348 from excentric charge in cavity, 349 gradient from quadrupole, 349 in dipole fluctuation, 515 www.pdfgrip.com 594 Index in MD, 164, 165 with ionic strength, 347 reaction flux, 448 reaction potential from charge in cavity, 344 from charges in cavity, 348 from excentric charge in cavity, 349 from image charge, 350 in dielectric medium, 344 with ionic strength, 345 reciprocal lattice vectors, 332 reciprocal lattice, Fourier space, 332 reciprocal relations, 450 Redfield limit, 125 reductant, 439 relative dielectric permittivity, 337 relativistic correction, 29 relativistic quantum mechanics, 25 relaxation methods quantum, 85 relevant degrees of freedom, 249 REM, see replica exchange method REMD, see replica exchange molecular dynamics replica exchange method, 205, 211 replica exchange molecular dynamics, 205 replica Monte Carlo, 205 RESPA, 191 reversible processes, 425 rhombic dodecahedron, 142 rigid body motion, 405 Rodrigues parameters, 413 rotation matrix and quaternions, 416 definition, 406 time derivative, 409 rotation Euler angles, 412 quaternions, 413 unit vectors, 411 rotational constraint, 145 rotational partition function, 471 rotation, 404 rovibrational states, 107 saturation pressure, 435 scaled particle hydrodynamics, 306 SCF, see self-consistent eld Schră odinger equation time-independent, 78 imaginary time, 86 time-dependent, 23 Schwarz inequality, 319 second (unit), xxiv second virial coefficient definition, 436 quantum corrections, 72 quantum statistics, 469 water, 175 self-consistent field, 101 self-energy, 340 semi-empirical methods, 106 semipermeable membrane, 442 SETTLE, 422 SHAKE, 420 Shannon, information, 455 shooting method, 80 SI units, xvii, xv simple Langevin equation, 267 simple pure Langevin equation, 267 simulated annealing, 205 simulated tempering, 205 Slater-type orbitals, 106 slow growth, in thermodynamic integration, 223 smart Monte Carlo, 273 SMD, see steering molecular dynamics smooth particle mesh Ewald, 168, 374 soft-core potentials, 225 solvated electron dynamics, 118 solvent accessible surface area, 352 SPC, see water SPC/E, see water specific heat isobaric, 426 isochoric, 426 spectrum, Python program, 329 SPH, see scaled particle hydrodynamics spherical harmonic functions, 361 spherical harmonics addition theorem, 361 spin glass, replica Monte Carlo, 205 spin matrices, 28 splines algorithms, 542 B-splines, 548 cardinal B-splines, 548 cubic, 526 Euler exponential, 553 fitting, 530 for tabulation, 539 periodic, 529 quadratic, 526 quartic, 526 SPME, see smooth particle mesh Ewald SRstep, Python program, 87 standard hydrogen electrode, 440 standard reduction potential, 441 state functions definition, 425 extensive, 425 intensive, 425 table, 427 stationary state, 450 steady state, 450 steering molecular dynamics, 237 step disturbance, 508 Stirling approximation, 455 STO, see Slater-type orbitals stochastic equation mathematical incorrectness, 253 stochastic thermostat, 196 stoichiometric coefficients, 437 www.pdfgrip.com Index Stokes’ law, 275 Stratonovich, 254 stress tensor and pressure, 292 coarse-grained, 292 fluid dynamics, 283 Irving and Kirkwood localization, 484 localization, 483 quantum, 69 substantive derivative, 281 superatom approach, 249 superatom model, 249, 256 surface hopping, 113, 134 susceptibility electric, 336 magnetic, 336 switching function, 160 symmetry factor, statistics, 471 symplectic algorithm, 492 mapping, 492 notation, 492 transformation, 495 systematic force in Langevin dynamics, 251 temperature, SI unit of, xxiv temperature thermodynamic definition, 426 tempering, 205 ternary phase diagram, 434 thermal conductivity, from NEMD, 521 thermal wavelength, 469 thermodynamic cycles, 212 thermodynamic forces, 444 thermodynamic integration constraints, 226 definition, 223 for potential of mean force, 234 four-dimensional dynamics, 225 slow growth, 223 soft-core potentials, 225 vanishing particle, 226 thermodynamic perturbation, 235 thermodynamic potential excess, 219 particle insertion, 219 standard, 435 thermodynamics data sources, 444 degrees of freedom, 433 first law, 429 second law, 424, 432 thermostat Andersen, 198 comparison, 203 extended system, 201 Gauss, 198 Nos´ e–Hoover, 201 stochastic, 196 weak-coupling, 199 tight-binding approximation, 107 time correlation function, 511 time, SI unit of, xxiv time-dependent DFT, 105 tin-foil boundary conditions, 166 TIP, see water TIP4P, see water torque, generalized, 402 torque, 405 total differential, 426 transformation canonical, 495 coordinate, 401 diagonalization, 385 Hamiltonian, 495 Jacobian, 495 of basis set, 384 symplectic, 495 unitary, 384 transition dipole moment, 36 transition rate multi-level system, 128 quantum dynamics, 124 two-level system, 126 translation, 403 triclinic box, 142 triple point, 433 Trotter formula, 387 Trotter–Suzuki splitting, 118 Trotter-Suzuki expansion, 193 truncated octahedron, 142 tunneling states, 82 two-form, 495 two-level system, 122 typographic conventions, xv umbrella sampling, 235 uncertainty relation, 22, 318 unitary transformation, 384 units atomic, xvii, xvii molecular, xviii, xvii rationalized, xvi SI, xv technical, xvi van ’t Hoff equation, 442 van der Waals gas, 433 vanishing particle, in thermodynamic integration, 226 variational quantum Monte Carlo, 84 vector potential, 337 vector space complete, 381 complex, 380 definition, 380 n-dimensional, 380 normed, 380 real, 380 vector definition, 380 www.pdfgrip.com 595 596 Index inner product, 381 normalized, 381 orthogonal pair, 381 orthonormal set, 381 representation, 379 scalar product, 381 velocity four vector, 26 velocity-Verlet algorithm, 192 Verlet algorithm, 191 vibration Morse function, vibrational partition function, 472 vibrational-rotational-tunneling, 108 virial expansion, 436 virial theorem, 485, 504 virial Clausius, 485 periodic conditions, 486 virtual interaction sites, 157 viscosity bulk, 285 by NEMD, 520 from fluctuation, 516 shear, 285 viscous stress tensor, 284 Voronoi tesselation, 306 VRT, see vibrational-rotational-tunneling walkers, Python program, 94 walkers, 89 water MCDHO model, 179 point-charge models, 179 quantum effects, 41 relativistic correction, 30 second virial coefficient, 175 SPC model, 179 SPC/E model, 179 TIP3 model, 179 TIP4P model, 179 wave four-vector, 26 wave group velocity, 24 weak-coupling thermostat, 199 wedge product, 494 Widom’s particle insertion, 219 Wiener process, 253 Wiener–Khinchin theorem, 327 Wiener–L´evy process, 253 Wigner correction, 72 Wigner rotation matrices, 407 Wood’s corrections, 168 Zeeman interaction, 34 zero-padding, 324 www.pdfgrip.com ... www.pdfgrip.com SIMULATING THE PHYSICAL WORLD The simulation of physical systems requires a simplified, hierarchical approach, which models each level from the atomistic to the macroscopic scale From quantum. .. 25 31 36 From quantum to classical mechanics: when and how 3.1 Introduction 3.2 From quantum to classical dynamics 3.3 Path integral quantum mechanics 3.4 Quantum hydrodynamics 3.5 Quantum corrections... neither to be complete nor to be rigorous Our aim is to show the beauty and simplicity of the basic quantum theory; relativistic quantum theory comprises such 19 www.pdfgrip.com 20 Quantum mechanics: