Thermostat Algorithms for Molecular Dynamics Simulations

Transcription

1 Adv. Polym. Sc. (005) 173: DOI: /b9947 Sprnger-Verlag Berln Hedelberg 005 Thermostat Algorthms for Molecular Dynamcs Smulatons Phlppe H. Hünenberger Laboratorum für Physkalsche Cheme, ETH Zürch, CH-8093 Zürch, Swtzerland Abstract Molecular dynamcs smulatons rely on ntegratng the classcal (Newtonan) equatons of moton for a molecular system and thus, sample a mcrocanoncal (constantenergy) ensemble by default. However, for compatblty wth experment, t s often desrable to sample confguratons from a canoncal (constant-temperature) ensemble nstead. A modfcaton of the basc molecular dynamcs scheme wth the purpose of mantanng the temperature constant (on average) s called a thermostat algorthm. The present artcle revews the varous thermostat algorthms proposed to date, ther physcal bass, ther advantages and ther shortcomngs. Keywords Computer smulaton, Molecular dynamcs, Canoncal ensemble, Thermostat algorthm 1 Introducton Ensembles Thermostat Algorthms TemperaturentheMonteCarloAlgorthm TemperatureRelaxatonbyStochastcDynamcs TemperatureRelaxatonbyStochastcCouplng TemperatureConstranng TemperatureRelaxatonbyWeakCouplng TemperatureRelaxatonbytheExtended-SystemMethod Generalzatons of the Prevous Methods Practcal Consderatons Appendx: Phase-Space Probablty Dstrbutons References Lst of Abbrevatons and Symbols CM MC center of mass Monte Carlo

2 106 Phlppe H. Hünenberger MD molecular dynamcs NMR nuclear magnetc resonance SD stochastc dynamcs δ Kronecker delta symbol or Drac delta functon h Heavsde step functon k B Boltzmann s constant U nstantaneous potental energy K nstantaneous knetc energy H Hamltonan E energy (thermodynamcal) H enthalpy L Hll energy R Ray enthalpy T temperature (nstantaneous) T temperature (thermodynamcal) T o reference temperature (heat bath) β β = (k B T o ) 1 V volume (nstantaneous) V volume (thermodynamcal) P pressure (nstantaneous) P pressure (thermodynamcal) N number of partcles (n-speces, nstantaneous) N number of partcles (n-speces, thermodynamcal) ν chemcal potental (n-speces, nstantaneous) µ chemcal potental (n-speces, thermodynamcal) p sys system lnear momentum L sys system angular momentum p box box lnear momentum L box box angular momentum N df number of nternal degrees of freedom N c number of geometrcal constrants N r number of external degrees of freedom m mass of atom ṙ o real velocty of atom ṙ pecular velocty of atom F force on atom p momentum of atom R stochastc force on atom γ frcton coeffcent of atom λ velocty scalng factor t tmestep ζ T temperature relaxaton tme α collson frequency (Andersen thermostat) τ B relaxaton tme (Berendsen thermostat) Q mass of the tme-scalng coordnate (Nosé-Hoover thermostat)

3 Thermostat Algorthms 107 τ NH L e H e E e effectve relaxaton tme (Nosé-Hoover thermostat) extended-system Lagrangan (Nosé-Hoover thermostat) extended-system Hamltonan (Nosé-Hoover thermostat) extended-system energy (Nosé-Hoover thermostat) 1 Introducton Classcal atomstc smulatons, and n partcular molecular dynamcs (MD) smulatons, have nowadays become a common tool for nvestgatng the propertes of polymer [1] and (bo-)molecular systems [, 3, 4, 5, 6, 7]. Due to ther remarkable resoluton n space (sngle atom), tme (femtosecond), and energy, they represent a powerful complement to expermental technques, provdng mechanstc nsght nto expermentally observed processes. However, drect comparson wth experment requres that the boundary condtons mposed on the smulated system are n adequaton wth the expermental condtons. The term boundary condton s used here to denote any geometrcal or thermodynamcal constrant enforced wthn the whole system durng the smulaton. One may dstngush between hard and soft boundary condtons. A hard boundary condton represents a constrant on a gven nstantaneous observable,.e. t s satsfed exactly at any tmepont durng the smulaton. A soft boundary condton represents a constrant on the average value of an observable,.e. the correspondng nstantaneous value s allowed to fluctuate around the specfed average. The defnton of a soft boundary condton generally also requres the specfcaton of a tmescale for whch the average observable should match the specfed value. There exst four man types of boundary condtons n smulatons: 1. Spatal boundary condtons nclude the defnton of the shape of the smulated system and the nature of ts surroundngs. In molecular smulatons, one typcally uses ether: () vacuum boundary condtons (solute molecule surrounded by vacuum); () fxed boundary condtons (solute-solvent system surrounded by vacuum, e.g. droplet [8, 9, 10, 11, 1, 13, 14, 15]); () perodc boundary condtons (solute-solvent system n a space-fllng box, surrounded by an nfnte array of perodc copes of tself [16, 17]). In the two former cases, the effect of a surroundng solvent can be rentroduced n an mplct fashon by a modfcaton of the system Hamltonan. Typcal modfcatons are the ncluson of: () solvaton forces accountng for the mean effect of the solvent [18, 19, 0, 1]; () stochastc and frctonal forces accountng for the effect of collsons wth solvent molecules [, 3, 4, 5, ]; () forces at the system boundary to mmck a system-solvent nterface [8, 9, 10, 11, 1, 13, 14, 15]. Spatal boundary condtons are hard boundary condtons, because they apply strctly to all confguratons durng a smulaton.. Thermodynamcal boundary condtons nclude the defnton of the n + thermodynamcal quanttes characterzng the macroscopc state of a (monoplastc) n-component system (for systems under vacuum boundary condtons, only n + 1 quanttes are requred because the volume s not defned whle

4 108 Phlppe H. Hünenberger the thermodynamcal pressure s zero). These quanttes can be selected form pars of extensve and ntensve quanttes ncludng: () the number of partcles (N {N = 1...n}) or chemcal potental (µ {µ = 1...n}) ofall speces; () the volume V or pressure P; () the energy E (or a related extensve thermodynamcal potental) or temperature T. The selected set of n + quanttes, together wth ther reference (macroscopc) values, defne the thermodynamcal ensemble that s sampled durng a smulaton (Table 1). By default, MD smulatons sample mcrostates n the mcrocanoncal (NVE) ensemble. By applyng specfc modfcatons to the system Hamltonan or equatons of moton, t s possble to mantan nstead a constant temperature, pressure or chemcal potental for the dfferent speces (or any combnaton of these changes). The thermodynamcal boundary condtons nvolvng extensve quanttes should be treated as hard boundary condtons, whle those nvolvng ntensve quanttes should be soft. 3. Expermentally derved boundary condtons are used to explctly enforce agreement between a smulaton and some expermental result. These may be appled to enforce, e.g., the reproducton of (average) electron densty maps from X-ray crystallography [7, 8, 9, 30], or the agreement wth (average) nteratomc dstances and J-couplng constants from NMR measurements [31, 3, 33, 30]. Snce experments always provde averages over a gven tme and number of molecules, expermentally derved boundary condtons should be handled as soft boundary condtons. 4. Geometrcal constrants can also be consdered as boundary condtons. A typcal example s the use of bond-length constrants n smulatons [34, 35, 36, 37, 38, 39], whch represent a better approxmaton to the quantum-mechancal behavor of hgh-frequency oscllators (hν k B T ) compared to the classcal treatment [40]. Snce they are satsfed exactly at every tmepont durng a smulaton, geometrcal constrants represent hard boundary condtons. The present artcle s concerned wth one specfc type of thermodynamcal boundary condton, namely the mposton of a constant (average) temperature durng MD smulatons by means of thermostat algorthms. The smultaneous enforcement of a constant (average) pressure [51, 5, 53, 54, 55, 46, 56, 57, 58, 59, 60, 61, 53, 6, 63, 64, 65, 66, 67, 68, 69, 70] or chemcal potental [71, 7, 73, 74] wll not be consdered here. The dscusson s also restrcted to systems under ether vacuum or perodc boundary condtons,.e., solated systems. Ths mples that the Hamltonan s tme-ndependent, and nvarant upon translaton or rotaton of the whole system. Ths Hamltonan may contan terms accountng for the mean effect of the envronment (e.g., mplct-solvaton term), as long as t stll satsfes the above condtons. The only excepton consdered here (whenever explctly stated) s the possble ncluson of stochastc and frctonal forces as appled n stochastc dynamcs (SD) smulatons, or of random collsonal forces as appled n the stochastccouplng (Andersen) thermostat. Fnally, t should be stressed that the ncluson of geometrcal constrants durng a smulaton affects the statstcal mechancs of the sampled mcrostates [75]. Ths s manly becausen the presenceof such constrants,

5 Thermostat Algorthms 109 Table 1. The eght thermodynamcal ensembles, and the correspondng ndependent and dependent varables. Intensve varables are the chemcal potental for all n speces (µ {µ = 1...n}), the pressure (P) and the temperature (T ). Extensve varables are the number of partcles for all speces (N {N = 1...n}), the volume (V ), and the energy (E), enthalpy (H = E + PV), Hll energy (L = E µ N ), or Ray enthalpy (R = E + PV µ N ). Note that grand-ensembles may be open wth respect to a subset of speces only (e.g., semgrand-canoncal ensemble). The generalzed ensemble s not a physcal ensemble, because ts sze s not specfed (no ndependent extensve varable). Isothermal ensembles are dscussed n many standard textbooks. Specfc references are gven for the (less common) adabatc ensembles Independent Dependent Ensemble NVE µpt Mcrocanoncal [41, 4, 43, 44, 45] NVT µpe Canoncal NPH µvt Isoenthalpc-sobarc [46, 47, 48, 45] NPT µvh Isothermal-sobarc (Gbbs) µvl NPT Grand-mcrocanoncal [49, 45] µvt NPL Grand-canoncal µpr NVT Grand-sothermal-sobarc [50, 45] µpt NVR Generalzed the knetc energy of the system cannot be wrtten n a confguraton-ndependent way (unless the constrants are exclusvely nvolved n fully-rgd atom groups, e.g., rgd molecules). Ths restrcton lmts the valdty of a number of equatons presented n ths artcle. However, many results are expected to reman approxmately vald for systems nvolvng a small proporton of constranted degrees of freedom, and no attempt s made here to derve forms ncludng explctly the effect of geometrcal constrants. Ensembles An solated system s characterzed by a tme-ndependent, translatonally nvarant and rotatonally nvarant Hamltonan. Integraton of the classcal equatons of moton for such a system leads, n the lmt of nfnte samplng, to a trajectory mappng a mcrocanoncal (NVE) ensemble of mcrostates 1. Assumng an nfnte numercal precson, ths s also what a standard MD smulaton wll delver. The laws of classcal mechancs also lead to two addtonal conserved quanttes, namely the lnear momentum p sys of the system, and the angular momentum L sys of 1 Thermodynamcal ensembles are generally defned wthout the constrant of Hamltonan translatonal and rotatonal nvarance, n whch case the prevous statement s not entrely correct. In the present artcle, however, the termnology of Table 1 wll be (loosely) retaned to encompass ensembles where ths nvarance s enforced. The statstcal mechancs of these latter ensembles must be adapted accordngly [76, 77, 78, 79, 80, 81]. Ths requres n partcular the ntroducton of a modfed defnton for the nstantaneous temperature, relyng solely on nternal degrees of freedom and knetc energy (Sect. 3).

6 110 Phlppe H. Hünenberger the system around ts center of mass (CM). In smulatons under perodc boundary condtons, the two quanttes refer to the nfnte perodc system. However, n ths case, f the lnear momentum p box of the computatonal box s also conserved, the correspondng angular momentum L box s not. Ths s because correlated rotatonal moton n two adjacent boxes exert frcton on each other, leadng to an exchange of knetc energy wth the other (nternal) degrees of freedom of the system. Note that the physcal propertes of a molecular system are ndependent of p sys.however, they depend on L sys, because the rotaton of the system leads to centrfugal forces. For ths reason, L sys should be added to the lst of ndependent varables defnng the ensemble sampled. Whenever L sys s not gven, t generally mplctly means that L sys = 0. The use of L sys 0 n smulatons under perodc boundary condtons (overall unform rotaton of the nfnte perodc system) s actually mpossble, because t would lead to non-perodc centrfugal forces. Fnally, t should be specfed that the total energy E of the system s defned here so as to exclude the knetc energy contrbutons correspondng to the overall translaton and rotaton of the system (so that E s ndependent of p sys and L sys ). Because the ndependent varables of the mcrocanoncal ensemble are all extensve, they should be strctly conserved (.e., tme-ndependent) durng the course of a smulaton. The correspondng dependent varables, namely the chemcal potental µ, the pressure P, and the temperature T, are not conserved. In a non-equlbrum smulaton, these quanttes may undergo a systematc drft. In an equlbrum smulaton, the correspondng nstantaneous observables (denoted by ν, P, andt ) wll fluctuate around well-defned average values µ, P,andT. Two mportant comments should be made concernng the prevous statement. Frst, the nstantaneous observables ν, P, andt are not unquely defned. The nstantaneous temperature s generally related to the total knetc energy of the system (Eq. (8)), and the nstantaneous pressure to the total vral and knetc energy. However, alternatve defntons are avalable (dfferng from the above by any quantty wth a vanshng equlbrum average), leadng to dentcal average values n equlbrum stuatons, but to dfferent fluctuatons. Second, a mcrocanoncal ensemble at equlbrum could equally well be specfed by statng that N, V,andE are conserved, and gvng the values of µ nstead of N, P nstead of V,orT nstead of E (as long as at least one extensve varable s specfed). However, such a specfcaton would be rather unnatural as well as napplcable to non-equlbrum stuatons. Furthermore, only the natural varables for defnng a gven thermodynamcal ensemble (Table 1) areether tme-ndependent or characterzed by vanshng fluctuatons n the lmt of a macroscopc system. Fnally, t should be stressed that computer smulatons cannot be performed at nfnte numercal precson. As a consequence, quanttes whch are formally tme-ndependent n classcal mechancs may stll undergo a numercal drft n smulatons. In mcrocanoncal smulatons, ths s typcally the case for E, as well as p sys and L sys (vacuum boundary condtons), or p box (perodc boundary condtons). Unfortunately, the mcrocanoncal ensemble that comes out of a standard MD smulaton does not correspond to the condtons under whch most experments are

7 Thermostat Algorthms 111 carred out. For comparson wth experment, the followng ensembles are more useful, whch nvolve one or more ntensve ndependent varables (Table 1): 1. In the canoncal ensemble (NVT), the temperature has a specfed average (macroscopc) value, whle the nstantaneous observable representng the total energyof the system (.e., the HamltonanH) can fluctuate. At equlbrum, the root-mean-square fluctuatons σ E of the Hamltonan around ts average value E are related to the system sochorc heat capacty, c V, through [16] σ E = H NVT H NVT = k BT c V. (1) The fluctuatons σ T of the nstantaneous temperature T (defned by Eq. (8)) n a canoncal ensemble are gven by [16] σt = T T NVT NVT = N 1 df T, () where N df s the number of nternal degrees of freedom n the system (Eq. (9)). These fluctuatons vansh n the lmt of a macroscopc system, but are often non-neglgble for the system szes typcally consdered n smulatons.. In the sothermal-sobarc (Gbbs) ensemble (NPT), the pressure has (just as the temperature) a specfed average value, whle the nstantaneous volume V of the system can fluctuate. At equlbrum, the root-mean-square fluctuatons σ V of the nstantaneous volume around ts average value V are related to the system sothermal compressblty, β T, through [16] σ V = V NPT V NPT = Vk BTβ T. (3) The root-mean-square fluctuatons σ H of the nstantaneous enthalpy H + PV around ts average value H are related to the system sobarc heat capacty, c P, through [16] σ H = (H + PV) NPT H + PV NPT = k BT c P. (4) Both the nstantaneous temperature T and the nstantaneous pressure P wll fluctuate around ther correspondng macroscopc values, the magntude of these fluctuatons vanshng n the lmt of a macroscopc system. 3. The grand-canoncal ensemble (µvt) has a constant volume and temperature (as the canoncal ensemble), but s open for exchangng partcles wth a surroundng bath. In ths case, the chemcal potental of the dfferent speces has a specfed average, whle the nstantaneous value N of the number of partcles can fluctuate. For a one-component system at equlbrum, the fluctuatons σ N of the nstantaneous number of partcles around ts average value N are related to the system sothermal compressblty, β T, through [16] σ N = N µvt N µvt = N V 1 k B Tβ T. (5)

8 11 Phlppe H. Hünenberger The root-mean-square fluctuatons σ L of the nstantaneous Hll energy H µn around ts average value L are gven by [16] σl = (H µn) ( ) L(µ, V, T ) H µvt µn µvt =k BT. (6) T Three other combnatons of varables are possble (Table 1), but the correspondng ensembles [45] are of more lmted practcal relevance. The last combnaton (generalzed ensemble) s not physcal, because ts sze s not specfed (no ndependent extensve varable). Note that although MD samples the mcrocanoncal ensemble by default, the basc Monte Carlo (MC; [8, 83, 84]) and stochastc dynamcs (SD; [, 3, 4, 5, ]) algorthms sample the canoncal ensemble. Performng a MD smulaton n an other ensemble than mcrocanoncal requres a means to keep at least one ntensve quantty constant (on average) durng the smulaton. Ths can be done ether n a hard or n a soft manner. Applyng a hard boundary condton on an ntensve macroscopc varable means constranng a correspondng nstantaneous observable to ts specfed macroscopc value at every tmepont durng the smulaton (constrant method). Remember, however, that the choce of ths nstantaneous observable s not unque. In contrast, the use of a soft boundary condton allows for fluctuatons n the nstantaneous observable, only requrng ts average to reman equal to the macroscopc value (on a gven tmescale). Typcal methods for applyng soft boundary condtons are the penalty-functon, weak-couplng, extended-system and stochastc-couplng methods [85]. These methods wll be dscussed n the followng sectons n the context of constant-temperature smulatons. Although there are many ways to ensure that the average of an nstantaneous quantty takes a specfed value, ensurng that the smulaton actually samples the correct ensemble (and n partcular provdes the correct fluctuatons for the specfc nstantaneous observable n the gven ensemble) s much more dffcult. µv 3 Thermostat Algorthms A modfcaton of the Newtonan MD scheme wth the purpose of generatng a thermodynamcal ensemble at constant temperature s called a thermostat algorthm. The use of a thermostat can be motvated by one (or a number) of the followng reasons: () to match expermental condtons (most condensed-phase experments are performed on thermostatzed rather than solated systems); () to study temperaturedependent processes (e.g., determnaton of thermal coeffcents, nvestgaton of temperature-dependent conformatonal or phase transtons); () to evacuate the heat n dsspatve non-equlbrum MD smulatons (e.g., computaton of transport coeffcents by vscous-flow or heat-flow smulatons); (v) to enhance the effcency of a conformatonal search (e.g., hgh-temperature dynamcs, smulated annealng); (v)

9 Thermostat Algorthms 113 to avod steady energy drfts caused by the accumulaton of numercal errors durng MD smulatons. The use of a thermostat requres the defnton of an nstantaneous temperature. Ths temperature wll be compared to the reference temperature T o of the heat bath to whch the system s coupled. Followng from the equpartton theorem, the average nternal knetc energy K of a system s related to ts macroscopc temperature T through K = K = 1 k B N df T (7) where k B s Boltzmann s constant, N df the number of nternal degrees of freedom of the system, and K ts nstantaneous nternal knetc energy. Defnng the nstantaneous temperature T at any tmepont as T = k B N df K, (8) one ensures that the average temperature T s dentcal to the macroscopc temperature T. Ths defnton s commonly adopted, but by no means unque. For example, the nstantaneous temperature could be defned based on the equpartton prncple for only a subset of the nternal degrees of freedom. It may also be defned purely on the bass of confguraton, wthout any reference to the knetc energy [87, 88]. In the absence of stochastc and frctonal forces (see below; Eq. (17)), a few degrees of freedom are not coupled (.e., do not exchange knetc energy) wth the nternal degrees of freedom of the system. These external degrees of freedom correspond to the system rgd-body translaton and, under vacuum boundary condtons, rgd-body rotaton. Because the knetc energy assocated wth these external degrees of freedom can take an arbtrary (constant) value determned by the ntal atomc veloctes, they must be removed from the defnton of the system nternal temperature. Consequently, the number of nternal degrees of freedom s calculated as three tmes the total number N of atoms n the system, mnus the number N c of geometrcal constrants,.e. N df = 3N N c N r. (9) The subtracton of constraned degrees of freedom s necessary because geometrcal constrants are characterzed by a tme-ndependent generalzed coordnate assocated wth a vanshng generalzed momentum (.e., no knetc energy). A more formal statstcal-mechancal justfcaton for the subtracton of the external degrees of A thermostat algorthm (nvolvng explct reference to a heat-bath temperature T o ) wll avod systematc energy drfts, because f the nstantaneous temperature s forced to fluctuate wthn a lmted range around T o, the energy wll also fluctuate wthn a lmted range around ts correspondng equlbrum value. To perform long mcrocanoncal smulatons (no thermostat), t s also advsable to employ an algorthm that wll constran the energy to ts reference value E o (ergostat algorthm [86]).

10 114 Phlppe H. Hünenberger freedom n the case of perodc boundary condtons can be found elsewhere [45]. A correspondng dervaton for vacuum boundary condtons has, to our knowledge, never been reported. When stochastc and frctonal forces are appled, as n SD, these forces wll couple the rgd-body translatonal and rotatonal degrees of freedom wth the nternal ones. In ths case all degrees of freedom are consdered nternal to the system. Thus, Eq. (9) s to be used wth N r = 0 n the presence of stochastc and frctonal forces, and otherwse wth N r = 3 under perodc boundary condtons or N r = 6 under vacuum boundary condtons. Smlarly, the nstantaneous nternal knetc energy s defned as K = 1 N m ṙ, =1 where the nternal (also called pecular) veloctes ṙ are obtaned from the real atomc veloctes ṙ o by excludng any component along the external degrees of freedom 3. These corrected veloctes are calculated as ṙ o f N r = 0 ṙ = ṙ o ṙ o CM f N r = 3, (11) ṙ o ṙ o CM I 1 CM (ro ) L o CM (ro r o CM ) f N r = 6 where r o CM s the coordnate vector of the system center of mass (CM), Lo CM the system angular momentum about the CM, and I CM s the (confguraton-dependent) nerta tensor of the system relatve to the CM. The latter quantty s defned as I CM (r) = (10) N m (r r CM ) (r r CM ), (1) =1 where a b denotes the tensor wth elements µ, ν equal to a µ b ν. Applcaton of Eq. (11) ensures that N m ṙ = 0 for N r = 3 or 6 (13) =1 and (rrespectve of the orgn of the coordnate system) N m r ṙ = 0 for N r = 6. (14) =1 Equaton (13) s a straghtforward consequence of the defnton of r o CM. Equaton (14) s proved by usng ω o CM (ro r o CM ) = ṙo ṙ o CM where ωo CM = I 1 CM (ro ) L o CM s the angular velocty vector about the CM. Thus, the lnear and angular momenta of the nternal veloctes vansh, as expected. 3 It s assumed that the veloctes ṙ o are already exempt of any component along possble geometrcal constrants.

11 Thermostat Algorthms 115 Because the nstantaneous temperature s drectly related to the atomc nternal veloctes (Eqs. (8) and(10)), mantanng the temperature constant (on average) n MD smulatons requres mposng some control on the rate of change of these veloctes. For ths reason, thermostat algorthms requre a modfcaton of Newton s second law 4 r (t) = m 1 F (t). (15) In the present context, ths equaton (and the thermostatzed analogs dscussed below) should be vewed as provdng the tme-dervatve of the nternal velocty ṙ defned by Eq. (11). In turn, ṙ s related to the real atomc velocty ṙ o through the nverse of Eq. (11), namely ṙ f N r = 0 ṙ o = ṙ + ṙ CM f N r = 3, (16) ṙ + ṙ CM + I 1 CM (ro ) L CM (ro r o CM ) f N r = 6 where r CM and L CM are constant parameters determned by the ntal veloctes ṙ o (0). Ths dstncton between real and nternal veloctes s often gnored n standard smulaton programs. Many programs completely dsregard the problem, whle others only remove the velocty component along the external degrees of freedom for the computaton of the temperature (but do not use nternal veloctes n the equatons of moton). However, as dscussed n Sect. 4, ths can have very unpleasant consequences n practce. In the followng dscusson, t s assumed that the equaton of moton (Eq. (15) or any thermostatzed modfcaton) s appled to the nternal veloctes defned by Eq. (11), whle the atomc coordnates are propagated smultaneously n tme usng the real veloctes ṙ o defned by Eq. (16). The prototype of most sothermal equatons of moton s the Langevn equaton (asusednsd;seesect.3.),.e. r (t) = m 1 F (t) γ (t)ṙ (t) + m 1 R (t), (17) where R s a stochastc force and γ a (postve) atomc frcton coeffcent. Many thermostats avod the stochastc force n Eq. (17) and use a sngle frcton coeffcent for all atoms. Ths leads to the smplfed form r (t) = m 1 F (t) γ(t)ṙ (t). (18) In ths case, γ loses ts physcal meanng of a frcton coeffcent and s no longer restrcted to postve values. A postve value ndcates that heat flows from the system to the heat bath. A negatve value ndcates a heat flow n the opposte drecton. Note that f Eq. (18) was appled to the real veloctes ṙ o (as often done n smulaton programs) nstead of the nternal veloctes ṙ, the lnear and angular momenta of the system would not be conserved (unless they exactly vansh). 4 It s assumed that the forces F are exempt of any component along possble geometrcal constrants.

12 116 Phlppe H. Hünenberger Any algorthm relyng on the equaton of moton gven by Eq. (18) s smooth (.e., generates a contnuous velocty trajectory) and determnstc 5. It s also tmereversble f γ s antsymmetrc wth respect to tme-reversal 6. Practcal mplementatons of Eq. (18) often rely on the stepwse ntegraton of Newton s second law (Eq. (15)), altered by the scalng of the atomc veloctes after each teraton step. In the context of the leap-frog ntegrator 7 [89], ths can be wrtten ṙ ( t + t ) = λ(t; t) ṙ = λ(t; t) ( t + t ) [ ṙ (t t ] ) + m 1 F (t) t, (0) where λ(t; t) s a tme- and tmestep-dependent velocty scalng factor. Imposng the constrant 8 λ(t; 0) = 1, one recovers Eq. (18) n the lmt of an nfntesmal tmestep t, wth λ(t; t) 1 λ(t; t) γ(t) = lm = t 0 t ( t) t=0. (1) Note that for a gven equaton of moton,.e., a specfed form of γ(t), Eq.(1) does not unquely specfy the scalng factor λ(t; t). It can be shown that Eq. (0) retans the orgnal accuracy of the leap-frog algorthm f the velocty-scalng factor appled to atom s chosen as [90] [ γ (t) λ (t; t) = 1 γ(t) t + + γ(t)f ] (t) ( t). () m ṙ (t) From a thermodynamcal pont of vew, some thermostats can be proved to generate (at constant volume and number of atoms) a canoncal ensemble n the lmt of nfnte samplng tmes (and wthn the usual statstcal-mechancal assumptons of equal aprorprobabltes and ergodcty). More precsely, some thermostats lead to a canoncal ensemble of mcrostates,.e., mcrostates are sampled wth a statstcal 5 The advantages of determnstc algorthms are that () the results can be exactly reproduced (n the absence of numercal errors), and () there are well-defned conserved quanttes (constants of the moton). In the case of Eq. (18), the constant of the moton s t C = K(t) + U(t) + dt K(t)γ (t). (19) 6 0 Consderng a gven mcrostate, tme-reversblty s acheved f the change dt dt (leadng n partcular to r r, ṙ ṙ, and r r) leaves the equaton of moton for the coordnates unaltered (whle the veloctes are reversed). Clearly, ths condton s satsfed for Eq. (18) only f the correspondng change for γ sγ γ. 7 The mplementaton of thermostats wll only be dscussed here n the context of the leapfrog ntegrator. However, mplementaton wth other ntegrators s generally straghtforward. 8 An algorthm wth λ(t; 0) 1 would nvolve a Drac delta functon n ts equaton of moton.

13 Thermostat Algorthms 117 weght proportonal to e βh where β = (k B T o ) 1. In ths case and n the absence of geometrcal constrants, expressng the Hamltonan n Cartesan coordnates as H(r, p) = U(r) + K(p), U beng the potental energy, the probablty dstrbuton of mcrostates may be wrtten ρ(r, p) = e βh(r,p) dr dp e βh(r,p) = e βu(r) dr e βu(r) e βk(p) dp e βk(p). (3) Integratng ths expresson over ether momenta or coordnates shows that the dstrbuton s also canoncal n both confguratons (.e., confguratons are sampled wth a statstcal weght proportonal to e βu ) and momenta (.e., momenta are sampled wth a statstcal weght proportonal to e βk ). In Cartesan coordnates, such a canoncal dstrbuton of momenta reads ρ p (p) = 3N e βk(p) dp e βk(p) = e β(m ) 1 pµ dpµ e β(m ) 1 pµ µ = 3N µ p(p µ ), (4) where Eq. (10) was used together wth p = m ṙ. Notng that p(ṙ µ ) = m p(p µ ) and evaluatng the requred Gaussan ntegral, ths result shows that nternal veloctes obey a Maxwell-Boltzmann dstrbuton,.e., the velocty components ṙ µ appear wth the probablty p(ṙ µ ) = ( ) 1/ βm e (1/)βm ṙµ. (5) π Note that the above statements do not formally hold n the presence of geometrcal constrants, but are generally assumed to provde a good approxmaton n ths case. Some other thermostats only generate a canoncal ensemble of confguratons, but not of mcrostates and momenta. Ths s generally not a serous dsadvantage for the computaton of thermodynamcal propertes, because the contrbuton of momenta to thermodynamcal quanttes can be calculated analytcally (deal-gas contrbuton). Fnally, there also exsts thermostats that generate dstrbutons that are canoncal nether n confguratons nor n momenta. From a dynamcal pont of vew, assessng the relatve merts of dfferent thermostats s somewhat subjectve 9. Clearly, the confguratonal dynamcs of a system wll be affected by the tmescale of ts nstantaneous temperature fluctuatons, and a good thermostat should reproduce ths tmescale at least qualtatvely. However, the drect comparson between expermental thermostats (e.g., a heat bath surroundng 9 An objectve queston, however, s whether the thermostat s able to produce correct tmecorrelaton functons (at least n the lmt of a macroscopc system). Snce transport coeffcents (e.g., the dffuson constant) can be calculated ether as ensemble averages (Ensten formulaton) or as ntegrals of a tme-correlaton functon (Green-Kubo formulaton), at least such ntegrals should be correct f the thermostat leads to a canoncal ensemble. When ths s the case, t has been shown that the correlaton functons themselves are also correct at least for some thermostats [91, 86].

14 118 Phlppe H. Hünenberger a macroscopc system, or the bulk medum around a mcroscopc sample of matter) and thermostats used n smulatons s not straghtforward. The reason s that expermental thermostats, because they nvolve the progressve dffuson of heat from the system surface towards ts center (or nversely), lead to nhomogenetes n the spatal temperature dstrbuton wthn the sample. On the other hand, the thermostats used n smulatons generally modfy nstantaneously and smultaneously the veloctes of all atoms rrespectve of ther locatons, and should lead to an essentally homogeneous temperature dstrbuton. One may nevertheless try to quantfy the tmescale of the temperature fluctuatons to be expected n a thermostatzed smulaton. Ths tmescale can be estmated based on a sem-macroscopc approach [55]. Consder a system characterzed by an average temperature T, n contact wth a heat bath at a dfferent temperature T o.byaverage temperature, t s meant that the quantty T s spacally-averaged over the entre system and tme-averaged over an nterval that s short compared to the expermental tmescale, but long compared to the tme separatng atomc collsons. The dfference between T and T o may result, e.g., from a natural fluctuaton of T wthn the system. From macroscopc prncples, the rate of heat transfer from the heat bath to the the system should be proportonal to the temperature dfference T o T and to the thermal conductvty κ of the system. Thus, the rate of change n the average temperature can be wrtten (at constant volume) T (t) = cv 1 E(t) = ζt 1 [T o T (t)] (6) wth the defnton ζ T = ξ 1 V 1/3 c v κ 1, (7) where c v s the system sochorc heat capacty, V the system volume, and ξ admensonless constant dependng on the system shape and on the temperature nhomogenety wthn the system. For a gven system geometry (e.g., sphercal) and ntal temperature dstrbuton (.e., T (x, 0)),a reasonable value for ξ could n prncple be estmated by solvng smultaneously the flux equaton J(x, t) = κ T (x, t), (8) where J(x, t) s the energy flux through a surface element perpendcular to the drecton of the vector, and the conservaton equaton T (x, t) t = Vc 1 v J(x, t). (9) Eq. (6) mples that, at equlbrum, the natural fluctuatons of T away from T o decay exponentally wth a temperature relaxaton tme ζ T,.e. 1 ζ T (t) = T o + [T (0) T o ] e T t. (30) Note that on a very short tmescale (.e., of the order of the tme separatng atomc collsons), the nstantaneous temperature T (t) s also affected by mportant stochastc varatons (see Sect. 3.). Only on an ntermedate tmescale does the mean effect

15 Thermostat Algorthms 119 of these stochastc fluctuatons result n an exponental relaxaton for T (t).however, because stochastc varatons contrbute sgnfcantly to the nstantaneous temperature fluctuatons, a thermostat based solely on an exponental relaxaton for T (t) leads to ncorrect (underestmated) temperature fluctuatons (see Sect. 3.5). To summarze, although assessng whether one thermostat leads to a better descrpton of the dynamcs compared to another one s largely subjectve, t seems reasonable to assume that: () thermostats permttng temperature fluctuatons are more lkely to represent the dynamcs correctly compared to thermostats constranng the temperature at a fxed value; () thermostats wth temperature fluctuatons are more lkely to represent the dynamcs correctly when these fluctuatons occur at a tmescale (measured n a smulaton, e.g., as the decay tme of the temperature autocorrelaton functon) of the order of ζ T (Eq. (6)), and when the dynamcs s smooth (contnuous velocty trajectory). These dfferences wll be more sgnfcant for small systems, where the temperature fluctuatons are of of larger magntudes (the correspondng root-mean-square fluctuatons scale as N 1/,seeEq.()) and hgher frequences (the correspondng relaxaton tmes scale as N 1/3,seeEq.(7)). A summary of the common thermostats used n MD smulatons, together wth ther man propertes, s gven n Table. The varous algorthms are detaled n the followng sectons. 3.1 Temperature n the Monte Carlo Algorthm Although the present dscusson manly focuses on thermostatzed MD, the smplest way to generate a thermodynamcal ensemble at constant temperature s to use the MC algorthm [8, 83, 84]. Ths algorthm does not nvolve atomc veloctes or knetc energy. Random tral moves are generated, and accepted wth a probablty p = mn{e β U, 1} (31) dependng on the potental energy change U assocated wth the move and on the reference temperature T o. Followng ths crteron, moves nvolvng rgd-body translaton and, under vacuum boundary condtons, rgd-body rotaton are always accepted because they do not change the potental energy. For ths reason, the correspondng degrees of freedom are external to the system. Note also that under vacuum boundary condtons, the centrfugal forces due to the rgd-body rotaton of the system, whch would be ncluded n a MD smulaton, are absent n the MC procedure. Therefore, MC samples by default an ensemble at zero angular momentum. It can be shown that the ensemble generated by the MC procedure represents (at constant volume) a canoncal dstrbuton of confguratons. The modfcaton of the MC scheme to sample other sothermal ensembles (ncludng the grand-canoncal ensemble [9]) s possble. Modfcatons permttng the samplng of adabatc ensembles (e.g., the mcrocanoncal ensemble [9, 93, 94]) have also been devsed. The MC procedure s non-smooth, non-determnstc, tme-rreversble, and does not provde any dynamcal nformaton.

16 10 Phlppe H. Hünenberger Table. Characterstcs of the man thermostat algorthms used n MD smulatons. MD: molecular dynamcs (generates a mcrocanoncal ensemble, only shown for comparson); MC: Monte Carlo (Sect. 3.1); SD: stochastc dynamcs (wth γ > 0 for at least one atom; Sect. 3.); A: MD wth Andersen thermostat (wth α>0; Sect. 3.3); HE: MD wth Hoover-Evans thermostat (Sect. 3.4); W: MD wth Woodcock thermostat (Sect. 3.4); HG: MD wth Hale-Gupta thermostat (Sect. 3.4); B: MD wth Berendsen thermostat (wth t <τ B < ; Sect. 3.5); NH: MD wth Nosé-Hoover thermostat (wth 0 < Q < ; Sect. 3.6). MD s a lmtng case of SD (wth γ = 0 for all atoms), A (wth α = 0), B (wth τ B ), and NH (wth Q, γ(0) = 0). HE/W s a lmtng cases of B (wth τ B = t) and s a constraned form of NH. HG s also a constraned form of NH. Determnstc: trajectory s determnstc; Tme-reversble: equaton of moton s tme-reversble; Smooth: velocty trajectory s avalable and contnuous. Energy drft: possble energy (and temperature) drft due to accumulaton of numercal errors; Oscllatons: possble oscllatory behavor of the temperature dynamcs; External d.o.f.: some external degrees of freedom (rgd-body translaton and, under vacuum boundary condtons, rotaton) are not coupled wth the nternal degrees of freedom. Constraned K: no knetc energy fluctuatons; Canoncal n H: generates a canoncal dstrbuton of mcrostates; Canoncal n U: generates a canoncal dstrbuton of confguratons. Dynamcs: dynamcal nformaton on the system s ether absent ( ) or lkely to be unrealstc ( ; constraned temperature or non-smooth trajectory), moderately realstc (+; smooth trajectory, but temperature fluctuatons of ncorrect magntude), or realstc (++; smooth trajectory, correct magntude of the temperature fluctuatons). The latter apprecaton s rather subjectve and depends on an adequate choce of the adjustable parameters of the thermostat MD MC SD A HE W HG B NH Determnstc Tme reversble Smooth Energy drft + + Oscllatons + External d.o.f Constraned K Canoncal n H Canoncal n U Dynamcs Eqn. of moton ,79 3. Temperature Relaxaton by Stochastc Dynamcs The SD algorthm reles on the ntegraton of the Langevn equaton of moton [, 3, 95, 96, 97, 98, 99, 100]asgvenbyEq.(17). The stochastc forces R (t) have the followng propertes 10 : () they are uncorrelated wth the veloctes ṙ(t ) and systematc forces F (t ) at prevous tmes t < t; () ther tme-averages are zero; () ther mean-square components evaluate to m γ k B T o ; (v) the force component R µ (t) 10 More complex SD schemes can be used, whch ncorporate tme or space correlatons n the stochastc forces. It s also assumed here that the frcton coeffcents γ are tmendependent.

17 Thermostat Algorthms 11 along the Cartesan axs µ s uncorrelated wth any component R jν (t ) along axs ν unless = j, µ = ν, andt = t. The two last condtons can be combned nto the relaton R µ (t)r jν (t ) =m γ k B T o δ j δ µν δ(t t). (3) It can be shown that a trajectory generated by ntegratng the Langevn equaton of moton (wth at least one non-vanshng atomc frcton coeffcent γ ) maps (at constant volume) a canoncal dstrbuton of mcrostates at temperature T o. The Langevn equaton of moton s smooth, non-determnstc and tmerreversble. Under vacuum boundary condtons and amng at reproducng bulk propertes, t may produce a reasonable pcture of the dynamcs f the mean effect of the surroundng solvent s ncorporated nto the systematc forces, and f the frcton coeffcents are representatve of the solvent vscosty (possbly weghted by the solvent accessblty). If SD s merely used as a thermostat n explct-solvent smulatons, as s the case, e.g., when applyng stochastc boundary condtons to a smulated system [101, 11, 1], some care must be taken n the choce of the atomc frcton coeffcents γ. On the one hand, too small values (loose couplng) may cause a poor temperature control. Indeed, the lmtng case of SD where all frcton coeffcents (and thus the stochastc forces) are set to zero s MD, whch generates a mcrocanoncal ensemble. However, arbtrarly small atomc frcton coeffcents (or even a non-vanshng coeffcent for a sngle atom) are suffcent to guarantee n prncple the generaton of a canoncal ensemble. But f the frcton coeffcents are chosen too low, the canoncal dstrbuton wll only be obtaned after very long smulaton tmes. In ths case, systematc energy drfts due to accumulaton of numercal errors may nterfere wth the thermostatzaton. On the other hand, too large values of the frcton coeffcents (tght couplng) may cause the large stochastc and frctonal forces to perturb the dynamcs of the system. In prncple, the perturbaton of the dynamcs due to stochastc forces wll be mnmal when the atomc frcton coeffcents γ are made proportonal to m. In ths case, Eqs. (17) and(3) show that the root-mean-square acceleraton due to stochastc forces s dentcal for all atoms. In practce, however, t s often more convenent to set the frcton coeffcents to a common value γ. The lmtng case of SD for very large frcton coeffcents (.e., when the acceleraton r can be neglected compared to the other terms n Eq. (17)) s Brownan dynamcs (BD), wth the equaton of moton ṙ (t) = γ 1 m 1 [F (t) + R (t)]. (33) Although the magntude of the temperature fluctuatons s n prncple not affected by the values of the frcton coeffcents (unless they are all zero), the tmescale of these fluctuatons strongly depends on the γ coeffcents. In fact, t can be shown [54] that there s a close relatonshp between the frcton coeffcents n SD (used as a mere thermostat) and the temperature relaxaton tme ζ T n Eq. (6). Consder the case where all coeffcents γ are set to a common value γ. Followng from Eqs. (8)and(10), the change T of the nstantaneous temperature over a tme nterval from t = 0to τ can be wrtten

18 1 Phlppe H. Hünenberger T = k B N df τ 0 dt K(t) = k B N df Insertng Eq. (17), ths can be rewrtten T = N { τ dt [ F (t) γ m ṙ (t) ] ṙ (t) + k B N df R (t) =1 0 0 [ ṙ (0) + t 0 N τ m dt r (t) ṙ (t). (34) =1 0 τ 0 dt (35) [ dt m 1 F (t ) γ ṙ (t ) + m 1 R (t )] ]}. Usng Eq. (3) and the fact that the stochastc force s uncorrelated wth the veloctes and systematc forces at prevous tmes, ths smplfes to T = N k B N df { τ dt [F (t) ṙ (t) γ m ṙ (t)] =1 0 τ t + m 1 dt R (t) dt R (t )} τ 0 = N dt [F (t) ṙ (t) γ m ṙ k B N (t)] df = Ndf 1 Nγ T o τ. (36) Ths expresson can be rewrtten 11 T τ = k B N df N F ṙ + γ [T o T ], (37) =1 where F ṙ and T stand for averages over the nterval τ. The frst term represents the temperature change caused by the effect of the systematc forces, and would be unaltered n the absence of thermostat (Newtonan MD smulaton). Thus, the second term can be dentfed wth a temperature change arsng from the couplng to a heat bath. Ths means that on an ntermedate tmescale (as defned at the end of Sect. 3), the mean effect of thermostatzaton can be wrtten T (t) = γ [T o T (t)]. (38) Comparng wth Eq. (6) allows to dentfy γ wth the nverse of the temperature relaxaton tme ζ T n Eq. (6),.e., to suggest γ = (1/)ζT 1 as an approprate value for smulatons. Ths dscusson also shows that the sem-macroscopc expresson of Eq. (6) s only vald on an ntermedate tmescale, when the stochastc fluctuatons occurng on a shorter tmescale (.e., of the order of the tme separatng atomc collsons) are averaged out and only ther mean effect s retaned. 11 In the absence of constrants N df = 3N due to Eq. (9) wth N c = 0andN r = 0(as approprate for SD). In the presence of constrants, the dervaton should nclude constrant forces.

19 Thermostat Algorthms Temperature Relaxaton by Stochastc Couplng The stochastc-couplng method was proposed by Andersen [55]. In ths approach, Newton s equaton of moton (Eq. (15)) s ntegrated n tme, wth the modfcaton that at each tmestep, the velocty of all atoms are condtonally reassgned from a Maxwell-Boltzmann dstrbuton. More precsely, f an atom s selected for a velocty reassgnment, each Cartesan component µ of the new velocty s selected at random accordng to the Maxwell-Boltzmann probablty dstrbuton of Eq. (5). The selecton procedure s such that the tme ntervals τ between two successve velocty reassgnments of a gven atom are selected at random accordng to a probablty p(τ) = αe ατ,whereα s a constant reassgnment frequency. In prncple, one can select at random and for each atom a seres of successve τ-values (obeyng the specfed probablty dstrbuton) before startng the smulaton. Ths seres s then used to determne when the partcle s to undergo a velocty reassgnment. In practce, a smpler procedure can be used when t α 1 (nfrequent reassgnments). At each tmestep and for each atom n turn, one generates a random number between 0 and 1. If ths number s larger than α t for a gven atom, ths atom undergoes a velocty reassgnment. Ths procedure leads to a probablty dstrbuton p(τ) t = (1 α t) τ/ t α t (39) for the ntervals τ wthout velocty reassgnment. Ths mples ln α 1 p(τ) = (τ/ t) ln(1 α t) = ατ + O[(α t) ]. (40) Thus, when t α 1, p(τ) = αe ατ, as expected. If the condton s not satsfed, ths second method wll not work because the probablty of multple reassgnments wthn the same tmestep becomes non-neglgble. The equaton of moton for the Andersen thermostat can formally be wrtten ( ) n r (t) = m 1 [ṙ F (t) + δ t τ,m,n (t) ṙ (t) ], (41) n=1 m=1 where {τ,n n = 1,,...} s the seres of ntervals wthout reassgnment for partcle, andṙ,n the randomly-reassgned velocty after the nth nterval. Ths approach mmcks the effect rregularly-occurrng stochastc collsons of randomly chosen atoms wth a bath of fcttous partcles at a temperature T o. Because, the system evolves at constant energy between the collsons, ths method generates a successon of mcrocanoncal smulatons, nterrupted by small energy jumps correspondng to each collson. It can be shown [55] that the Andersen thermostat wth non-zero collson frequency α leads to a canoncal dstrbuton of mcrostates. The proof [55] nvolves smlar arguments as the dervaton of the probablty dstrbuton generated by the MC procedure. It s based on the fact that the Andersen algorthm generates a Markov chan of mcrostates n phase space. The only requred assumpton s that every mcrostate s accessble from every other one wthn a fnte tme (ergodcty). Note

20 14 Phlppe H. Hünenberger also that the system total lnear and angular momenta are affected by the velocty reassgnments, so that these degrees of freedom are nternal to the system, as n SD. The Andersen algorthm s non-determnstc and tme-rreversble. Moreover, t has the dsadvantage of beng non-smooth,.e., generatng a dscontnuous velocty trajectory where the randomly-occurrng collsons may nterfere wth the natural dynamcs of the system. Some care must be taken n the choce of the collson frequency α [55, 10]. On the one hand, too small values (loose couplng) may cause a poor temperature control. The same observatons apply here as those made for SD. The lmtng case of the Andersen thermostat wth a vanshng collson frequency s MD, whch generates a mcrocanoncal ensemble. Arbtrarly small collson frequences are suffcent to guarantee n prncple the generaton of a canoncal ensemble. But f the collson frequency s too low, the canoncal dstrbuton wll only be obtaned after very long smulaton tmes. In ths case, systematc energy drfts due to accumulaton of numercal errors may nterfere wth the thermostatzaton. On the other hand, too large values for the collson frequency (tght couplng) may cause the velocty reassgnments to perturb heavly the dynamcs of the system. Although the magntude of the temperature fluctuatons s n prncple not affected by the value of the collson frequency (unless t s zero), the tmescale of these fluctuatons strongly depends on ths parameter. In fact, t can be shown [55] that there s a close relatonshp between the collson frequency and the temperature relaxaton tme ζ T n Eq. (6). Each collson changes the knetc energy of the system by (3/)k B [T o T (t)] onaverage, and there are Nα such collsons per unt of tme. Thus, one expects T (t) = cv 1 E(t) = (3/)cv 1 Nαk B [T o T (t)], (4) where T and E stand for averages over an ntermedate tmescale (as defned at the end of Sect. 3). Comparng wth Eq. (6) allows to dentfy (3/)cv 1 Nαk B wth the nverse of the temperature relaxaton tme ζ T n Eq. (6),.e., to suggest α = (/3)(Nk B ) 1 c v ζt 1 as an approprate value for smulatons. Note that because ζ T scales as N 1/3 (Eq. (7)), the collson frequency for any partcle scales as N /3, so that the tme each partcle spends wthout reassgnment ncreases wth the system sze. On the other hand, the collson frequency for the whole system, Nα, scales as N 1/3, so that the length of each mcrocanoncal sub-smulaton decreases wth the system sze. 3.4 Temperature Constranng Temperature constranng ams at fxng the nstantaneous temperature T to the reference heat-bath value T o wthout allowng for any fluctuatons. In ths sense, temperature constranng represents a hard boundary condton, n constrast to the soft boundary condtons employed by all other thermostats mentoned n ths artcle. Note that constranng the temperature,.e., enforcng the relaton T (t) = T o (or Ṫ (t) = 0) represents a non-holonomc constrant. Holonomc constrants are those