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Calculation of Effective Interaction Potentials From Systematic Coarse-Graining of Molecular Models by the Inverse Monte Carlo: Theory, Practice and Software Alexander Lyubartsev Division of Physical Chemistry Department of Material and Environmental Chemistry Stockholm University Modeling Soft Matter: Linking Multiple Length and Time Scales Kavli Institute of Theoretical Physics, UCSB, Santa Barbara 4-8 June 2012 Soft Matter Simulations length model method time 1Å electron w.f + ab-initio 100 ps 1 nm nuclei BOMD, CPMD 10 nm atomistic classical MD 100 ns coarse-grained Langevine MD, 100 nm µ more DPD, etc 100 s coarse-grained µ 1 m continuous The problem: Larger scale ⇔ more approximations Coarse-graining – an example Original size - 900K Compressed to 24K Coares-graining: reduction degrees of freedom All-atom model Coarse-grained model Large-scale 118 atoms 10 sites simulations We need to: 1) Design Coarse-Grained mapping: specify the important degrees of freedom 2) For “important” degrees of freedom we need interaction potential Question: what is the interaction potential for the coarse-grained model? Formal solution: N-body mean force potential Original (FG = fine grained system) FG ( ) =θ( ) H r 1, r 2,... ,rn R j r1, ...r n n = 118 j = 1,...,10 Usually, centers of mass of selected molecular fragments Partition function : n =∫∏ (−β ( ))= Z dr i exp H FG r 1, ...,r n i=1 n N =∫∏ ∏ δ( −θ ( )) (−β ( ))= dr i dR j R j j r1, ...rn exp H FG r1, ... ,rn = i1 j 1 N =∫∏ (−β ( )) dR j exp H CG R1, ... , RN j =1 1 where β= k B T n N ( )=−1 ∫∏ ∏ δ( −θ ( )) (− ( )) H CG R1, ..., R N β ln dr i R j j r1, ...,r n exp βH FG r1, ...r n i=1 j=1 is the effective N-body coarse-grained potential = potential of mean force = free energy of the non-important degrees of freedom N-body potential of mean force (CG Hamiltonian) HCG(R1,...RN) : Structure: all structural properties are the same 〈 〉 =〈 〉 for any A(R ,..R ) : A A 1 N FG CG Thermodynamics: in principle yes (same partition function) β but: HCG = HCG(R1,...RN , , V) already N-body mean force potential is state point dependent T-V dependence should be in principle taken into account in computing thermodynamic properties Dynamics: can be approximated within Mori-Zvanzig formalism (e.g within DPD: Eriksson et all, PRE, 77, 016707 (2008)) Problem with HCG(R1,...RN) : simulation with N-body potential is unrealistic. We need something usable – e.g. pair potentials! (or any other preferably one-dimensional functions) H (R R ,... , R )≈∑ V ( R ) =∣ − ∣ CG 1, 2 N ij ij Rij Ri R j i> j How to approximate ? - minimize the difference (Boltzmann averaged) : Energy matching - minimize the difference of gradient (force) Force matching - minimize the relative entropy - provide the same canonical averages, e.g RDF-s - Inverse MC - Iterative Boltzmann inversion .. and may be some other properties “Newton inversion” These approaches are in fact interconnected …. The method (A.P.Lyubartsev, A.Mirzoev, L.J. Chen, A.Laaksonen, Faraday Discussions, 144, 2010) λ λ λ Assume: H( 1, 2,... M): Hamiltonian (potential energy) depending λ λ λ on a set of parameters 1, 2,... M . We wish to reproduce M canonical averages <S1>,<S2>,...,<SM> Set of λα , α=1,…,M ↔ Space of Hamiltonians direct {λα} {<Sα>} inverse In the vicinity of an arbitrary point in the space of Hamiltonians one can write: ∂〈 S 〉 〈 〉=∑ α 2 Δ Sα ∂ Δλγ O Δλ γ λγ where ∂ 〈 〉 ∂ ∂ ∂ Sα =〈 Sα 〉−β(〈 H 〉−〈 H 〉〈 〉) ∂ ∂ λ ∂ Sγ ∂ S γ λγ γ λα λα This allows us to solve the inverse problem iteratively Algorithm: (0) Choose trial values λα Direct MC or MD e c n e g r (n) e Calculate <Sα> and differences v (n) (n) n ∆<Sα> = <Sα > - Sα* o c l i t n u Solve linear equations system t a e (n) p Obtain ∆λα e R (n+1) (n) (n) New Hamiltonian: λα =λα +∆λα Newton method of solving non-linear equation: ∂ S ∂ if no convergence, a S λ λ S( ) regularization procedure S* can be applied: (n) (n) ∆<Sα> = a(<Sα > - Sα*) λ* λ λ λ with a<1 1 0 Inverse Monte Carlo: Reconstruction of pair potentials from RDFs A.Lyubartsev, A.Laaksonen,Phys,Rev.E, 52, 1995) Vα We consider Hamiltonians in the form: =∑ ( ) H V α S α ri α =∑ ( ) Any pair potential H V r ij can be i , j | | | | | | | written in this way within the grid approximation:: α=1,…,M Rcut α α λ Vα=V(Rcut /M) - potential within -interval = - parameters Sα - number of particle’s pairs with distance between them within α-interval = 1 V gr = 〈S 〉 estimator of RDF: α 2 2 / α 4πr α Δ r N 2 ∂〈 〉 S =− 〈 〉−〈 〉 〈 〉 Inversion matrix became: ∂ S S S S V we can reconstruct pair potential from RDF Additional comments: ● Relationsip “ pair potential ⇔ RDF ” is unique (Hendeson theorem: R. L. Henderson, Phys. Lett. A 49, 197 (1974)). ● In practice the inverse problem is often ill-defined, that is noticeable different potentials yield RDFs not differing by eye on a graph ● Another scheme to correct the potential (Iterative Boltzmann inversion): V(n+1)(r) = V(n)(r) + kT ln(g(n)(r)/gref(r)) - may yield different result from IMC though RDFs are similar - may not completely converge in multicomponent case - may be reasonable to use in the beginning of the IMC iteration process ● Analogy with Renormalization group Monte Carlo (R.H.Svendsen, PRL, 42, 859 (1979)) Molecular (multiple-site) systems 1) a set of different site-site potentials 2) intramolecular potentials: bonds, angles, torsions We use the same expression: =∑ H V α Sα α But now α runs over all types of potentials: that is, over all site-site potentials and over all intramolecular potentials, and within each type of potential: over all relevant distances If α corresponds to a intramolecular potential, then <Sα> is the corresponding bond, angle or torsion distribution Treatment of electrostatics: If sites are charged, we separate electrostatic part of the potential as: ( )= ( )+ qi q j V tot r ij V short r ij πε ε 4 0 rij q - sum of charges from atomistic model i ε - dielectric permittivity: either experimental; or extracted from fitting of asymptotic behaviour of effective potentials) Electrostatic part: computed by Ewald summation (or PME) V - updated in the inverse procedure short Test example: Single site water model +0.41 -0.82 +0.41 O-O RDF Potential Problems: P ≈ 9000 bar ; E ~ + 1kJ/mol - not a liquid state,... Thermodynamic corrections We can try to fit some thermodynamics properties (P, E or µ) keeping the potential (approximately) consistent with RDFs Pressure corrections: 1) add function ∆V = Α(1-r/r ) (Wang et al, Eur.Phys.J. E, 28, 221 (2009) cut 2) compute RDF and invert it in two-phase system (Lyubartsev et al, Faraday Discussions, 144, 43, (2010)) 3) “Variational” inverse MC: ∑ (〈 〉− * )2 Minimize α S α S α under constraints P = P*, µ = µ* ... (without constraints – equivalent to the standard IMC) Pressure-corrected 1-site water CG potential P= 5 +/-10 bar RDF coincide within thickness of line But: E = -12.5 kJ/mol (exp: -41) µ = -9.6 kJ/mol (exp: -24) From ab-initio to atomistic: ab-intio derived 3-site water model Reference simulation: ab-initio (CPMD) simulation of 32 water Some details: Box sixe 9.86 Å , BLYP functional Vanderbilt pseudopotentials 25 Ha cutoff timestep 0.15 fs 25 ps simulation at 300 C 3.0 O-O 250 O-O CPMD Ab-initio eff. pot 2.5 200 Exp SPC model SPC 150 2.0 T k 100 / H-H l F H-H a 50 i 1.5 t D O-H n e R 0 t 1.0 o p -50 . f f E -100 0.5 O-H -150 0.0 -200 2 4 6 r (Å) 2 4 6 r (Å) Some insight Average energy is -25 kJ/mol Experimental: -41 kJ/mol. Comparison with CPMD: E(32 H O) – 32 E(H O,opt) = y 2 2 g r -29 kJ/mol e n e D M P C Difference about 4 kJ/mol due to many body effects effective potential energy Li+ - water ab-initio potentials Input RDF-s obtained from 5+20 ps Car-Parrinello MD for 1 Li+ ion in 32 water molecules (A.P.Lyubartsev, K.Laasonen and A.Laaksonen, J.Chem.Phys., 114,3120,2001) IMC procedure: SPC model for water; only LiO potential varied to match ab-initio LiO RDF 60 Inversion RDF Fit Non-electrostatic part of Dang, J.Chem.Phys.96,6970(1992) Periole et al, J.Phys.Chem.B,102,8579(1998) the LiO potential may be 40 Heinzinger, Physica BC, 131, 196 (1985) ) well fitted by: M / J k ( t o p 20 . f U = A exp(-br) f LiO E 0 where A = 37380 kJ/M -1 2 3 4 b = 3.63 Å r ( Å ) Atomistic simulations: Egorov et al, J.Phys.Chem. B, 107, 3234(2003) From atomistic to coarse-grained: solvent-mediated potentials. Atomistic: All-atom simulations (MD) with explicit solvent (water). State of art: < 100000 atoms - box size ~ 80-100 Å + Time scale problem CG: Coarse-grain solutes; use continuum solvent (McMillan-Mayer level), From 60-ties: primitive electrolyte model: ions - hard spheres interacting by Coulombic potential with suitable ε, ion radius - adjustable parameter We can now build effective solvent-mediated potential, which takes into account molecular structure of the solvent. RDF from atomistic MD Implicit solvent effective potentials NaCl ion solution Ion-ion RDFs Ion-ion effective (from atomistic MD) potentials 4,0 6 3,5 NaCl, L=39Å NaCl, L=39Å NaCl, L=24Å NaCl, L=24Å NaNa, L=39Å 4 NaNa, L=39Å 3,0 NaNa, L=24Å NaNa, L=24Å ClCl, L=39Å T ClCl, L=39Å k 2,5 ClCl, L=24Å ClCl, L=24Å / l 2 Prim.
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