Inseting this into the left hand side of the equation of motion above gives the most commonly used algoithm in classical molecula dynamics simulations

Transcription

1 Chem465 in 2000 Univesity of Washington Lectue notes Hannes Jonsson Classical dynamics When we ae dealing with heavy atoms and high enough enegy o tempeatue, it is often suciently accuate to neglect quantum eects and calculate the time evolution of a system (such as an atom o a collection of atoms) using the classical equation of motion, Newton's second law, F ~ = d ~P dt. Hee F ~ is the foce acting on the classical paticle and P ~ is the momentum given by P ~ = m~v. Since the velocity is the st deivative of position with espect to time, the equation of motion is a second ode dieential equation fo the position of the paticle as a function of time (the second deivative being the acceleation) d 2 x(t) dt 2 = F=m : In most cases of inteest in chemisty, the foce is consevative, i.e. it can be witten as the dieential of a scala function, the potential enegy, V (x). In one dimensional systems F =,dv=dx. The foce is then only a function of the coodinate of the paticle, F (x), and not, fo example, dependent on the velocity. The poblem of calculating the time evolution of such a system, theefoe, educes to the poblem of nding a good epesentation of the potential enegy as a function of the coodinates of the atoms and then solving the second ode dieential equation fo the classical tajectoies of the atoms. We will st focus on the second aspect of this poblem. We will use a nite dieence appoximation to calculate numeically the tajectoy fo a given initial condition and a given potential enegy function. The nite dieence algoithm is called the Velet algoithm. Then, we will examine a few potential functions which ae commonly used to epesent appoximately the inteaction between atoms. Simple deivation of the Velet algoithm Hee, time will be discetized, i.e. time evolves in discete steps t = h, but the spacial coodinates will not be discetized. To begin with, we will focus on a one-dimensional poblem fo simplicity, and then genealize the algoithm to thee-dimensional space. Let x k be the position afte k timesteps, i.e. x k = x(t = kh). The cental nite dieence appoximation fo the second deivative is x k = x k+1 + x k,1, 2x k h 2 : 16

2 Inseting this into the left hand side of the equation of motion above gives the most commonly used algoithm in classical molecula dynamics simulations, the Velet algoithm x k+1 = 2x k, x k,1 + h 2 F (x k )=m : Since the dieential equation is second ode, we need to know two points to get the ecusion stated. Deivation using Taylo's expansions The Velet algoithm can also be deived using Taylo expansions. Then it becomes cleae what the accuacy of the algoithm is, in paticula how the eo scales with the size of the timestep. Taylo expanding x(t + h) about x(t) gives x(t + h) = x(t) + h _x(t) + h2 2 x(t) + h3 3 x(3) + O(h 4 ) : Expanding x(t, h) gives a simila expession, except the odd tems have a minus sign x(t, h) = x(t), h _x(t) + h2 2 When the two expessions ae added, the odd tems cancel x(t), h3 3 x(3) + O(h 4 ) : x(t + h) + x(t, h) = 2x(t) + h 2 x(t) + O(h 4 ) : This expession is geneal, valid fo any function x(t) that is dieentiable enough times. The equation of motion fo classical dynamics can now be used to eliminate the second deivative x(t + h) + x(t, h) = 2x(t) + h 2 F (x(t))=m + O(h 4 ) : The Velet algoithm is obtained by neglecting the fouth and highe ode tems. It tuns out that this algoithm is `stable', i.e. the eo made at any given step tends to decay athe than magnify late on. Thee is no guaantee fo such behaviou in nite dieence algoithms. Some ae stable and some ae not. The easiest way to nd out whethe an algoithm is stable is to ty it and see if it woks. Deivation of the Velocity Velet algoithm In classical mechanics it is possible to know simultaneously both the position and velocity of paticles (paticles being atoms, fo example). A classical system is often epesented as a point in `phase space', a space of all the coodinates and momenta of paticles in the system. Knowing the position and velocity (o momentum) at one point 17

3 in time completely species the state of the system and, though the equation of motion, the futue (and past) motion of the paticles, as long as the foce acting on the paticles ae known. The velocity, howeve, does not appea explicitly in the Velet algoithm. If it is needed, fo example, to calculate the kinetic enegy o to implement some kind of tempeatue contol (modications of the kinetic enegy), a cental nite dieence estimate can be used, _x k =(x k+1, x k,1 )=2h. It is a bit awkwad that _x k cannot be found until x k+1 has been calculated. In stating up a calculation, we typically know whee the paticle is and we have some idea of what velocity it has (fo example by knowing the kinetic enegy), but this is not enough to stat the Velet algoithm because the positions at two adjacent timesteps ae needed. Given the coodinate x(0) and velocity _x(0), the coodinate at a pevious step x(,h) needs to be constucted. A dieent algoithm, the Velocity Velet algoithm, explicitly includes the velocity at each step and is `self-stating' fom the position and velocity at the initial time. It is mathematically identical to the oiginal Velet algoithm in the sense that it geneates the same tajectoy in the absence of oundo eos in the compute. Any n,th ode dieential equation can be educed to a set of n st ode dieential equations. In paticula, the classical equation of motion x(t) = F (x)=m is second ode and can be educed to two st ode equations. Let v(t) = _x(t) = dx=dt and the two equations ae _x(t) = v(t) (1) and _v(t) = F (x(t))=m (2): A nite dieence algoithm can again be deived using Taylo expansions. Stating with x(t + h) x(t + h) = x(t) + h _x(t) + h2 2 x(t) + O(h3 ) : Using v(t) to eliminate _x and F=m to eliminate x gives x(t + h) = x(t) + hv(t) + h2 2 Then, expanding the second function, v(t + h) F (x(t)) m + O(h 3 ) (1b): v(t + h) = v(t) + h _v(t) + h2 2 v(t) + O(h3 ) : We can use F=m to eliminate _v, but we need to develop an expession fo v in tems of known quantities. This can be done by expanding _v(t + h) _v(t + h) = _v(t) + hv(t) + O(h 2 ) : 18

4 It is enough to go up to ode h 2 hee because we only need an appoximation that is good to ode h 3 to the quantity h2 v(t). Multiplying by h=2 and eaanging, gives 2 so, the expession fo v(t + h) becomes h 2 2 v(t) = h 2 (_v(t + h), _v(t)) + O(h3 ) v(t + h) = v(t) + h _v(t) + h 2 (_v(t + h), _v(t)) + O(h3 ) : Using the equation of motion, this can nally be ewitten as v(t + h) = v(t) + h 2m (F (x(t + h)) + F (x(t))) + O(h3 ) (2b): Schematically, the velocity Velet algoithm is as follows Given x k and v k and an expession fo F (x) Step 1: calculate x k+1 = x k + hv k + h 2 F (x k )=2m Step 2: evaluate F (x k+1 ) Step 3: calculate v k+1 = v k + h 2m (F (x k)+f (x k+1 )) Now all quantities fo the new step, k +1, have been found, go back to step 1. Genealization to many atoms in thee dimensions: The Velet algoithm can easily be genealized to highe dimensions and many atoms. Fo N atoms in thee dimensional space, the potential function depends on 3N coodinates, V (x1;y1;z1;x2;y2;z2;x3;:::;xn;yn;zn) = V (~ 1 ;~ 2 ;:::;~ N ). The foce on atom numbe i is ~F i =, i V (~ 1 ;~ 2 ;:::;~ i ;:::;~ N ) whee the subscipt i on the gadient opeato means that the deivatives should be taken with espect to coodinates of atom i. Witing each catesian component sepaately Fxi =, d dxi V (x1;y1;z1;x2;y2;:::;xi;:::) 19

5 Fyi =, d dyi V (x1;y1;z1;x2;y2;:::;xi;:::) Fzi =, d dzi V (x1;y1;z1;x2;y2;:::;xi;:::) : A ecusion elation given by the Velet algoithm o the velocity Velet algoithm can be used fo each [coodinate,velocity] pai, but they all get coupled togethe though the foce, which geneally depends on coodinates of all the atoms. Let x1 k be the x coodinate of atom 1 at timestep k and y1 k be the y coodinate of atom 1 at timestep k, etc. Futhemoe, let v x denote the x component of the velocity and v y denote the y component and v z the z component. The foce components ae Fxi k Fxi(x1 k ;y1 k ;:::;yn k ): Then a schematic algoithm fo N atoms in thee dimensions can be witten as follows: Given x1 k ;y1 k ;:::;yn k and u1 k ;v1 k ;:::;vn k and an expession fo Fx1 k ;Fy1 k ;:::;FyN k Step 1: calculate x1 k+1 = x1 k + hv x 1 k + h 2 Fx1 k =2m y1 k+1 = y1 k + hv y 1 k + h 2 Fy1 k =2m z1 k+1 = z1 k + hv z 1 k + h 2 Fz1 k =2m ::: zn k+1 = zn k + hv z N k + h 2 FzN k =2m Step 2: evaluate Fx1 k+1 ;Fy1 k+1 :::;FzN k+1 Step 3: calculate v x 1 k+1 = v x 1 k + h 2m (Fx1 k + Fx1 k+1 ) v y 1 k+1 = v y 1 k + h 2m (Fy1 k + Fy1 k+1 ) ::: v z N k+1 = v z N k + h 2m (FzN k + FzN k+1 ) Now all quantities fo the new step, k +1, have been found, go back to step 1. Potential enegy functions In calculating the dynamics of atoms and molecules, it is vey impotant to choose a potential function that accuately mimics the system of inteest. The potential function descibes how the potential enegy of a system of atoms depends on the coodinates of the 20

6 atoms. It is assumed hee that the electons adjust to new atomic positions much faste than the motion of the atomic nuclei (this is called the Bon-Oppenheime appoximation) and the potential function that is needed fo the nuclea motion is stictly the enegy obtained afte calculating the electonic wavefunction keeping the atomic coodinates xed. The calculation of the electonic wavefunction fo a system of many atoms is vey dif- cult. Most often, a simple functional fom is assumed fo the potential function and the paametes adjusted to epoduce some expeimental o theoetical data. Those ae called empiical o semi-empiical potential functions. Although we use the oppotunity hee to biey discuss potential functions in the context of classical simulations, the same consideations and functions apply when the motion of the atoms and molecules is teated quantum mechanically. In pinciple, the inteaction potential of N atoms can be expanded in a many-body expansion V ( 1 ; 2 ;:::; N ) = X i v i (~ i )+ X i X j>i v 2 (~ i ;~ j )+ X i X j>i X k>j>i v 3 (~ i ;~ j ;~ k )+ ::: whee the st sum is ove one-body tems, the second sum ove paiwise inteactions, the thid one ove thee-body contibutions, etc. The one-body tems aise if an extenal eld is applied (fo example, the inteaction with an extenal electical eld, o the potential descibing the wall of a containe). The paiwise inteactions ae usually most impotant and ae at shot ange epulsive due to epulsion of the two electon clouds and at long ange attactive due to the induced-dipole/induced-dipole inteactions. The thee-body tems aise because the inteaction of a pai of atoms is modied by the pesence of a thid atom. Fo ae gases, the paiwise potentials alone descibe quite well the potential enegy function, but even thee the thee-body coections ae signicant (amounting to ca. 10% in the binding enegy of the cystals of heavie ae gases). Fo systems with stong, bonding inteactions, the many-body expansion is a vey poo appoach, because the expansion conveges vey slowly. The most commonly used pai potential is the Lennad-Jones potential v() = , whee the st tem descibes the epulsion at shot ange as the electon clouds ovelap moe than is optimal and the second tem epesents the long ange attaction. The fom of the second tem,6 has the coect behaviou fo the induced-dipole/induced-dipole inteaction. The fom of the epulsive tem is, howeve, not suppoted by theoetical calculations and is simply chosen fo convenience. A moe accuate pai potential fom, fo example, fo descibing the inteaction of ae gas atoms is v() = Ae,, f () C 6 : 21

7 The ovelap of the closed shell electon clouds is usually well descibed with an exponential fom. The function f () is a switching function, which is 1 at long ange but goes to zeo at shot ange and pevents the attactive enegy tem fom diveging. Bonded inteactions, whee the attactive inteaction comes fom the fomation of a chemical bond, is bette descibed with a Mose potential. v() = D e,2(,0), 2e,(, 0) : The attaction decays exponentially as the ovelap of the electon clouds deceases. In addition to this exponential attaction, a longe ange van de Waals attaction should be added, but in compaison to chemical bond enegies, this can often be neglected. When bonding constaints need to be taken into account, fo example, when the numbe of bonded neighbos is small and the angle between bonds has a pefeed value, the many-body tems ae vey impotant and the potential function becomes quite complicated. One good example is cabon, which only bonds to at most 4 othe atoms if sp 3 hybidized (as opposed to ae gas atoms that can suound themselves with up to 12 nea neighbos), but in dieent hybidization states will pefe 2 o 3 neighbos. A geat deal of eot in theoetical chemisty goes into the development of accuate and yet convenient functional foms that take into account many-body eects. One functional fom that has been used extensively in modeling chemical eactions involving fomation and beaking of chemical bonds is the LEPS fom. As an example, the `extended LEPS' potential fo a thee atom system A; B and C, capable of descibing a substitution eaction A + BC! AB + C is of the fom E( ab ; bc ; ac ) = Q ab 1+a + Q bc 1+b + Q ac 1+c, J 2 ab (1 + a) 2 + J 2 bc (1 + b) 2 + J 2 ac (1 + c) 2, J ab J bc (1 + a)(1 + b), J bc J ac (1 + b)(1 + c), J ab J ac (1 + a)(1 + c) The paametes can be chosen in such away that the enegy is lage when all thee atoms ae close to each othe, i.e. when only one bond is allowed at a time. The paametes a; b and c can be adjusted to change the height and location of the potential baie fo the eaction. The functional fom is inspied by appoximate calculations of the inteaction of thee H atoms. The Qs stand fo Coulomb inteactions between the electon clouds and the nuclei and the Js stand fo exchange inteactions (tems that aise because of the indistinguishability of the quantum mechanical electons). A typical fom fo the distance dependence of the Q and J tems is simila to the Mose potential Q() = D e,2(, 0), e,(, 0)

8 and J() = D 4 e,2(,0), 6e,(, 0) : The bonded inteaction of two atoms is Q+J (in the singlet state, whee unpaied electons on each atom have opposite spin and can pai up as the atoms appoach each othe) and the non-bonded inteaction is Q, J (puely epulsive aswould be the case if the unpaied electons have the same spin, a tiplet). Deivation of the foce fo L-J potentials: In calculating the classical dynamics of atoms it is necessay to be able to evaluate the foce acting on the atoms. Given a potential enegy suface, the task is to calculate the gadient ~F i =, ~ i V (~ 1 ;~ 2 ;:::;~ i ;:::) with espect to the coodinates of each atom. This equation says that the foce acting on atom i is obtained by taking the potential function, which depends on the coodinates of all the atoms in the system, and dieentiate it with espect to the coodinates of atom i. This can be a bit involved when the potential function is complicated. Fotunately, Mathematica and simila pogams can be of geat help. As a simple example, a deivation of the foce acting on atoms inteacting via a paiwise additive potential is given below. The total potential enegy of the system is a sum ove a potential, v(), fo each distinct pai of atoms V (~ 1 ;~ 2 ;:::;~ N ) = NX i NX j<i v( ij ) whee ij is the distance between atoms i and j. 23

9 The distance can be obtained fom the coodinates of the atoms (which efe to some abitay oigin of the coodinate system) q q ij = (~ i, ~ j ) (~ i, ~ j ) = x 2 ij + y2 ij + z2 ij whee x ij = x i, x j. The foce on atom k is a vecto pointing in the diection of the steepest descent of the potential enegy. The expession is ~F k =, ~ k V (~ 1 ;~ 2 ;:::;~ k ;:::) =, X j6=k =, X k ~ k v( kj ) k k v( kj kj Using the chain ule fo dieentiation, we can see that k = 1. Theefoe, the deivative of the pai potential above can be witten in tems of v 0, i.e. the deivative with espect to the agument of the function. Fo the x component of the foce Fo all thee kj kj = v 0 ( kj kj ~ k v( kj ) = v0 ( kj ) kj = x kj kj v 0 ( kj ) : (x kj ^x + y kj ^y + z kj ^z) = (~ k, ~ j ) kj v 0 ( kj ) = ^ kj v 0 ( kj ) whee ^ kj is a unit vecto in the diection fom atom j to k. we get In the paticulaly simple case whee the pai poential is a Lennad-Jones potential The foce on atom k becomes v() = 4 v 0 () = 4 ~F k =, X j6=k 12 6,, (~ k, ~ j ) kj kj : : 24