CE 53 Molcular Simulation Lctur 8 Fr-nrgy calculations David A. Kofk Dpartmnt of Chmical Enginring SUNY Buffalo kofk@ng.buffalo.du
2 Fr-Enrgy Calculations Uss of fr nrgy Phas quilibria Raction quilibria Solvation Stability Kintics Calculation mthods Fr-nrgy prturbation Thrmodynamic intgration Paramtr-hopping Histogram intrpolation
3 Ensmbl Avrags Simpl nsmbl avrags ar of th form M dπ ( M( To valuat: sampl points in phas spac with probability π( at ach point, valuat M( simpl avrag of all valus givs <M> Prvious xampl man squar distanc from origin in rgion R 2 s( 2 ds( r d r ( insid R s outsid R sampl only points in R, avrag r 2 Principl applis to both MD and MC
Ensmbl Volums Entropy and fr nrgy rlat to th siz of th nsmbl.g., S k lnω(e,v,n No ffctiv way to masur th siz of th nsmbl no phas-spac function that givs siz of R whil sampling only R imagin bing plac rpatdly at random points on an island what could you masur at ach point to dtrmin th siz of th island? Volum of nsmbl is numrically unwildy.g. for hard sphrs r., Ω 5 33 Ω numbr of stats of givn E,V,N 4 r.5, Ω 3 7 r.9, Ω 5-42 Shap of important rgion is vry complx cannot apply mthods that xploit som simpl gomtric pictur
Rfrnc Systms All fr-nrgy mthods ar basd on calculation of frnrgy diffrncs Exampl volum of R can b masurd as a fraction of th total volum 5 Ω R s( Ω sampl th rfrnc systm kp an avrag of th fraction of tim occupying targt systm what w gt is th diffrnc SR S kln ( ΩR / Ω Usfulnss of fr-nrgy diffrnc it may b th quantity of intrst anyway if rfrnc is simpl, its absolut fr nrgy can b valuatd analytically.g., idal gas, harmonic crystal
Hard Sphr Chmical Potntial Chmical potntial is an ntropy diffrnc S/ k βµ + N UV, For hard sphrs, th nrgy is zro or infinity any chang in N that dos not caus ovrlap will b chang at constant U To gt ntropy diffrnc [ SUV (,, N SUV (,, N ] simulat a systm of N+ sphrs, on non-intracting ghost occasionally s if th ghost sphr ovrlaps anothr rcord th fraction of th tim it dos not ovrlap S/ k V Δ βµ Ω N + V f 3 V 3 NΛ Ω 3 N NΛ NΛ non-ovrlap Hr is an applt dmonstrating this calculation N+ N 6
Fr-Enrgy Prturbation Widom mthod is an xampl of a fr-nrgy prturbation (FEP tchniqu FEP givs fr-nrgy diffrnc btwn two systms labld, Working quation ( A A Q Q d d π ( d d ( U U U d ( U U ( U U U U U Fr-nrgy diffrnc is a ratio of partition functions 7
Fr-Enrgy Prturbation Widom mthod is an xampl of a fr-nrgy prturbation (FEP tchniqu FEP givs fr-nrgy diffrnc btwn two systms labld, Working quation ( A A Q Q d d d ( U U d π ( d ( U U ( U U U U U U Add and subtract rfrnc-systm nrgy 8
Fr-Enrgy Prturbation Widom mthod is an xampl of a fr-nrgy prturbation (FEP tchniqu FEP givs fr-nrgy diffrnc btwn two systms labld, Working quation ( A A Q Q d ( U U U ( U U d π ( ( U U d d d U U U 9 Idntify rfrncsystm probability distribution
Fr-Enrgy Prturbation Widom mthod is an xampl of a fr-nrgy prturbation (FEP tchniqu FEP givs fr-nrgy diffrnc btwn two systms labld, Working quation ( A A Q Q Sampl th rgion important to systm, masur proprtis of systm d d d ( U U U d ( U U d π ( ( U U U U U Writ as rfrncsystm nsmbl avrag
Chmical potntial For chmical potntial, U - U is th nrgy of turning on th ghost particl call this u t, th tst-particl nrgy ( A A µ V NΛ 3 u t tst-particl position may b slctd at random in simulation volum for hard sphrs, -βu t is for ovrlap, othrwis thn (as bfor avrag is th fraction of configurations with no ovrlap This is known as Widom s insrtion mthod
Dltion Mthod Th FEP formula may b usd also with th rols of th rfrnc and targt systm rvrsd Original: à Modifid: à ( A A ( U U sampl th systm, valuat proprtis of systm Considr application to hard sphrs + β( A A + β( U U 3 + βµ N + β V βu t is infinity for ovrlap, othrwis but ovrlaps ar nvr sampld tru avrag is product of tchnically, formula is corrct u t in practic simulation avrag is always zro mthod is flawd in application many tims th flaw with dltion is not as obvious as this
4 Othr Typs of Prturbation Many typs of fr-nrgy diffrncs can b computd Thrmodynamic stat tmpratur, dnsity, mixtur composition Hamiltonian for a singl molcul or for ntir systm.g., valuat fr nrgy diffrnc for hard sphrs with and without lctrostatic dipol momnt Configuration distanc/orintation btwn two soluts.g, protin and ligand Ordr paramtr idntifying phass ordr paramtr is a quantity that can b usd to idntify th thrmodynamic phas a systm is in.g, crystal structur, orintational ordr, magntization
Gnral Numrical Problms 5 Sampling problms limit rang of FEP calculations Targt systm configurations must b ncountrd whn sampling rfrnc systm Two typs of problm aris targt-systm spac vry small targt systm outsid of rfrnc first situation is mor common although dltion FEP provids an avoidabl xampl of th lattr
Staging Mthods 6 Multistag FEP can b usd to rmdy th sampling problm dfin a potntial U w intrmdiat btwn and systms valuat total fr-nrgy diffrnc as Each stag may b sampld in ithr dirction yilding four staging schms choos to avoid dltion calculation A A ( A A + ( A A w w W Umbrlla sampling W Bnntt's mthod W Stagd dltion W Stagd insrtion Us stagd insrtion Us umbrlla sampling Us Bnntt s mthod W W W
Exampl of Staging Mthod Hard-sphr chmical potntial Us small-diamtr sphr as intrmdiat β Δ( A A β u( σ w N t In first stag, masur fraction of tim random insrtion of small sphr finds no ovrlap N ( A N Aw ( u( u( t Δ + β σ σ In scond stag, small sphr movs around with othrs. Masur fraction of tim no ovrlap is found whn it is grown to full-siz sphr w 7 ( A N A N u ( t ( u( u( t β Δ + β σ σ σ N w
8 Multipl Stags W 3 W 2 W 2 W 3 W W 2 W W 3 W Multistag insrtion Multistag umbrlla sampling Multistag Bnntt s mthod W W2 W3 W2 W W3 W2 W W3
9 Non-Boltzmann Sampling Th FEP mthods ar an instanc of a mor gnral tchniqu that aims to improv sampling Unlik biasing mthods, improvmnt ntails a chang in th limiting distribution Apply a formula to rcovr th corrct avrag Q M dm( Q ( U U U d M( Q Q W W W M W ( U U W ( U U W U W W W
2 Thrmodynamic Intgration. Thrmodynamics givs formulas for variation of fr nrgy with stat d( βa Udβ βpdv + βµ dn βa βa U P β V VN, TN, Ths can b intgratd to obtain a fr-nrgy diffrnc drivativs can b masurd as normal nsmbl avrags βa( V βa( V P( V dv 2 V V 2 this is usually how fr nrgis ar masurd xprimntally
Thrmodynamic Intgration 2. TI can b xtndd to follow uncommon (or unphysical intgration paths much lik FEP, can b applid for any typ of fr-nrgy chang Formalism Lt λ b a paramtr dscribing th path th potntial nrgy is a function of λ nsmbl formula for th drivativ thn β A ln Q N N U( r ; λ dr 3N Λ N! λ λ Q λ Q N N U( r ; λ N + dr βu ( r ; λ 3N Λ βu λ N! 2 βu βa( λ2 βa( λ + dλ λ λ λ λ 2
Thrmodynamic Intgration Exampl 22 Th soft-sphr pair potntial is givn by n ur ( σ ε r Exhibits simplifying bhavior bcaus εσ n is th only potntial paramtr Softnss and rang varis with n larg n limit lads to hard sphrs small n lads to Coulombic bhavior Thrmodynamic intgration can b usd to masur fr nrgy as a function of softnss s /n Intgrand is βa s βε ur (ln( σ / r 2 s U/ε 3. 2.5 2..5..5...5 2. Sparation, r/σ n 2 n 2 n 6 n 3 2.5
Paramtr Hopping. Thory 23 Viw fr-nrgy paramtr λ as anothr dimnsion in phas spac N E E( p, r, λ Partition function N Q dp dr λ N N E( p, r, λ Mont Carlo trials includ changs in λ Probability that systm has λ λ or λ λ N N N N E N N E( p, r, λ N N β ( p, r, λ dp dr + dp dr Q + Q π( λ π( λ N N N N N E( p, r, λ Q d d Q+ Q p r Q+ Q N N N N E( p, r, λ Q d d Q+ Q p r Q+ Q N λ λ
Paramtr Hopping. Implmntation Mont Carlo simulation in which l-chang trials ar attmptd Accpt trials as usual, with probability min[, -βδu ] 24 Rcord fractions f, f of configurations spnt in λ λ and λ λ Fr nrgy is givn by ratio ( A A Q Q ( Q + Q f In practic, systm may spnd almost no tim in on of th valus Can apply wighting function w(λ to ncourag it to sampl both Accpt trials with probability min[,(w n /w o -βδu ] Fr nrgy is Q Q ( Q + Q f Good choic for w has f f Multivalu xtnsion is particularly ffctiv l taks on a continuum of valus ( A A w f wf
25 Summary Fr nrgy calculations ar ndd to modl th most intrsting physical bhaviors All usful mthods ar basd on computing fr-nrgy diffrnc Four gnral approachs Fr-nrgy prturbation Thrmodynamic intgration Paramtr hopping Distribution-function mthods FEP is asymmtric Dltion mthod is awful Four approachs to basic multistaging Umbrlla sampling, Bnntt s mthod, stagd insrtion/dltion Non-Boltzmann mthods improv sampling