CME 599 / MSE 620 Fall 2008 Statistical Thermodynamics and Introductory Simulation Concepts

Transcription

1 CM 599 / MS 60 Fall 008 Statstcal hrmodynamcs and Introductory Smulaton Concpts S.. Rann ssocat Profssor Chmcal and Matrals ngnrng Unvrsty of Kntucy, Lxngton Sptmbr 9, 008

2 Outln Introducton nd for statstcal thrmo Probablty & statstcs concpts rvw Kntc thory xampl Statstcal thrmodynamcs Postulats of statstcal thrmo Proprts drvd from V, V, µv and PV nsmbls xampls: Idal gas, monatomc crystal Stat. mch. of classcal (vs. quantum systms Usful smulaton concpts rgodcty Intracton potntals and smulatng bul matrals

3 Lmtatons of Quantum Mthods Sm-mprcal MO dos not modl nducd dpols wll bst for ntramolcular ffcts Sm-mprcal MO lmtd to hundrds of atoms b nto modls lctronc ntractons wll (ncludng nducd dpols but lmtd to tns of atoms du to computatonal cost Consquncs: quantum mthods ar bst for solatd molculs or small clustrs Molculs n vacuum by dfault, but solvaton can b modld (.g. COSMO CM/MS ntrstd n collctv bhavor

4 Statstcal Mchancs radtonal thrmodynamcs: macroscopc nrgy consrvaton (1st law ntropy producton (nd law hory of nondal mxturs, phas dagrams, tc. Stat. Mch. lns mcroscopc & macroscopc Molcular ntrprtaton of macro. masurmnts Macroscopc mplcatons of ntratomc ntractons nswrs qustons of why phnomna obsrvd Most matur for systms at qulbrum (onqulbrum xtnsons latr

5 Rvw of Probablty Statstcal mchancs / molcular smulatons us statstcs of larg numbr of partcls o ntrprt and dvlop das, us probablty Probablty dfnd for larg numbr of trals Dscrt cas: fnt numbr of stats P ormalzaton: lm Whr no. tms stat obsrvd P 1 Stat vrag valus of a functon, ( 1 ( P ( If probablty of stats nown, any proprty can b calculatd

6 Rvw of Probablty ( Fluctuatons also of grat mportanc Varanc (σ vry usful: ( ( δ ( 0 ( ( P P P δ ( 0 ( ( ( ( + + P P P P σ δ

7 Rvw of Probablty (3 If stat X s contnuous, dfn probablty dnsty d f ( X dx Whr d no. tms stat btwn X and X + X obsrvd Probablty dfnd ovr ntrvals ( X P X1 X f ( X1 vragng: ( X f ( X dx Gaussan dstrbuton common f G ( X 1 πσ xp ( X X X σ X X dx X Standard dvaton ~ wdth of Gaussan dstrbuton

8 Kntc hory xampl Consdr molculs of mass m n box, lngth l Calculat prssur forc/ara on walls v nx ssum lastc collsons wth walls xampl x-drcton -v nx Wall n x-drcton: momntum chang mv nx Forc p/ t whr t tm btwn collsons l/v nx F n mv t mv n x n Smlar magntud of forc n y and z drctons l x

9 Kntc hory xampl ( Prssur xrtd by n th partcl P n forc ara For partcls ( v + v + v m l n n m P v 3V Snc spds contnuous, n 6l mv V x y z 3 n m 3V v v 0 v f n ( v dv vrag translatonal ntc nrgy: m m 3PV 3 U trans v f ( v dv m v 0 m From dal gas law, U trans 3 PV

10 Quantum or classcal? t atomc lvl, nrgy s quantzd Implcatons mportant only for crtan cass Wll usually mploy classcal approxmaton Lump lctronc ffcts nto contnuum ntratomc potntal functons (~contnuous nrgy lvls Do ths for both molcular fluds and solds (but potntal nrgy functons dffrnt Must paramtrz to fnd constants Data from spctroscopy, phas dagram, FM, tc. Vald for larg numbr of waly ntractng partcl W us quantum drvaton whn smplr

11 Postulats of Statstcal Mchancs qual a pror probablts ll confguratons wth sam nrgy assumd to occur wth qual probablty gvn nrgy lvl usually s dgnrat rgodc hypothss vragng ovr tm nsmbl avragng nsmbl: rplcas of systm, ach wth sam spcfcatons but dffrnt confguratons Spcfcatons dtrmn typ of nsmbl.g. for canoncal nsmbl, V ny confguraton can b rachd from anothr For smulatons, not guarantd - must chc!

12 Implcatons of Postulats For smplcty, consdr quantum stat Schrödngr qn: ( s th Hamltonan Dgnracy o. stats w/ Ω(,V, V spcfd mcrocanoncal nsmbl Dvd nto two subsystms w/ 1 + Choos 1 to maxmz Ω (most lly ln(, ln(1, ln 1(1 ( ln 1(1 + ln ( ln Ω Ω 0 1 (1 ln ( 1, V, 1 xt, assum that S(,V, 1,V1, V ln(,v, β

13 Mcrocanoncal nsmbl solatd systms, ach wth partcls, volum V (rgd walls and nrgy Confguraton of ach dffrnt Intrstd n lmt as approachs nfnty

14 Implcatons of Postulats ( For manng of β, borrow df. from thrmo: 1 S ln Ω(, V, β 1/(, V, V ow, mov to canoncal nsmbl (V Small subsystm n contact wth larg hat bath so that + and (- ~ constant Prob. of obsrvng stat proportonal to dgnracy, so: xpandng about 0: ln ( ln Ω ( ( ln + O( 1/ ( OLZM xp( / P DISRIUIO ( P xp( / xp( / j Ω( Ω ( j j j

15 Canoncal nsmbl closd systms, ach wth partcls, volum V (rgd walls at tmpratur Confguraton of ach dffrnt Intrstd n lmt as approachs nfnty

16 Rlaton to Macroscopc Varabls Knowng nrgy dstrbuton, avrag of any proprty from valus at stats xp( / P xp( / ul proprts drctly or from ach othr vrag nrgy: xp( / xp( / whr Q (canoncal partton functon Q dfnd by stats or by nrgy lvls Q stats xp( / Ω xp( / lvls lnq ( 1/

17 Othr Proprts from hrmo Hlmoltz fr nrgy, F, rlatd to Q: y rlatng Q to Hlmholtz fr nrgy, othr thrmodynamc proprts can b drvd: Statstcal thrmo: fnd Q, drv all ls hs turns out to b mpossbl n smulatons Q Q F S V V ln ln,, + V Q V F p,, ln j j V V Q F,,,, ln µ / ( (F/ 1 Q F ln

18 Idal Monatomc Gas For ndstngushabl, ndpndnt molculs, q (assumng ach molcul n dffrnt stat Q 1! Whr q partton functon of sngl molcul Partcl n 3D box nrgs: Dnsty of stats w/ nrgy < ε: 3/ π8mε Φ( ε L 6 h o. stats ε to ε +dε: 3 3/ ε l x l y l z h (l x + l y 8mL dφ π8m 1/ ω( ε dε dε Vε dε dε 4 Fgur L. Hll, from h + l z n Introducton to Statstcal Mchancs, Dovr, p. 75.

19 Idal Monatomc Gas ( pproxmat nrgy lvls as contnuous ow can asly calculat partton functon nd arrv at macroscopc OS for prssur Can fnd many othr functons: 3 0 / 1 / ( xp Λ V d / ( q,l,l l l l l z y x z y x ε ε ω ε d rogl wavlngth / Λ m h π V! Q 3 1 V V q V Q p ln ln, / ln( / ln( ln 3, Λ V q Q V µ

20 Idal Monatomc Crystal. nstn ndpndnt atoms clos to qulbrum lattc postons, harmonc potntals Rasonabl approxmaton at low tmpratur u( r u(0 + (1/ fr caus partcls ar ndpndnt, RL POIL PPROXIM POIL nrgy lvls for harmonc potntal ar ε ( n +1/ hν n + { u(0 / } q( V /, Q V 3 (,, ν f / m π

21 Idal Monatomc Crystal ( So, q n 0 ε / hν / n Θ n 0 hν Θ/ ( hν / n Θ 1 / hrfor, Q u(0 / Θ / Othr thrmo. varabls: Hlmholtz fr nrgy F lnq u(0 / 3 Θ/ 1 3 ln 1 Θ/ Θ /

22 Monatomc Crystal Rsults vrag nrgy Low tmpratur Hgh tmpratur Hat capacty: lnq, V Θ u(0 / + 3hν / + 3 C 3 C V lnq ( Θ / Θ / / 1 Θ u( 0 / + 3hν / 3 Hgh : V Low : 3 Θ, V Θ/ ( Θ /, V 1 C V 3 Θ Θ /

23 Grand Canoncal nsmbl (smprmabl systms, ach wth partcls, volum V & chmcal potntal µ j j j V P V S G F,,,,,, µ

24 Grand Canoncal nsmbl Subsystms ar opn ; only µ,v, fxd Can drv grand partton functon Ξ (,V, xp( / xp 1 1Q(,V, xp(/ Probablty of obsrvng stat wth partcls: On can drv for ntropy: xp( P ( ; µ, V, so, and µ µ S + lnξ + (/ (, V / xp( µ / Ξ hs and othr nsmbls chosn for props. S pv, P( ln P( pv ln(,v,

25 Isobarc nsmbl Closd systms, but comprssbl and thrmally conductv: constant,p, (,p,, xp( / xp( pv / V j xp( (, V / P ( V ;, p, j xp( pv /

26 Fluctuatons If proprty can b valuatd nstantanously, can dtrmn fluctuatons, and mor proprts For nstanc, Hat Capacty: V V C, / / + / / / [ ] / / / 1 C V

27 Classcal Statstcal Mchancs Frnl and Smt :quantum tratmnt W now that w lac computatonal rsourcs to solv Schrödngr qn. for partcls Instad of nrgs found as gnvalus of th Hamltonan of th systm, us classcal: 1 cl + ( px + py + pz + m j 1 For partton functon, rplac ε wth cl Intgrat ovr all momnta and postons,..., z Partcls ar ndstngushabl, so dvd by! ( x 1

28 Classcal Statstcal Mchancs ( Dvd by h (Planc s constant for ach momntum/poston par (consstnt wth Hsnbrg uncrtanty prncpl For partcls wth momnta p and postons r and dmnsonalty d, Q classcal r xp( β H h { [ ]} p /( m U( r 1 dp dr xp +! d dp dr dp xp dr { [ ]} β p /( m + U( r ( p { [ ]} xp β p /( m + U( r, r

29 Classcal Stat. Mch. (3 rms nvolvng momntum can b solvd f m s constant for all partcls n 3D: dp Substtutng, xp( Q p / m classcal ( m In gnral, nrgy xprssd as sum, so f Intrnal contrbutons can b sparatd out q trans dp xp( 3 / p / m 3 1 dr xp 3 Λ! ε ε + ε + ε + ε vb β ( ε + ε + ε + ε rot trans vb rot lc q trans lc q vb { ( r } q rot q lc

30 Idal Gas st For dal gas, molculs don t ntract 0 Q V classcal (1 3 1 d 3 r Λ! grs wth quantum rsult (3D box Can drv vlocty probablty dnsty by sparatng out potntal nrgy Λ! f ( v dv xp( βm[ v m π x y z [ ] dvxp( mv / 3/ + v + v xp( βm[ v ]/ dv x + v x 3 y dv + v y z dv z ]/ dv x dv y dv z

31 Smulatons: Prlmnary Dtals

32 rgodc Hypothss ru nsmbl avragng: ach ralzaton s ndpndnt and unqu Smulatons: almost always volv n tm Molcular dynamcs: Solv quatons of moton 1 t lm dt t 0 t [ p(t, r(t ] If a larg numbr of ntal condtons sampld, (r ntal condtons t 1 lm t t t 0 dt o. t 0 [ ; p( 0, r( 0, t ] [ ; p( 0, r( 0, t ] ( r V V r 1 lm t dt r nt condtons

33 rgodcty Ptfalls Mont Carlo nsmbl basd, but n practc, also tracs trajctors n phas spac Must avod loops, naccssbl rgons Som mthods dvlopd to avod problms Fgur M.P. lln and D.J. ldsly 1987, from Computr Smulaton of Lquds, Oxford U. Prss p. 36

34 Intracton Potntals Can go byond classcally solvabl problms (dal gass, harmonc solds ( r, p ( p + ( r Kntc nrgy always th sam whr s partcl numbr and α s coordnat Hart of modls ar ntracton potntals 1 r + j> ( r, rj + j> > j> 3( r 1 : xtrnal fld : Parws ntractons (most mportant 3 : hr-body potntals (bttr, but costly 1 α α p / m (, rj, r

35 Common Par Potntals Hard sphr Purly rpulsv Squar wll Idalzd mnmum Soft sphr HS Contnuous, rpulsv u u σ1 σ (r (r 0 (r SS u r u < σ σ SW ν (r ar (r u σ r (r < σ1 ε ( σ r < σ1 0 (r σ u r Lnnard-Jons (most wdly usd u LJ (r 4ε rpulsv 1 6 ( σ / r ( σ / r attractv u Optmal poston r

36 Mor Complx Par Potntals Coulombc Ionc systms, unusually long-rang Molculs Untd atoms for rgd unts (.g. CH Mayb parws potntals wth rgd bonds Partal chargs at on pont or throughout Harmonc, rotatonal ntrnal dgrs of frdom? Lattc or contnuum? Lattc much fastr; contnuum mor accurat Lattcs lmt ntrnal dgrs of frdom (polymrs u zz (r j z z j 4πε r 0 j Prmttvty of fr spac

37 Small Systms: oundars? Want to smulat larg sampl (bul fluds Lmtd to thousands of atoms Us prodc boundary condtons ox must b larg nough to prvnt ntrfrnc (L ~ 6σ for LJ Can consdr boxs othr than squar Potntals usually cut off to sav tm, prvnt partcl flng tslf (~.5σ for LJ Partcl lavng on sd mrgs on othr Rplcas of systm at ach boundary Partcls fl phantom mags from across box