Configurational-Bias Monte Carlo

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1 Configurational-Bias Monte Carlo Thijs J.H. Vlugt [2] Random Sampling versus Metropolis Sampling (2) N interacting particles in volume V at temperature T Configurational-Bias Monte Carlo Thijs J.H. Vlugt Engineering Thermodynamics, Process & Energy Department Faculty of Mechanical, Maritime and Materials Engineering (3mE) Delft University of Technology Delft, The Netherlands t.j.h.vlugt@tudelft.nl Q(N,V,T) = 1 Λ 3N N! F (N,V,T) = k B T ln Q(N,V,T) N 1 N U(r N ) = i=1 dr N exp [ βu(r N ) ] j=i+1 u(r ij )= i<j u(r ij ) January 9, 29 Configurational-Bias Monte Carlo Thijs J.H. Vlugt [3] Random Sampling versus Metropolis Sampling (3) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [1] Random Sampling versus Metropolis Sampling (1) Computing the ensemble average ofacertainquantitya(r N ) N interacting particles in volume V at temperature T Random Sampling of r N : A = lim n n i=1 A(rN i )exp[ βu(r N i )] n i=1 exp [ βu(r N i )] Metropolis sampling; generate n configurations r N with probability proportional to exp [ βu(r N i )], therefore: A = lim n n i=1 A(rN i ) n vector representing positions of all particles in the system: r N total energy: U(r N ) statistical weight of configuration r N is exp[ βu(r N )] with β =1/(k B T )

2 Configurational-Bias Monte Carlo Thijs J.H. Vlugt [6] Simulation Technique (3) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [4] Simulation Technique (1) What is the ratio of red wine/white wine in the Netherlands? Configurational-Bias Monte Carlo Thijs J.H. Vlugt [7] Simulation Technique (4) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [5] Simulation Technique (2) Bottoms up

3 Configurational-Bias Monte Carlo Thijs J.H. Vlugt [1] Metropolis Monte Carlo (1) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [8] Simulation Technique (5) How to generate configurations r i with a probability proportional to N (r i )=exp[ βu(r i )]??? N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth. A.H. Teller and E. Teller, Equation of State Calculations by Fast Computing Machines, J. Chem. Phys. 21 (1953) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [11] Metropolis Monte Carlo (2) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [9] Simulation Technique (6) Whatever our rule is to move from one state to the next, the equilibrium distribution should not be destroyed Bottoms up Shoe shop Liquor store

4 Configurational-Bias Monte Carlo Thijs J.H. Vlugt [14] Detailed Balance (2) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [12] Move from the old state (o) to a new state (n) andback new 1 N (o)α(o n)acc(o n) =N (n)α(n o)acc(n o) new 5 new 2 α(x y); probability to select move from x to y old acc(x y); probability to accept move from x to y often (but not always); α(o n) =α(n o) new 4 new 3 Therefore: acc(o n) α(n o)exp[ βu(n)] α(n o) = = acc(n o) α(o n)exp[ βu(o)] α(o n) exp[ βδu] N (o) n leaving state o = entering state o [α(o n)acc(o n)] = n [N (n)α(n o)acc(n o)] N (i) : probability to be in state i (here: proportional to exp[ βe i ]) α(x y) : probability to attempt move from state x to state y acc(x y): probability to accept move from state x to state y Configurational-Bias Monte Carlo Thijs J.H. Vlugt [15] Metropolis Acceptance Rule General: acc(o n) acc(n o) = X Choice made by Metropolis (note: infinite number of other possibilities) acc(o n) =min(1,x) Note than min(a, b) =a if a<band b otherwise always accept when X 1 when X<1, generate uniformly distributed random number from <, 1 > and accept or reject according to acc(o n) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [13] Detailed Balance (1) Requirement (balance): N (o) n [α(o n)acc(o n)] = n Detailed balance: much stronger condition for every pair o,n [N (n)α(n o)acc(n o)] N (o)α(o n)acc(o n) =N (n)α(n o)acc(n o) new 5 old new 1 new 2 new 4 new 3

5 Configurational-Bias Monte Carlo Thijs J.H. Vlugt [18] Chain Molecules Configurational-Bias Monte Carlo Thijs J.H. Vlugt [16] Monte Carlo Casino Configurational-Bias Monte Carlo Thijs J.H. Vlugt [19] Self-Avoiding Walk on a Cubic Lattice Configurational-Bias Monte Carlo Thijs J.H. Vlugt [17] Smart Monte Carlo: α(o n) α(n o) Not a random displacement Δr uniformly from [ δ, δ], but instead Δr = r(new) r(old) = A F + δr F A δr : force on particle : constant : taken from Gaussian distribution with width 2A so P (δr) exp[ (δr 2 )/4A] 3D lattice; 6 lattice directions only 1 monomer per lattice site (otherwise U = ) interactions when r ij =1and i j > 1 P (r new ) exp [ (r new (r old + A F (o))) 2 ] 4A ] (Δr A F (o))2 α(o n) exp [ α(n o) = 4A [ ] exp (Δr+A F (n))2 4A

6 Configurational-Bias Monte Carlo Thijs J.H. Vlugt [22] Rosenbluth Sampling (1) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [2] Simple Model for Protein Folding 1. Place first monomer at a random position 2. For the next monomer (i), generate k trial directions (j =1 k) each with energy u ij 2 by 2 interaction matrix Δ ij YPDLTKWHAMEAGKIRFSVPDACLNGEGIRQVTLSN (E. Jarkova, T.J.H. Vlugt, N.K. Lee, J. Chem. Phys., 25, 122, 11494) 3. Select trial direction j with a probability P j = exp[ βu ij ] k j=1 exp[ βu ij] b) Continue with step 2 until the complete chain is grown (N monomers) z (b) 1 prot1 5 prot2 prot3 prot Force (k B T/b) (dz) Force (k B T/b) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [23] Rosenbluth Sampling (2) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [21] Number of Configurations without Overlap P j = exp[ βu ij ] k j=1 exp[ βu ij] Random Chains: n i=1 R = lim R i exp[ βu i ] n n i=1 exp[ βu i] Fraction of chains without overlap decreases exponentially as a function of chainlength (N) N total (= 6 N 1 ) without overlap fraction no overlap

7 Configurational-Bias Monte Carlo Thijs J.H. Vlugt [26] Intermezzo: Ensemble Averages at Different Temperatures Configurational-Bias Monte Carlo Thijs J.H. Vlugt [24] Rosenbluth Sampling (3) U β = = = = dr N U(r N )exp[ βu(r N )] drn exp[ βu(r N )] dr N U(r N )exp[ β U(r N )] exp[(β β) U(r N )] drn exp[ β U(r N )] exp[(β β) U(r N )] U(r N ) exp[(β β) U(r N )] β exp[(β β) U(r N )] β U(r N ) exp[δβ U(r N )] β exp[δβ U(r N )] β Probability to insert monomer i (at trial direction j ) P j = exp[ βu ij ] k j=1 exp[ βu ij] Probability to grow the chain (N monomers, k trial directions) N i=1 P chain = exp[ βu ij (i)] N k i=1 j=1 exp[ βu ij] = exp[ βu chain] W chain Configurational-Bias Monte Carlo Thijs J.H. Vlugt [27] Rosenbluth Distribution Differs from Boltzmann Distribution Configurational-Bias Monte Carlo Thijs J.H. Vlugt [25] Rosenbluth Sampling (4) end to end distance r Probability to grow the chain (N monomers, k trial directions).4.3 N =1 N = 1 Rosenbluth distribution Boltzmann distribution.1.8 Rosenbluth distribution Boltzmann distribution N i=1 P chain = exp[ βu ij (i)] N k i=1 j=1 exp[ βu ij] = exp[ βu chain] W chain Therefore, weightfactor for each chain i is the Rosenbluth factor W i : P(r).2 P(r).6 n i=1 R Boltzmann = lim W i R i n n i=1 W i The unweighted distribution is called the Rosenbluth distribution: r r n i=1 R Rosenbluth = lim R i n n Of course: R Rosenbluth R Boltzmann

8 Configurational-Bias Monte Carlo Thijs J.H. Vlugt [3] Pruned-Enriched Rosenbluth Method (2) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [28] Distribution of Rosenbluth Weights enriching successful pruning unsuccessful pruning 2 N=5 N=1 N=2 N=3 ln(p(w)) ln(w) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [31] Pruned-Enriched Rosenbluth Method (3) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [29] Pruned-Enriched Rosenbluth Method (1) Example: N = 2, k =2, βμ ex = ln W = RR W min = 2 W max = 5 W min = 15 W max = 5 Grassberger (1997); grow chains using Rosenbluth Method: W = 6 j=1 exp [ βu 2j ] 6 N 5 i=3 j=1 exp [ βu ij ] 5 = N 5 i=3 j=1 exp [ βu ij ] 5 p(ln(w)) ln(w) Two additional elements: Enriching. If W>W max during the construction of the chain, k copies of the chain are generated, each with a weight of W/k. This is a deterministic process. The growth of those chains is continued. Pruning. If W<W min during the construction of the chain, with a probability of 1/2 the chain is pruned resulting in W =. If the chain survives, the Rosenbluth weight is multiplied by 2 and the growth of the chain is continued.

9 Configurational-Bias Monte Carlo Thijs J.H. Vlugt [34] Random Insertion of Chains is Useless! Configurational-Bias Monte Carlo Thijs J.H. Vlugt [32] Pruned-Enriched Rosenbluth Method (4) Chain Length Probability without overlaps βμ ex = ln W, Ann. Rev. of Phys. Chem. 1999, 5, Configurational-Bias Monte Carlo Thijs J.H. Vlugt [35] Configurational-Bias Monte Carlo Configurational-Bias Monte Carlo Thijs J.H. Vlugt [33] Static versus Dynamic Generate configurations using the Rosenbluth scheme Accept/Reject these configurations in such a way that detailed balance is obeyed Split potential energy into bonded (bond-stretching, bending, torsion) and non-bonded (i.e. Lennard-Jones or Coulombic) interactions Generate (k) trial positions according to bonded interactions (unbranched chain: l, θ, φ are independent) U bonded = U stretch (l)+u bend (θ)+u tors (φ) P (l) dl l 2 exp[ βu(l)] P (θ) dθ sin(θ)exp[ βu(θ)] P (φ) dφ exp[ βu(φ)] Static create chain conformations, and use correct weight factor (random insertion, Rosenbluth scheme, PERM) single chain only; no Markov chain Dynamic create Markov chain, accept new configuration, acceptance rule should obey detailed balance multi-chain system, usable for all ensembles (i.e. Gibbs, μv T ) Configurational-Bias Monte Carlo (CBMC), Recoil Growth (RG), Dynamic PERM (DPERM)

10 Configurational-Bias Monte Carlo Thijs J.H. Vlugt [38] Generate a Random Number from a Distribution (2) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [36] Generate Trial Configurations: Linear Alkane F (y) = y f(y )dy f(x) >p(x) CH2 φ θ CH2 l CH3 CH3 u(l) = k l 2 (l l ) 2 u(θ) = k θ 2 (θ θ ) 2 u(φ) = 5 c i cos i (φ) i= P (l) dl l 2 exp[ βu(l)] P (θ) dθ sin(θ)exp[ βu(θ)] P (φ) dφ exp[ βu(φ)] Configurational-Bias Monte Carlo Thijs J.H. Vlugt [39] Generate a Random Number from a Distribution (3) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [37] Generate a Random Number from a Distribution (1) subroutine bend-tors lready=.false do while (.not.lready) call ransphere(dx,dy,dz) x = xold + dx y = yold + dy z = zold + dz call bend(ubend,x,y,z) call tors(utors,x,y,z) if(ranf().lt.exp(-beta*(ubend+utors))) + lready=.true enddo generate appropriate θ and φ random vector on unit sphere monomer position bending energy torsion energy accept or reject F (y) = y p(y )dy

11 Configurational-Bias Monte Carlo Thijs J.H. Vlugt [42] Super-Detailed Balance (2) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [4] CBMC Algorithm Notation: K(o n) =N (o)α(o n)acc(o n) K(o n) = K(o n b n b o ) b nb o K(n o) = K(n o b n b o ) b nb o Detailed balance requires K(o n) =K(n o) Possible solution (Super-detailed balance): K(o n b n b o )=K(n o b n b o ) for each b n b o. Generate a trial configuration using the Rosenbuth scheme. k trial segments {b} k = {b 1 b k }, each trial segment is generated according to P (b) exp[ βu bonded (b)] Compute non-bonded energy, select configuration i with a probability P (b i )= exp[ βu non b (b i )] k j=1 exp[ βu non b(b j )] = exp[ βu non b(b i )] w l Continue until chain is grown, W (n) = n l=1 w l Similar procedure for old configuration, generate k 1 trial positions (trial position 1 is the old configuration itself), leading to W (o) Accept/reject according to acc(o n) =min(1,w(n)/w (o)) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [43] Super-Detailed Balance (3) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [41] Super-Detailed Balance (1) Detailed balance for each set b n b o : K(o n b n b o ) = N(o) α(o n b n b o ) acc(o n b n b o ) = exp[ βu(o)] C exp[ βu bonded (n)] exp[ βu non b (n)] P (b n b o ) acc(o n b n b o ) W (n) K(n o b n b o ) = exp[ βu(n)] C exp[ βu bonded (o)] exp[ βu non b (o)] P (b n b o ) acc(n o b n b o ) W (o) As therefore U = U non b + U bonded acc(o n) acc(n o) = W (n) W (o)

12 Configurational-Bias Monte Carlo Thijs J.H. Vlugt [46] Trick: Dual-Cutoff CBMC Configurational-Bias Monte Carlo Thijs J.H. Vlugt [44] Branched Molecules (1) 1..8 b1 η r cut */[A o ] Grow chain with approximate potential; W.2 c1 c2 b2 Correct for difference later (δu, difference real and approximate potential for selected configuration) ( acc(o n) =min 1, W ) (n) W exp[ β[δu(n) δu(o)]] (o) P (b 1, b 2 ) exp[ β[u bend (c 1, c 2, b 1 )] + u bend (c 1, c 2, b 2 )] + u bend (b 1, c 2, b 2 )]] Configurational-Bias Monte Carlo Thijs J.H. Vlugt [47] Application 1: Adsorption in MFI-type zeolite (1) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [45] Branched Molecules (2) Zeolites: microporous channel structure crystalline, SiO 2 building blocks substitution of Si 4+ by Al 3+ and a cation (Na + or H + ) typical poresize: 4 12Å synthetic and natural; >17 framework types Use CBMC to generate internal configurations: Generate n t random trial positions and select one (i) with a probability P int (i) = exp[ βu bonded (i)] nt j=1 exp[ βu bonded(j)] = exp[ βu bonded(i)] W int (n) Repeat until k trial orientations are found; these are fed into CBMC leading to W (n) Repeat procedure for old configuration, leading to W int (o) and W (o). Accept or reject according to ( acc(o n) =min 1, W (n) W int ) (n) W (o) W int (o)

13 Configurational-Bias Monte Carlo Thijs J.H. Vlugt [5] Application 1: Adsorption in MFI-type zeolite (4) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [48] Application 1: Adsorption in MFI-type zeolite (2) 2. butane exp. butane sim. isobutane exp. isobutane sim. N/[mmol/g] 1. n C p/[kpa] i C 4 y x z Vlugt et al, J. Am. Chem. Soc., 1998, 12, 5599 Configurational-Bias Monte Carlo Thijs J.H. Vlugt [51] Application 1: Adsorption in MFI-type zeolite (5) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [49] Application 1: Adsorption in MFI-type zeolite (3) n-c 4, low loading i-c 4, low loading i-c 4,highloading Grand-canonical ensemble system is coupled to particle reservoir at chemical potential μ and temperature T,statisticalweight exp[βμn βu(r N )]/N! additional trial moves to exchange particles between the zeolite and the reservoir equilibrium: μ gas = μ zeolite ; measure average N for given μ and β μt Vlugt et al, J. Am. Chem. Soc., 1998, 12, 5599

14 Configurational-Bias Monte Carlo Thijs J.H. Vlugt [54] Application 1: Adsorption in MFI-type zeolite (8) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [52] Application 1: Adsorption in MFI-type zeolite (6) Research Octane Number (RON) 8 n-c 6 6 (a) Pure component isotherms 4 3MP Loading, θ /[molecules per unit cell] (b) 5-5 mixture of C 6 isomers at 362 K, n-c 6 CBMC simulations DSL model fits > MP Partial pressure, pi /[Pa] Configurational-Bias Monte Carlo Thijs J.H. Vlugt [55] Application 1: Adsorption in MFI-type zeolite (9) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [53] Application 1: Adsorption in MFI-type zeolite (7) blue = branched (i-c 6 ) red = linear (n-c 6 ) Flux n C 6 i C 6 selectivity pure %-5% Experiments by J. Falconer, Univ. Colorado

15 Configurational-Bias Monte Carlo Thijs J.H. Vlugt [58] Efficiency of CBMC Configurational-Bias Monte Carlo Thijs J.H. Vlugt [56] Application 2: Gibbs Ensemble Monte Carlo (1) k: number of trial directions B. Smit, S. Karaborni, and J.I. Siepmann, J. Chem. Phys. 12, 2126 (1995) Heptane (n-c 7 ) a: probability that trial direction has a favorable energy growth can continue as long as at least 1 trial direction is favorable 55 Experimental data Scaling relations Simulations chain length N P success =(1 (1 a) k ) N =exp[ cn] increasing k means increasing CPU time. T/[K] ρ/[g/cm 3 ] Configurational-Bias Monte Carlo Thijs J.H. Vlugt [59] Dead-End Alley Configurational-Bias Monte Carlo Thijs J.H. Vlugt [57] Application 2: Gibbs Ensemble Monte Carlo (2) B. Smit, S. Karaborni, and J.I. Siepmann, J. Chem. Phys. 12, 2126 (1995) 1 9 simulations experiments.24 T c [K] ρ c [gram/cm 3 ].22.2 simulations experiments (1) experiments (2) N c N c

16 Configurational-Bias Monte Carlo Thijs J.H. Vlugt [62] Recoil Growth Algorithm (2) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [6] Recoil Growth: Avoiding Dead-End Alley Example for k =2and l =3 E F C I D B A G H O Configurational-Bias Monte Carlo Thijs J.H. Vlugt [63] Super Detailed Balance Configurational-Bias Monte Carlo Thijs J.H. Vlugt [61] Recoil Growth Algorithm (1) In general, N(o) α(o n) acc(o n) =N(n) α(n o) acc(n o) Therefore, ( ) α(n o) acc(o n) =min 1, exp[ βδu] α(o n) What about α(o n)? Generate a tree t n. Decide which parts of the tree are open or closed (O n ). Make a random walk on the tree (rw n ) Place first bead at random position Position (i) can be open or closed depending on the environment (energy u i ); here we use p open =min(1, exp[ βu i ]) and toss a coin. If open, continue with next segment If closed, try another trial direction up to a maximum of k If all k directions are closed, retract by one step Maximum retraction length: l max l +1 l: recoil length, l max : maximum length obtained during the growth of the chain Computed weight W (n) and repeat procedure for old configuration Accept or reject using acc(o n) =min(1,w(n)/w (o))

17 Configurational-Bias Monte Carlo Thijs J.H. Vlugt [66] Computing the Weight (k =2, l =3) Configurational-Bias Monte Carlo Thijs J.H. Vlugt [64] Random Walk on a Tree (k =2, l =3) E D F B A C G H I E D F B O A C m =2 1 m =1 2 G H I open closed O Configurational-Bias Monte Carlo Thijs J.H. Vlugt [67] Efficiency of RG Compared to CBMC Configurational-Bias Monte Carlo Thijs J.H. Vlugt [65] Super-Detailed Balance η CBMC RG; l max =1 RG; l max =2 RG; l max = k K(o n t n t o O n O o ) = K(n o t n t o O n O o ) α(o n t n t o O n O o ) = P (t n )P (O n t n )P (rw n t n O n ) P (t o rw o )P (O o t o rw o ) α(n o t n t o O n O o ) = P (t o )P (O o t o )P (rw o t o O o ) P (O n t n ) P (O n t n rw n ) = P (rw n t n O n ) = P (t n rw n )P (O n t n rw n ) n i=1 p i 1 n i=1 m i n i=1 acc(o n) = min 1, exp[ βδu] m i (n) p i (n) n i=1 m i (o) p i (o)

18 Configurational-Bias Monte Carlo Thijs J.H. Vlugt [68] The End Questions??