Accelerating the convergence of path integral dynamics with a generalized Langevinequation

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

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

David E. ManolopoulosPhysical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, UK

The quantum nature of nuclei plays an important role in the accurate modelling of light atomssuch as hydrogen, but it is often neglected in simulations due to the high computational overheadinvolved. It has recently been shown that zero-point energy effects can be included comparativelycheaply in simulations of harmonic and quasi-harmonic systems by augmenting classical moleculardynamics with a generalized Langevin equation (GLE). Here we describe how a similar approachcan be used to accelerate the convergence of path integral (PI) molecular dynamics to the exactquantum mechanical result in more strongly anharmonic systems exhibiting both zero point energyand tunnelling effects. The resulting PI-GLE method is illustrated with applications to a double-welltunnelling problem and to liquid water.

I. INTRODUCTION

Atomistic computer simulations have become an im-portant complement to experimental measurements instudies of a wide variety of physical, chemical and biolog-ical systems. However, in order to reduce computationaleffort that is required for larger systems, a number ofapproximations are typically made in these simulations.One that is commonly employed is to assume that theatomic nuclei behave as classical particles, even when theinteractions between them are obtained from an ab initiocalculation. This is a reasonable assumption for heavyatoms at high temperatures. But for lighter atoms – andhydrogen in particular – significant deviations are to beexpected from classical behavior even at room tempera-ture.

When empirical interaction potentials are used to com-pute the forces acting on the nuclei, nuclear quantumeffects are often implicitly accounted for, because forcefields are typically parameterised so as to agree withexperimental data when used in classical molecular dy-namics simulations. When ab initio methods are usedto describe the interactions between the nuclei, there isno such parameterisation for nuclear quantum effects,and the error that results from assuming purely classicalbehavior of the nuclear motion is often comparable tothat stemming from an approximate treatment of elec-tron correlation.1

The conventional approach to including nuclear quan-tum effects exploits the isomorphism between the quan-tum mechanical partition function of the physical systemand the classical partition function of an extended prob-lem consisting of a necklace (or ring polymer) of repli-cas of the system in which corresponding atoms are con-nected by harmonic springs.2 As the number of replicas(or beads of the necklace) is increased, this imaginarytime path integral3 (PI) method samples an ensemblethat converges systematically to that of the quantum

problem with distinguishable particles, at the expenseof a computational effort that increases in proportion tothe number of beads.

Methods have been devised to reduce this effort bysplitting the calculation of forces into an inexpensive,short-ranged contribution that is computed on everybead, and a long-range contribution that is evaluated ona contracted ring polymer with fewer beads.4–6 Thesemethods are straightforward to implement in simula-tions with empirical force fields. However they cannotpresently be used in ab initio simulations, where thesplitting of the forces into an inexpensive short-rangecontribution plus a more expensive long-range contribu-tion is significantly more difficult to arrange. Approx-imate quantum methods such as the Feynman-Hibbs3,7

approach are also difficult to use in ab initio simulations,because they require the computation of the Hessian.

A more general approach to reducing the effort of pathintegral simulations is therefore called for, and there arecertain indications in the literature that this might nowbe possible. In particular, it has recently been shown thatcolored noise, generalized Langevin equation (GLE) ther-mostats can be used not only to enhance the samplingof classical and path integral molecular dynamics,8,9 butalso to modify conventional molecular dynamics so asto include nuclear quantum effects in mildly anharmonicsystems in which zero-point energy plays a significantrole.10 This approach involves negligible overhead withrespect to purely classical dynamics, provides very nat-urally the proton momentum distribution (which is rel-evant for comparison with inelastic neutron scatteringexperiments), and can be applied with the same ease toempirical and ab initio force fields. However, it does notprovide a satisfactory description of more subtle quan-tum effects such as tunnelling, and its accuracy cannotbe systematically improved.

In this paper, we will show that it is possible to com-bine a GLE thermostat with path integral molecular dy-

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namics (PIMD) in such a way as to exploit the best fea-tures of both techniques. In particular, by tuning theproperties of the correlated noise in an appropriate GLE,we will show that the systematic convergence of PIMD tothe exact quantum mechanical result can be greatly ac-celerated, leading to significant computational savings fora given level of accuracy. The resulting PI+GLE schemeprovides an equally reliable description of zero point en-ergy and tunnelling effects, it is equally applicable to sim-ulations with empirical and ab initio force fields, and un-like the original single-bead “quantum thermostat” dis-cussed above it can be systematically improved simplyby increasing the number of path integral beads.

The outline of the paper is as follows. In Section IIwe briefly recall a few concepts from path integral andgeneralized Langevin equation methods, and describe ourstrategy to have them work in synergy. In Section III wepresent a systematic study of a one-dimensional quarticdouble well potential, discussing the effect of zero pointenergy and tunnelling on the convergence of our method.In Section IV we examine how the method performs forliquid water, and in Section V we draw our conclusions.

II. COMBINING PATH INTEGRALS WITHTHE GENERALIZED LANGEVIN EQUATION

A. Path integral methods

Let us first recall the basic principles of imaginary timepath integral methods, so as to introduce our notation.We will only discuss the one-dimensional case and referthe reader to the literature2,3,11–15 for a more detaileddiscussion.

Consider the Hamiltonian for a particle in a one-dimensional potential,

H =1

2p2 + V (q), (1)

where p and q are the mass-scaled momentum and posi-tion operators and the potential V (q) is assumed to besuch that the quantum mechanical partition function

Z = tr[e−H/kBT

](2)

is well defined. The path integral formalism avoids thesolution of the Schrodinger equation for the Hamiltonianin Eq. (1) and allows one instead to sample configura-tions consistent with the quantum mechanical equilib-rium distribution by exploiting an isomorphism with anextended classical problem.2 Indeed a standard Trotter-product15 approximation to the Boltzmann operator inEq. (2) yields the following extended phase space expres-sion for the partition function

Z ≈ ZP =1

(2π~)P

∫dPp

∫dPq e−HP (p,q)/PkBT , (3)

where HP (p,q) is the classical Hamiltonian of a ficti-tious ring polymer composed of P replicas of the systemconnected by harmonic springs

HP (p,q) =

P−1∑j=0

[1

2p2j + V (qj) +

1

2ω2P (qj − qj+1)2

],

(4)with ωP = PkBT/~ and qP ≡ q0. The error in thisapproximation is O(P−2) and so vanishes to leave theexact quantum mechanical result in the limit as P →∞.15

The momenta are often integrated out of Eq. (3)to leave a purely configurational integral which formsthe basis of the path integral Monte Carlo (PIMC)technique.11 One can however retain the momenta, anddescribe the dynamical evolution of the ring polymerby means of Hamilton’s equations. This is the PIMDapproach,12,13 which provides a particularly efficient wayto sample the configuration space in situations such asmolecular liquid simulations in which an effective PIMCcalculation would require complicated collective moves.A fully converged PIMD calculation produces the exactthermodynamic and structural properties of the quan-tum mechanical system. Although it is something ofan aside to the present work, in which we shall be con-cerned exclusively with static equilibrium properties, itis also now well established that the centroid moleculardynamics16,17 and ring polymer molecular dynamics18,19

generalizations of PIMD can be used to provide quitereasonable estimates of dynamical properties such as dif-fusion coefficients and chemical reaction rates.

The extension to three dimensions and to an arbi-trary number of interacting distinguishable particles isstraightforward. But rather than describe this extensionhere, let us consider instead the simple case of a one-dimensional harmonic potential, V (q) = ω2q2/2. The re-sult will be used in what follows. For this simple model,the integral in Eq. (3) can be evaluated by transforming

to the P normal modes qkP−10 of the ring polymer withfrequencies

ωk =√ω2 + 4ω2

P sin2(kπ/P ). (5)

It is then easy to show that the equilibrium position dis-tribution of each bead of the ring polymer will be a Gaus-sian centred on q = 0, with a variance

⟨q2⟩P

=1

P

P−1∑k=0

⟨q2k⟩

= kBT

P−1∑k=0

1

ω2k

, (6)

which converges to the correct quantum mechanical ther-mal expectation value

⟨q2⟩

=~

2ωcoth

~ω2kBT

(7)

in the limit as P →∞.

3

B. Generalized Langevin equations

It has recently been shown that an appropriate gener-alized Langevin equation (GLE) thermostat can be usedto sample configurations for a harmonic oscillator thatare consistent with the quantum mechanical variance inEq. (7) using just a single replica of the system, therebyavoiding the overhead of a P -bead PI simulation.10

The basic idea behind GLE thermostats is to con-struct a linear, Markovian stochastic differential equa-tion (SDE) in an extended momentum space which iscoupled to the Hamiltonian dynamics in such a way thatthe equations of motion read

q = p(ps

)=

(−V ′(q)

0

)−(app aTpap A

)(ps

)+

(bpp bTpbp B

)(ξ

),

(8)

where ξ is a vector of n+1 uncorrelated Gaussian randomnumbers with 〈ξi (t) ξj (0)〉 = δijδ (t). It can be shownthat the resulting trajectories are statistically equivalentto those obtained from a non-Markovian Langevin equa-tion involving only p and q,20

q = p

p =− V ′(q)−∫ t

−∞K(t− s)p(s)ds+ ζ(t),

(9)

where the friction kernel K(t) and the noise correlationfunction H(t) = 〈ζ(t)ζ(0)〉 are related by analytical ex-pressions to the matrices that appear in Eq. (8). Theserelationships, together with a more detailed discussion,can be found in Ref.8

The only non-linear term in Eq. (8) is the force V ′(q).If the potential is harmonic, the force depends linearly onq, and the whole set of equations describe an Ornstein-Uhlenbeck process,21 which can be solved analyticallyto yield closed-form expressions for static and dynamicproperties of the trajectory. Based on these expressions,one can iteratively refine the parameters ap, a

Tp , ap, A,

bp, bTp , bp and B in Eq. (8) subject to certain positivity

constraints until the desired response of the thermostatis obtained.8

When considering the generalisation of this strategy toa multi-dimensional harmonic problem, one realises that,because of the linear nature of the SDE and of the con-sequent rotational invariance, the dynamics will conformto the analytical predictions obtained from the Ornstein-Uhlenbeck process even if the thermostats are applied toCartesian coordinates, without the need to diagonalisethe Hessian and transform to normal modes. Moreover,it has been demonstrated in previous work that the ana-lytical properties of the thermostat obtained in the har-monic limit provide a meaningful prediction of its be-havior for anharmonic problems with a similar range offrequencies.8

By employing this strategy it is possible to obtain anumber of useful effects, such as efficient sampling of thecanonical distribution in constant-temperature MD. Inthis application, a fluctuation-dissipation theorem musthold, which relates the friction kernel and the memoryof the noise in Eq. (9) through H(t) = kBTK(t). Moregenerally, when this condition is not imposed, one ob-tains a non-equilibrium dynamics in which a frequency-dependent effective temperature is enforced. Either way,one can calculate the stationary covariance of the har-monic dynamics in the (q, p, s) extended phase space,and the mean squared fluctuations of the positions andmomenta, which we will label cqq(ω) and cpp(ω), respec-tively. It is then possible to design a GLE dynamicswhich behaves as a “quantum thermostat”;10 i.e., whichenforces probability distributions of positions and mo-menta that are consistent with those expected for a quan-tum harmonic oscillator,

cqq(ω) =1

ω2cpp(ω) =

~2ω

coth~ω

2kBT, (10)

and does so over a wide range of frequencies.When applying this idea to a multi-dimensional sys-

tem one faces the problem of zero-point energy leakage.22

Anharmonic coupling causes a flow of heat from high-frequency to low-frequency vibrations, and a departurefrom the desired behavior in Eq. (10). This problemcan be mitigated to some extent by exploiting the flex-ibility of the GLE in Eq. (8) to enhance the couplingstrength of the thermostat and ensure that all vibra-tions are maintained at the correct effective temperature.This approach has been shown to give satisfactory resultsin a number of realistic, condensed-matter applications,ranging from the calculation of diamond-graphite coex-istence curves23 to the proton momentum distribution inhydrogen-storage materials.24

Being simple to implement and computationally inex-pensive, this “quantum thermostat” is an important steptowards a routine treatment of nuclear quantum effectsin molecular dynamics. It lacks however two desirablefeatures; namely, the possibility of treating more subtlequantum effects such as tunnelling, and the possibilityof increasing the accuracy in a systematic way. Sinceboth these requirements are met by PI methods, one sus-pects that it might be possible to develop a synergisticapproach which combines path integrals with the GLEthermostat so as to obtain accurate results without theeffort of a fully-converged PI simulation. The implemen-tation of such a PI+GLE approach is the subject of thenext section.

C. A synergistic combination

When a GLE thermostat is applied to a path integralmolecular dynamics simulation, the internal frequenciesof the ring polymer necklace will be present along withthe physical vibrations of the system. It is therefore once

4

again instructive to consider a harmonic model with fre-quency ω, for which both the ring polymer frequenciesand the stochastic dynamics can be treated analytically.

In order to construct a non-equilibrium Langevindynamics that enforces the quantum mechanical equi-librium distribution corresponding to the frequency-dependent fluctuations in Eq. (10), one must allow inthe path integral context for the fact that the average ofq2 is obtained from a sum over the ring polymer normalmodes,

⟨q2⟩P

=1

P

P−1∑k=0

⟨q2k⟩

=1

P

P−1∑k=0

cqq(ωk), (11)

where the normal mode frequencies ωk are given inEq. (5). Here the last equality holds if a GLE whichresults in the frequency-dependent position fluctuationcqq(ω) has been applied separately to each bead of thering polymer.

It follows from this that one cannot simply tune theparameters in the GLE acting on each bead of the neck-lace so that cqq(ω) is given by Eq. (10). Instead, theappropriate frequency-dependent distribution to be en-forced depends on the number of beads in the necklace,and can be obtained by solving

1

P

P−1∑k=0

cqq(ωk) =~

2ωcoth

~ω2kBT

. (12)

The first task is therefore to solve this functional equationfor cqq(ω), recalling again that the frequencies ωk arerelated to the frequency ω of the harmonic oscillator byEq. (5); the solution will be a universal function of ω thatis equally applicable to any harmonic oscillator, just asin the case of the original quantum thermostat10 thatenforces the condition on cqq(ω) in Eq. (10).

Before we describe our approach to solving Eq. (12),let us transform it into dimensionless form by defin-ing x = ~ω/2kBT , h(x) = x cothx, and gP (x) =(kBT/P )(2x/~)2cqq(2xkBT/~). With these definitions,Eq. (12) becomes

P−1∑k=0

gP (xk)

x2k/x2

= h(x), (13)

where

x2k = x2 + P 2 sin2 kπ

P. (14)

The solution to Eq. (13) is not unique, and one must picka particular solution by means of appropriate boundaryconditions. In particular, one would like to enforce aphysical behavior on the solution, with no discontinuitiesand reasonable asymptotic forms in the x → 0 and x →∞ limits. The tentative solution

g(0)P (x) = h(x/P ) (15)

i=01

24

16

32

aL

10-9

10-6

10-3

È1-

g 8HiLHx

Lg8

HxLÈ

P=1

24

816

32

bL

10-3 10-2 10-1 11 101

x

1

1.2

1.4

1.6

1.8

g PHx

L

FIG. 1. (a) Relative error in the estimates of g8(x) at dif-ferent iterations of the self-consistent procedure in Eq. (17),with α = 1/8. The wiggles which appear after many itera-

tions occur because each g(i)P (x) is approximated as a spline

interpolation before computing g(i+1)P (x). These wiggles can

be systematically reduced by using a denser grid in x. (b)Converged gP (x) curves for different bead numbers. Notethat the curves with larger values of P are flat up to largervalues of x.

satisfies these requirements, and it is a good approxima-tion to gP (x) from several points of view. First of all,it is the exact solution in the one-bead case (which cor-responds to the bare quantum thermostat) and in theinfinite bead limit, where it yields a constant effectivetemperature on all frequencies (which is correct, as inthis case PIMD alone is sufficient to converge to the ap-propriate distribution). It also provides the appropriatelarge-x limit for a smooth solution to Eq. (13), for arbi-trary P .

In order to refine this tentative solution, one can castEq. (13) into a fixed-point iteration, by singling out thek = 0 term:

gP (x) = h(x)−P−1∑k=1

gP (xk)

x2k/x2. (16)

We have found empirically that a self-consistent itera-tive procedure that yields an exact solution gP (x) toEq. (13) to arbitrary precision can be obtained by stabil-ising Eq. (16) with a mixing strategy,

g(0)P (x) = h(x/P )

g(i+1)P (x) = α

[h(x)−

P−1∑k=1

g(i)P (xk)

x2k/x2

]+ (1− α)g

(i)P (x).

(17)

5

01000200030004000

c qq

Ω2,c

pp@K

D cpp

cqqΩ2

exact

10-3

10-2

10-1

11

Κ V

11 101 102 103 104

Ω @cm-1D

10-210-1

11101102103

KHΩ

LΩ

11 101 102 103 104

Ω @cm-1D

PI+GLE HP=1L QT

FIG. 2. Comparison between the properties of the stochas-tic dynamics which result from the fitting strategy used inthe present work (left column) and that used for the origi-nal quantum thermostat10 (right column), for simulations atT = 298 K. Note that by sacrificing the fit to the fluctuationsof momenta [cpp(ω)] it is possible to obtain a perfect fit tocqq(ω) and a better sampling efficiency κV (ω) = 1/[τV (ω)ω],where τV (ω) is the correlation time of the potential energyfor a harmonic oscillator with frequency ω. In the case ofPI+GLE it is less essential to enforce strong coupling to avoidzero point energy leakage, and it is therefore possible to avoidthe overdamping which hinders the sampling in the case of theoriginal quantum thermostat. This overdamping is clear fromthe bottom right-hand panel, which shows that K(ω) ω atlow frequencies where K(ω) is the Fourier transform of thememory kernel in Eq. (9).

In particular, the mixing parameter α = 1/P was foundto give a sufficiently fast and convergent iteration for allof the numbers of beads we tried. The convergence of

g(i)P (x) to gP (x) is shown in Fig. 1 for P = 8, along

with examples of the resulting converged solutions forother values of P . Tabulated values of these solutionsfor P ranging from 1 to 128 can be downloaded from anon-line repository25. Once one has a converged gP (x),the frequency-dependent position fluctuation cqq(ω) inEq. (11) is given by

cqq(ω) = (PkBT/ω2)gP (~ω/2kBT ), (18)

and the only remaining problem is to design a GLE thatcan be used to enforce this fluctuation on each ring poly-mer bead.

D. Fitting and implementation

The design of a GLE consistent with cqq(ω) in Eq. (18)consists of optimising the matrices in Eq. (8) until thedesired response of the thermostat is obtained. As hasbeen found for a variety of other applications of the

GLE,8–10,26 the efficacy of the resulting PI+GLE schemewill depend on the strategy by which the optimisationis performed, and on the additional criteria besides fit-ting cqq(ω) that are used to define the merit functionfor the optimisation. For applications of the quantumthermostat to anharmonic problems the strength of thecoupling between the thermostat and the Hamiltoniandynamics and the efficiency of the sampling can be justas important as the agreement between cqq(ω) and thetarget function in Eq. (18). The possibility of improv-ing the accuracy systematically by increasing the numberof beads makes zero-point energy leakage and other an-harmonic effects a lesser concern in the present context.However, if systematic convergence is to be achieved, it isadvisable that the sampling efficiency does not differ dra-matically between the fits for different numbers of beads.

In the case of the original quantum thermostat de-scribed in Ref.10, we chose to constrain cpp(ω) =ω2cqq(ω) as in Eq. (10). This had the advantage of pro-viding ready access to the quantum mechanical momen-tum distribution, which is a quantity of direct relevanceto recent deep inelastic neutron scattering experiments.However, by constraining cpp(ω) in this way, we foundthat we had to enforce a strongly overdamped regimeat low frequencies in order to avoid zero point energyleakage in applications to multi-dimensional systems.10

In the present PI+GLE scheme, as in PIMD itself, thering polymer momenta lose all physical significance assoon as there is more than one bead, and extracting thequantum mechanical momentum distribution requires aconsiderably more intricate calculation.27 When it comesto fitting the GLE, it is therefore more natural to regardthe momenta simply as a sampling device, and to focusexclusively on the fluctuations of configurations cqq(ω),as we have already done in our description of PI+GLEin Sec. III.C.

As shown in Fig. 2, this leaves significantly more free-dom in the fit, even in the case of just one bead. Theextra freedom allows us to reproduce the desired cqq(ω)with a maximum discrepancy smaller than 0.5% and toachieve high sampling efficiency over a broad range offrequencies, yielding a sampling performance compara-ble to that of an “optimal sampling” GLE.8,9 To giveanother example involving more beads, the self-diffusioncoefficient for a model of liquid water (see Sec. IV) -which in this context can only be regarded as a mea-sure of the sampling efficiency for slow, collective mo-tion - is reduced by less than 50% in a well-converged (8bead) PI+GLE simulation compared to NVE dynamics,whereas the original (one bead) quantum thermostat de-creases the diffusion coefficient of the same model by afactor of 10.

Having obtained a nearly-constant sampling efficiencyis also beneficial to the transferability of the fitted param-eters, as discussed in Refs.8,9. As a matter of fact, thevery same parameterization has been adopted for bothof the examples given below, which are as different asa one-dimensional quartic double well and liquid water.

6

With an appropriate scaling,8,10 these GLE parameterscan be safely adopted in all circumstances where the max-imum physical frequency present in the system is smallerthan 35kBT/~. The GLE parameters we have used inthe present study are available up to P = 16 and may bedownloaded from an online repository25.

The details of how we actually optimised the GLEmatrices for the present PI+GLE application are rathertechnical, and less important than the criteria employedfor the optimisation that we have just described. A com-prehensive discussion of the optimisation of GLE ma-trices has recently been given elsewhere,8 and we usedexactly the same techniques in the present study. Theimplementation of a GLE thermostat into a PIMD codehas also been discussed in detail in a recent paper,9 whereit was shown to be essentially no more complicated thanadding a thermostat to a classical molecular dynamicssimulation. However, in the case of PI+GLE there are afew additional points that we do need to make.

First of all, the dynamics in PI+GLE must be per-formed using physical bead masses, as opposed to thealternative masses that are sometimes used in path inte-gral schemes. Unless physical bead masses are used thefrequencies of the necklace will be different from thosein Eq. (5), and Eq. (14) will have to be modified ac-cordingly; this will change the functional equation forgP (x) in Eq. (13) and the new equation will have to besolved from scratch. The equations of motion can be inte-grated efficiently with physical bead masses by perform-ing a normal mode transformation for the free ring poly-mer evolution9 and/or employing a multiple time stepscheme.28,29

Another important observation is that, at variancewith canonical sampling GLE schemes in which the freeparticle propagation of the GLE preserves the equilib-rium distribution, the stationary probability distributionof the PI+GLE scheme requires an accurate integrationof the stochastic differential equations of motion on thetimescale of the fastest modes. For this reason, whenevera multiple time step integrator is used, the thermostatshould be applied in the inner loop. As a consequence,the integration of the GLE introduces a sizeable compu-tational overhead, which can be significant in cases whenthe calculation of physical forces is inexpensive. In thecase of ab initio molecular dynamics, which is the pri-mary target for the present method, the overhead will becompletely negligible.

III. THE QUARTIC DOUBLE WELL

As mentioned in the Introduction, one of thefundamental shortcomings of the original quantumthermostat10 is its inability to provide a realistic descrip-tion of tunnelling effects. The one-dimensional double

0

1

2

3

0 0.2 0.4 0.60

1

2

3

0 0.2 0.4 0.6

P=1 P=2

P=4 P=8 PIMDPI+GLE

QM

q @ÞD

pHqL

FIG. 3. Probability density for a hydrogen atom in a quar-tic double well potential with the minima separated by 0.6 Aand a barrier height of 1000 K. All simulations were performedwith a target temperature of 300 K. The exact quantum me-chanical result (dashed black line) was obtained by numeri-cal solution of the Schrodinger equation, the contributions ofthe various eigenstates being averaged with the appropriateBoltzmann weight. The four panels compare this exact resultwith the PIMD and PI+GLE results with increasing beadnumbers.

well potential,

V (q) = h

[(2q

d

)2

− 1

]2, (19)

in which both zero-point energy and tunnelling are im-portant, and can be tuned by adjusting the height h ofthe barrier and the distance d between the minima, there-fore provides an ideal first example with which to bench-mark the performance of our combined PI+GLE method.

In Fig. 3 we report the particle density p(q) obtainedin PIMD and PI+GLE simulations of this potential withd = 0.6 A and h = 1000 K at a temperature of 300K, using different numbers of beads. By comparing thecurves with the exact finite-temperature density

ρ(q) =∑i

e−εi/kBT |ψi(q)|2 /∑i

e−εi/kBT , (20)

where εi and ψi are the eigenvalues and eigenfunctionsof the Schrodinger equation, one sees that the use of aGLE dramatically improves the results even when P = 1.By increasing P systematic convergence to exact resultis achieved, and the convergence is greatly accelerated bythe Langevin dynamics.

In order to characterise quantitatively the convergenceof the density, we computed the distance between p(q)and ρ(q), as a function of the parameters of the potentialand the bead number. For the distance metric we em-

7

ployed the square root of the Jensen-Shannon divergence

d2JS(p, ρ) =1

2

∫ ∞−∞

[p(q) ln

2p(q)

p(q) + ρ(q)+ ρ(q) ln

2ρ(q)

p(q) + ρ(q)

]dq,

(21)which is a metric that is widely used to compare proba-bility distributions.30

In Fig. 4 we present the convergence of p(q) to ρ(q) asthe number of beads is increased. The simulations wereperformed using the mass of a hydrogen atom, differentvalues of d and h as indicated in the figure, and a timestep of 0.1 fs. A very small time step was needed becausewe integrated the PI equations of motion directly usingthe velocity Verlet method, without exploiting the exactfree ring polymer evolution that becomes possible in thenormal mode representation.9 For each set of parameterswe ran PIMD and PI-GLE trajectories for 20 ns.

It is clear from Fig. 4 that the PI-GLE simula-tions yield a significantly better estimate of the finite-temperature density of the quartic double well than thePIMD simulations, even in the regime of small d andlarge h where tunnelling plays an important role. Theaddition of the GLE provides a given level of accuracywith an effort that is between two and eight times smallerthan with a standard PI, depending on the parametersof the potential and on the accuracy required.

As the density approaches the exact ρ(q), all physi-cal observables that depend on the position of the par-ticle converge to their quantum mechanical expectationvalues. In Fig. 5 we have plotted the average potentialenergy computed from the same trajectories that wereused to compute the densities in Fig. 4. Again, the useof a GLE thermostat consistently improves the quality ofresults.

IV. LIQUID WATER

In the previous section we have demonstrated, ina simple one-dimensional case, that an appropriately-constructed GLE can be used to accelerate the conver-gence of path integral molecular dynamics to the exactquantum mechanical probability distribution. However,as discussed in earlier publications,8,10 when applying thequantum thermostat to a multi-dimensional problem, onemust be wary of zero point energy leakage. The GLE setsthe various normal modes to different effective tempera-tures, and energy may flow from hot (high frequency) tocold (low frequency) modes because of anharmonic cou-plings.

To assess the behavior of the combined PI+GLE strat-egy in this respect, and to evaluate the computationalsavings that can be expected from the GLE in a more typ-ical application, we have performed some additional sim-ulations for liquid water. For these simulations, we used arecently-developed flexible water model that was fit to re-produce a wide variety of properties of the liquid in pathintegral simulations.31 This avoids the double counting of

1 2 4 8 16 3210-3

10-2

10-1

1 2 4 8 16 32

10-3

10-2

10-1

10-3

10-2

10-1

Number of beads

d JS

@pHq

L,ΡHq

LD

PIMDPI+GLE

d=0.6 Þ d=2 Þ

h=30

0K

h=10

00K

h=30

00K

FIG. 4. Distance between the exact density ρ(q) and thedensities p(q) obtained from PIMD and PI+GLE simulationswith different bead numbers, for the quartic double well po-tential in Eq. (19). Six different combinations of parametersof the double well potential have been tested. Note that inall cases the use of a GLE leads to a significant improvementin the estimate of the density.

1 2 4 8 16 32200400600800

1000

1 2 4 8 16 32150200250300

200300400500

160180200220

120140160180200

115120125130135140

Number of beads

<V

>@K

D

PIMDPI+GLE

d=0.6 Þ d=2 Þ

h=30

0K

h=10

00K

h=30

00K

FIG. 5. Average potential energy for a hydrogen atomin a quartic double well potential computed by PIMD andPI+GLE, as a function of the number of beads and for dif-ferent parameters of the potential. The exact, quantum me-chanical expectation value is marked with a line.

nuclear quantum effects that would result from the use ofa force field fit to reproduce experimental data in classicaldynamics. We performed simulations with both conven-tional PIMD and PI+GLE, using different numbers ofbeads. The refined ring polymer contraction scheme ofMarkland et al.5 was used to reduce the computationaleffort, with the long-range electrostatic interactions be-yond 5 A contracted to the ring polymer centroid. Theuse of this scheme has been shown previously to providesignificant computational savings in empirical force fieldsimulations without affecting the accuracy of the results.5

8

1 2 4 8 16 32 64 128Number of beads

5

10

15

c Vk

B

15

20

25

30

35

<T

>@k

Jm

olD

-40

-30

-20<

V>

@kJ

mol

D

PIMDPI+GLE

QT

FIG. 6. The average value of the potential energy andthe virial kinetic energy for a simulation of a flexible watermodel31 at T = 298 K, plotted as a function of the numberof beads. The results obtained with conventional PIMD andPI+GLE are compared, and the value of 〈V 〉 obtained withthe original quantum thermostat10 (QT) is also reported.

Simulations were performed with a target temperatureT = 298 K. We used a multiple-time step scheme with anouter time step of 0.75 fs and covalently bonded interac-tions computed every 0.15 fs. We ran 3.15 ns trajectoriesfor each set of parameters, with the first 150 ps usedfor equilibration. Ergodic constant-temperature sam-pling was achieved in the PIMD simulations by applyinga targeted stochastic scheme to the internal modes of thenecklace9 and a global stochastic velocity rescaling to thecentroid.32

In Fig. 6 we report the expectation values of thepotential energy and the centroid virial kinetic en-ergy, which were computed using standard path inte-gral estimators.14 In this case, the GLE is seen to beextremely useful, and leads to much faster convergenceof averages compared to conventional PIMD. Interest-ingly, we found that to converge average energies to anerror smaller than 1 kJ/mol in the PIMD simulations,P should be increased well beyond the 32 beads thatare generally adopted for room-temperature water. Con-versely, PI+GLE yields results in perfect agreement witha 128 bead PIMD simulation using fewer than 16 beads.

As a more sensitive benchmark of the convergenceof the properties of quantum water we have also com-puted the constant-volume specific heat cV , which isknown to require very large values of P for an accurate

determination.33 The value of and statistical uncertaintyin cV were obtained from a quadratic fit to the valuesof 〈V 〉 and 〈T 〉 computed at 293, 298 and 303 K. ThePI+GLE method is again seen to perform exceedinglywell in this test (see Fig. 6). The convergence is some-what accelerated by a fortuitous cancellation between theerrors in the PI+GLE estimates of the potential and ki-netic energy contributions to cV , which are however bothvery well converged by the time P ' 8.

Comprehensive results for the convergence of struc-tural properties of water are reported in Fig. 7, in whichwe systematically examine the radial distribution func-tions that are obtained with different bead numbers.Here again the GLE helps tremendously in acceleratingthe convergence of the PI calculation, in particular in theregion of the intramolecular peaks, which are strongly af-fected by nuclear quantum effects. The only case in whichthe PI+GLE simulation is in worse agreement agreementwith the exact result than the PIMD simulation is whenthe oxygen-oxygen g(r) is computed with too few beads(P = 1 and P = 2). This is a consequence of the zero-point energy leakage in the PI+GLE simulation, whichheats up low frequency degrees of freedom and washes outthe long-range features from gOO(r). By increasing thestrength of the thermostat in the low-frequency region(as is done for instance in the case of the original quan-tum thermostat – see Sec. II.D) it is possible to mitigatethis effect, at the expense however of a greater distur-bance on diffusive modes, and hence on the efficiency ofsampling. The PI+GLE strategy allows one to counterthe effect of zero point energy leakage with a moderateincrease in the number of beads, and indeed the PI+GLEradial distribution functions for liquid water are seen tobe significantly more accurate than the PIMD radial dis-tribution functions in Fig. 7 for all P ≥ 4.

Combining the results in Figs. 6 and 7, one sees thatPI+GLE provides the same accuracy in the thermody-namic and structural properties of liquid water with 8beads as PIMD provides with 32 beads. This is thelevel of accuracy that is commonly accepted for room-temperature water simulations, and adding an appropri-ate GLE to the PI simulation allows it to be achievedwith a four-fold reduction in the number of beads. Thecomparison becomes even more favourable if higher ac-curacy is required, with a 16-bead PI+GLE simulationbeing as accurate as a 128-bead PIMD simulation for allof the observables we have considered.

V. CONCLUSIONS

In the present paper we have discussed and thor-oughly demonstrated how a properly-designed general-ized Langevin equation can be used to accelerate theconvergence of path integral molecular dynamics to theexact quantum mechanical thermal expectation values.This leads to substantial savings in computational effort,the number of beads required to obtain a given level of

9

2 3 4 5 6

-0.04

-0.02

0

0.02

1 1.5 2 2.5 3

0

0.5

1

1 2 3 4 5-0.04

-0.02

0

0.02

0.04

0.06

-0.1

-0.05

0

0.05

-1

0

1

2

3

-0.1

0

0.1

-0.2

-0.1

0

0.1

-2

0

2

4

6

8

-0.2

0

0.2

0.4

-0.2

-0.1

0

0.1

-5

0

5

10

15

-0.5

0

0.5

1

-0.4

-0.2

0

0.2

0

10

20

-0.5

0

0.5

1

1.5

DgOO DgOH DgHH

P=16

P=8

P=4

P=2

P=1PIMDPI+GLE

QT

r @ÞD

FIG. 7. Errors in the radial distribution functions of liquid water at T = 298 K obtained in low-P simulations using PIMD andPI+GLE, relative to a fully-converged PIMD reference simulation with 128 beads. Note that the scale of the y-axis is greatlymagnified as P is increased and the ∆g’s become smaller. The results obtained using the original quantum thermostat10 (QT)are also reported in the panels with P = 1. The statistical error in the distribution functions is of the order of 10−3.

accuracy being reduced by a factor of 4 or more. Theoriginal (one-bead) quantum thermostat10 provides aneven cheaper way to include nuclear quantum effects inmildly anharmonic systems, but it is an inherently ap-proximate technique. The present PI+GLE combinationcaptures tunnelling effects as well as zero point energyeffects, and it can be systematically improved to give theexact quantum mechanical result simply by increasingthe number of path integral beads. We expect that thiscombination will be particularly valuable when used inconjunction with ab initio molecular dynamics, in whichthe forces acting on the nuclei are so expensive to evalu-ate that nuclear quantum effects have only seldom been

considered in the past.

One final observation is that we have concentrated ex-clusively in this paper on the standard second order Trot-ter product path integral in Eqs. (3) and (4). Anotherway to reduce the number of path integral beads is touse a higher-order imaginary time propagator, and someinteresting progress has been made in this direction overthe years.34–37 This approach is fundamentally differentfrom the approach we have investigated here, and poten-tially complementary to it. One could in principle de-velop a GLE scheme to accelerate the convergence of anyimaginary time propagator, including the more promisingof the higher-order propagators that have been suggested

10

in the literature. In this way, one might hope to be ableto reduce the number of beads even further, and makethe inclusion of nuclear quantum effects in ab initio simu-lations almost routine. In any event, the results we havepresented in this paper clearly demonstrate the potentialof the generalized Langevin equation as a computationaltool for accelerating the convergence of PIMD.

VI. ACKNOWLEDGMENTS

We would like to thank Oliver Riordan for a helpfuldiscussion about the functional equation for gP (x) in

Eq. (13) and the non-uniqueness of its solutions, and Gio-vanni Bussi and Thomas Markland for stimulating andinsightful conversations.

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