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4/19/2005 CHEM 408 - Sp05L14 - 1Thermal Averaging & DynamicsThermal Averaging & Dynamics• emphasis so far on characterizing energy minima of isolated molecules; many properties of interest can be computed in this manner if only few low E structures presentG-G+T~4 kJ/molωω / deg• relevant questions are how do energies compare to kBT; how harmonic?•H2O(g) @ 300 K but not n-butane(g), not (H2O)2(g), not H2O(l) …• but some many problems require thermal averaging that can’t be accomplished by simple means4/19/2005 CHEM 408 - Sp05L14 - 2Example: what is the r.m.s. C1-C4distance (<l2>)1/2in n-butane at 25ºC?Info: TG+G-ε / (kJ/mol)044l / Å4334/19/2005 CHEM 408 - Sp05L14 - 3Example: what is the r.m.s. C1-C4distance (<l2>)1/2in n-butane at 25ºC?TG+G-ε / (kJ/mol) 0 4 4l /Å433Qexp(-e/kT) 1 0.20 0.20 1.40Pi0.71 0.14 0.14l2 /Å211.41 1.29 1.29 3.744/19/2005 CHEM 408 - Sp05L14 - 44/19/2005 CHEM 408 - Sp05L14 - 5• we’ll discuss three methods for obtaining thermally averaged data:¾ Molecular Dynamics (MD): dynamical simulation of classical trajectory via F=ma (Leach Ch. 7)¾ Monte Carlo (MC): statistical sampling of phase space based on Metropolis algorithm (Leach Ch. 8)¾ Stochastic Dynamics (SD): dynamical simulation where “unimportant” degrees of freedom treated as random variables (Leach 7.8)• we’ll also discuss some of the underlying statistical mechanicalideas and some basic tricks of the trade as we go (Leach Ch. 6)4/19/2005 CHEM 408 - Sp05L14 - 6Ideas of Statistical Mechanics• the classical state of an N-particle system is specified the 6Ncoordinates and momenta of all particles:},...,,,...,,,,{},{111211133zNzyxNNppppxzyxpq =• the 6N-dimensional space spanned by these coordinates is termed “phase space”• given some initial condition, {q3N(0), p3N(0)}, the time evolution of a system, determined by the classical equations of motion of thesystem, for example,completely determine the future trajectory {q3N(t), p3N(t)} in phase space2)(2)()(2tqmmtptFiiiii&&rrr==• trajectories in phase space cannot cross4/19/2005 CHEM 408 - Sp05L14 - 7tpqSimple Example: 1d Harmonic Oscillatork = force constantm = mass}0,{)}0(),0({ bpq =• phase space only 2d {q, p}• for initial conditions:the trajectory is:)cos()( tbtqω=)sin()(2tbmtpωω−=mkπω21=withFigure adapted from Cramer Fig. 3.1all trajectories are ellipses of varying size but constant aspect4/19/2005 CHEM 408 - Sp05L14 - 8From: Allen & Tildesley, Computer Simulation of Liquids (Oxford, 1987)Schematic of Phase Space Trajectories in a Complex System4/19/2005 CHEM 408 - Sp05L14 - 9• MD and MC methods reflect two fundamentally different ways of thinking about measured properties of thermal systems: dttptqAANNt)}(),({1lim303∫∞→=><τττNNNNNNdpdqpqpqAA333333),(),(ρ∫∫>=<trajectory average(MD simulation)ensemble average(MC simulation & statistical mechanics)where ρ(q3N,p3N) is the probability density function. • For a system at constant (N, V, T):}/),(exp{1),(3333TkpqEQpqBNNNN−=ρ∫∫−=NNBNNNdpdqTkpqEhNTVNQ33333}/),(exp{1!1),,(Q(N,V,T) is termed the (canonical) partition function4/19/2005 CHEM 408 - Sp05L14 - 10• basis of statistical mechanics is the “ergodic hypothesis”:>=<>< AAtthe average over the dynamical trajectory of a complex system inequilibrium is equivalent to the ensemble average using the appropriate phase space density function ρ• for example, imagine a collection of 100 coupled harmonic oscillatorsρ(q1,p1)p1q1q1(t), p1(t)• idea is that trajectory will “fill out” equilibrium ρ(q1,p1) over time4/19/2005 CHEM 408 - Sp05L14 - 11• Molecular Dynamics (MD) methods approximately solve the classical equations of motion given the force laws governing a particular system• Monte Carlo (MC) methods perform the phase space integrals by sampling the distribution functions via random walk algorithm; kinetic energy need not be simulated • dynamical as well as equilibrium information can be obtained• in both methods averages are over finite number of steps sampled:)}(),({11iMiittptqAMA∑==><∑=>=<MiiqAMA1)(1• mechanical properties, properties that are defined in terms of (q, p) like internal energy, pressure, density are easily computed inthis manner• entropy and free energies require special


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PSU CHEM 408 - Thermal Averaging

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