CE 530 Molecular Simulation

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1 CE 530 Molecular Simulatio Lecture 5 Log-rage forces ad Ewald sum David A. Kofke Departmet of Chemical Egieerig SUNY Buffalo kofke@eg.buffalo.edu

2 Review Itermolecular forces arise from quatum mechaics too complex to iclude i legthy simulatios of bulk phases Empirical forms give simple formulas to approximate behavior itramolecular forms: bed, stretch, torsio itermolecular: va der Waals, electrostatics, polarizatio Ulike-atom iteractios weak lik i quatitative work

3 3 Trucatig the Potetial Bulk system modeled via periodic boudary coditio ot feasible to iclude iteractios with all images must trucate potetial at half the box legth (at most) to have all separatios treated cosistetly Cotributios from distat separatios may be importat These two are same distace from cetral atom, yet: Black atom iteracts Gree atom does ot These two are earest images for cetral atom Oly iteractios cosidered

4 Trucatig the Potetial 4 Potetial trucatio itroduces discotiuity Correspods to a ifiite force Problematic for MD simulatios ruis eergy coservatio Shifted potetials Removes ifiite force Still discotiuity i force Shifted-force potetials Routiely used i MD ur () ur ( c) r r () r = 0 r > r For quatitative work eed to re-itroduce log-rage iteractios u s du ur () ur ( ) ( r r) r r usf () r = dr 0 r > r c c c c c c

5 Leard-Joes example r c =.5σ Trucatig the Potetial 5 u s () r ur () ur ( c) r r = 0 r > r c c du ur () ur ( ) ( r r) r r usf () r = dr 0 r > r c c c c u(r) f(r) u(r) u-shift u(r) f-shift f(r) f-shift..6.0 Separatio, r/σ Separatio, r/σ.4.5

6 6 Radial Distributio Fuctio Radial distributio fuctio, g(r) key quatity i statistical mechaics quatifies correlatio betwee atom pairs Defiitio gr () = ρ() rdr ρ id () rdr Number of atoms at r i actual system dr 4 3 Hard-sphere g(r) Low desity High desity Number of atoms at r for ideal gas id N ρ () rdr = dr V Here s a applet that computes g(r)

7 8 Radial Distributio Fuctio. Java Code public class MeterRDF exteds MeterFuctio /** * Computes RDF for the curret cofiguratio */ public double[] curretvalue() { iterator.reset(); //prepare iterator of atom pairs for(it i=0; i<poits; i++) {y[i] = 0.0;} //zero histogram while(iterator.hasnext()) { //loop over all pairs i phase double r = Math.sqrt(iterator.ext().r()); //get pair separatio if(r < xmax) { it idex = (it)(r/delr); //determie histogram idex y[idex]+=; //add oce for each atom } } it = phase.atomcout(); //compute ormalizatio: divide by double orm = */phase.volume(); //, ad desity*(volume of shell) for(it i=0; i<poits; i++) {y[i] /= (orm*vshell[i]);} retur y; }

8 9 Simple Log-Rage Correctio Approximate distat iteractios by assumig uiform distributio beyod cutoff: g(r) = r > r cut Correctios to thermodyamic properties Iteral eergy Virial N Ulrc = ρ u()4 r πr dr cut Chemical potetial r du Plrc = ρ r 4πr dr 6 dr r cut U µ lrc = ρ ur ()4πrdr= N r cut lrc U Expressio for Leard-Joes model P LJ lrc LJ lrc σ σ πnρσ ε 3 r c r c 8 = 9 3 πρ σ ε σ 3 σ = 9 r c r c For r c /σ =.5, these are about 5-0% of the total values

9 Coulombic Log-Rage Correctio Coulombic iteractios must be treated specially very log rage /r form does ot die off as quickly as volume grows r c fiite oly because + ad cotributios cacel Methods Full lattice sum Here is a applet demostratig direct approach Ewald sum 4 r π rdr = Treat surroudigs as dielectric cotiuum Leard-Joes Coulomb 4 0

10 Aside: Fourier Series Cosider periodic fuctio o - L/, +L/ A Fourier series provides a equivalet represetatio of the fuctio f (x) = a 0 + ( a cosx + b si x) = The coefficiets are + L / / + L / / a = f ( x)cos( π x / L) dx b = f ( x)si( π x / L) dx f(x) Oe period L/ x +L/

11 Fourier Series Example f(x) is a square wave f(x) + L / / a = f ( x)cos( π x / L) dx 0 + L / πx L dx / 0 = cos( / ) cos( πx / L) dx = 0 + L / / b = f ( x)si( π x / L) dx = 4 π odd 0 eve x

12 Fourier Series Example 3 f(x) is a square wave = f(x) + L / / a = f ( x)cos( π x / L) dx 0 + L / πx L dx / 0 = cos( / ) cos( πx / L) dx = 0 + L / / b = f ( x)si( π x / L) dx = 4 π odd 0 eve x

13 Fourier Series Example 4 f(x) is a square wave =, 3 f(x) + L / / a = f ( x)cos( π x / L) dx 0 + L / πx L dx / 0 = cos( / ) cos( πx / L) dx = 0 + L / / b = f ( x)si( π x / L) dx = 4 π odd 0 eve x

14 Fourier Series Example 5 f(x) is a square wave =, 3, 5 f(x) + L / / a = f ( x)cos( π x / L) dx 0 + L / πx L dx / 0 = cos( / ) cos( πx / L) dx = 0 + L / / b = f ( x)si( π x / L) dx = 4 π odd 0 eve x

15 Fourier Series Example 6 f(x) is a square wave =, 3, 5, 7 f(x) + L / / a = f ( x)cos( π x / L) dx 0 + L / πx L dx / 0 = cos( / ) cos( πx / L) dx = 0 + L / / b = f ( x)si( π x / L) dx = 4 π odd 0 eve x

16 7 Fourier Represetatio The set of Fourier-space coefficiets b cotai complete iformatio about the fuctio f (x) = a 0 + Although f(x) is periodic to ifiity, b is o-egligible over oly a fiite rage Sometimes the Fourier represetatio is more coveiet to use. = ( ) a cosx + b si x b

17 8 Covergece of Fourier Sum If f(x) = si(πkx/l), trasform is simple b = for = k Coverges very quickly! b = 0 otherwise f(x) b x

18 9 Observatios o Fourier Sum Smooth fuctios f(x) require few coefficiets b Sharp fuctios (square wave) require more coefficiets Large- coefficiets describe high-frequecy behavior of f(x) large = short wavelegth Small- coefficiets describe low-frequecy behavior small = log wavelegth e.g., = 0 coefficiet is simple average of f(x)

19 Fourier Trasform As L icreases, f(x) becomes less periodic Fourier trasform arrives i limit of L à Compact form obtaied with expoetial form of cos/si f (x) = a 0 + a = π b = π + L/ L/ + L/ L/ e = iθ ( + b ) a cos πx L f (x)cos(π x / L) dx f (x)si(π x / L) dx Useful relatios derivative = cosθ + isiθ covolutio si πx L iverse forward + ˆ π ikx f( x) = f( k) e dk + ˆ( ) ( ) π ikx f k = f x e dx a = real part of trasform b = imagiary part f ( x) ( k) = ( πik) f( k) f() t g( x t) dt ( ) ˆ k = f( k) gˆ ( k) ( m) m ˆ 0

20 Fourier Trasform Example Gaussia α x f( x) = exp α π Trasform is also a Gaussia! f(x) / ˆ( α k f k ) = exp π α x Width of trasform is reciprocal of width of fuctio k-space is reciprocal space sharp f(x) requires more values of F(k) for good represetatio δ(x-x o ) trasforms ito a sie/cosie wave of frequecy x o : ˆ( ) o δ k = e πikx F(k) x k

21 Fourier Trasform Relevace May features of statistical-mechaical systems are described i k-space structure trasport behavior electrostatics This descriptio focuses o the correlatios show over a particular legth scale (depedig o k) Macroscopic observables are recovered i the k à 0 limit Correspodig treatmet is applied i the time/frequecy domais

22 Review of Basic Electrostatics 3 Force betwee charges I terms of electric field Static electric field satisfies Er () = 4 πρ() r Er () = 0 Charge desity ρ(r) for poit charge q : Electrostatic potetial zero curl implies E ca be writte potetial eergy of charge q at r, relative to positio at ifiity u() r = qφ() r Poisso s equatio φ = 4πρ qq F= r r ˆ Fr () = q Er () ρ() r = q δ() r Er () = φ() r Mass aalogy: u(z) = m gz = mφ(z)

23 4 Ewald Sum We wat to sum the iteractio eergy of each charge i the cetral volume with all images of the other charges express i terms of electostatic potetial U q = qφ i ri charge i i cetral volume ( ) the charge desity creatig the potetial is ρ(r) = q δ (r r ),image vectors i = q δ r (r + L) ( ) this is a periodic fuctio (of period L), but it is very sharp Fourier represetatio would ever coverge

24 5 Ewald Sum: Fourier. Compute field istead by smearig all the charges 3/ ρ() r = q( α/ π) exp α r ( r + L) Electrostatic potetial via Poisso equatio direct space form reciprocal space iclude = 0 φ() r = 4 πρ() r k φ( k) = 4 πρ( k) Discrete Fourier trasform the charge desity V V ρ( k) = de r ρ( r) = V qe ikr ikr k e /4α Large α takes ρ back to δ fuctio a = L b = L + L/ / + L/ / f (x)cos(π x / L) dx f (x)si(π x / L) dx

25 6 Ewald Sum. Fourier. Use Poisso s equatio for electrostatic potetial 4π φ( k) = ρ( k) k Ivert trasform to recover real-space potetial φ() r = φ( k) e k 0 = V k 0 ikr 4π q e k ik ( r r ) k /4α e f (x) = a 0 + ( a cosx+b si x) i priciple requires sum over ifiite umber of wave vectors k but reciprocal Gaussia goes to zero quickly if α is small (broad Gaussia, large smearig of charge) =

26 Ewald Sum. Fourier 3. 7 The electrostatic eergy ca ow be obtaied for poit charges i potetial of smeared charges U q = q iφ ri i = = Two correctios are eeded self iteractio k /4α i k 0 k i, V k 0 k ( ) 4πV e 4πV e k correct for smearig /4α qq ρ( k) e ik ( r r ) i product of idetical sums i ρ( k) = qe kr V φ(r) = V k 0 4πq k e ik (r r ) e k /4α

27 Ewald Sum. Self Iteractio. 8 I Ewald sum, each poit charge is replaced by smeared Gaussia cetered o that charge this is doe to estimate the electrostatic potetial field x x All poit charges iteract with the resultig field to yield the potetial eergy This meas that the poit charge iteracts with its smeared represetatio We eed to subtract this x

28 Ewald Sum. Self Iteractio. We work i real space to deal with the self term Poisso s equatio for the electrostatic potetial due to a sigle smeared charge 9 φ() r = 4 πρ() r The solutio is q ( α ) φ() r = erf r r ρ() r = ( α/ π) exp α r r 3/ q = r r r r I particular, at r = 0 φ(0) = ( α/ π) q / The self-correctio subtracts this for each charge U self = = α ( ) π q φ(0) q idepedet of cofiguratio

29 30 Ewald Sum. Smearig Correctio. We add the correct field ad subtract the approximate oe to correct for the smearig p G Δ φ () r = φ () r φ () r q q = erf r r r r q = erfc r r ( α r r ) ( α r r ) This field is short raged for large α (arrow Gaussias) ca view as poit charges surrouded by shieldig coutercharge distributio x

30 3 Ewald Sum. Smearig Correctio. Sum iteractio of all charges with field correctio coveiet to stay i real space Usually α is chose so that sum coverges withi cetral image Total Coulomb eergy Δ U = qδφ ( r ) = i i i i qq i erfc r i ( α r i ) U = U ( α) U ( α) +Δ U( α) c q self each term depeds o α, but the sum is idepedet of it if eough lattice vectors are used i the reciprocal- ad real-space sums Here is a applet that demostrate the Ewald method

31 3 Ewald Method. Commets Basic form requires a O(N ) calculatio efficiecy ca be itroduced to reduce to O(N 3/ ) good value of α is 5L, but should check for give applicatio ca be exteded to sum poit dipoles Other methods are i commo use reactio field particle-particle/particle mesh fast multipole

32 33 Summary Cotributios from distat iteractios caot be eglected potetial trucated at o more tha half box legth treat log-rage assumig uiform radial distributio fuctio Coulombic iteractios require explicit summig of images too costly to perform direct sum Ewald method is more efficiet smear charges to approximate electrostatic field simple correctio for self iteractio real-space correctio for smearig

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of