Molecular dynamics algorithm for multiple time scales: Systems with long range forces

Transcription

1 Molecular dynaics algorith for ultiple tie scales: Systes with long range forces Mark E. Bruce J. Berne Departent of Cheistry, Colubia University, New York, New York Glenn J. Martyna Departent of Cheistry, University of Pennsylvania, Philadelphia, Pennsylvania (Received 16 October 1990; accepted 23 January 1991) A frequently encountered proble in olecular dynaics is how to treat the long ties that are required to siulate condensed systes consisting of particles interacting through long range forces. Stard ethods require the calculation of the forces at every tie step. Because each particle interacts with all particles within the interaction range of the potential the longer the range of the potential the larger the nuber forces that ust be calculated at each tie step. In this note we present a variant of the RESPA (reference syste propagator algorith), which we developed for hling systes with ultiple tie scales like disparate ass ixtures. This version of RESPA greatly reduces the nuber of forces that ust be coputed at each tie step thereby leads to a draatic acceleration of such siulations. The RESPA ethod uses ideas siilar to NAPA, an algorith we invented to treat high frequency oscillators interacting with low frequency bath. The ethod is based on a choice of a reference syste in which the particles interact through short range forces. The reference syste is nuerically integrated for n tie steps St the error incurred by using short range forces is corrected by solving a rigorous set of equations once every At = &it. This ethod reduces the cpu tie draatically. It is shown that this approach suitable generalizations should be very useful for future siulations of quantu classical condensed atter systes. 1. INTRODUCTION Consider a syste consisting of Nparticles in which the particles interact through forces with a cutoff distance R,. Each particle then feels the forces fro LV, apr 2 neighbors the cpu tie required to advance the syste one tie step St is proportional to the nuber of forces, NN,/2, that ust be calculated. Clearly the siulation tie grows as the cube of the cutoff distance. In this short note a ethod for accelerating the siulation of such systes is presented. This ethod, which is related to the NAPA algorith that we invented for treating the proble of high frequency oscillators coupled to low frequency oscillators, is a variant of the RESPA algorith that we invented for treating systes with ultiple tie scales like disparate ass ixtures. The gist of the ethod is to define a dynaical reference syste involving only short range forces to derive equations of otion for the deviation, S(t), of the coordinates of the syste fro those of the syste containing the full forces. The reference syste equations of otion are integrated for n tie steps r&t. The tie dependence of the reference syste is then fed into the coupled equations for S(t) the resulting equations are nuerically integrated for one large tie step At = n&. The initial conditions for each large tie step are chosen so that this deviation S(t) is zero at the start of each new tie step with the consequence that the deviation is always kept sall. The only approxiation in this algorith springs fro the nuerical integrator used to integrate the equations of o- * Ph.D. student in the Departent of Physics, Colubia University. tion of the reference syste the coupled equations. Otherwise the ethod is self correcting exact. It should be noted that as the range of the interaction increases, the ratio of the nuber of long range forces to the nuber of short range forces which ust be coputed will increase. We thus expect that RESPA will grow ore efficient as the range of the interaction grows. By studying two systes with different ranges, we verify that this expectation is in accordance with our results. II. METHOD To illustrate the ethod, consider a set of differential equations of the for Z=LF(x), (2.1) where x are, respectively, the ass positions of the particles in the syste. We ust solve this set of equations subject to the initial conditions {x( 0)) jc( 0) 1. To proceed we subdivide the forces F(x) into short long range coponents F, F,, respectively. We define a reference syste equation of otion Z. =IF,(x,) which ust be solved subject to the initial conditions x0 (0) = x(o), i. (0) = i(0). (2.3) The solution of the reference syste equations of otion Eq. (2.2) denoted J. Che. Phys. 94 (lo), 15 May I Aerican institute of Physics 6811

2 6812 Tuckeran, Berne, Martyna: Molecular dynaics algorith x,(t) =x0 (t;x(o),~(o) 1. (2.4) The true position of the syste deviates fro the reference syste position. This can be expressed as x(t) =x0(t) + S(t). (2.5) Substitution of this into Eq. (2.1) eliination ing Eq. (2.2) results in the set of equations &=f [F,(xo(t) +a) +4(x,(t) +S> -F,(x,(t))]. of Z. us- (2.6) If FI is derived fro the long range part of the potential, it will be slowly varying copared to I;;. To solve Eq. (2.6) subject to the initial conditions [which follow fro Eq. (2.3)] S(0) = 0, h(o) = 0 (2.7) we propose the following schee. ( 1) Nuerically integrate Eq. (2.2) for a sequence of n sall tie steps 6t (At = n&) generating x0 (t) for O( t<at. (2) Substitute the nuerical solution for x0 (t) into Eq. (2.6). (3) Solve Eq. (2.6) subject to the initial conditions, Eq. (2.7), for one tie step At using a suitable integrator to obtain S( At) 8( At). (4) Calculate x(at) =x,(at) +&At), i(at) = f, (At) +&At). (2.8) This process is repeated using x( At),,k( At) as initial conditions. In general, at each step, the output is used as the initial conditions for the next step. The advantage of this ethod lies in the resetting of the initial conditions on S(t) 8(t) to 0 at every step. Since S(t) s(t) never deviate uch fro 0 in a given step, the force ter F, (x0 + 6) - F, (x0 ) in Eq. (2.6) is prevented fro becoing too large, thus allowing the use of a larger tie step in the nuerical integration than could be used in the stard integration schees. It should be ephasized that the equations of otion Eqs. (2.2) (2.6) forx, (t) S( t) are exact, i.e., no approxiation has been ade related to the disparity in range of the forces that the physical properties of the syste (e.g., pressure, internal energy, etc.) will be the sae for RESPA as for straightforward integration of the equations of otion. The integration of Eq. (2.6) can be done by any integrator suitable for equations with explicit as well as iplicit tie dependence.3 The Runge-Kutta (or predictor-corrector) ethod are such an integrators. In particular, the velocity Verlet3s4 integrator can be adapted straightforwardly for Eq. (2.6)) as we will later deonstrate. The ajor reason that this ethod saves so uch tie is that only at the end of a large tie step At Eq. (2.2) does one have to copute all of the forces. Thus it is not necessary to recopute the long range forces after every short tie step. These have to be updated only when Eq. (2.6) is integrated. This saving in force coputations can be draatic. Let z, z[ denote the average nuber of particles that interact with a particle through short long range forces, respectively. The nuber of forces that ust be calculated during the n tie steps St during which the reference syste is evolving is nn(zj2) the nuber that ust be calculated for the integration of Eq. (2.6) is Nz,/2. Thus the total cpu tie for one large tie step using the new ethod, assuing that the calculation of the forces is the tie consuing part of the calculation, is proportional to [ nrvz, + Nz, 1. On the other h if the calculation is done by stard integrators to copute n steps of length St oving all of the particles during each cycle the total nuber of forces that ust be calculated is nn(z, + z,)/2. This leads to the prediction that the ratio ofcpu ties for the stard ethod to that of the RESPA ethod will be r= n(z, + z[) n(pr :I (nz, +z,lz(nz, +pr:) (2.9) where R, is the cutoff distance for the long range force, p is the nuber density. It should be noted that li r = n. RJR,- (2.10) Thus the new ethod will be at best n ties faster than straightforward ethods. The size of n will be deterined by the agnitude of F, copared to F,. The longer the range of the potential the saller F, will be copared to F, the larger will be the nuber of tie steps n for which the reference syste trajectory will be close to the true trajectory. The subdivision of the forces into slow fast coponents can be ade in a variety of ways. One possibility is to base the subdivision on the WCA approxiation. If r is the position of the iniu of the pair potential, then the subdivision is V(r) = V,(r) where V,(r) = + V,(r), VW - Ur,), rgr, 0, r> r, Ur, 1, r<r, V,(r) = l V(r), r>r, so that the corresponding forces are I;,(r) = F(r), r<r,,, 0, r>r, (2.11) (2.12) (2.13) (2.14) 0, r<r, F,(r) = (2.15) F(r), Or,. As will be seen in the foregoing, the choice of subdivision leads to a considerable reduction in cpu tie over stard ethods, but ay not be optiu because the long range force can still be large in the neighborhood of r,. In this eventuality, n ust be chosen sall in order to insure accurate integration of the equations of otion. A ore flexible subdivision is to introduce a switching J. Che. Phys., Vol. 94, No. lo,15 May 1991

3 Tuckeran, Berne, Martyna: Molecular dynaics algorith 6613 function S(r) which varies onotonically between 1 0 as r increases to express the interparticle force as F(r) = S(r)F(r) -+- (1 - S(?))F (Y). (2.16) Then the short long range coponents of the force are taken to be F,(r) = S(r)F(r) (2.17) F,(r) = (1 -S(r) )F((r). (2.18) One can vary the position of the inflection point width of S(r) so as to iniize the cpu tie for the given odel. Let us now consider the ipleentation of this ethod using the velocity Verlet integrator. Although the discussion outlined before is perfectly acceptable it is possible to iprove it. It would be useful to have the algorith satisfy the following two requireents: ( 1) For F, -0 the algorith should reduce to the stard velocity Verlet algorith for F+[ using a tie step At = n&. (2) The ethod should work for long range forces even when they are large, but slowly varying. The ethod as outlined does not satisfy condition ( 1) becoes inefficient when the long range force is large. An algorith that satisfies these two conditions is the following. Replace the reference syste equation, Eq. (2.2), by a,(t) =LF,(X,w) +-1-F,w)), (2.19) where the last ter is constant in tie for the integration interval. If the long range force is strong but slowly varying this ter will ake an iportant contribution to the reference syste s acceleration, thereby allowing a larger choice of n for the sae energy conservation. Following the sae steps leading to Eq. (2.6) leads to 6=$ [F,(x,(t) +a -F,(xo(t)) +F,(Xo(t) + S(t)) - F,wN)], (2.20) an equation which contains the constant F, (x(o) ). One then carries out the sae proceedure outlined following Eq. (2.6). When the velocity Verlet integrator is used to solve this new set of equations it is a siple atter to show that for F, = 0 condition 1 is satisfied. It is also a siple atter to show that the use of velocity Verlet on Eq. (2.20) gives S( At) = 0. In general, velocity Verlet applied to Eqs. (2.19) (2.20) gives the following iterative equations for coputing the trajectory x(t) k(t) fro Eq. (2.8) : x( At) = x0 (At), itat) =&(A0 +g [Fs(x(At)) - F,(x,(At))] III. RESULTS +z [f,mat)) ---F/(x(O))]. (2.21) In the following exaples we require a easure of the accuracy of the algorith. One good easure is the degree to which energy is conserved. In a run of length T we define A&L T (it E(t) - E(O) To s I E(O) (3.1) where E(0) E(t) are the initial energy the energy at tie t. Clzarly, the ore accurate the algorith the saller will be AE. We shall copare cup tiekrequired for different algoriths for the sae values of AE. Although we study fluids with different potential odels here, the general approach will be to choose a stard integrator (e.g., velocity Verlet) with a given tie step 6t ch2sen to give a specified accuracy in energy conservation AE. For coparison, the new algorith RESPA is used to integrate the equations of otion for the sae syste, first using a WCA subdivision then using a switching function 1, r<r, -/z S(r) = 1 + R *(2R - 3), r, - R<r<r,, (3.2) i 0, r, cr where R = [ r - (r, - R) ] /;1 where r, is the cutoff distance,l is the healing length (the effective width of S (r) ). Of course, the choice of switching function is arbitrary, the above having been used in a different context by Watanabe Reinhardt. For the WCA subdivision n is chosen to give the sae energy conservation used in the full velocity Verlet integration. For the switching function subdivision, r,, il, n are chosen to give the sae energy conservation to optiize the cpu tie. In each of these siulations, the sall tie step St is taken to be the tie step used in the velocity Verlet algorith. A. Lennard-Jones (12-6) fluids First we study the well worn proble of the LJ ( 12-6) potential. Clearly, this is not a long range force proble. Nevertheless, even for this odel, we find an ipressive reduction in cpu tie when RESPA is used. For this study, the LJ fluid consists of 864 particles interacting with a pairwise additive LJ (12-6) potential (3.3) in a box of size L. The syste was equilibrated for each of four therodynaic states discussed below. Using the stard velocity Verlet integrator with periodic boundary conditions a cutoff of in (3a, L /2), the cpu tie required to run a total length of f;eal tie T = 3OOSt with energy conservation tolerance AE = 10-5 is deterined. Th,e four therodynaic states studied here are (T,p- ) = (1.0,0.8), (1.5, 0.9), (2.0, l.o), (2.5, l.l), where ^r = k, T/E is the reduced teperature pd is the reduced nuber density. RESPA, using both the WCA the switching function subdivisions defined above is copared with the stard velocity Verlet integrator for all of these states. In this coparison the sae short tie step St is used in the stard velocity Verlet algorith, RE- SPA( WCA) RESPA( SWITCH) for a given therodynaic state. The parazter n is chosen to give the specified energy conservation AE = 10-5 the tie required by J. Che. Phys., Vol. 94, No. IO,15 May 1991

4 6614 Tuckeran, Berne, Martyna: Molecular dynaics algorith RESPA to run the sae real tie is deterined for each of the two force subdivisions. In addition a search is ade of the paraeter space of the switching function (naely, r, /2) to obtain the sallest cpu tie. We find that r, = 1.7~ /z = 0.1~ is a very good choice. Although a detailed optiization ight lead to a different set of paraeters for each therodynaic state, we find that the results are quite insensitive to the value of il. The results of this coparison are shown in Table I. It is clear fro this study that RESPA can accelerate the siulation of even short range forces by as uch as a factor of 3.7. This is a significant iproveent. It can be seen fro the table that the WCA subdivision iproves as the fluid density is increased whereas the switching function subdivision is not strongly density dependent at high densities gives superior perforance at all densities studied. In the switching function algorith there is a trade off between two effects. Because the short range force is switched off at greater distances than in the WCA subdivision ore forces ust be coputed for the short tie steps therebye increasing the tie required to integrate the short cycles over the WCA ethod. On the other h the long range forces will be weaker when they are switched on than in the WCA truncation aking it possible to achieve the sae energy conservation for larger values of n, thereby reducing the nuber of cpu cycles to achieve a given aount of real tie. Fro Eq. (2.9) rswitch rwca =- %WITCH nwcazs,wca + zl.wca nwca [ %WITCHZr,SWITCH + l,swltch ]. (3.4) If the nuerator of the bracketed expression Zi,WCA % nwcazs,wca in the denoinator Zl,SWITCH % nswitchzs.switch v SWITCH -- rwca -n nswitchzl,wca WCAzl,SWITCH (3.5) Since the short range cutoff for the switching function is typically chosen to be larger than the WCA cutoff, ZlWCA >Z,,SWITCH nswitch >nwca, it follows that rswitch /rwca ) 1 so that the switching function ethod will be superior to the WCA ethod. 6. Lennard-Jones (12-3) fluids To investigate the power of RESPA we study the LennardJones ( 12-3) potential TABLE I. Coparison of RESPA with velocity Verlet algoriths for LennardJones ( 126) fluids.? PC nwca rwca SWITCH rswitch TABLE II. Coparison of RESPA with velocity Verlet algoriths algoriths for Lennard-Jones ( 12-3) fluids. ^r PQ3 nwca rwca nswitch rswitch (r) = 4ea, K:~ 2-(:~l 9 (3.6) a long range potential. The constant a3 is chosen to give the sae well depth E as the ( 12-6) potential. As before the syste was equilibrated for each of the four sae therodynaic state using the stard velocity Verlet integrator with periodic boundary conditions, only for this syste a cutoff of in(5a, L /2)^is used, the energy conservation tolerance is set at AE = 10w6. Otherwise, the sae coparison is carried out as for the LJ ( 12-6) syste, the sae type of search through the switching function paraeter space is done. For this syste we find that r, = 1.9~ ;1 = is a very good choice. The results of this coparison are shown in Table II. It is clear fro this study that RESPA can accelerate the siulation of this long range syste by as uch as a factor of 4.3. This is a rearkable saving in cpu tie deonstrates the efficiency of RESPA in hling long range forces. C. Lennard-Jones (12-1) fluids: Ewald suation To study the ipleentation of RESPA with the Ewald suation technique, we consider the Lennard-Jones ( 12-1 ) potential given by 4(r) =4ea,[(:) *-(+)I. (3.7) As before, the constant ai is chosen so that this potential has the sae well depth E as the ( 12-6) potential. Only the l/r part of this potential is written as an Ewald su. This gives a force which has the structure F(r) = F (rp) + F(recip)(T;a,k,,,). (3.8) That is, the force consists of a real space part a reciprocal space su. We indicate the explicit dependence of these coponents on the convergence paraeter a the cutoff in reciprocal space denoted k,,,. There are several possibilities for ipleenting RESPA with the force in Eq. (3.8). One is to use the switching function on the real space part of the force. This would give a reference syste force F,(r) = S(r)F ~(r;a )) (3.9) a long range residual force F,(r) = (1 -S(r))(F )(r,cr ) + F (recip) (r;a,k,$, ) ). (3.10) Given the different ranges of the reference syste force the residual, we expect that a different convergence paraeter reciprocal space cutoff should be used for each piece, we indicate this dependence by the s I super- J. Che. Phys., Vol. 94, No. lo,15 May 1991

5 Tuckeran, Berne, Martyna: Molecular dynaics algorith 6815 TABLE III. Coparison of RESPA with velocity Verlet algoriths for Lennard-Jones (12-1) fluids with Ewald su. r Pd bvltch rswitch scripts on CY k,,,. The choice of acs, a(, k 2:X) k ax (I) will depend on the syste size. Given the large syste studied here, we choose acs = acos&pk, = k,,, in Eqs. (3.9) (3.10). Since the reference syste force is purely real space, the use of RESPA becoes extreely efficient. To illustrate the iproveent given by RESPA with Ewald suation, we study the sae four therodynaic states as in the prev@s two cases using an energy conservation tolerance of AE = 10w5. The sae values of r, ;1 were used as in the ( 12-3) case, in addition to these, we choose k,,, = 6 al = 6. A cutoff of in (5a, L /2) was used as in the ( 12-3) case. The results are suarized in Table III. We see that the tiing ratios are ore draatic than for the (12-3) case even though the sae cutoff was used. This iproveent is clearly due to the fact that in the ordinary Verlet siulations, the reciprocal space su ust be evaluated at every step in addition to the real space part, while in the RESPA siulations, this su ust only be evaluated every n steps when doing the long range part of the force. The larger k,,, ust be, the ore these tiing ratios will iprove. IV. CONCLUSION In this paper we have presented a ethod which accelerates olecular dynaics siulations of systes with long range forces. This ethod, which is a variant of the RESPA ethod, is based on a set of exact equations which greatly reduces the nuber of force calculations that are required for the siulation. Since it is the coputation of the forces that doinates the cpu tie required to siulate systes, this reduction in the nuber of pair forces that have to be evaluated leads to a saving in cpu tie. The ratio of the tie required for the siulation using this new ethod to that for a stard ethod is given by r in Eq. (2.9). For systes with very long range forces r = n, where n is the nuber of tie steps for which the short range forces are integrated. The approach taken here is siilar in spirit to the ethod that we presented for solving the proble with a very stiff oscillator buried in a slow fluid, naely, the NAPA algorith. We have introduced two of any possible schees for subdividing the forces into short anfd long range coponents. We find that an optiized switching function is to be preferred over the WCA subdivision but it would not be surprising to find that a different subdivision schee works even better than the above. By studying the Lennard-Jones (12-6) (12-3) potentials, we saw that the cpu saving gained by RESPA increases as the range of the interaction increases. ACKNOWLEDGMENTS We wish to thank Dr. Joseph Hautan Ji Barean for useful discussions about Ewald suation techniques. M. Tuckeran, G. Martyna, B. J. Berne, J. Che. Phys. 93, 1287 (1990). M.Tuckeran, B. J. Berne, A. Rossi, J. Che. Phys. 94,1465 ( 1991). M. P. Allen D. J. Tildesley, Coputer Siulation of Liquids (Oxford University, Oxford, 1989). 4 W. C. Swope, H. C. Andersen, P. H. Berens, K. R. Wilson, J. Che. Phys. 76,637 (1982). M. Watanabe W. P. Reinhardt, Phys. Rev. Lett. 65, 3301 (1990). J. Che. Phys., Vol. 94, No. lo,15 May 1991