3.320: Lecture 18 (4/14/05)Non-Boltzmann sampling and Umbrella samplingCases where non-Boltzmann sampling may be usefulNon-Metropolis Monte CarloExample: Force-bias Monte CarloCase Study: Studying Surface segregation in Cu-NiEquilibration problems in Monte Carlo (not unlike real systems)This problem is similar to problems in MDCould get free phase transitions from free energies, but F and S are difficult to compute …Could we get the free energy ?Methods to Obtain Free Energy DifferencesOverlapping Distribution MethodsThermodynamic Integration: You will be so proud you remember thermodynamicsExample: Integrate from T=0 in Ising modelExamples of TD integration in Ising-like ModelsIssues with Thermodynamic IntegrationWhy stop at integrating with physical parameters; The wonders of computationsEffect of dipole in waterTurn Lead into Gold ?Monte CarloReferencesMethods with multiple time scales:Coarse-grain fast one awayEquilibration of Chemical Composition and Structure: A slow time-scale problemCase Study: First Principles Predication of Alloy Phase StabilityNeed to Equilibrate all Time Scales -> Free EnergyCoarse-graining: The conceptChange coordinatesApproximations to F({s}) determine which excitations (entropies) are included in the total free energySummary so far4/14/05 3.320 Atomistic Modeling of Materials G. Ceder and N Marzari3.320: Lecture 19 (4/14/05)S(T) = S(Tref) + CVTTrefT∫dTF(µ,T) = F(µref,T) + <σ> dµµrefµ∫Free Energies and physical Coarse-grainingT<σ>4/14/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariNon-Boltzmann sampling and Umbrella samplingSimple Sampling Importance SamplingSample with Boltzmann weightSample randomly< A > = Aνν=1M∑< A > = exp(−βHν)exp(−βHν)ν=1M∑Aνν=1M∑Non Boltzmann Sampling∆H = H - HoSample with some Hamiltonian Ho< A > = exp(−β(Hν− Hυo)) Aνν=1M∑exp(−β(Hν− Hυo))ν=1M∑4/14/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariCases where non-Boltzmann sampling may be useful1) To sample part of phase space relevant for a particular property2) To sample phase space more efficiently4/14/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariNon-Metropolis Monte CarloAllow non-equal a-priori probabilities to get less possible moves that are not acceptedWijo= f ∆Hij[]Wjio= f ∆Hji[]In Metropolis this is symmetricDetailed balancePiWijoPij= PjWjioPjiPijPji=f ∆Hij[]f ∆Hji[]exp(−β∆Hij)4/14/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariExample: Force-bias Monte CarloFδri=AFi+δrirandomGo downhill faster, but requires force calculationPangali et al., Chem. Phys. Lett., 55, 413 (1978)4/14/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariCase Study: Studying Surface segregation in Cu-NiSee Foiles, S. M. “Calculation of the surface segregation of Ni-Cu alloys with the use of the embedded atom method.” Physical Review B 32, no. 12 (1985): 7685–7693.Embedded atom for energy modelSupercells that are 24x15 to 48x25 atoms with vacuumGrand canonical Hamiltonian4/14/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariCu distributionCu in first layerCu in second layerCu in third layerEquilibration problems in Monte Carlo (not unlike real systems)4/14/05 3.320 Atomistic Modeling of Materials G. Ceder and N Marzari4/14/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariThis problem is similar to problems in MDMay overshoot the melting point or fluctuate into the melt when below Teqe.g. solidification temperatureEquillibriumtemperatureCoolHeatDifficult to form solidMay settle into amorphous state4/14/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariCould get free phase transitions from free energies, but F and Sare difficult to compute …F and S are not simple averagesF = U – TSS =−kBPνν∑ln(Pν)U = Pνν∑EνNeed relative probabilities Need absolute probabilitiesF as an integrated quantityF = PνEν+ kBT ln(Pν)[]ν∑F = Pν−kBT ln(Z)[]ν∑Quantity that needs to be integrated is flat (but unknown)4/14/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariCould we get the free energy ?Problem: F is not an average. Free energy does not exist in a microstate, it is a property of the distribution function. Same for entropy F =−kT ln(Q) = − kT ln exp(−βHν)ν∈e∑⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Can write F as an average, but not over the important statesF = − kT ln1exp(βHν)⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ − kT ln[M]Proof exp(βHν) = exp(−βHν)Qν∈e∑exp(βHν) = 1Q 1ν∈e∑ =MQ4/14/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariMethods to Obtain Free Energy Differences1) Free energy integration (including λ-integration)2) Overlapping distribution methods3) Others4/14/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariOverlapping Distribution Methods ∆F = − kT lnQIIQI = − kT lnexp(−βHνII)ν∈e∑exp(−βHνI)ν∈e∑ ∆F = − kT ln exp(−β(HνII− HνI)ν∈e∑⎡ ⎣ ⎢ ⎤ ⎦ ⎥ exp(−βHνI)QI∆F = − kT ln exp(−β(HνII− HνI)I[]Forward projecting: Using the states sampled in state I to get the free energy difference with II.4/14/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariExample: free energy difference between two different temperatureOverlapping distribution methods will fail when the distributions do not overlap much. E.g. Low temperature simulation may not sample much of the excitations that would be present at high temperatureE<E>IEquilibrium distribution<E>II4/14/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariThermodynamic Integration: You will be so proud you remember thermodynamicsAnd now for an important message …A(λ2) − A(λ1) = ∂A∂λλ1λ2∫dλExample: Entropy as function of TS(T2) − S(T1) = ∂S∂TT1T2∫dT = CVTT1T2∫dTCan be obtained from Monte CarloS(T ) = S(Tref) + CVTTrefT∫dTNeed to find reference state in which we know entropyExample: Ising model from T = 04/14/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariExample: Integrate from T=0 in Ising model32100 54T / T0321C / (N k)Figure by MIT OCW.4/14/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariExamples of TD integration in Ising-like ModelsS(T) = S(Tref) + CVTTrefT∫dTF(µ,T ) = F(µref,T) + <σ> dµµrefµ∫< E > =