Molecul Dynmics Simulions Lech 7.-7.5 compue hemlly eged popeies by smpling epesenie phse-spce jecoy i numeicl soluion of he clssicl equions of moion fo sysem of N inecing picles he coupled e.o.m. e: d i d d x i Fi i N F xi mi d mi whee he foce on picle i is eled o he s deiie of he inemolecul poenil Φ by: Fi ΦR F R i xi Φ x ll 3N picle coodines i 4/6/005 CHEM 408 - Sp05 L6 -
some simple exmples: d hmonic oscillo m k Φ x x k x 0 F x k x x0 x/k, Fx/k 0.4 0. 0.0-0. F Φ inecing LJ picles -.0-0.5 0.0 0.5.0 x-x 0 i ij j σ Φ ij 4ε ij 4ε σ F ˆ i ij σ ij σ ij 3 6 σ ij 7.0.5.0.5 3.0 / σ 4/6/005 CHEM 408 - Sp05 L6 - Φ/ε, F/ε/σ 0.0-0.5 -.0 -.5 Φ F
4/6/005 CHEM 408 - Sp05 L6-3 jecoy is popged using finie diffeence mehods, d use Tylo expnsion... 3!! 3 x f x f x f f x f on o find :... 3!! 3 b d d whee d d 3 3 d d b 4 4 d d c i elociy cceleion ec. jek
4/6/005 CHEM 408 - Sp05 L6-4 conside expnding boh nd -:... 3 6 b... 3 6 b b b 4 O hese esul is he oiginl Vele lgoihm: he Vele lgoihm is coec o 4 h ode in posiions; elociies e implici bu cn be ppoximed by: is deemined by F{}; so clculion sequence is: clc F{}/m clc fom nd - clc if desied fom nd -
4/6/005 CHEM 408 - Sp05 L6-5 ins o he oiginl Vele lgoihm poide fo moe ccue elociies: Vele Lep-Fog lgoihm: elociy Vele lgoihm: nd known he sme se of imes closely eled o hese Vele mehods is he Beemn lgoihm moe compliced Ge pedico-coeco mehods lso used boh nd known wih good ccucy
Schemic Summy of Vele Vins elociy lep-fog oiginl Figue fom Allen & Tildesley, Compue Simulion of Liquids Oxfod, 987 4/6/005 CHEM 408 - Sp05 L6-6
choosing n inego is ply me of se; key poin is o minimize # foce eluions he cosly p nd minin ccucy moe compliced lgoihms inole moe compuion ofen moe F eluions bu llow fo lge fo gien ccucy fo gien inego, choice of ime sep is impon smll lge A - A collision jus igh 4/6/005 CHEM 408 - Sp05 Figues fom Lech L6-7
Digession on Enegies he enegy of clssicl sysem cn be descibed in ems of is phse spce coodines by: 3N 3N 3N 3N H q, p Φ q K p dependence of sysem enegy on phse spce coodine clssicl Hmilonin poenil enegy kineic enegy he hemodynmic inenl enegy E is n ege quniy: E < H >< Φ > < K > whee <x> denoes phse spce ege of x eihe oe dynmic jecoy MD o sic ensemble MC K in simple N,V,E MD simulions H is consn of he moion whees Φ nd K flucue in ime 0 0 0 30 40 50 Time 4/6/005 CHEM 408 - Sp05 L6-8 Enegy 0 - - H V
Schemic Exmples d Hmonic Oscillo Φq Enegy A Liquid Φq 3N q 3N K E Enegy 0 K Φ q Coodine q E Φ Coodine q 3N 4/6/005 CHEM 408 - Sp05 L6-9
. Enegy Conseion in MD Liquid A N56; T~00 K E Effec of Sep Size Time Sep ~8 K.0 elociy Vele Enegy Flucuion G4 Φ G5 Figue fom Allen & Tildesley, Compue Simulion of Liquids Oxfod, 987 4/6/005 Figue fom Lech CHEM 408 - Sp05 L6-0 G6
clculing exc jecoies is impossible compe efeence sysem 0 o ones in which picles e displced by 0-3, 0-6, 0-9 σ N 0 i i N i sele fo ccue jecoies fo imes > molecul coelion imes ps enegy conseion such h E << K Liquid A N~08? posiions enegies K/K Figue fom Allen & Tildesley, Compue Simulion of Liquids Oxfod, 987 4/6/005 CHEM 408 - Sp05 L6 -
Lech s ecommendions on sep size: efficien simulion is mde difficul by he pesence of moions hing diffeen chceisic ime consns; one exmple encouneed in mny poblems is he fc h H-om ibions e much fse hn mos ohe ibions wo soluions e: muliple ime sep lgoihms: clcule foces diffeen depending on how quickly hey chnge wih ime -RESPA consin dynmics: feeze he moion of fses degees of feedom SHAKE, RATTLE 4/6/005 CHEM 408 - Sp05 L6 -
3. Iniilizing & Running Simulions iniil coodines fom: - pio simulion - lice configuion - cysl sucue poeins elociies smpled fom Mxwell- Bolzmnn disiuion pope T: P ix mi πk B T / exp mi k T equilibe eole sysem, pehps chnging K unil popeies he eched ppen equilibium B ix Al ne TP sed fom fcc lice N08 ρ k N pessue N i cos k i H K V P nslionl ode ms displcemen Figue fom Allen & Tildesley, Compue Simulion of Liquids Oxfod, 987 4/6/005 CHEM 408 - Sp05 L6-3
4. Clculing Some Simple Popeies N,V,E inenl enegy E is consn hemodynmic empeue T is popoionl o he ege kineic enegy <K> by: K N pi T 3N 3Nk m B i i he hemodynmic pessue P is eled o he iil P NkBT V 3V N N j> i i ij F ij he dielecic consn ε is eled o he flucuions in he ne smple dipole momen Σµ by: 4π 9Vk T B N ε Ds µ i i 3 Ds ε expession depends on elecosic BCs; his equion pplies o ecion field BCs wih D s he dielecic consn of he suounding medium. 4/6/005 CHEM 408 - Sp05 L6-4
5. Chceizing Ode in Disodeed Sysems sucul ode is ofen chceized by dil disibuion funcion df, defined by N g ρ 4π d whee <N> is he ege numbe of picles found in spheicl shell of dius nd hickness d bou cenl picle, nd ρ is he ege picle densiy N/V fo ndom disibuion of picles N 4πρ d g of odeed cysl nd g. Thus, deiions of g fom uniy mesue he mgniude of spil ine-picle coelions Figue fom D. Chndle, Inoducion o Moden Sisicl Mechnics Oxfod, 987 4/6/005 CHEM 408 - Sp05 L6-5
schemic g of liquid & gs phses Figue fom D. Chndle, Inoducion o Moden Sisicl Mechnics Oxfod, 987 in simple sysems g cn be deemined fom X-y & neuon sceing expeimens hemo pops cn ofen be expessed in ems of g, e.g. P Φ πρ g d NkBT V 0 πρ 3k T B 0 g d d 3 d g of liquid & solid A T.P. 84 K 4/6/005 CHEM 408 - Sp05 L6-6
dynmic ode is chceized using ime coelion funcions cfs, defined by: C AA A0 A A0 A A whee Aq 3N,p 3N is some obseble popey nd AA-<A> is he flucuion in his popey one cn lso conside coss coelions beween quniies A nd B, i.e. A0 B A0 B A B C AB signl pocessing nlogy Hile helps illuse uiliy of cfs: A - signl B - coded signl A 0 A nd A0 B Figues fom J. M. Hile, Molecul Dynmics Simulion, Elemeny Mehods Wiley, 99 4/6/005 CHEM 408 - Sp05 L6-7
exmple: elociy uocoelion funcion in Al C 0 nomlized elociy uocoelion fns. elociy of om C /C 0 elociy x 0 ime how C is compued ege oe N molecules nd M ime oigins Figues fom Lech 4/6/005 CHEM 408 - Sp05 L6-8
Aside: Diffusion Consns lim D 3 0 d 0 0 6 6. MD in Ohe Ensembles lhough mos nul unde consn N,V,E condiions one cn pefom MD simulions unde condiions of consn T nd o consn P bes mehods couple sysem of inees o imginy eseois h exchnge E fo consn T o V fo consn P 7. Wh is Accessible o Simple MD? ypicl sysems e ~0 3 oms fo imes of ~0-9 s heoic simulions e ~0 6 oms, imes o ~0-6 s muli-scle mehods cn gely exend nge of use 4/6/005 CHEM 408 - Sp05 L6-9
exmple of heoic MD: micelle fomion Mink e l. J. Phys. Chem. B 04, 65 000 4/6/005 CHEM 408 - Sp05 L6-0
4/6/005 CHEM 408 - Sp05 L6 -