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Kramers Theory of Reaction Kinetics

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Kramers Theory of Reaction Kinetics Kramers theory of reaction kinetics Winter School for Theoretical Chemistry and Spectroscopy Han-sur-Lesse, 7-11 December 2015 Hans Kramers Hendrik Anthony Kramers 1894 – 1952 1911 HBS, 1912 staatsexamen, 1916 doctoraal wis- en natuurkunde 1916 Arrived unannounced in Kopenhagen to do PhD with Niels Bohr 1919 promotion in Leiden (Ehrenfest), married in 1920 Lector in Kopenhagen 1926 Professor theoretical physics in Utrecht 1934 Professor in Leiden (chair of Ehrenfest), professor in TU-Delft Physics 7(4), April 1940, Pages 284-304 - 8 references Outline • Intro Kramers theory • Transition State Theory • The Langevin equation • The Fokker-Planck equation • Kramers reaction kinetics • high-friction vs low-friction limit • memory, Grote-Hynes theory • Bennet-Chandler approach • Reactive Flux • Transmission coefficient calculation • Free energy methods • Summary • Bibliography Molecular Transitions are Rare Events chemical reactions phase transitions protein folding defect diffusion self-assembly stock exchange crash Rare Events • Molecular transitions are not rare in every day life • They are only rare events with respect to the femto- second timescale of atomic motions • Modeling activated transitions by straightforward simulation would be extremely costly (takes forever) • Advanced methods: Transition state theory (TST), Bennet-Chandler (Reactive Flux) approach, Transition Path Sampling (TPS), Free energy methods, Parallel Tempering, String Method,.... A very long trajectory • A and B are (meta-)stable states, i.e. A B attractive basins • transitions between A and B are rare • transitions can happen fast • system looses memory in A and B Free energy landscape Rare Events Transition State The free energy has local minima separated by barriers Many possible transition paths via meta-stable states Projection on reaction coordinate shows FE profile P.G. Bolhuis, D. Chandler, C. Dellago, P.L. Geissler with transition state barrier - G/kT k = k0 e Δ Annu. Rev. Phys. Chem 2002 Reaction rate depends Arrhenius equation exponentially on the barrier height A long trajectory Reaction coordinate Transition State Theory • Derived by Henry Eyring and by Meredith Gwynne Evans and Michael Polanyi in 1935 E /RT • Explains Arrhenius equation (1889) k = Ae− a (van’t Hoff 1884) Henry Eyring • Activated complex is in quasi equilibrium with reactants • All reactants that reach the activated complex state, continue to products [AB]‡ probability to reach A + B [AB]‡ C K‡ = the transition state $ ! [A][B] kBT ∆G‡/RT - rate is frequency of unstable k = k‡K‡ = e− h mode times the TS probability - transmission factor is κ=1 in TST frequency (correction for barrier recrossings) Transition State Theory • what is a transition state? activated structure, critical nucleus, entropic bottleneck,… • where does the refactor (h dependence) come Henry Eyring from? • TST rate is an upper limit for the true rate • The actual rate is slower that the TST estimate due to recrossing • recrossing occur due to a bad estimate for the TS (dividing surface) / hidden slow variables (aka the reaction coordinate problem) and/or due to noisy dynamics from the environment • is kappa just a cheating factor or does it have a physical meaning? Separation of time scales • Suppose a coordinate X describes the reaction progress • X is coupled to environment • the coupling makes the dynamics of X stochastic (not deterministic) • to escape from the reactant well, X (=random walker) must acquire enough energy • The escape time depends on from the bath the size of the fluctuations • this energy must be lost (X(t)-<X>)2 again after passing the • Enoise << Ebarrier => Ea/RT >> 1 barrier relaxation time τs escape time τe = 1/k 2 d V a ⌧s 2⇡ me↵ / 2 << ⌧e = ⌧s exp (E /kBT ) ⇡ r dX ⇣ ⌘ Diffusion Density of particles: ⇢(r,t) Current density: j(r,t)=⇢(r,t)v(r,t) A gradient in the density leads to a current: Diffusion of particles due to a concentration gradient j = D ⇢ Fick’s law − r diffusion coefficient Mean square displacement: 2 - also end-to-end distance x (t) =2Dt - ballistic motion: x2(t) t2 h i ! Velocity auto correlation function: Self-diffusion of a particle seen as a random walk 1 D = d⌧ vx(⌧)vx(0) 0 h i ensemble average ... Z h i Langevin equation mass x acceleration = force 2 d x dU(x) dx Paul Langevin m = γ + ⇣(t) 1872-1946 dt2 − dx − dt U is external viscous drag random potential γ is friction collisions (dissipation) Describes Brownian motion random motion of small particles (pollen grains) in Robert Brown a solvent due to thermal fluctuations (Brown 1828) 1773-1858 Illustration with Octave/Matlab script d2x dU(x) dx m = γ + ⇣(t) dt2 − dx − dt % brownian_2d.m simulates a two-dimensional Brownian motion N=10000; % number of steps to take T=70; % maximum time h=T/N; % time step t=(0:h:T); % t is the vector [0 1h 2h 3h ... Nh] sigma = 1.0; % strength of noise x=zeros(size(t)); % place to store x locations y=zeros(size(t)); % place to store y locations x(1)=0.0; % initial x location y(1)=0.0; % initial y location for i=1:N % take N steps x(i+1)=x(i)+sigma*sqrt(h)*randn; y(i+1)=y(i)+sigma*sqrt(h)*randn; end; plot(x,y); grid on % add a grid to axes % axis([0 T -3 8]); % set axis limits Illustration with Octave/Matlab script d2x dU(x) dx m = γ + ⇣(t) dt2 − dx − dt % brownian_2d.m simulates a two-dimensional Brownian motion N=10000; % number of steps to take T=70; % maximum time h=T/N; % time step t=(0:h:T); % t is the vector [0 1h 2h 3h ... Nh] sigma = 1.0; % strength of noise x=zeros(size(t)); % place to store x locations y=zeros(size(t)); % place to store y locations x(1)=0.0; % initial x location y(1)=0.0; % initial y location for i=1:N % take N steps x(i+1)=x(i)+sigma*sqrt(h)*randn; y(i+1)=y(i)+sigma*sqrt(h)*randn; end; plot(x,y); grid on % add a grid to axes % axis([0 T -3 8]); % set axis limits Langevin equation - laminair flow (small velocities) Stokes formula for viscous drag: - spherical particle with radius α - γ is the friction γ = 6⇡⌘↵ - 1/γ is the mobility − - η is the dynamic viscosity f = γv - v is the velocity of the particle d − Equation of motion with only drag and random forces: dv(t) - Langevin equation without ∇V m = 6⇡⌘↵v(t)+⇣(t) - stochastic differential equation dt − - what do we know about ζ(t)? - ⇣(t) =0 h i 2 - ⇣(t)⇣(t0) =2Dγ δ(t t0) Taking the ensemble average: - h(white noise,i Wiener process)− - (Markovian dynamics vs memory) d v(t) m h i = 6⇡⌘↵ v(t) - since average random force is zero dt − h i t/⌧ - relaxation time of initial perturbation which implies: v(t) = v(0) e− h i ⌧ = m/(6⇡⌘↵) strong friction Particle in a very viscous solvent: d2x dU(x) dx dx - m = γ + ⇣(t) γ = ⇣(t) dt2 − dx − dt dt - particle is driven by fluctuating solvent - no inertial motion -> m=0 - no external potential solving the differential equation: t 1 - since ⇣(t) =0 h i x(t)=x(0) + dt0⇣(t0) - unsurprisingly: x(t) = x(0) γ - h i h i Z0 consider many paths from x(0) what about the variance? - x(0) to the left and square both sides t t 2 1 x(t) x(0) = dt0 dt0 ⇣(t0 )⇣(t0 ) − γ2 1 2 1 2 Z0 Z0 ⌦ ↵ ⌦ ↵ 2 - particle gets 1016 (gas) to 1020 (liquid) kicks per sec. - kicks are very fast and considered uncorrelated x(t) x(0) =2Dt 2 - ⇣(t)⇣(t0) =2Dγ δ(t t0) − h i − ⌦ ↵ weaker friction Particle in a less viscous solvent: d2x dU(x) dx γt/m - m = γ + ⇣(t) v(t) = v(0) e− dt2 − dx − dt h i - inertial motion -> m > 0 - still no external potential For the position we use: - τ=m/γ t x(t)=x(0) + dt0v(t) Z0 γ=0.001 take the ensemble average position γ=0.1 t x(t) = x(0) + dt0 v(t) γ=1.0 γ=2.0 h i 0 h i Z time m γt/m x(t) = x(0) + v(0) 1 e− h i γ − Fluctuation-dissipation dv(t) m = 6⇡⌘↵v(t)+⇣(t) dt − conversion of heat dissipation of kinetic into kinetic energy energy into heat of the bath of the particle Velocity auto correlation in weak friction: - decays to zero with τ=m/γ Dγ γ(t2 t1)/m v (t )v (t ) = e− − - i.e. correlated when t2 – t1 < m/γ h x 1 x 2 i m Dγ - v (t)v (t) = h x x i m Gives also the kinetic energy: 3 - equipartition theorem: E = k T kin 2 B 1 2 3Dγ Ekin = m v(t) = - fluctuation-dissipation theorem: 2 h i 2 there is a relation between the kicking and the friction! kBT - combine with MSD -> determine kB ! Einstein relation: k T D = x2 = B t γ h i ⇡⌘↵ - Jean Baptiste Perrin 1926 Nobel prize Fokker-Planck Integrating the Langevin equation from an initial point (x0,t0) results in a stochastic trajectory. What is the probability to arrive at point x at time t? —> P(x,t | x0,t0) - we could integrate over all possible trajectories conditional probability - or try to predict how the probability evolves dx d2x dU(x) dx E.g. strong friction case: γ = ⇣(t) m = γ + ⇣(t) dt dt2 − dx − dt 2 x(0) =0, x (t) =6Dt - (now in 3 dimensions) h i h i The distribution that gives this variance is a Gaussian: 2 1 3/2 x /4Dt 3 P (x,t)=( ) e− - prefactor from: d xP(x,t)=1 4⇡Dt Z @P Solution of diffusion equation: = D 2P - simplest Fokker- @t r Planck equation Diffusion equation P Fick’s law: j = D P - current probability (or − r current density) j j(x,t)=P (x,t)v(x,t) volume flux in flux out with P @P @j Conservation law: = j j @t −@x ∆x @P Combine: = D 2P @t r Solution of diffusion equation: 1 3/2 x2/4Dt P (x,t)=( ) e− 4⇡Dt Illustration with Octave/Matlab script 1 3/2 x2/4Dt P (x,t)=( ) e− 4⇡Dt % Script tp plot diffusion equation clear; clf; N=1000; % number of steps to take T=50; % maximum time h=T/N; % time step nx=100; % size of xgrid xmin=-5.0; xmax=5.0; dx=(xmax-xmin)/nx; x=zeros(size(nx)); % place to store y locations y=zeros(size(nx)); % place to store y locations sigma = 0.01; % strength of noise D=1.0*4*3.14159265;
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