Statistial mechanics & free energy methods Tutoriel CPMD/CP2K Paris, 6–9 avril 2010 François-Xavier Coudert Molecular simulation Statistical mechanics Choice of statistical ensemble (N, V, T) or (N, P, T) or ... Partition function Q = ∑i exp(-Ei/kT) Molecular simulation Q = ∑i ... = ... exp(–Ei/kT) Sampling phase space + Evaluating the energy Single point calculation Quantum methods: Energy minimization Hartree–Fock & post Hartree–Fock, or transition state searching density functional theory (DFT), quantum Monte Carlo Integrating Newton’s equation (molecular dynamics = MD) Classical atomistics forcefields: coulombic interactions, Stochastic sampling (random moves) (Monte Carlo = MC) dispersion (London, van der Waals, etc.), polarisation, intramolecular terms... Friction and random forces (Langevin dynamics) Coarse-grained forcefields Molecular simulation Sampling phase space to calculate observables as averages on a large number of configurations Molecular Monte-Carlo dynamics Random Trajectories moves Many statistical Dynamical ensembles processes Nonlocal sampling Collective moves βE ai a(τ)dτ ∂V e− A= a = = lim m ¨r = P ∝ N t t − ∂r →∞ Free energy Free energy is a most important concept in chemistry It governs the physical and chemical behaviour of systems “Free energy methods” covers two (overlapping) concepts: ✓ The function of state (A, G, Ω, ...), and its change during a process: ∆A, ∆G, ... ✓ Profiles of free energy with respect to an internal constraint, e.g. a reaction coordinate We’ll be talking today about: ✓ solvation ✓ ligand association ✓ reactions & reaction paths Free energy Different types of free energy for each ensemble: A short (but necessary) clarification on vibrational entropy, solvation models, and other approximations “You can easily and pretty accurately calculate the solvation entropy using ChemSol” (2008, CCL.net list) the solvation entropy is best calculated as (H – G)/T, and G itself is painful to compute UnforeseenFree Stationary energy States ... or what if the potential energy landscape is very rugged? • When you’re using local approximations for the free energy, Dynamicalare you positive bottleneck your and potential diffusive energy barriers surface doesn’t look like this? • Many irrelevant saddle points • First Prev Next Last Go Back Full Screen Close Quit • • • • • • • • Free energy methods We will talk today about: ✓ Thermodynamic perturbation & thermodynamic integration ✓ Umbrella sampling ✓ Adiabiatic free energy sampling ✓ Metadynamics One word about collective variables They can be: ✓ distances ✓ angles ✓ simulation box parameters ✓ coordination numbers ✓ etc. Thermodynamic perturbation Zwanzig, 1954: Some conditions have to be met: ✓ States A and B have to be “close” ✓ i.e. they should have widely overlapping phase space basins ✓ i.e. EA – EB should always be ≤ kT As such, not very useful... How do we solve that? Two approaches of free energy calculationq Thermodynamic perturbation thermodynamicalPerturbation Thermodynamical Intégration ) ! > Introduce intermediate points... i+1 ! ! " -> H/ i Usually in the form of a coupling parameter: ! $" A( λ ∈ [0,1] # #A Embarrasingly parallel! Two approaches of free energy calculationq! ! Thermodynamical integration mmethodethod (TI) AnotherThermodynamical method isthermodynamicalPerturbation the thermodynamicintegration mmethodethod integration (TI) Thermodynamical Intégration !""11#A ! #U ! $"A !""11! dd! " ! #Utot(!rp, ) ! ) %%#A! tot(rp, ) ! ##!! > $"A !!""00dd! " i+1 ! ! %%! !!""00##!! " -> H/ i ! $" !!A( &$! ! #'UUtot(rp, ) to# t(rp, ) Utot(rp, ) #A #'UU!!" &$! U ! ##to!!t(rp, ) " tot(rp, ) tot(rp, ) ##!!! ! ! ! ! ! S 2 reaction F +CH Cl CH F + Cl Thermodynamic perturbationN − 3 → 3 − Mugnai, Cardini & Schettino, JCP 2003 Autoionization of Liquid Water Examples Left: (Free) energy profile along the reaction paths at 0 and 300 K Contraint: O∗ H∗ distance − • Right: Dipole moment CH X and Y− along the 0 K reaction path • 3 ionisation of water – – SN2 reaction: F +CH3Cl → CH3F + Cl (O–H distance constrainted)First Prev Next Last Go Back Full Screen Close Quit • • • • • • • • Mugnai, Cardini & Schettino, JCP 2003 First Prev Next Last Go Back Full Screen Close Quit • • • • • • • • Umbrella sampling The basic idea: ✓ divide reaction pathway into windows ✓ constrain the system in each window (biasing potential) ✓ determine the free energy, correct it for the biasing potential ✓ paste everything altogether Calculation of average quantities Average values obtained through biased simulations are ccorrectedorrected biais N N ()V !r "% A!r "exp& # '& kT $# b A * U )V ()V biais !"r N % exp& # & kT # ' $ U )V b Example Calculation of the free Umbrellaenergy variation sampling along the internal rotation of the chromophore Example:In Green FluorPMF eforscent internal PrProteinotein rotation of the chromophore in GFP Practical aspects PracticalSuccessive biased aspects simulations Successive biased simulations courtesy of I. Demachy, Paris-Sud 11 Adiabatic free energy sampling The basic idea: ✓ define reaction coordinates and environment coordinates ✓ give the reaction coordinates a large effective mass (the environment coordinates will follow adiabatically) ✓ thermostate the reaction coordinates at high temp. (so that the free energy profile is fully sampled) Conversion of 2-BromoethanolAdiabatic free to Dibromoethaneenergy sampling The heavy atoms of the bromoethanol molecule are taken as the reactive sub- system (masses scaled by 100, coupled to a Nose-Hoover at T=2000 K). Solute and H at 300 K, QM/MMExample: setup from 2-bromoethanol protonated torsion bromonium ion to dibromoethane Br, C, C, O coordinates are the reaction coordinates QM/MM simulation Free exploration of configurations Theavy = 2000 K Mheavy: scaled by 100 110 C-C-O dissociation angle and Br-C-C bromonium 60 concerted 1 C-O distance 4 J. VandeVondele and U. Rothlisberger, J. Phys. Chem. B 2002 First Prev Next Last Go Back Full Screen Close Quit • • • • • • • • Metadynamics The basic idea: ✓ choose collective variables s ✓ history-dependent potential biasing the simulation not to come back too often to visited microstates Metadynamics alanine dipeptide ψ/φ cyclohexane conformation changes movies by Vojtěch Spiwok Metadynamics Example: SN2 vs E2 vs ... d͑A Ϫ B͒ n 1 Ϫ ͩ d ͪ 0 nc͑A Ϫ B͒ ϭ m. [1] ͸ d͑A Ϫ B͒ SPECIAL FEATURE B 1 Ϫ ͩ d ͪ 0 This function estimates the number of atoms B within the cutoff radius d0 of atom A (n and m are positive integers that determine how ‘‘fast’’ the function goes from 1 to 0 at the cutoff radius r0). Previously, we have computed successfully the 3D free energy surface for the E2 reaction by using the three coordination numbers as the CV (6). The reactant complex state is described by the values nc(Fa–C) ϭ 1, nc(Fb–H) ϭ 0, nc(Cb–H) ϭ 3, and the product complex state is defined by the values nc(Fa–C) ϭ 0, nc(Fb–H) ϭ 1, nc(Cb–H) ϭ 2. We applied the hills method to construct the 3D free energy surface. The LFEP and the US potential were obtained from the sum of hills after having filled up the reactant and product wells with hills. The 1D free energy profile was converged by using US. Alternatively, the free energy surface can in principle be converged to arbitrary accuracy within the hills method, by using small enough hills and sampling many barrier recrossings (10). In practice, however, not only is converging the 3D surface more costly than converging the 1D profile, but also, the efficiency of the hills method decreases as the accumulating hills grant access to more and more of the higher-energy regions of configuration space. For the E2 reaction, eventually this hills simulation ended when the free energy exploration stranded in a new local minimum by means of the SN2attackofFb on Ca while Fa was Fig 1. Mechanistic spectra of possible TS of the bimolecular elimination (E2) leaving and forming a hydrogen bond with one of the ␤-hydro- reaction (A) and of the competing bimolecular nucleophilic substitution (SN2) reaction vs. E2 reaction channels (B). gens. Escaping from this minimum requires the additional CV nc(Fb Ϫ C)tobreakthenewlyformedFb–Ca bond. Note that the SN2channelleadstoaproductstatethatischemicallyidentical the US technique to converge the free energy profile along the to the initial state; only Fa and Fb have been exchanged. Fig 2. Dynamics of the six CV shown as a function of the number of hills. (Top) CHEMISTRY LFEP (6). Returning via the SN2channelfromthisproductstateis In the current study, the set of CV is extended with another therefore less probable thanThe coordination proceeding with numbers a ‘‘new’’ of C around E2 Fa (black line) and Fb (red line). (Middle) three coordinates, which allows us also to map out the competing reaction, now with Fb asmetadynamics the leavingThe coordination group and F numbersa as the attackingwith of H around 6 collective Fa (black line) and variables: Fb (red line). SN2reactionchannelintheexplorationofthefreeenergy base. A proper description(Bottom of this) step The thus coordination requires also numbers the of H around Ca (black line) and Cb (red landscape and understand why traditional methods fail to obtain additional nc(Fa Ϫ Hcoordination)CV.Finally,arotationoftheE2leavingline). The background numbers colors blue, yellow, for green, F anda–C, red emphasize Fb–C, the fourFa–H, groups FH and FϪ around C H in the product state allows the free energy profiles of the E2 and SN2reactionchannels. distinct2 4 E2 reactions (labeled A, B, C, and D at the top) that are sampled during Disregarding the change of the C–C bond, from single into exchange of Ca and Cb,sothatalsothethe simulation,nc(CF witha bϪ–H,H the)needstobe darker Ca shades–H, indicatingand C theb product–H states and the double, during the E2 reaction between fluoroethane and the added to the set as the sixthlighter CV. shades indicating the reactant states (also denoted R and P on top). In the present work, we compute the six-dimensional (6D) free fluoride ion, there are three chemical bonds that are broken or The numbers 1–10 at the top label the transitions and match the numbers in energy surface by using the coordination numbers nc(Fa–C), formed during the reaction, namely, the Fa–Ca and a Cb–H bonds the flow chart in Fig.