arX

iv:1

204.

0822

v1 [

phys

ics.

com

p-ph

] 3

Apr

201

2

Colored-noise thermostats a la carte∗

Michele Ceriotti† and Michele ParrinelloComputational Science, Department of Chemistry and Applied Biosciences,

ETH Zurich, USI Campus, Via Giuseppe Buffi 13, CH-6900 Lugano, Switzerland

Giovanni BussiUniversita di Modena e Reggio Emilia and INFM-CNR-S3, via Campi 213/A, 41100 Modena, Italy

(Dated: October 31, 2018)

Recently, we have shown how a colored-noise Langevin equation can be used in the context ofmolecular dynamics as a tool to obtain dynamical trajectories whose properties are tailored to displaydesired sampling features. In the present paper, after having reviewed some analytical results forthe stochastic differential equations forming the basis of our approach, we describe in detail theimplementation of the generalized Langevin equation thermostat and the fitting procedure usedto obtain optimal parameters. We discuss in detail the simulation of nuclear quantum effects, anddemonstrate that, by carefully choosing parameters, one can successfully model strongly anharmonicsolids such as neon. For the reader’s convenience, a library of thermostat parameters and somedemonstrative code can be downloaded from an on-line repository.

Stochastic differential equations (SDE) have been usedto model the time evolution of processes characterized byrandom behavior, in fields as diverse as physics and eco-nomics. In particular, the Langevin equation (LE) hasbeen regularly applied in the study of Brownian motion,and has been used extensively in molecular dynamics(MD) computer simulations as a convenient and efficienttool to obtain trajectories which sample the constant-temperature, canonical ensemble1,2.

In its original form, the Langevin equation is basedon the assumption of instantaneous system-bath interac-tions, which correspond to the values of the random forcebeing uncorrelated at different times. A non-Markovian,generalized version of the LE arises in the context ofMori-Zwanzig theory3,4. In this theory, if one consid-ers a harmonic system coupled with a harmonic bath,it is possible to integrate out the degrees of freedom ofthe bath. This leaves one with a linear stochastic equa-tion where both the friction and the noise have a finitememory. The conventional LE is recovered in the limit ofa clear separation between the characteristic time-scaleof the system’s dynamics and that of the system-bathinteraction.

This class of non-Markovian SDEs has been extensivelyused to model the dynamics of open systems interactingwith a physically-relevant bath (see e.g. Refs.5–7). In-stead, our recent works8,9 have used colored(correlated)-noise SDEs as a device to sample efficiently statisticaldistributions in molecular-dynamics (MD) simulations.These works aimed to show how a stochastic thermo-stat suitable for Car-Parrinello-like dynamics8 could beconstructed, and to include nuclear quantum effects in alarge class of problems at a fraction of the cost of path-integrals calculations9. In these applications the real dy-

∗Reprinted with permission from J. Chem. Theory Comput., 2010,6 (4), pp 11701180. Copyright 2010 American Chemical Society.

namics is lost, and one focuses only on the efficient cal-culation of static ensemble averages.In this Paper we discuss the practical implementation

of the generalized Langevin equation (GLE) thermostatthat we used in the two cases mentioned above. Wealso provide the reader with the analytical and numeri-cal tools needed to construct SDEs tailored to their ownsampling needs. Throughout we take advantage of thedimensional reduction scheme, which allows to exploitthe equivalence between a non-Markovian dynamics anda Markovian dynamics in higher dimensionality. In doingthis, we supplement the physical coordinates with addi-tional degrees of freedom4, whose equations of motionare taken as linear, so as to simplify the formalism andanalytical derivations.In the Appendices we recall some of the properties of

multidimensional stochastic processes5,10–12, which areuseful to our discussion, and present a short comparisonof the GLE thermostat and the widely used massive Nose-Hoover chains13–16. A simple FORTRAN90 code imple-menting our method to the dynamics of an harmonic os-cillator and a library of optimized thermostat parameterscan be downloaded from an on-line repository17.

I. GENERALIZED LANGEVIN THERMOSTAT

A. Markovian and non-Markovian formulations

The Langevin equation for a particle with position qand momentum p, subject to a potential V (q), can bewritten as

q =p

p =− V ′(q)− appp+ bppξ(t).(1)

where ξ(t) represent an uncorrelated, Gaussian-distributed random force with unitary variance and zeromean [〈ξ〉 = 0, 〈ξ(t)ξ(0)〉 = δ(t)]. Here and what fol-lows we use mass-scaled coordinates. Furthermore, for

2

consistency, the friction coefficient (usually denoted byγ) is here given the symbol app, while bpp is the intensityof the random force. In this notation, the fluctuation-dissipation theorem (FDT) reads b2pp = 2appkBT . If thisrelation holds, the dynamics generated by Eq. (1) willsample the canonical ensemble at temperature T 4,18.As explained in the Introduction, in order to bypass the

complexity of dealing with a non-Markovian formulationdirectly, we supplement the system with n additional de-grees of freedom s = si which are linearly coupled tothe physical momentum and between themselves. Theresulting SDE can be cast into the compact form

q =p(

ps

)

=

(

−V ′(q)0

)

−

(

app aTpap A

)(

ps

)

+

(

bpp bTp

bp B

)(

ξ

)

,

(2)

Here, ξ is a vector of n+1 uncorrelated Gaussian randomnumbers, with 〈ξi (t) ξj (0)〉 = δijδ (t). Clearly, Eq. (1)is recovered when n = 0. For an harmonic potentialV (q) = 1

2ω2q2, Eqs. (2) are linear, and an Ornstein-

Uhlenbeck process is recovered whose time propagationcan be evaluated analytically. In the non-linear case onecan use the Trotter-decomposition to split the dynamicsinto a linear part, which evolves the (p, s) momenta, and anon-linear part, which evolves the Hamilton equations19.This is facilitated by the fact that the dynamics of (p, s)alone is linear, and its exact finite-time propagator canbe analytically evaluated (see Subsection I E).Here and in the rest of the paper, we adopt the same

notation introduced in Ref.9 to distinguish between ma-

trices acting on the full state vector x = (q, p, s)T or onparts of it, as illustrated below:

q p s

q mqq mqp mTq

p mqp mpp mTp

s mq mp M

Mp

Mqp

(3)

The Markovian dynamical equations (2) are equivalentto a non-Markovian process for the physical variablesonly. This is best seen by first considering the evolu-tion of the (p, s) variables in the free-particle analogueof Eqs. (2). The additional degrees of freedom s can beintegrated away, and one is left with (see Ref.4 and Ap-pendix A)

p = −

∫ t

−∞

K(t− s)p(s)ds+ ζ(t) (4)

where the memory kernel K(t) is related to the elementsof Ap by

K(t) = 2appδ(t)− aTp e−|t|Aap. (5)

Based on the fact that the the free-particle dynamicsof (p, s) is an OU process, one also finds than the re-lationship between the static covariance matrix Cp =

⟨

(p, s)T (p, s)⟩

, the drift matrixAp and the diffusion ma-

trix Bp is given by:42

ApCp +CpATp = BpB

Tp . (6)

In Appendix A we show that setting Cp = kBT is suf-ficient to satisfy the FDT. In this case, Eq. (6) fixes Bp

once Ap is given. FDT also implies that the colored-noise autocorrelation function H(t) = 〈ζ(t)ζ(0)〉 is equalto kBTK(t), whereas the more complex relation betweenK(t) and H(t), valid in the general case, is reported inEq. (A5).Since there is no explicit coupling between the position

q and the additional momenta s, one can check that ex-actly the same dimensional reduction can be performedin the case of an arbitrary potential coupling p and q, andthat Eqs. (2) correspond to the non-Markovian process

q = p

p = −∂V

∂q−

∫ t

−∞

K(t− s)p(s)ds+ ζ(t).(7)

In the memory kernel (5), A can be chosen to be a generalreal matrix, and can have complex eigenvalues, providedthey have a positive real part. This results in a K(t) thatis a linear combination of exponentially damped oscilla-tions. Therefore, a vast class of non-Markovian dynamicscan be represented by Markovian equations such as (2).

B. Exact solution in the harmonic limit

The thermostats typically used in MD simulations havea few parameters, that are chosen by trial and error. Athermostat based on Eqs. (2) depends on a much largernumber of parameters, and hence the fitting procedure ismore complex. It is therefore important to find ways tocompute a priori analytical estimates so as to guide thetuning of the thermostat.To this end, we examine the harmonic oscillator, which

is commonly used to model physical and chemical sys-tems. By choosing V (q) = 1

2ω2q2 the force term in (2)

becomes linear, and the dynamics of x = (q, p, s)T is theOU process x = −Aqpx+Bqpξ. In Eqs. (2) the s degreesof freedom are coupled to the momentum only. There-fore, most of the additional entries in Aqp and Bqp arezero, and the equations for x read

qps

= −

0 −1 0

ω2 app aTp0 ap A

qps

+

0 0 0

0Bp0

0

ξ

. (8)

The exact finite-time propagator for Eqs. (8) can becomputed, and so it is possible to obtain any ensem-ble average or time-correlation function analytically. Ofcourse, one is most interested in the expectation valuesof the physical variables q and p. In particular, one canobtain the fluctuations

⟨

q2⟩

and⟨

p2⟩

and correlation

3

functions of the form⟨

q2(t)q2(0)⟩

, which can be usedto measure the coupling between the thermostat and thesystem. The resulting expressions are simple to evaluatebut lengthy, and we refer the reader to Appendix B fortheir explicit form.

One can envisage, using the estimates computed for anoscillator of frequency ω, to predict and hence optimizethe response of a normal mode of a similar frequency inthe system being studied. Furthermore, thanks to theproperties of Eq. (8), one does not need to perform anormal-modes analysis to turn this idea into a practi-cal method. Consider indeed a perfect harmonic crystal,and apply an independent instance of the GLE thermo-stat to the three cartesian coordinates of each atom. Itis easy to see that, since Eq. (8) is linear, and containsGaussian noise, the thermostatted equations of motionare invariant under any orthogonal transformation of thecoordinates. Therefore, the resulting dynamics can bedescribed on the basis of the normal modes just as in or-dinary Hamiltonian lattice dynamics. As a consequence,each phonon will respond independently as a 1-D oscilla-tor with its own characteristic frequency. Thus, to tunethe GLE thermostat, one only needs the analytical resultsin the one-dimensional case, evaluated as a function of ω.The parameters can then be optimized for a number ofdifferent purposes, based solely on minimal informationon the vibrational spectrum of the system under inves-tigation, without any knowledge of the phonons eigen-modes.

The invariance properties of the GLE thermostat leadto additional advantages. For instance, we can contrastits behavior with that of Nose-Hoover (NH) chains, basedon equations which are quadratic in p (see Appendix C).As a consequence of the nonlinearity, the efficiency of anNH chains thermostat for a multidimensional oscillatordepends on the orientation of the eigenmodes relative tothe cartesian axes, an artefact which is absent in our case.

Having set the background, we now turn to the de-scription of the various applications of Eqs. (2).

C. Efficient canonical sampling

We first discuss the design of a GLE which can op-timally sample phase space. In this case, the targetstationary distribution is the canonical ensemble, so theequations of motion need to satisfy the detailed-balancecondition. Still, there is a great deal of freedom availablein the choice of the autocorrelation kernel or, equiva-lently, in the choice of Ap and Bp matrices. These freeparameters can be used to optimize the sampling effi-ciency. To this end, we must first define an appropriatemerit function. Standard choices are the autocorrelationtimes of the potential and total energy (V and H respec-

tively):

τV =1

〈V 2〉

∫ ∞

0

〈(V (t)− 〈V 〉)(V (0)− 〈V 〉)〉dt

τH =1

〈H2〉

∫ ∞

0

〈(H(t)− 〈H〉)(H(0)− 〈H〉)〉 dt.

(9)

In the harmonic case, these can be readily computed interms of correlation times of q2 and p2 (see Appendix B),and will depend on Ap and the oscillator’s frequencyω. For example, one easily finds that in the white-noiselimit, with no additional degrees of freedom as in Eq. (1),

τH (ω) =1

app+

app4ω2

, τV (ω) =1

2app+

app2ω2

, . (10)

Both response times are constant in the high-frequencylimit, and increase quadratically in the low-frequency ex-treme of the spectrum. For a given frequency one canchoose app so as to minimize the correlation time - thusenhancing sampling. It should be noted that Eqs. (10)contain a “trivial” dependence on ω, as one expects thatsampling a normal mode would require at least a timeof the order of its vibrational period. One can thus de-fine a renormalized κ(ω) = [τ(ω)ω]

−1as a measure of

the efficiency of the coupling. In the white-noise case,κ = 1 for the optimally-coupled frequency (ωH = app/2and ωV = app, respectively), and decreases linearly forlower and higher values of ω.While this result in itself provides a guide to choose

a good value of the friction coefficient in conventional(white-noise) Langevin dynamics, we can enhance thevalue of κ(ω) over a broader frequency range, by using acolored-noise SDE. If we want to obtain canonical sam-pling, the FDT has to hold, so that Cp = kBT . Wetherefore consider the entries of Ap as the only indepen-dent parameters, since Bp is then determined by Eq. (6).In practice, we set up a fitting procedure, in which

we choose a set of frequencies ωi, distributed over abroad range (ωmin, ωmax). For an initial guess for thethermostat matrix Ap we compute κ(ω) for each ofthese frequencies. We then vary Ap, so as to optimizemini κ(ωi), and aim at a sampling efficiency on the range(ωmin, ωmax) which is as high and frequency-independentas possible. We will discuss this fitting procedure in moredetail in Section II.In Figure 1 we compare the optimized κ(ω) for differ-

ent frequency ranges and number of additional degrees offreedom. We find empirically that κ(ω) = 1 is the bestresult which can be attained, and that nearly-optimalefficiency can be reached over a very broad range of fre-quencies. This constant efficiency decreases slightly asthe fitted range is extended, regardless of the numbern of si employed. For a given frequency range, however,increasing n has the effect of making the response flatter.Clearly this scheme will work optimally in harmonic

or quasi-harmonic systems, and anharmonicity will in-troduce deviations from the predicted behavior. In theextreme case of diffusive systems such as liquids, one has

4

10-4 10-3 10-2 10-1 11 101 102 103 10410

10

10

110

10

10

1

optimize10

10

10

1

Ω @arb.unitsD

Κ=

1Τ

VΩ

FIG. 1: Sampling efficiency as estimated from Eq. (9) for anharmonic oscillator, plotted as a function of the frequency ω.The κ(ω) curve for a white-noise Langevin thermostat opti-mized for ω = 1 [black, dotted lines, Eq.(10)] is contrastedwith those for a set of optimized GLE thermostats. The pan-els, from bottom to top, contain the results fitted respectivelyover a frequency range spanning two, four and six orders ofmagnitudes around ω = 1. Dark, continuous lines correspondto matrices with n = 4, and dashed, lighter lines to n = 2.The GLE curves correspond to the sets of parameters kv_4-2,kv_2-2, kv_4-4, kv_2-2, kv_4-6, kv_2-6, which can be down-loaded from an on-line repository17.

to ask the question of how much diffusion will be affectedby the thermostat, especially since in an overdampedLE equation the diffusive modes are considerably sloweddown (see e.g. Ref.20). To estimate the impact of thethermostat on the diffusion, we define the free-particlediffusion coefficient D∗ as that calculated switching offthe physical forces. Its value when a GLE thermostat isused is

mD∗

kBT=

1

〈p2〉

∫ ∞

0

〈p(t)p(0)〉dt =

=[

A−1p

]

pp=

(

app − aTp A−1ap

)−1.

(11)

where we have assumed the FDT to hold. In practicalcases, if an estimate of the unthermostated (intrinsic)diffusion coefficient D is available, one should choose thematrix Ap in such a way that D∗ ≫ D, so that the ther-mostat will not behave as an additional bottleneck fordiffusion. Equation (11) has the interesting consequencethat D∗ can be enhanced either by reducing the over-all strength of the noise, as in white-noise LE, but alsoby carefully balancing the terms in the denominator ofEq. (11).We have found empirically that for an Ap matrix

fitted to harmonic modes over the frequency range(ωmin, ωmax), the diffusion coefficient computed by (11)

FIG. 2: A cartoon representing a two-thermostat setup, whichwe take as the simplest example of a stochastic process violat-ing the fluctuation-dissipation theorem. If the relaxation timeversus frequency curves for the two thermostats are different,a steady-state will be reached in which normal modes corre-sponding to different frequencies will equilibrate at differenteffective temperatures.

is D∗ ≈ kBT/ωmin. This latter expression gives a usefulrecipe for choosing the minimal frequency to be consid-ered when fitting a GLE thermostat for a system whosediffusion coefficient can be roughly estimated.

D. Frequency-dependent thermostatting

The ability to control the strength of the thermostat-system coupling as a function of the frequency – demon-strated above – points quite naturally at more sophis-ticated applications. For instance, one can apply twothermostats with distinct target temperatures and dif-ferent efficiencies κ(ω) (see Figure 2). Obviously, sucha simulation is not an equilibrium one, since energy issystematically injected in some modes and removed fromothers, but leads to a steady state that has useful prop-erties. Indeed, the normal modes will couple differentlyto the two thermostat, so that the effective temperatureof each mode can be controlled as a function of ω. Thistwo-thermostats example is just an instance of a broaderclass of stochastic processes, for whom the FDT is vi-olated. In general, we can relax the assumption thatCp = kBT , and for a given drift matrix we can choose aBp which is suitable to our purpose.Returning to the harmonic oscillator case, one can

solve exactly the dynamics for a given choice of Ap, Bp

and frequency ω. The resulting dynamics is performedin the n + 2-dimensional space defined by the variables(q, p, s), according to Eq. (8). For a compact notation,we used the full matrices Aqp and Bqp. The full Cqp(ω),

5

which defines the stationary distribution in the steadystate, can be computed solving an equation analogousto (6):

AqpCqp +CqpATqp = BqpB

Tqp (12)

One can tune the free parameters (Ap and Bp) so as tomake the cqq(ω) and cpp(ω) elements of the extended co-variance matrix as close as possible to the desired targetfunctions

⟨

q2⟩

(ω) and⟨

p2⟩

(ω).

In a previous paper9 we applied this method to ob-tain

⟨

q2⟩

(ω) and⟨

p2⟩

(ω) in agreement with the valuesappropriate for a quantum harmonic oscillator, and ob-tained a good approximation to the quantum-correctedstructural properties in quasi-harmonic systems. Manyother applications can be envisaged, which take advan-tage of frequency-dependent thermostatting. For in-stance, one could use this technique in accelerated sam-pling methods21–23, which work by artificially heating thelow-frequency modes, whilst keeping the other modes atthe correct temperature.

E. Implementation

The implementation of a GLE thermostat inmolecular-dynamics simulations is straightforward.Here, we consider the case of a velocity-Verlet integra-tor, which updates positions and momenta by a timestep ∆t, according to the scheme:

p←p+ V ′(q)∆t/2

q ←q + p∆t

p←p+ V ′(q)∆t/2.

(13)

Eqs. (13) can be obtained using Trotter splitting in aLiouville operator formalism24. In the same spirit onecan introduce our GLE thermostat by performing twofree-particle steps by ∆t/2 on the (p, s) variables19:

(p, s)← P [(p, s) ,∆t/2]

p← p+ V ′(q)∆t/2

q ← q + p∆t

p← p+ V ′(q)∆t/2.

(p, s)← P [(p, s) ,∆t/2]

(14)

At variance with thermostats based on second-orderequations of motion such as Nose-Hoover, where a mul-tiple time-step approach is required to obtain accuratetrajectories25,26, this free-particle step can be performedwithout introducing additional sampling errors. The ex-act finite-time propagator for (p, s) reads:

P [(p, s) ,∆t]T = T(∆t) (p, s)T + S(∆t)ξT (15)

where ξ is a vector of n+ 1 uncorrelated Gaussian num-bers, and the matrices T and S can be computed once,at the beginning of the simulation and for all degrees of

freedom10,27. The relations between T, S, Ap, Cp and∆t read

T = e−∆tAp ,SST = Cp − e−∆tApCpe−∆tAT

p .

It is worth pointing out that when FDT holds, thecanonical distribution is invariant under the actionof (15), whatever the size of the time-step. A useful con-sequence of this property is that, in the rare cases whereapplying (15) introduces a significant overhead over theforce calculation, the thermostat can be applied everym steps of dynamics, using a stride of m ∆t. This willchange the trajectory, but does not affect the accuracyof sampling.The velocity-Verlet algorithm (13) introduces finite-∆t

errors, whose effect needs to be monitored. In micro-canonical simulations, this is routinely done by checkingconservation of the total energy H . Following the workof Bussi et al.28 we introduce a conserved quantity H ,which can be used to the same purpose:

H = H −∑

i

∆Ki (16)

where ∆Ki is the change in kinetic energy due to theaction of the thermostat at the i-th time-step, and thesum is extended over the past trajectory. In cases wherethe FDT holds, such as that described in Section IC,the drift of the effective energy quantitatively measuresthe violation of detailed balance induced by the velocity-Verlet step, similarly to Refs.19,28. In the cases wherethe FDT does not hold, such as the frequency dependentthermostating described in Section ID, the conservationof this quantity just measures the accuracy of the inte-gration, similarly to Refs.29,30.

II. FITTING OF COLORED-NOISE

PARAMETERS

A key feature of our approach resides in our abilityto optimize the performance of the thermostat based onanalytical estimates, making the method effectively pa-rameterless. Such optimization, however, is not trivial,even if computationally inexpensive. The relationshipbetween Ap, Bp and the correlation properties of the re-sulting trajectory is highly nonlinear. Furthermore, wefound empirically that many local minima exist whichgreatly hinder the optimization process. With these dif-ficulties in mind, we provide a downloadable library offitted parameters17 which can be adapted to most ofthe foreseeble applications, according to the prescriptionsgiven in Section IID. Details about the fitting procedureare given in the following three subsections.

A. Parameterization of GLE matrices

A number of constraints must be enforced on the driftand diffusion matrices in order to guarantee that the re-

6

sulting SDE is well-behaved. It is therefore importantto find a representation of the matrices such that duringfitting these conditions are automatically enforced, andthat the parameters space is efficiently explored. A firstcondition, required to yield a memory kernel with expo-nential decay, is that all the eigenvalues of Ap must havepositive real part. A second requirement is that the ker-nel K(ω) is positive for all real ω. This ensures that thestochastic process will be consistent with the second lawof thermodynamics31.Finding the general conditions for Ap to satisfy this

second constraint is not simple. However, we can statethat a sufficient condition for K(ω) > 0 is that Ap +AT

p

is positive definite. For simplicity we shall assume sucha positivity condition to hold, since we found empiricallythat this modest loss of generality does not significantlyaffect the accuracy or the flexibility of the fit. Moreover,in the case of canonical sampling, Ap + AT

p > 0 is alsorequired in order to obtain a real diffusion matrix, sinceBpB

Tp = kBT

(

Ap +ATp

)

according to Eq. (6).One would like to find a convenient parameterization,

which enforces automatically these constraints. This is

best done by writing Ap = A(S)p + A

(A)p , the sum of a

symmetric and antisymmetric part. Since any orthogonaltransform of the s degrees of freedom would not changethe dynamics (see Appendix A), one can assume without

loss of generality that the A(S) block in A(S)p is diagonal

(see Eq. (3) for the naming convention). Since in the

general case the antisymmetric A(A)p does not commute

with A(S)p , we will assume it to be full, while A

(S)p can

be written in the form

A(S)p =

a a1 a2 · · · ana1 α1 0 · · · 0

a2 0 α2. . . 0

......

. . .. . .

...an 0 0 · · · αn

. (17)

In order to enforce the positive-definiteness, one use an

analytical Cholesky decomposition A(S)p = QpQ

Tp , with

Qp

Qp =

q q1 q2 · · · qn0 d1 0 · · · 0

0 0 d2. . . 0

......

. . .. . .

...0 0 0 · · · dn

(18)

and αi = d2i , ai = diqi, and a = q2 +∑

i q2i . Such a

parameterization guarantees that Ap will generate a dy-namics with a stationary probability distribution, andrequires 2n + 1 parameters for the symmetric part (theelements ofQp, Eq. (18)), and n(n+1)/2 for the antisym-

metric partA(A)p . If we want the equilibrium distribution

to be the canonical, one we must enforce the FDT, andBpB

Tp is uniquely determined.

If we aim at a generalized formulation, which allows forfrequency-dependent thermalization, there are no con-straints on the choice of Bp other than the fact that bothBpB

Tp and the covariance Cp must be positive-definite.

Clearly, a real, lower-triangular Bp is the most gen-eral parameterization of a positive-definite BpB

Tp , and

amounts at introducing (n+1)(n+2)/2 extra parameters.

Together with the assumption that A(S)p > 0, the con-

dition BpBTp > 0 is sufficient to ensure that the unique

symmetric Cp which satisfies (6) is also positive-definite.

B. Fitting for canonical sampling

Armed with such a robust and fairly general parame-terization, one only needs to define a merit function tobe optimized. Again, we first consider the simpler caseof canonical sampling. Here, we want to obtain a flat re-sponse over a wide, physically-relevant frequency range(ωmin, ωmax). We have chosen the form

χ1 =

[

∑

i

|log κ(ωi)|m

]1/m

, (19)

where ωis are equally spaced on a logarithmic scale overthe fitted range. If a large value of m is chosen, the ωi

which yields the lowest efficiency is weighted more, anda flat response curve is obtained. We found empiricallythat values of m larger than 10 lead to a proliferation oflocal minima, and hinder efficient optimization. To re-solve this, one can use the optimal parameters for m = 2as input for further refinement at larger m, until conver-gence is achieved.This procedure can be modified so as to provide an

efficient thermostat which can be used in Car-Parrinello-like dynamics. In this case, the GLE has to act as alow-pass filter in which only the low ionic frequencies areaffected, and fast electronic modes are not perturbed.To obtain this effect, we compute (19) only for the ωi’swhich are smaller than a cutoff frequency ωCP , and weintroduce an additional term

χ2 =

[

∑

ωi>ωCP

|max [log κ(ωi)− k(ωCP − ωi), 0]|m

]1/m

.

(20)χ2 enforces a steep decrease of κ(ω) above ωCP , with aslope k on a logarithmic scale. Values of k as large as 9can be used, which guarantee an abrupt drop in thermal-ization efficiency for the fast modes (see Figure 3).

C. Non-thermal noise and quantum thermostat

We now discuss the case in which the thermostat ispermitted to violate FDT, in order to achieve frequency-dependent equilibration. For these applications, onemust also fit the fluctuations cpp(ω) and cqq(ω) to some

7

optimize filter

10-3 10-2 10-1 11 101 102

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

11

Ω @arb. unitsD

Κ=

1Τ

HΩ

FIG. 3: Thermostatting efficiency, as estimated from Eq. (9),for a colored-noise thermostat optimized for Car-Parrinellodynamics. Sampling efficiency is optimized for ω ∈ (10−3, 1),and an abrupt drop in efficiency is enforced for ω ∈ (1, 10),using the penalty function (20) in the fitting. The continuous(dark red) curve corresponds to k = 9, the dashed (orange)curve to k = 6 and the dotted (light orange) curve to k = 3.The κ(ω) curve for a white-noise thermostat centered on theoptimized range is also reported for reference (dotted blackcurve). The three curves correspond to the parameters setcp-9_4-3, cp-6_4-3 and cp-3_4-3

17.

target function cpp and cqq. We shall not treat the generalcase, but rather investigate the example of the quantumthermostat (Ref.9). The procedure followed provides aclear guide for future extensions to different applications.In order to reproduce quantum ions effects, one must

selectively heat high-frequency phonons, for which zero-point energy effects are important, without affecting thelow-frequency modes which behave classically. The re-quired frequency dependence of the variance for this caseis that of a quantum oscillator, i.e. cpp(ω) = ω2cqq(ω) =~ω2 coth ~ω

2kBT The ω → 0, classical limit can be proved tocorrespond to two conditions on the elements of the free-particle covariance matrix Cp; namely, cpp = kBT andaTp A

−1cp = 0. One could enforce such constraints ex-actly, by considering the entries of Cp as independent fit-ting parameters, and obtaining the diffusion matrix fromEq. (6). We found however that this choice makes it dif-ficult to obtain a positive-definite BpB

Tp , and that the

fitting becomes more complex and inefficient.As an alternative, we decided to enforce the low-

frequency limit with an appropriate penalty function,

χ3 = (cpp/kBT − 1)2 +(

aTp A−1cp/kBT

)2, (21)

to be optimized together with the sampling efficiency (19)and a term which measures how well the finite-frequencyfluctuations were fitted:

χ4 =

[

∑

i

∣

∣

∣

∣

logcqq(ωi)

cqq(ωi)

∣

∣

∣

∣

m

+

∣

∣

∣

∣

log cpp(ωi)

cpp(ωi)

∣

∣

∣

∣

m]1/m

(22)

Since the low-frequency limit is already enforced by (21),we compute (22) on a set of points equally spaced be-tween the maximum frequency ωmax and one half of theonset frequency for quantum effects ωq = kBT/~.

D. Transferability of fitted parameters

The scheme described in the previous Sections allowedus to obtain matrices suitable for all the applicationsdiscussed in previous works. Furthermore, it providesa starting point for obtaining matrices which one mightdeem useful for novel applications. However, the readeris advised that the fitting is still far from being a black-box procedure. It is thus necessary to experiment with acombination of different initial parameters and minimiza-tion schemes. We found the downhill simplex methods32

to be particularly effective, but resorted to simulated an-nealing when the optimization got stuck in a local mini-mum. There is a great deal of arbitrariness in the choiceof the terms (19-22), and in their weighted combinationχ =

∑

wiχi. To make the procedure even more delicate,we observe that in high-n cases the parameters tend tocollapse into “degenerate” minima, where the full dimen-sionality of the search space is not exploited. This phe-nomenon can be successfully circumvented by enforcingan even spacing of the eigenvalues ofA over the frequencyrange of interest, and slowly releasing this restraint dur-ing the later stages of optimization.However, the problems mentioned above have no major

practical consequences, as the computation of analyticalestimates is inexpensive and one can afford a great deal oftrial-and-error during the optimization. Moreover, fittedparameters can be reused, since the optimized parame-ters can be easily transferred to similar problems becauseof the scaling properties of the dynamics (8).In fact, one can see that if the drift and covariance

matrices (Ap,Cp) lead to the efficiency curves κ(ω)and fluctuations cpp(ω), the scaled matrices (αAp, βCp)will yield κ(α−1ω), and the fluctuations βcpp(α

−1ω).This means that if Ap is optimized for sampling overthe range (ωmin, ωmax), αAp will be optimal over(αωmin, α ωmax). We also remark that if (Ap,Cp) arefitted to the quantum harmonic oscillator fluctuations attemperature T , (αAp, αCp) will be suitable for tempera-ture αT . Care must be taken in this case to ensure thatthe scaled frequency range still encompasses the wholevibrational spectrum of the system being studied.

III. UNDERSTANDING THE QUANTUM

THERMOSTAT

As discussed in Ref.9, one must pay a great deal of at-tention when using a “quantum thermostat”, because en-ergy is transferred between modes of different frequency,as a consequence of the anharmonic coupling. This isreminiscent of zero-point energy (ZPE) leakage which

8

plagues semiclassical approaches to the computation ofnuclear quantum effects33,34. In the cases we exploredso far, empirical evidence suggests that quasi-harmonicsolids, can be treated with good accuracy down to tem-peratures as low as 10% of the Debye temperature ΘD.Clearly, the ultimate test to assess of the accuracy ofthe method is a comparison with path-integral calcula-tions, to be performed on a similar but computationallycheaper model, such as a smaller-size box or a simplerforce field.

One would like however to obtain some qualitativemeasure of the quality of the fit, and to gauge the trans-ferability of a given set of parameters. To this end, wefirst state a couple of empirical rules, and then validatethem on two fairly different real systems. A first obser-vation is that it is useless to push the fitting of the fluc-tuations cpp(ω) and cqq(ω) to very high accuracy, if thiscomes at the expense of the coupling efficiency. In fact,we would be trading a small, controlled fitting error witha possibly larger, uncontrollable and system-dependenterror stemming from anharmonicity. Secondly, we ob-served that in order to contrast more effectively the flowof energy between different phonons, one should try to re-duce the correlation time of the kinetic energy τK , ratherthan focus solely on the terms (9), which are better suitedto measure sampling efficiency. In fact, a low τK(ω) cor-responds to a slightly overdamped regime, where sam-pling efficiency is sub-optimal, but ZPE is enforced moretightly.

To demonstrate these concepts in a real system, weperformed some calculations with a Tersoff model of di-amond at a temperature T = 200 K. At this low tem-perature, slightly below 0.1ΘD, quantum effects are verystrong, and we therefore expect to have problems main-taining the large difference in temperature between thestiff and soft phonons. Using a very harmonic systemsuch as diamond is particularly useful, since one can mon-itor directly the efficiency of the thermostat by project-ing the atomic velocities on a selection of normal modes.Hence, a projected kinetic temperature T ′(ω) can becomputed, and its value checked against the predictionsin the harmonic limit, in the same spirit as in Ref.9. InFigure 4 we report the results with a matrix fitted tak-ing into account only the terms (21) and (22). Even ina harmonic system such as diamond there are major er-rors due to ZPE leakage from the high-frequency to thelow-frequency modes, which the thermostat compensatesonly partially. These poor results should be comparedwith those of Figure 5. Here, we have also introducedin the fit a term analogous to (19) to reduce the valueof τK(ω). The projected kinetic temperature now agreesalmost perfectly with the analytical predictions cpp(ω)for most of the modes. The only ones displaying signif-icant deviations are the faster ones, for whom the valueof τK(ω) is slightly larger. The cpp(ω) curve deviatesby nearly 10% from the exact, quantum-mechanical ex-pectation value. However, thanks to the more efficientcoupling, the errors due to anharmonicities are better

optimize

classical limit

10-1 11 101 102 103 104

0.01

0.1

1

10

0.9

0.95

1

1.05

1.11.15

500 1000 1500200400600800

10001200

OOHL

Ω @cm-1D

ΤKHΩL@p

s-1D

Hcpp

,Ω2c q

qL

exac

t

HaL

HbL

Ω @cm-1D

T’@

KD

FIG. 4: (a): ω-dependence of the kinetic energy correlationtime τk(ω) (light, dotted line) and of the ratio of the fit-ted fluctuations cpp(ω) (dashed line) and of ω2cqq(ω) (fullline) with the exact, quantum-mechanical target function.(b): normal-mode-projected kinetic temperature for a few,selected phonons. The dashed line is the value expectedfrom the fitting cpp(ω), while the full line is the exact,quantum-mechanical expectation value for a harmonic oscil-lator. Calculations have been performed with the parametersqt-20_6_BAD

17.

optimize

classical limit

10-1 11 101 102 103 104

0.01

0.1

1

10

0.9

0.95

1

1.05

1.11.15

500 1000 1500200400600800

10001200

OOHL

Ω @cm-1D

ΤKHΩL@p

s-1D

Hcpp

,Ω2c q

qL

exac

t

HaL

HbL

Ω @cm-1D

T’@

KD

FIG. 5: (a): ω-dependence of the kinetic energy correla-tion time τK(ω) (light, dotted line) and of the ratio ofthe fitted fluctuations cpp(ω) (dashed line) and of ω2cqq(ω)(full line) with the exact, quantum-mechanical target func-tion. (b): normal-mode-projected kinetic temperature fora few, selected phonons. The dashed line is the value ex-pected from the fitting cpp(ω), while the full line is the exact,quantum-mechanical expectation value for a harmonic oscil-lator. Calculations have been performed with the parametersqt-20_6

17.

9

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

r*

0

1

2

3

gHr*L

FIG. 6: Radial distribution function as computed fromfully-converged path-integral calculations35 (black, dottedline), and from a quantum-thermostat MD trajectory for aLennard-Jones model of solid neon at T = 20 K. Distancesare in reduced units. Full line corresponds to the parametersset qt-20_6 (cfr. Figure 5), and lighter, dashed line to theset qt-20_6_BAD (cfr. Figure 4).

compensated, and in actuality, the overall error is muchsmaller than for the parameters presented in Figure 4.To test whether these prescriptions work for less har-

monic problems, we now turn to a completely differentsystem; namely, the structural properties of solid neonat 20 K. At variance with diamond, quantum-ions effectsare less pronounced, but the system is close to its melt-ing temperature, and it is significantly anharmonic. Asshown in Figure 6, the agreement between our resultsand those of accurate path-integral calculations35 is al-most perfect if the parameters of Figure 5 are used. Asexpected, large errors are present if qt-20_6_BAD is used.Further improvements on the fitting strategy, and the ap-plication to strongly anharmonic systems is currently be-ing investigated, and will be the subject of further work.

IV. CONCLUSIONS

In this paper we have discussed in detail the use ofcolored-noise dynamics, based on Ornstein-Uhlenbeckprocesses, as a tool for performing molecular dynam-ics. Applications range from enhanced sampling, whichwe demonstrate in the harmonic limit and which willbe applied to real systems in forthcoming publications,to thermostats for adiabatically separated problems andfrequency-dependent thermalization.Our idea exploits the linear nature of the OU stochas-

tic differential equations, which allows one to use theone-dimensional harmonic oscillator as a simple butphysically-motivated reference model. On the basis of theanalytical prediction obtained in that case, we describe arecipe for fitting the thermostat parameters so as to ob-tain the desired response properties in real systems. Theprocedure is not simple, and we are considering differentapproaches to make it more robust and effective. Fortu-nately however, fitted matrices can be easily transferred

from one system to another. With this in mind we haveprovided an extensive library of optimized parameters17,which makes fitting unnecessary for most applications.We also comment on practical issues concerning the

implementation of the generalized-Langevin thermostatin a molecular-dynamics program and its use in appli-cations. In particular, we discuss in detail how one canuse colored noise to model nuclear quantum effects9. Weprovide some empirical rules to guide the fitting in thisdifficult case, and we demonstrate that a normal-modeanalysis in a quasi-harmonic system is a valuable toolfor assessing the quality of a set of parameters. We be-lieve that further investigation will find many other appli-cations for colored-noise in molecular-dynamics, and incomputer simulations of molecular systems in general. Asan example, we are currently investigating using a zero-temperature, optimal-sampling GLE thermostat in orderto perform structural optimization. On similar lines, andtaking inspiration from “quantum annealing”36,37, onecan envisage using frequency-dependent thermalizationto improve the performance of simulated annealing.

V. ACKNOWLEDGEMENTS

The authors would like to thank David Manolopoulosfor important suggestions and fruitful discussion, Mar-cella Iannuzzi for having implemented the GLE thermo-stat in CP2K38, Alessandro Curioni for implementationin CPMD39, Grigorios Pavliotis and Michela Ottobre fordiscussion and references on stochastic processes. Wealso acknowledge Stefano Angioletti-Uberti, Paolo El-vati, Hagai Eshet, Kuntal Hazra, Rustam Khaliullin andTom Markland for discussion and for having preliminar-ily tested the thermostat in real applications, providingvaluable feedback. We are especially in debt with GarethTribello, who contibuted to testing and greatly helped usimproving the manuscript.

Appendix A: Memory kernels for the

non-Markovian formulation

The connection between the Markovian (2) and non-Markovian (7) formulations of the colored-noise Langevinequation can be understood using techniques similar tothose adopted in Mori-Zwanzig theory4,11. Let us firstconsider a very general, multidimensional OU process,where we single out some degrees of freedom (y) that wewish to integrate out, leaving only the variables markedas x.(

x

y

)

= −

(

Axx Axy

Ayx Ayy

)(

x

y

)

+

(

Bxξ

Byξ

)(

ξ

)

(A1)

Assuming that the dynamics has finite memory, onecan safely take y(−∞) = 0, and the ansatz

y(t) =

∫ t

−∞

e−(t−t′)Ayy [−Ayxx(t′) +Byξξ(t

′)] dt′. (A2)

10

Substituting into (A1), one sees that y can be eliminatedfrom the dynamics of x, and arrives at

x(t) =−

∫ t

−∞

K(t− t′)x(t′)dt′ + ζ(t)

K(t) =2Axxδ(t)−Axye−tAyyAyx (t ≥ 0)

ζ(t) =Bxξξ(t)−

∫ t

−∞

Axye−(t−t′)AyyByξξ(t

′).

(A3)

One can see that (A3) are invariant under any orthogo-nal transformation of the y dynamical variables, meaningthat such a transformation leaves the dynamics of the x’sunchanged.The colored noise is better described in terms of its

time-correlation function, H(t) =⟨

ζ(t)ζ(0)T⟩

. Let

us first introduce the symmetric matrix D = BBT ,whose parts we shall label using the same schemeused for A in Eq. (A1). We shall also need Zyy =∫∞

0e−AyytDyye

−ATyytdt. With these definitions in mind,

one finds

H(t) = δ(t)Dxx+Axye−tAyy

[

ZyyATxy −Dyx

]

(t ≥ 0) .(A4)

Note that the value of H(t) for t < 0 is determined bythe constraint H(−t) = H(t)T ; the value of K(t) in-stead, is irrelevant for negative times: we will assumeK(−t) = K(t)T to hold, since this will simplify somealgebra below.Let’s now switch to the case of the free-particle coun-

terpart of Eqs. (2), which is relevant to the memory func-tions entering Eqs. (7). Here, we want to integrate awayall the s degrees of freedom, retaining only the momen-tum p. Hence, we can transform Eqs. (A3) and (A4) tothe less cumbersome form

K(t) =2appδ(t) − aTp e−|t|Aap

H(t) =dppδ(t)− aTp e−|t|A [Zap − dp]

(A5)

This compact notation hides certain relevant propertyof the memory kernels, which are more apparent whenthe kernels are written in their Fourier representation. IfDp = BpB

Tp is transformed according to Eq. (6). K(ω)

and H(ω) read

K(ω) =2app − 2aTpA

A2 + ω2ap

H(ω) =K(ω)

(

cpp − aTpA

A2 + ω2cp

)

+

+ 2ω2

(

aTp1

A2 + ω2cp

)(

1 + aTp1

A2 + ω2ap

)

.

(A6)

It is seen that the memory functions (hence the dynam-ical trajectory) are independent of the value of C, thecovariance of the fictitious degrees of freedom. More-over, a sufficient condition for the FDT to hold is readilyfound. By setting cpp = kBT and cp = 0, one obtainsH(ω) = kBTK(ω), which is precisely the FDT for a non-Markovian Langevin equation. Since the value of C is

irrelevant we can take Cp = kBT , which simplifies thealgebra and leads to numerically-stable trajectories.

Appendix B: Covariance matrix and correlation

times for the harmonic oscillator

Given A and C matrices (the drift term and the staticcovariance for a generic OU process), one can find the dif-fusion matrix B by an expression analogous to Eq. (6).The same relation can be used to obtain the elements ofC given the drift and diffusion matrices, by solving thelinear system. However, the covariance matrix can becomputed more efficiently by finding the eigendecompo-sition of A = O diag(αi) O

−1, and computing

Cij =∑

kl

Oik

[

O−1BBTO−1T]

klOjl

αk + αl. (B1)

Now, let x be the vector describing the trajectoryof the OU process. In order to compute τH or τV(Eq. (9)) one needs time correlation functions of theform 〈xi(t)xj(t)xk(0)xl(0)〉. The corresponding, non-normalized integrals

τijkl =

∫ ∞

0

[〈xi(t)xj(t)xk(0)xl(0)〉 − 〈xixj〉 〈xkxl〉] dt

(B2)can be computed in terms of the tensorial quantity

Xijkl =∑

mn

Oim

[

O−1C]

mlOjn

[

O−1C]

nk

αm + αn(B3)

as τijkl =14 (Xijkl +Xijlk +Xklij +Xlkij). For example

– if we consider the full OU process in the harmonic case– one computes

τH =ω4τqqqq + 2ω2τqqpp + τpppp

ω4c2qq + 2ω2c2qp + c2pp, τV =

τqqqqc2qq

(B4)

where we use an obvious notation for the indices in τijkl .

Appendix C: A comparison with Nose-Hoover

Chains

The most widespread techniques for canonical sam-pling in MD are probably white-noise Langevin and Nose-Hoover chains (NHC). White-noise Langevin can be con-sidered as a limit case of the thermostatting method wedescribe in this work, but NHC is based on a redicallydifferent philosophy. It is therefore worth performing abrief comparison between the latter and the GLE ther-mostat.In the “massive” version of the NH thermostat13,14,

each component of the physical momentum is coupledto an additional degree of freedom with a fictitious massQ, by means of a second-order equation of motion. The

11

resulting dynamics ensures that the physically-relevantdegrees of freedom will sample the correct, constant-temperature ensemble, with the advantage of having de-terministic equations of motion, and a well-defined con-served quantity. However, in the harmonic case, trajecto-ries are poorly ergodic. This problem can be addressed bycoupling the fictitious momentum to a second bath vari-able with a similar equation of motion. By repeating thisprocess further a “Nose-Hoover chain” can be formed,which ensures that the dynamics is sufficiently chaotic toachieve efficient sampling15,40. The drawback of this ap-proach is that the thermostat equations are second-orderin momenta. It is therefore difficult to obtain analyti-cal predictions for the properties of the dynamics, andthe integration of the additional degrees of freedom mustbe performed with a multiple time-step approach, whichmakes the thermostat more expensive.To examine the performances of NHC and GLE, one

could envisage comparing the sampling efficiency as de-fined by the correlation times (9). Obtaining such esti-mates is not straightforward, not only because the theharmonic case cannot be treated analytically, but alsobecause in the multidimensional case the properties ofthe trajectory will not be invariant under an orthogonaltransformation of coordinates, as discussed in Section I.The simplest model we can conceive for comparing NHCand GLE is therefore a two-dimensional harmonic oscil-

lator, with different vibrational frequencies on the twonormal modes and adjustable relative orientations of theeigenvectors with respect to the thermostatted coordi-nates.

The resulting τV is reported in Figure 7: in the highlyanisotropic cases, the efficiency of the NH chains dependsdramatically on the orientation of the axes, while forwell-conditioned problems is almost constant. The lin-ear stochastic thermostat, on the other hand, has a pre-dictable response, which is completely independent onorthogonal transforms of the coordinates. In the one-dimensional case – or when eigenvectors are perfectlyaligned with axes – NH chains are very efficient for all

modes with frequency ω <√

kBTQ . One should how-

ever consider that, in the absence of an exact propaga-tor, choosing a small Q implies that integration of thetrajectory for the chains will become more expensive.

Obviously, such a simple toy model does not give quan-titative information on the behavior in real-life cases,where modes of different frequencies coexist with an-harmonicity and diffusive behavior. However, it demon-strates that the colored-noise Langevin thermostat per-forms almost as well as the axis-aligned NH chains. Fur-thermore, unlike the NHC, there are no unpredictablefailures for anisotropic potentials.

† Electronic address: michele.ceriotti@phys.chem.ethz.ch1 T. Schneider and E. Stoll, Phys. Rev. B 17, 1302 (1978).2 S. A. Adelman and C. L. Brooks, J. Phys. Chem. 86, 1511

(1982).3 R. Zwanzig, Phys. Rev. 124, 983 (1961).4 R. Zwanzig, Nonequilibrium statistical mechanics (Oxford

University Press, New York, 2001).5 C. C. Martens, J. Chem. Phys. 116, 2516 (2002).6 J.-S. Wang, Phys. Rev. Lett. 99, 160601 (2007).7 L. Kantorovich, Phys. Rev. B 78, 094304 (pages 12)

(2008).8 M. Ceriotti, G. Bussi, and M. Parrinello, Phys. Rev. Lett.102, 020601 (pages 4) (2009).

9 M. Ceriotti, G. Bussi, and M. Parrinello, Phys. Rev. Lett.103, 030603 (pages 4) (2009).

10 C. W. Gardiner, Handbook of Stochastic Methods(Springer, Berlin, 2003), 3rd ed.

11 J. Luczka, Chaos 15, 026107 (2005).12 F. Marchesoni and P. Grigolini, J. Chem. Phys. 78, 6287

(1983).13 S. Nose, J. Chem. Phys. 81, 511 (1984).14 W. G. Hoover, Phys. Rev. A 31, 1695 (1985).15 G. J. Martyna, M. E. Tuckerman, and M. L. Klein, J.

Chem. Phys. 97, 2635 (1992).16 D. J. Tobias, G. J. Martyna, and M. L. Klein, J. Phys.

Chem. 97, 12959 (1993).17 gle4md, http://gle4md.berlios.de.18 R. Kubo, Reports on Progress in Physics 29, 255 (1966).19 G. Bussi and M. Parrinello, Phys. Rev. E 75, 056707

(2007).

20 G. Bussi and M. Parrinello, Comp. Phys. Comm. 179, 26(2008).

21 L. Rosso, P. Minary, Z. Zhu, and M. E. Tuckerman, J.Chem. Phys. 116, 4389 (2002).

22 J. VandeVondele and U. Rothlisberger, J. Phys. Chem. B106, 203 (2002).

23 L. Maragliano and E. Vanden-Eijnden, Chem. Phys. Lett.426, 168 (2006).

24 M. Tuckerman, B. J. Berne, and G. J. Martyna, J. Chem.Phys. 97, 1990 (1992).

25 M. E. Tuckerman, D. Marx, M. L. Klein, and M. Parrinello,J. Chem. Phys. 104, 5579 (1996).

26 S. Jang and G. A. Voth, J. Chem. Phys. 107, 9514 (1997).27 R. F. Fox, I. R. Gatland, R. Roy, and G. Vemuri, Phys.

Rev. A 38, 5938 (1988).28 G. Bussi, D. Donadio, and M. Parrinello, J. Chem. Phys.

126, 014101 (2007).29 F. Bruneval, D. Donadio, and M. Parrinello, J. Phys.

Chem. B 111, 12219 (2007).30 B. Ensing, S. O. Nielsen, P. B. Moore, M. L. Klein, and

M. Parrinello, J. Chem. Theory Comput. 3, 1100 (2007).31 G. W. Ford, J. T. Lewis, and R. F. O’ Connell, Phys. Rev.

A 37, 4419 (1988).32 J. A. Nelder and R. Mead, Comp. J. 7, 308 (1965).33 R. Alimi, A. Garcıa-Vela, and R. B. Gerber, J. Chem.

Phys. 96, 2034 (1992).34 S. Habershon and M. D. E., J. Chem. Phys 131, 024501

(2009).35 K. Singer and W. Smith, Molecular Physics 64, 1215

(1988).

12

Φ

0 0.05 0.1 0.15 0.2 0.25

ΦΠ

51101

5 ×101102

5 ×102103

5 ×103Τ

VΩ

max

FIG. 7: Correlation time for the potential energy of a 2-Dharmonic oscillator, as a function of the angle between theeigenmodes and the cartesian axes. τV is computed for dif-ferent values of the condition number ωmax/ωmin, from bot-tom to top 10, 31.6 and 100. Thin lines serve as an aid forthe eye, connecting the results obtained in the three casesusing a massive NH chains thermostat, with four additionaldegrees of freedom and Q = kBT/ω

2

max. Error bars are alsoshown for individual data points. Thick lines correspond tothe (constant) result predicted for a GLE thermostat, usingrespectively the thermostat parameters kv_2-1, centered on0.32ωmax, kv_4-2, centered on 0.18ωmax, and kv_4-2 cen-tered on 0.1ωmax. The values obtained in actual GLE simu-lations agree with the predictions within the statistical error-bar, and are not reported.

36 Y.-H. Lee and B. J. Berne, J. Phys. Chem. A 104, 86(2000).

37 G. E. Santoro and E. Tosatti, J. Phys. A 39, R393 (2006).38 CP2K, http://cp2k.berlios.de.39 CPMD, Copyright IBM Corp 1990-2006, Copyright MPI

fur Festkorperforschung Stuttgart 1997-2001.40 M. E. Tuckerman, B. J. Berne, G. J. Martyna, and M. L.

Klein, J. Chem. Phys. 99, 2796 (1993).41 D. Marx and J. Hutter, Modern Methods and Algorithms

of Quantum Chemistry 1, 301 (2000).42 Note the remarkable formal analogy between Eq. (6) and

the equations for the orthogonality constraints in Car-Parrinello dynamics, see e.g. Ref.41