Constant pressure and temperature discrete-time Langevin molecular dynamics

Transcription

1 Constant pressure and temperature dscrete-tme Langevn molecular dynamcs Nels Grønbech-Jensen and Oded Farago Ctaton: The Journal of Chemcal Physcs 141, (14); do: 1.163/ Vew onlne: Vew Table of Contents: Publshed by the AIP Publshng Artcles you may be nterested n Advanced multple tme scale molecular dynamcs J. Chem. Phys. 131, 1411 (9); 1.163/1.319 Molecular dynamcs smulaton of planar elongatonal flow at constant pressure and constant temperature J. Chem. Phys. 16, 4456 (7); 1.163/ Molecular dynamcs of a dense flud of polydsperse hard spheres J. Chem. Phys. 113, 473 (); 1.163/ Symplectc algorthm for constant-pressure molecular dynamcs usng a Nosé Poncaré thermostat J. Chem. Phys. 11, 3474 (); 1.163/1.485 A large-scale and long-tme molecular dynamcs study of supercrtcal Lennard-Jones flud. An analyss of hgh temperature clusters J. Chem. Phys. 17, (1997); 1.163/

2 THE JOURNAL OF CHEMICAL PHYSICS 141, (14) Constant pressure and temperature dscrete-tme Langevn molecular dynamcs Nels Grønbech-Jensen 1, and Oded Farago 3,4 1 Department of Mechancal and Aerospace Engneerng, Unversty of Calforna, Davs, Calforna 95616, USA Department of Mathematcs, Unversty of Calforna, Davs, Calforna 95616, USA 3 Department of Bomedcal Engneerng, Ben Guron Unversty of the Negev, Be er Sheva 8415, Israel 4 Ilse Katz Insttute for Nanoscale Scence and Technology, Ben Guron Unversty of the Negev, Be er Sheva 8415, Israel (Receved 1 August 14; accepted 9 October 14; publshed onlne 18 November 14) We present a new and mproved method for smultaneous control of temperature and pressure n molecular dynamcs smulatons wth perodc boundary condtons. The thermostat-barostat equatons are bult on our prevously developed stochastc thermostat, whch has been shown to provde correct statstcal confguratonal samplng for any tme step that yelds stable trajectores. Here, we extend the method and develop a set of dscrete-tme equatons of moton for both partcle dynamcs and system volume n order to seek pressure control that s nsenstve to the choce of the numercal tme step. The resultng method s smple, practcal, and effcent. The method s demonstrated through drect numercal smulatons of two characterstc model systems a onedmensonal partcle chan for whch exact statstcal results can be obtaned and used as benchmarks, and a three-dmensonal system of Lennard-Jones nteractng partcles smulated n both sold and lqud phases. The results, whch are compared aganst the method of Kolb and Dünweg [J. Chem. Phys. 111, 4453 (1999)], show that the new method behaves accordng to the objectve, namely that acqured statstcal averages and fluctuatons of confguratonal measures are accurate and robust aganst the chosen tme step appled to the smulaton. 14 AIP Publshng LLC. [ I. INTRODUCTION Molecular Dynamcs (MD) computer smulatons have become a standard tool for nvestgatng a varety of atomc and molecular systems rangng from solds to smple fluds to complex bomolecular assembles. 1 They are partcularly attractve for dynamcs and for equlbrum samplng n hgh densty condensed matter systems where large scale collectve modes may be sgnfcant. These modes may not be easly excted (and relaxed) by the alternatve approach to phase space samplng, namely, Monte Carlo (MC) smulatons, because: () MC evoluton s dffusve n nature, and () MC tends to have low acceptance rates n hgh densty regons. The MC method, however, possesses one sgnfcant advantage over MD the ablty to sample, at least n prncple (.e., for suffcently long runs), almost any statstcal ensemble n a farly straghtforward manner. Ths task s accomplshed by performng MC moves that are ergodc and satsfy the detaled balance condton. In the canoncal (N,V,T) ensemble (where N, V, and T denote the number of partcles, volume, and temperature, respectvely), the latter requrement s usually fulflled by usng the Metropols crteron for the acceptance probablty: p acc = mn [1, exp ( U/k B T)], where k B s the Boltzmann constant and U s the change n the potental energy between the two states whch are approached va opposte moves. Smlarly, the sothermal-sobarc (N, P, T) (where P s the pressure) ensemble can be smulated by ncludng coordnate dsplacements that change the volume of the system and scale the coordnates of the partcles accordngly, and by redefnng the potental energy to U eff = U + PV Nk B T ln V [see, later, Eq. (1)]. Thngs become more complcated when t comes to MD smulatons, whch attempt to follow the dynamcs of a molecular system by numercally solvng Newton s classcal equatons of moton for the consttuent partcles. Ths, supposedly, generates trajectores wthn the mcrocanoncal ensemble (N,V,E) (where E s the nternal energy of the system, whch s the sum of potental and knetc energes) although, due to truncaton errors, one should not expect the computed trajectores to actually follow the real-tme dynamcs n manypartcle systems. 3 The most commonly used dscrete-tme ntegrator for MD smulatons s the Störmer-Verlet algorthm whch (n ts so-called velocty-verlet form) reads 4 r n+1 = r n + v n dt + dt m f n, (1) v n+1 = v n + dt m (f n + f n+1 ), () where r n, v n, and f n denote, respectvely, the coordnate, velocty, and the force actng on the partcle wth mass m at tme t n, and t n+1 = t n + dt. Notce that r n, v n, and f n represent Cartesan components, whch means that for a system of N partcles n a space wth dmensonalty d, the number of equatons one needs to compute per tme step dt, sdn /14/141(19)/19418/1/$3. 141, AIP Publshng LLC

3 N. Grønbech-Jensen and O. Farago J. Chem. Phys. 141, (14) The Verlet algorthm results n a trajectory whch s accurate to second order n the tme step dt. Ths devaton between the computed and the correct trajectores should not be a matter of concern f the smulatons properly sample the correct statstcal ensemble, or otherwse retan the measures of nterest. Thus, the crtcal test for the performance of any numercal ntegrator must be ts accuracy n measurng mportant thermodynamc quanttes and the varatons of the results wth dt. Remarkably, the Verlet algorthm suffers from the problem that the total knetc energy of a smulated system (whch s supposed to be proportonal to the temperature) becomes progressvely depressed for ncreasng tme step dt 5, 6 compared to the potental energy (see also the Appendx for a harmonc oscllator demonstraton of how the velocty n dscrete tme s not precsely the velocty of the correspondng spatal coordnate). Other thermodynamc observables also exhbt varatons wth dt, whch makes a strkng contrast wth MC smulatons n whch the thermodynamc confguratonal averages are nsenstve to the step szes. The mcrocanoncal ensemble sampled n MD smulatons does not provde the best representaton of expermental condtons, where the most common condton s that of constant temperature and pressure. Therefore, consderable effort has been devoted to the development of MD algorthms for smulatons of the sothermal-sobarc ensemble. In the smplest method, proposed by Berendsen, the system s weakly coupled to external heat ( thermostat ) and pressure ( barostat ) baths, usng the prncple of least local perturbaton. 7 Ths method has been crtczed for falng to correctly sample the statstcal ensemble, due to ts tendency to suppress fluctuatons n knetc energy and volume. A second, more relable method, poneered by Andersen for fxed pressure, 8 extended by Parrnello and Rahman 9 and by Nosé, 1 and revsed by Hoover 11 to fxed temperature MD smulatons, s the extended Lagrangan formalsm. The method s based on the dea of ncludng addtonal degrees of freedom, correspondng to the volume and/or the knetc energy of the system, together wth ther conjugate momenta varables. The new varables are coupled to the system n a manner whch guarantees that the trajectory correctly samples the sothermal-sobarc ensemble. The latter consttutes a sub-space of the confguraton space of the extended system. Wthn the extended phase space, the statstcs s mcrocanoncal and the equatons of moton can be derved from the extended Hamltonan, whch s conserved n tme. In prncple, these Hamltonan equatons of moton can be ntegrated numercally usng the Verlet algorthm. In practce, the mplementaton of the dscrete-tme Verlet algorthm rases several sgnfcant challenges and dffcultes. Specfcally, for the barostat part, the couplng between the partcles degrees of freedom and the pston (ntroduced to control the volume fluctuatons) s the source of the followng problems: 1. When the pston moves, the partcle coordnates must be rescaled, whch leads to a metrc problem wth the algorthm. Ths problem has been addressed n Refs The method s extremely senstve to the value assgned to the mass of the pston. A low mass wll result n rapd box sze oscllatons whch are not attenuated very effcently by the motons of the molecules, whle a large mass wll gve rse to a slow adjustment of the volume and may therefore be computatonally neffcent. 3. The force on the pston [see Eq. (8) below] depends on the nternal pressure of the system, the value of whch depends on the nstantaneous knetc energy of the partcles. Ths means that the velocty v n+1 s the soluton to an mplct equaton, whch therefore must be solved teratvely. Ths has several consequences, ncludng that the computed trajectory s no longer tme reversble a feature that jeopardzes the (extended) energy conservaton n long smulatons. A set of explct reversble ntegrators for the dynamcs has been developed by Martyna and co-workers The dependence of the nternal pressure on the knetc energy leads to naccurate determnaton of the pressure, snce the knetc contrbuton s derved from the partcles veloctes, whch, as shown n the Appendx, devate from the actual veloctes. The thrd approach to constant pressure and temperature MD smulatons employs the Andersen extended Lagrangan formalsm,.e., t couples the system to a global pston whch governs the volume fluctuatons of the system. However, nstead of usng a Nosé-Hoover thermostat and solvng the Hamltonan equatons of moton, the temperature s set by solvng the Langevn equaton: 18 ṙ = v, (3) m v = f (r, t) αv + β(t). (4) The Langevn equaton descrbes Newtonan dynamcs where the conservatve force feld f(r, t) s augmented by: () a frcton force proportonal to the velocty wth frcton coeffcent α, and () thermal whte ( delta-functon correlated ) nose, β(t). The frcton and nose terms represent the nteractons wth the mplct degrees of freedom of the heat bath. In order to satsfy Ensten s fluctuaton-dsspaton theorem that relates the frcton and nose to each other, t s usually assumed that the nose s Gaussan dstrbuted and has the followng statstcal propertes: 19 β(t) =, (5) β(t)β(t ) =αk B Tδ(t t ). (6) Hstorcally, Langevn stochastc thermostats have been developed n parallel to the Nosé-Hoover determnstc thermostat, n the early 198s. However, t was only n 1995, when Feller et al. proposed to smulate sothermal-sobarc condtons by consderng Langevn dynamcs for the pston s equaton of moton n Andersen s extended system. 1 Ths approach was mproved a few years later by Kolb and Dünweg, who consdered Langevn dynamcs for both the partcles and pston, and who developed an ntegrator for ths purpose. Whle many of the problems assocated wth the applcaton of the Nosé-Hoover thermostat for (N, P, T) smulatons remaned unsolved (especally those orgnatng from the couplng between the movement of the pston and the partcles),

4 N. Grønbech-Jensen and O. Farago J. Chem. Phys. 141, (14) the dea of smulatng the extended system wthn the framework of Langevn dynamcs appears to offer shorter correlaton tmes and mproved samplng. In ths paper we present a new method for Langevn dynamcs smulatons at constant pressure and temperature. The method, whch s both effectve and smple to mplement, provdes mprovements compared to the method of Kolb and Dünweg (KD). Key dstnctons between our algorthm and others, ncludng KD, le n the manner by whch the dsplacements of the partcles nto physcal and scaled components are decoupled. Wthn the tradtonal methods these two dsplacements are defned and separated pror to tme-dscretzaton, whle our method s based on formulatng the equatons of moton for an already temporally dscretzed set of coordnates. Another change that we ntroduce n the method, s to replace the knetc energy term n the nstantaneous pressure wth ts known thermodynamc average, whch s precsely the deal gas pressure Nk B T/V.Ths change does not only resolve the aforementoned problems n mplementatons of Verlet-type ntegrators to the pston s equaton of moton, t also makes the extended Lagrangan dynamcs more consstent wth the statstcal mechancs of the sothermal-sobarc ensemble that the smulatons ams to sample. Fnally, we take advantage of the recent advances n numercal ntegrators for Langevn dynamcs and replace the old BBK (Brooks, Brünger, and Karplus) thermostat wth the recently ntroduced G-JF (Grønbech-Jensen and Farago) thermostat. 6 Whle the former has a smulated temperature that dffers by O(dt) from the correct one, the latter exhbts no detectable changes n the confguratonal samplng statstcs as the tme step s vared n the entre numercal stablty 6, 3 range. The paper s organzed as follows: In Sec. II we derve the new method for sothermal-sobarc MD smulatons. Ths secton contans both a detaled dscusson of the theoretcal aspects of the method, as well as a dervaton of the algorthm for sothermal-sobarc smulatons. The new algorthm s tested aganst the method of Kolb and Dünweg n Sec. III. For ths purpose we present smulaton results of both a onedmensonal toy model that can be solved analytcally, and a three-dmensonal Lennard-Jones system. We conclude the paper n Sec. IV. II. ISOTHERMAL-ISOBARIC LANGEVIN DYNAMICS A. Statstcal mechancal consderatons In hs semnal paper on the extended Lagrangan formalsm, Andersen studed the statstcal mechancs of N partcles wthn a box wth a fluctuatng volume V subject to a constant external pressure P. 8 He assocated the volume fluctuatons wth the moton of a pston, and consdered an extended phase space of (dn + 1) degrees of freedom, ncludng () the Nd coordnates of the partcles, r, and ther Nd conjugate momenta p, and () the volume V representng the coordnate of a pston along wth ts conjugate momentum. The dervaton of the extended Lagrangan formalsm was done n the rather uncommon soenthalpc-sobarc ensemble (N, P, H), where the enthalpy s H = E + PV. Ths s the equvalent of the mcrocanoncal ensemble (N,V,E) for fxed pressure. The degrees of freedom r and V n the extended system are not ndependent of each other because the partcle coordnates are adjusted to volume fluctuatons va smple scalng durng MD smulatons. In order to have ndependent statstcal varables, one needs to defne the scaled coordnates s s,μ = r,μ /Lμ, (7) where μ = x, y, z, s = (s,x,s,y,s,z ) T, and L μ s the lnear sze of the smulatons box along the μ-axs. For smplcty, we here assume that the smulaton box s orthorhombc wth fxed aspect ratos, such that μ L μ = V, and that all the partcles have dentcal mass m, except for the pston, whch s consdered a coordnate wth nertal constant Q. The force actng on the pston s derved from the extended Hamltonan, whch s obtaned from the extended Lagrangan va Legendre transformaton. It s gven by (see Eq. (3.14C) n Ref. 8) f P = 1 Vd ( f r + p m ) P. (8) The transton from the soenthalpc-sobarc nto the sothermal-sobarc ensemble requres the ntroducton of a thermostat, and as noted n Sec. I, the thermostat can be ether determnstc (Nosé-Hoover) or stochastc (Langevn). In terms of the coordnates s and V,the sothermal-sobarc partton functon reads Z = = dv V N 1 dv 1 N μ=x,y,z N μ=x,y,z ds,μ e [U({L μ s,μ })+PV]/k B T ds,μ e [U({L μ s,μ })+PV Nk B T ln V ]/k B T. Ths partton functon can be nterpreted as f governng the canoncal ensemble of a system consstng of N partcles confned to a three-dmensonal unt cube ( s,μ 1), and a pston movng along an nfnte lne ( <V < ), wth the potental energy gven by U eff ({s,μ }, {L μ }) = U({L μ s,μ }) + PV Nk B T ln V. (1) Notce that the partton functon defned by Eq. (9) ncludes summaton only over the spatal degrees of freedom (of the partcles and pston), but not over ther conjugate momenta. Ths devaton from Andersen s extended Lagrangan formalsm, where both the coordnates and momenta were ncluded n the partton sum, deserves an explanaton. Andersen s method descrbes Newtonan dynamcs wthn a mcrocanoncal ensemble. In ths ensemble, the knetc and potental energes are coupled by energy conservaton. In contrast, Langevn dynamcs occurs wthn an open system n contact wth a heat bath. In ths canoncal ensemble, the knetc and potental energes are decoupled, and the degrees of freedom (9)

5 N. Grønbech-Jensen and O. Farago J. Chem. Phys. 141, (14) of the coordnates can be ntegrated separately from ther assocated momenta. The momenta degrees of freedom follow a Maxwell-Boltzmann Gaussan dstrbuton, whle the coordnates ( s and V ) are governed by the Boltzmann dstrbuton correspondng to U eff (1). The separaton of the ensemble nto two sub-spaces, correspondng to the coordnates and ther assocated momenta, s mportant because the goal of constant temperature and pressure smulatons s to sample the phase space of the coordnates correctly. The momenta,.e., the veloctes, are only used n these smulatons as a mean to assess the smulated knetc temperature. The average knetc energy s a reasonable measure of the temperature, but not a good one n dscrete-tme because of the second order (n dt) devaton between the measured velocty relatve to the trajectory of the correspondng coordnate (see the Appendx). Thus, numercal measures nvolvng velocty are not relable for non-vanshng tme steps. In constant volume smulatons, ths problem s avoded f the Langevn dynamcs s computed usng the accurate G-JF ntegrator, whch exhbt no changes n the confguratonal samplng statstcs n response to varatons n dt. Moreover, the aforementoned closely-related problem of constant pressure smulatons resultng from the dependence of the pston force on the veloctes [see Eq. (8)] s elmnated as well, because, n the confguraton phase space of nterest (whch does not nclude momenta degrees of freedom), the pston force (pressure) s derved from U eff (1) f P = U eff V = 1 Vd f r + Nk B T P V = P P, (11) where we have defned the nternal pressure P. Ths last mportant pont was nether ncluded by Andersen n hs orgnal paper, nor n other later contrbutons on Langevn dynamcs at constant pressure. B. Dervaton of the method 1. Dynamcs of the volume Followng Andersen s dea, we ntroduce the nertal coeffcent Q for a pston wth a coordnate that concdes wth the volume V of the system. The regular force (pressure), f P, actng on ths partcle, s gven by Eq. (11). The pston coordnate moves wth velocty V = V n a medum wth frcton coeffcent α at constant temperature T. The Langevn dynamcs of ths partcle s Q V + αv = f P + β(t). (1) Ths equaton wll be ntegrated usng the G-JF thermostat, whch (n the velocty-verlet form) s expressed by the followng equatons to calculate the coordnate (.e., volume) V n+1 and velocty V n+1 at tme t n+1 = t n + dt (See Eqs. (4) (8) n Ref. 3): V n+1 = V n + bdtv n + bdt Q f n P + bdt Q β n+1, (13) where V n+1 = ãv n + dt Q (ãf n P + f n+1 ) b P + Q β n+1, (14) ã = 1 ãdt Q 1 + ãdt, (15) Q b = ãdt Q, (16) and β n s a normally dstrbuted random number wth zero mean, and autocorrelaton β n β m = αk B Tdtδ m,n.. Dynamcs of the partcles The varaton of the volume causes complcatons for the dynamcs of the partcles, whch resde wthn the confnes of the defned, yet varable, volume. These complcatons are partcularly apparent n systems wth perodc boundary condtons snce the smulated volume s assocated wth a lattce constant of a smulaton box and not wth a physcal locaton of an actual pston. Thus, n order to preserve the translatonal nvarance of the equatons of moton n a bulk system wth perodc boundary condtons, t s necessary to globally couple the dynamcs of the volume to all the partcles, regardless of partcle locaton n the smulaton cell, 8 such that relatve dstances n the system are preserved. Ths s accomplshed through the scaled (normalzed) coordnate s,μ = r,μ /L μ, whch s understood to be constant for a smple expanson or contracton of L μ. However, the physcal velocty and acceleraton of the coordnate r,μ then cannot be translatonal nvarant wthout modfcatons. Andersen s soluton to the problem s to nvestgate the dervatve ṙ,μ = ṡ,μ L μ + s,μ L μ. (17) In Eq. (17) we can see the separaton of two dentfable components to the moton: () the dynamcs of the partcle relatve to the smulaton cell (frst term on the rhs), and () the dynamcs due to the moton of the smulaton cell (second term on the rhs). Thus, defnng v,μ = ṡ,μ L μ (18) as the relevant physcal velocty of the coordnate r,μ = s,μ L μ makes the partcle dynamcs nvarant to the orgn of the coordnate system an essental necessty for meanngful dynamcs. Whle ths elegant observaton has led to the advanced formulatons of both determnstc 1 1, 14, 15 and 1, stochastc methods for NPT smulatons, the nherent problem of tme dscretzaton perssts. We now arrve at the core of the dervaton of the new method. We smplfy the notaton for brevty n the rest of ths subsecton, such that, e.g., the coordnate r refers to r,μ, L to L μ, etc., unless specfcally ndcated otherwse. We reevaluate the partcle equatons of moton n dscrete-tme, startng wth the defnton of the scaled coordnate, Eq. (7). The total partcle dsplacement

6 N. Grønbech-Jensen and O. Farago J. Chem. Phys. 141, (14) r n+1 = r n+1 r n n one tme step s gven by r n+1 = t n d dt (sl) dt = s n+1 L n+1 s n L n (19) = (s n+1 s n ) Ln+1 + L n + sn+1 + s n (L n+1 L n ). () We use the analogy between Eqs. (17) and () to defne the relevant physcal, spatally nvarant, dscrete-tme partcle dsplacement q n+1 from the frst term n Eq. () q n+1 = (s n+1 s n ) Ln+1 + L n. (1) Notce that q ṡldt for dt, consstent wth the usual contnuous-tme defnton of the relevant velocty mentoned above v = ṡl (18). Thus, we conclude that the dscrete-tme partcle dynamcs must nvolve the physcal coordnate q and an assocated velocty v, whch must relate to the dscretetme dsplacement through vdt. () q n+1 = t n The correspondng dscrete-tme velocty change v n+1 = v n+1 v n s obtaned through the dt-ntegrated Langevn equaton [m v + αv] dt = [ f (r, t ) + β(t ) ] dt, (3) t n t n whch, usng () and wth no approxmaton, can be wrtten m v n+1 + α q n+1 = fdt + β n+1, (4) t n where we have defned the Wener process such that β n+1 = t n β(t ) dt (5) β n =, β n β m = αk B Tdtδ n,m. (6) (The nose autocorrelaton reads wth full notaton: β,μ n βm j,ν = αk B Tdtδ n,m δ,j δ μ,ν.) Notce that the ntroducton of the dscrete-tme Langevn equaton n Eq. (4), for lnkng the coordnate dsplacement q wth ts velocty change v, ensures physcally meanngful dscrete-tme evoluton. Startng from Eq. (), and followng our prevous work, 6 we now choose the tme-reversble relatonshp between the relatve dsplacement and change n the assocated velocty: q n+1 = t n vdt dt (vn+1 + v n ) = dt vn+1 + dtv n. (7) Insertng (4) nto (7) yelds q n+1 = bdtv n + bdt m t n fdt + bdt m βn+1, (8) where 1 b = 1 + αdt. (9) m Equatons (4) and (8) consttute a set of equatons for determnng q n+1 and v n+1. We then approxmate the dtntegrals over the determnstc force f such that all terms n the equatons become at least second order correct n dt (.e., consstent wth the tradtonal Verlet methods), whch yelds q n+1 = bdtv n + bdt m f n + bdt m βn+1, (3) v n+1 = α m qn+1 + dt m (f n + f n+1 ) + 1 m βn+1. (31) These are explct dscrete-tme equatons for evaluatng the evoluton of the coordnates q n and v n. In order to express the equatons n the most useful form for molecular smulatons, we use the relatonshp r n = s n L n (Eq. (7)) to combne Eqs. (1) and (3) for a drect expresson of the dynamcs of the physcal coordnate r n : r n+1 = Ln+1 L n rn [ + Ln+1 L n+1 + L bdt v n + dt n m f n + 1 m βn+1]. (3) We also nsert Eq. (3) nto Eq. (31) n order to obtan an explct equaton for the dynamcs of the velocty v n : v n+1 = av n + dt m (af n + f n+1 ) + b m βn+1, (33) where a = 1 αdt m 1 + αdt. (34) m Equatons (3) and (33) are the Verlet-type equatons for partcle updates n the stochastc G-JF thermostat/barostat, gven a change L n+1 = L n+1 L n n smulaton box dmenson durng the tme step dt = t n+1 t n. Notce that the velocty equaton depends only ndrectly on the change n the smulaton dmenson L through the force f n = f (r n, t n, L n ). To summarze, gven r n, v n, f n, fp n, V n, and V n,the dscrete-tme dynamcs evolves accordng to the followng protocol: 1. Compute V n+1 (and L n+1 μ )usngeq.(13).. Compute r n+1 (all r n+1 )usngeq.(3). 3. Evaluate the new forces f n+1 = f(r n+1, t n+1, L n+1 )(all f n+1 ), and f P ({ r n+1 },t n+1, {L n+1 μ }). 4. Compute V n+1 usng Eq. (14) and v n+1 (all v n+1 )usng Eq. (33). We reemphasze that the coordnates r n, v n, and f n here refer to each Cartesan coordnate of each partcle, and that L refers to L μ, such that V = μ L μ s the volume of an sotropcally varyng, orthorhombc smulaton box.

7 N. Grønbech-Jensen and O. Farago J. Chem. Phys. 141, (14) III. TESTING THE ALGORITHM In order to test the method we have appled t to two characterstc systems wth a specfc eye on the robustness aganst tme-step varatons. We compare our results wth those generated by the KD method, whch represents state-of-the-art of a sound approach to Langevn dynamcs NPT smulatons n atomc and molecular ensembles. The frst system s a partcular non-trval, one-dmensonal partcle model for whch we can analytcally derve measurable thermodynamc quanttes. Ths model therefore serves as a strct benchmark for the statstcal accuracy of a numercal test smulaton. The second system s the foundatonal model system n computatonal statstcal mechancs, namely the three-dmensonal ensemble of partcles nteractng wth a Lennard-Jones force feld. In ths latter case, we do not have analytcal expressons for the statstcal measures, but we nvestgate the measures for dfferent values of the dscrete tme step, and from that nfer the qualty of the appled numercal methods. A. One-dmensonal model system We consder a one-dmensonal system of normalzed length L (characterstc length r ) wth perodc boundary condtons. N dentcal partcles are located n order at {x 1, x,..., x N ; x < x + 1 } such that the perodc boundary condtons ensure two neghbors for each partcle (.e., x ± N = x ± L). Each partcle nteracts wth ts two neghbors va a parpotental that depends on the normalzed par dstance r. Expressng the energy n unts of the thermal energy (E = k B T), the normalzed par-potental u(r) [related to the physcal potental va U(r r) = E u(r)] reads u(r) = ɛ r + 1 ln r. (35) The par potental u(r) conssts of two contrbutons: a repulsve part (ɛ >), nversely proportonal to r, and an attractve logarthmc part. The latter may represent an entropc potental of mean force resultng from mplct degrees of freedom. Consderng the sobarc-sothermal ensemble (N, P, T) (where P denotes the one-dmensonal pressure,.e., the force, and s expressed n unts of E /r ), the partton functon of the system s gven by N [ ] Z = dl dx exp u(x +1 x ) PL. (36) Swtchng to the set of varables r = x + 1 x, the partton functon reads N [ ] Z = dr exp u(r ) P r [ ] dr N = e (ɛ/r+pr) r = [ dy e (ɛ/y +Py )] N, (37) where the last equalty has been obtaned by settng y = r. The value of the last ntegral s known, gvng [ ] π N Z = Pɛ P e. (38) The normalzed Gbbs free energy s gven by G = ln Z, and the mean nearest neghbor partcle normalzed dstance, l,s then derved by l L N = 1 G N P = 1 ɛ P + P. (39) The varance of the normalzed length dstrbuton s gven by σl (L L ) N = l P = 1 ɛ P + P 3. (4) For a system of N = 1 partcles, we smulate the evoluton for a normalzed transent tme of unts before producng statstcal averages of l and σ l over the next normalzed tme unts. Fgures 1(a) and 1(b) (a) <l> σ l (b) P=4. P=. P=1. G-JF KD dt G-JF KD P=4. P=. P= dt FIG. 1. Results for ɛ = 1. Smulated average length (a) and standard devaton σ l (b) for several values of appled 1D pressure (force) P. Markers represent the G-JF method of ths paper (sold marker ) and the KD method (open marker ). At small tme steps both methods produce the correct analytcal values gven by Eqs. (39) and (4). All smulatons were done wth Q = 1 and α = 1. Lnes serve as gudes to the eye.

8 N. Grønbech-Jensen and O. Farago J. Chem. Phys. 141, (14) p(l).3..1 G-JF Exact l FIG.. Results for ɛ = 1, P = 1anddt =.6. The computed length dstrbuton (open crcles) compared wth the analytcally calculated exact dstrbuton (sold curve). show the resultng data for both the G-JF method of ths paper (sold markers, dotted lne) and the KD method (open markers, dashed lne) for three dfferent values of the external, onedmensonal pressure P, wth partcle mass and dsspaton normalzed parameters m = 1 and α = 1, respectvely. By nspectng the convergence of l to ts equlbrum value, we can fnd values for the normalzed pston parameters that provde effcent relaxaton. For the model system dscussed heren, we choose Q = α = 1. The acqured data clearly show that the G-JF method s extremely accurate. The computed values of both the average and fluctuatons of the length agree wth the predctons of Eqs. (39) and (4). The accuracy of the method s also demonstrated n Fg., where the full length dstrbuton p(l) s plotted for P = 1 and dt =.6. The agreement wth the analytcally calculated exact dstrbuton s perfect. Another mportant feature of the method, demonstrated n Fgure 1, s ts robustness aganst tme step varatons. In comparson, we observes n Fg. 1 that the KD method yelds the correct result for small dt, but that the stablty range s generally consderably smaller than for the G-JF procedure. In a dfferent set of smulatons (data not shown), we used Q = 1 4 and α =. Ths choce of parameters made the KD barostat unstable for all the smulated values of dt (dt.1), whle keepng almost unchanged the stablty range of the G-JF method. The relatve robustness of the latter aganst varatons n the pston parameters s yet another mert of ths method. reads r 1 r 6, <r r s u(r) = a 4 (r r c ) 4 + a 8 (r r c ) 8, r s <r<r c, r c r (41) where ( ) 13 1/6 r s = , (4) 7 r c = r s 3u(r s ) , (43) 11u (r s ) a 4 = 8u(r s ) + (r c r s )u (r s ) 4(r c r s ) 4, (44) a 8 = 4u(r s ) + (r c r s )u (r s ). (45) 4(r c r s ) 8 Ths functon (see Fg. 3) s a short-range splned Lennard- Jones potental wth contnuty through the second dervatve at r = r s and contnuty through thrd dervatve at r = r c. Conductng NPT smulatons on a cubc system wth N = 864 partcles, we optmze the relaxaton of the barostat degree of freedom V by choosng small values for the nerta Q. By nspecton, we fnd that values n the range Q = 1 4 and Q = 1 5 represent effcent relaxaton. We also, by nspecton, conclude that a small frcton coeffcent α = 1 4 helps relax the system (although ths seems to be a weak effect) and, therefore, choose ths value for our smulatons. We have further chosen two characterstc normalzed temperatures, for both sold (k B T/E =.3) and lqud (k B T/E =.7) phases. Fnally, we have studed three dfferent appled pressures (P =.1,.1, 1.) (expressed n unts of E /r 3 ), and vared the dscrete normalzed tme step dt n the entre range of stablty to observe the behavor of the numercal methods. We only show the P =.1 data here snce the results of all three appled pressures exhbt the same characterstcs. All statstcal data are obtaned by ntatng the system n a hexagonal closepacked crystal near a zero-temperature ground state. We then smulate at least 1 5 normalzed tme unts before averages are acqured over the next 1 5 unts. The normalzed B. Three-dmensonal Lennard-Jones model system We now consder the smplest possble well-known system n the modelng of materals and lquds, namely, a threedmensonal ensemble of dentcal sphercal partcles. Each partcle has a normalzed mass m = 1 (n unts of m ) and normalzed frcton coeffcent α = 1, and they all nteract through the normalzed potental u(r) gven by the physcal par-potental U({r r}) = E u(r), where r = r s the normalzed par-dstance (n unts f the characterstc length r ) and E s the characterstc energy. The normalzed par-potental FIG. 3. Partcle nteracton as gven n Eq. (41). Upper plot shows u(r), lower plot shows u (r). Splne pont and cut-off dstance are ndcated by arrows.

9 N. Grønbech-Jensen and O. Farago J. Chem. Phys. 141, (14) FIG. 4. For k B T/E =.3 (sold phase): Smulated average volume V [(a) and (b)] and standard devaton σ V [(c) and (d)] for Q = 1 4 [(a) and (c)] and Q = 1 5 [(b) and (d)]. Markers represent the G-JF method of ths paper (sold marker ) andthe KD method (openmarker ). Horzontal dotted lnes are leveled at V for Q = 1 4 and dt =.1 [(a) and (b)], and at σ V for Q = 1 4 and dt =.1 [(c) and (d)]. All fgures show axes wth absolute quanttes on the left and percentage devaton on the rght axes. FIG. 5. For k B T/E =.3 (sold phase): Smulated average potental energy E p [(a) and (b)] and standard devaton σ Ep [(c) and (d)] for Q = 1 4 [(a) and (c)] and Q = 1 5 [(b) and (d)]. Markers represent the G-JF method of ths paper (sold ) and the KD method (open ). Horzontal dotted lnes are leveled at E p for Q = 1 4 and dt =.1 [(a) and (b)], and at σ Ep for Q = 1 4 and dt =.1 [(c) and (d)]. All fgures show axes wth absolute quanttes on the left and percentage devaton on the rght axes. tme τ s gven by E τ = m r. All the left axes of the fgures dsplay absolute results, whle all the rght vertcal axes dsplay the percentage devaton from the dt value of the quantty shown n the plot. Fgure 4 shows the data for the volume V of the smulaton box (Q = 1 4 :Fg.4(a); Q = 1 5 :Fg.4(b)) and ts fluctuatons (Q = 1 4 :Fg.4(c); Q = 1 5 :Fg.4(d)) asa functon of the tme step for a sold phase at k B T/E =.3 and external pressure P =.1. The new G-JF barostat results are dsplayed as sold markers ( ), whle the comparson KD method results are shown wth open markers ( ). The data clearly shows that the G-JF results are nearly ndependent of the tme step dt for both the average volume and the correspondng fluctuatons. In comparson, the KD method exhbts a consstent, albet weak, ncrease n average volume. More dramatc s the ncreasng devaton of the volume fluctuatons n the KD method. For Q = 1 4, ths can be n excess of 1%, whle we observe up to 7% dscrepancy for Q = 1 5. Such dscrepances clearly change not only measured thermodynamc propertes such as the elastc bulk modulus and heat expanson coeffcent, but also the structure of the materal under nvestgaton. For example, close nspecton (not shown) of the KD smulaton shows that the excessve volume fluctuatons nduce crystal defects nto the materal for large dt >.16, before the numercal nstablty s found for dt.19. Notce that the results of both methods converge to the same numbers for small dt throughout the smulaton data, ndcatng that any devaton from small dt consttutes a measure of the error nduced exclusvely by the dscrete tme step. The data for the total potental energy, E p = u(r j ) + PV, (46) <j are shown n Fg. 5. The results of the G-JF method are also here unmpressed wth the smulated tme step throughout the stablty ranges, whle the KD method shows a characterstc postve devaton. Snce the KD method concdes wth the BBK thermostat when the volume s constant, ths result s entrely expected n lght of our prevous work on the G-JF thermostat and comparsons 3 to other thermostats, ncludng BBK. These results are also consstent wth the knetc temperature T k measurements shown n Fgure 6, where T k s defned as T k = 1 m ( v n ). (47) 3Nk B In the G-JF method, the knetc temperature decreases wth ncreasng dt. Ths s antcpated snce, as mentoned n the Introducton, t s known that the momentum mv n s not the conjugate varable to r n for dt > (see, e.g., Refs. 5, 6, and 3 as well as the Appendx below), and that the dscrete-tme second order approxmatons (v n and V n )to FIG. 6. For k B T/E =.3 (sold phase): Smulated average knetc temperature T k [(a) and (b)] (from Eq. (47)) and standard devaton σ Tk [(c) and (d)] for Q = 1 4 [(a) and (c)] and Q = 1 5 [(b) and (d)]. Markers represent the G-JF method of ths paper (sold ) and the KD method (open ). Horzontal dotted lnes are leveled at T k =.3 for [(a) and (b)], and at σ Tk for Q = 1 4 and dt =.1 [(c) and (d)]. All fgures show axes wth absolute quanttes on the left and percentage devaton on the rght axes.

10 N. Grønbech-Jensen and O. Farago J. Chem. Phys. 141, (14) the velocty varables from the central dfference approach n the Verlet formalsm leaves knetc and confguratonal measures mutually nconsstent. Thus, the very good confguratonal samplng propertes of the G-JF method seen from the measurements of, e.g., volume and enthalpy (along wth ther fluctuatons) nevtably mean that a measurement [such as T k n Eq. (47)] derved from the (ncorrect) veloctes wll be ncorrect to second order n dt. Ths s a smple consequence of v n beng an approxmaton to the true velocty of r n, whch cannot be obtaned. Fgure 6 further dsplays the measured knetc temperature T k for the KD method, and t s apparent that ths quantty appears to confrm the requred temperature, whch s consstent wth the ncorrect confguratonal propertes of ths method seen for volume and enthalpy as dt s ncreased. Ths artfact of dscrete tme emphaszes that one should refran from usng knetc measures as relable quanttes n these types of smulatons. We now show results for a lqud phase at k B T/E =.7. Otherwse, all system and smulaton parameters are exactly as for the k B T/E =.3 results shown above. The lqud phase s valdated by structural analyss and through the measured dffuson constant, whch we derve from the Ensten defnton D = 1 ) n ( s lm N V 3 s. (48) ndt 6ndt We use tme averages over ndt = for all chosen values of dt, and s n s understood to extend beyond the nterval s < 1 n ths expresson. 4 Fgure 7 dsplays the non-zero measured dffuson coeffcent of the lqud state as a functon of the tme step. It s clear that the mgraton at ths temperature and pressure s weak, and that the dffuson measurement s nosy. Even so, the fgure demonstrates that both G-JF and KD methods exhbt dffuson coeffcents reasonably ndependent of the choce of the sze of tme step, although there may be a hnt of a slght ncrease for the KD method for ncreasng dt. Fgure 8 shows the k B T/E =.7 data for the volume V of the smulaton box. The KD results for both average and fluctuaton of the volumes exhbt sgnfcant ncreases wth dt, consstent wth the comparable k B T/E =.3 data. The G- JF results are much less mpressed by the tme step dt, but there s a small tendency for the volume and ts fluctuatons to FIG. 8. For k B T/E =.7 (lqud phase): Smulated average volume V [(a) and (b)] and standard devaton σ V [(c) and (d)] for Q = 1 4 [(a) and (c)] and Q = 1 5 [(b) and (d)]. Markers represent the G-JF method of ths paper (sold ) and the KD method (open ). Horzontal dotted lnes are leveled at V for Q = 1 4 and dt =.1 [(a) and (b)], and at σ V for Q = 1 4 and dt =.1 [(c) and (d)]. All fgures show axes wth absolute quanttes on the left and percentage devaton on the rght axes. decrease wth ncreasng dt. However, the overall mpresson s clearly that the G-JF method s sgnfcantly less dependent on varatons n dt than the KD method s. The k B T/E =.7 data for the potental energy are shown n Fg. 9. The KD results for ths lqud phase exhbt the typcal BBK behavor that was also seen n Fg. 5 for the sold phase at k B T/E =.3. In comparson, the potental energy shows only a slght decrease n both average and fluctuatons for the G-JF method. It s agan clear that the G-JF method produces smulated matter wth confguratonal propertes nearly ndependent of dt. The uncertanty on the acqured averages can be assessed from the assocated standard devatons and the averagng tme. We here also nclude a multple of smulaton data for the same parameters n order to ndcate the magntude of the statstcal error that should be assocated wth the presented standard devatons. FIG. 7. For k B T/E =.7 (lqud phase): Smulated dffuson coeffcent D from Eq. (48) for Q = 1 4 (a) and Q = 1 5 (b). Markers represent the G- JF method of ths paper (sold ) and KD method (open ). Horzontal dotted lnes are leveled at D for Q = 1 4 and dt =.1. Both fgures show axes wth absolute quanttes on the left and percentage devaton on the rght axes. FIG. 9. For k B T/E =.7 (lqud phase): Smulated average potental energy E p [(a) and (b)] and standard devaton σ Ep [(c) and (d)] for Q = 1 4 [(a) and (c)] and Q = 1 5 [(b) and (d)]. Markers represent the G-JF method of ths paper (sold ) and the KD method (open ). Horzontal dotted lnes are leveled at E p for Q = 1 4 and dt =.1 [(a) and (b)], and at σ Ep for Q = 1 4 and dt =.1 [(c) and (d)]. All fgures show axes wth absolute quanttes on the left and percentage devaton on the rght axes.

11 N. Grønbech-Jensen and O. Farago J. Chem. Phys. 141, (14) FIG. 1. For k B T/E =.7 (lqud phase): Smulated average knetc temperature T k [(a) and (b)] (from Eq. (47)) and standard devaton σ Tk [(c) and (d)] for Q = 1 4 [(a) and (c)] and Q = 1 5 [(b) and (d)]. Markers represent the G-JF method of ths paper (sold ) and the KD method (open ). Horzontal dotted lnes are leveled at T k =.3 [(a) and (b)], and at σ Tk for Q = 1 4 and dt =.1 [(c) and (d)]. All fgures show axes wth absolute quanttes on the left and percentage devaton on the rght axes. FIG. 11. For k B T/E =.3 (sold phase): Smulated average pressure P [(a) and (b)] and standard devaton σ P [(c) and (d)] for Q = 1 4 [(a) and (c)] and Q = 1 5 [(b) and (d)]. Markers represent the G-JF method of ths paper (sold ), KD method usng the nstantaneous pressure from Eq. (51) (open ), and KD method usng Eq. (5) (open ). Horzontal dotted lnes are leveled at P =.1 [(a) and (b)], and at σ P for Q = 1 4 and dt =.1 [(c) and (d)]. All fgures show axes wth absolute quanttes on the left and percentage devaton on the rght axes. We confrm that the knetc measurements of temperature and ts fluctuatons behave smlarly n the lqud and sold phases by comparng Fgs. 6 and 1. The latter shows the data for k B T/E =.7, whch agan demonstrates the sgnature of the momentum mv n not exactly beng the conjugate varable to r n. Thus, also for the lqud phase, we observe that the calculated dscrete-tme knetc temperature s progressvely short of the actual temperature of the confguratonal samplng statstcs that can be nferred from the potental energy measurements n Fgure 9. We fnally turn to nvestgatng the pressure. Clearly, one should expect that the average nternal pressure P s controlled and equal to the mposed external pressure P, snce ths s the prncple purpose of the barostat. It s mportant to note that the nternal pressure s defned dfferently n the G-JF and KD methods (see dscusson above n Sec. II). The former uses the canoncal ensemble defnton [see Eq. (11)] P can = 1 f r + Nk B T V, (49) whle the latter targets the mcrocanoncal ensemble expresson [see Eq. (8)] P mcro = 1 f r + 1 m v. (5) For each method we nspect the statstcs of the relevant nternal pressure. The results for the sold phase smulatons at k B T/E =.3 are dsplayed n Fgure 11. From the data, t s obvous that the mposed pressure s correctly adopted by the G-JF method presented here. The KD method, however, dsplays a curous and perhaps sgnfcant devaton from the expected. The orgn of these devatons s the tme reversble dscretzaton used n the KD method (see sequental steps (1) (7) n Sec. V of Ref. ), that apples a trapezodal approxmaton [(r n f n + r n+1 f n+1 )/] to the confguratonal pres- sure contrbuton [frst term on rhs of Eq. (5)], whle usng a md-pont approxmaton (v n+ 1 ) to the knetc part [second term on rhs of Eq. (5)]. Therefore, the dscrete-tme nstantaneous pressure PI n = 1 f n r n + 1 m ( v n ), (51) correspondng to the expresson (5) s dfferent from, and nconsstent wth, the enforced pressure n the KD method for dt >. Ths nconsstency s vsble n Fgure 11, where the marker shows the average of the nstantaneous pressure calculated from Eq. (51). The data exhbts a quadratcally ncreasng devaton between the enforced and measured nternal pressures as dt s ncreased. An measure of the nternal pressure, more consstent wth the enforced value P, s found from P n II = 1 r n + 1 f n m ( v n+ 1 ), (5) whch s shown by the markers. Ths measure of pressure seems properly enforced for all tme steps dt. The fluctuatons, defned as the standard devaton σ P,of the nternal pressure show very reasonable robustness of the G-JF method aganst dt varatons, although we do observe up to about 5% error for dt very close to the stablty lmt. In comparson, the KD method shows larger devatons, especally for the measure of the nternal pressure defned by Eq. (5). We thus conclude that wthn the KD method, nether PI n nor Pn II exhbt statstcs that s nsenstve to varatons n dt. Fgure 1 shows the acqured statstcs of the measured pressure and ts fluctuatons as a functon of the tme step for the lqud phase at k B T/E =.7. The overall behavor of the methods s the same for lqud and sold phases wth drect averages of the nstantaneous pressure, Eq. (51), beng sgnfcantly depressed for the KD method as dt s ncreased. We

12 N. Grønbech-Jensen and O. Farago J. Chem. Phys. 141, (14) FIG. 1. For k B T/E =.7 (lqud phase): Smulated average pressure P [(a) and (b)] and standard devaton σ P [(c) and (d)] for Q = 1 4 [(a) and (c)] and Q = 1 5 [(b) and (d)]. Markers represent the G-JF method of ths paper (sold ), KD method usng the nstantaneous pressure from Eq. (51) (open ), and KD method usng Eq. (5) (open ). Horzontal dotted lnes are leveled at P =.1 [(a) and (b)], and at σ P for Q = 1 4 and dt =.1 [(c) and (d)]. All fgures show axes wth absolute quanttes on the left and percentage devaton on the rght axes. also see that the fluctuatons of the KD pressure s farly ndependent of dt for Q = 1 4, whle the fluctuatons of the pressure PII n ncrease dramatcally for Q = 1 5. The G-JF method s generally robust, although we do observe some ncrease n pressure fluctuatons for Q = 1 5. We note that mproved statstcal accuracy of knetcs usng half-step veloctes v n+ 1 n the so-called leap-frog versons of the Verlet method have been nvestgated 5, 6 for determnstc Nosé-Hoover control of temperature and pressure. However, whle the half-step veloctes may be able to produce better consstency for the averaged knetc temperature n determnstc dynamcs, these approaches may nether translate to stochastc dynamcs nor resolve the fundamental queston of calculatng nstantaneous pressure (at tmes t n )for statstcal averages and fluctuatons, as llustrated above. IV. DISCUSSION We have presented and demonstrated a new thermostatbarostat par for smulatng atomc and molecular dynamcs wth perodc boundary condtons. The new G-JF method s smple and stable, and smulatons of thermodynamc propertes produce data wth very lttle dependency on the appled numercal tme step. We have nvestgated the method n the context of two characterstc models a one-dmensonal toy model wth known statstcal solutons, and the classcal three-dmensonal Lennard-Jones materal, smulated n both crystallne and lqud phases. In all cases the G-JF method behaves extremely well for measured averages as well as for ther fluctuatons. In comparson, the state-of-the-art KD method, whch s also representatve of other commonly used methods, may exhbt sgnfcant devatons n both averages and fluctuatons for ncreasng tme steps. As we have emphaszed throughout ths paper, and specfcally n the Appendx, t s crucal to apprecate that dscrete tme nvaldates the conjugate relatonshp between the coordnate r and ts smulated velocty v. Consequently, one cannot expect accurate smulaton measures for both confguratonal and knetc quanttes usng any gven method. Ths nterestng and essental feature becomes apparent when comparng the behavor of knetc and potental energes as a functon of tme step varatons. We submt that the G-JF thermostat and barostat are advantageous n that they consstently provde proper confguratonal propertes (such as Boltzmann dstrbutons, Ensten dffuson, potental energy, pressure, system volume, as well as ther fluctuatons), whle leavng knetc measures (such as measured knetc energy and the derved knetc temperature) wth predctable devatons. In contrast, most other methods (e.g., KD, Nosé-Hoover, etc.) enforce the expected knetc measures, thereby sacrfcng the accuracy of proper confguratonal samplng. The latter s unfortunate, snce most molecular smulatons are conducted n order to obtan confguratonal nformaton. We close by notng that the G-JF method s easly extended to non-sotropc volume adjustments, and that the algorthm s not only smple, but also n a form that makes t easy to mplement nto exstng molecular dynamcs codes that have thermodynamc temperature and pressure control. Specfcally, the method can also convenently be expressed n the so-called leap-frog and poston-verlet forms, as outlned for the thermostat n Ref. 3. ACKNOWLEDGMENTS Ths work was supported by the US Department of Energy, Project No. DE-NE536, and by the Israel Scence Foundaton (ISF), Grant No. 187/13. APPENDIX: DISCRETE-TIME RELATIONSHIP BETWEEN POSITION AND VELOCITY The contnuous-tme expectaton, that the momentum p n = mv n s the conjugate varable to the spatal coordnate r n,snot fulflled n dscrete-tme Verlet methods. Ths unfortunate consequence of tme dscretzaton has sgnfcant mplcatons for the use and nterpretaton of smulatons, and t can be llumnated by consderng a the smple analyss of a smulated harmonc oscllator, 7 m r = κr, (A1) where κ>sahooke s sprng constant. The contnuoustme soluton to ths equaton s, of course, r(t) exp (± t) and v(t) = ṙ =± r(t), where = κ/m. (Weuse for complex notaton n ths Appendx.) The dscrete-tme Verlet equatons for Eq. (A1) are found from Eqs. (1) and (): The soluton s r n+1 = r n r n 1 κdt m rn, (A) v n = rn+1 r n 1. (A3) dt r n exp(± V ndt) (A4)