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Improving the convergence of closed and open path integral molecular dynamics via higher order Trotter factorization schemes Alejandro Pérez and Mark E. Tuckerman Citation: J. Chem. Phys. 135, 064104 (2011); doi: 10.1063/1.3609120 View online: http://dx.doi.org/10.1063/1.3609120 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v135/i6 Published by the American Institute of Physics.

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THE JOURNAL OF CHEMICAL PHYSICS 135, 064104 (2011)

Improving the convergence of closed and open path integral molecular dynamics via higher order Trotter factorization schemes Alejandro Pérez1,a) and Mark E. Tuckerman2,b) 1

Department of Chemistry, New York University, New York, New York 10003, USA Department of Chemistry and Courant Institute of Mathematical Sciences, New York University, New York, New York 10003, USA 2

(Received 31 December 2010; accepted 18 June 2011; published online 10 August 2011) Higher order factorization schemes are developed for path integral molecular dynamics in order to improve the convergence of estimators for physical observables as a function of the Trotter number. The methods are based on the Takahashi-Imada and Susuki decompositions of the Boltzmann operator. The methods introduced improve the averages of the estimators by using the classical forces needed to carry out the dynamics to construct a posteriori weighting factors for standard path integral molecular dynamics. The new approaches are straightforward to implement in existing path integral codes and carry no significant overhead. The Suzuki higher order factorization was also used to improve the end-to-end distance estimator in open path integral molecular dynamics. The new schemes are tested in various model systems, including an ab initio path integral molecular dynamics calculation on the hydrogen molecule and a quantum water model. The proposed algorithms have potential utility for reducing the cost of path integral molecular dynamics calculations of bulk systems. © 2011 American Institute of Physics. [doi:10.1063/1.3609120] I. INTRODUCTION

Nuclear quantum effects play a significant role in the structural and dynamical properties of many physical systems. A successful approach to include these effects in molecular simulations is based on Feynman’s path integral formulation of quantum statistical mechanics.1–3 In this formulation, each quantum particle is represented by a ring polymer of P (known as the Trotter number) interacting quasi-particles (colloquially referred to as “beads”) subject to an attenuated classical potential.4 The path configurations can then be sampled either stochastically via Monte Carlo methods5, 6 or deterministically via molecular dynamics.7 The latter are particularly useful for ab initio or first-principles path integral molecular dynamics calculations to be discussed below and for approximate quantum dynamical calculations based on imaginary-time path integration.8–10 Some properties, such as the momentum distribution, require opening the ring polymer to obtain off-diagonal elements of the canonical density matrix and are, therefore, called open path-integral calculations.42 Generally, path integral molecular dynamics (PIMD) is a computational technique that allows nuclear quantum effects to be included in ordinary molecular dynamics (MD) simulations. Many natural processes are characterized by chemical events in which bonds are broken and new bonds are formed. Describing such events generally necessitates an explicit treatment of the electronic degrees of freedom for which first a) Electronic

mail: [email protected]. Present address: Nanobio Spectroscopy Group, Centro Joxe Mari Korta, Avenida de Tolosa, 72, E-20018 Donostia-San Sebastian, Spain. URL: http://nano-bio.ehu.es/users/alejandro. b) Electronic mail: [email protected]. URL: http://homepages.nyu. edu/~mt33/. 0021-9606/2011/135(6)/064104/17/$30.00

principles quantum chemical methods are often employed. In particular, an ab initio molecular dynamics (AIMD) calculation employs an electronic structure method to obtain the internuclear forces “on the fly” as the simulation proceeds.11–15 For such calculations, ab initio path integral molecular dynamics (AIPIMD) offers certain advantages over the Monte Carlo approach in terms of sampling efficiency and ease of implementation on parallel computing platforms. Overall, AIPIMD has all the necessary ingredients to describe chemical transformations when nuclear quantum effects cannot be neglected.16, 17 The AIPIMD technique has been successfully applied to predict the structure and the momentum distribution19 of bulk water,18, 19 to treat proton tunneling through hydrogen bonds in small gas phase systems,20–22 to suggest a mechanism of DNA stability,23 to elucidate the transport mechanisms of hydrated protons and hydroxide ions in water,24–26 and numerous other applications. However, despite the recent advent of massively parallel platforms, AIPIMD is still an expensive technique that often pushes computer resources to their limits. It is desirable, therefore, to devise new efficient path integral algorithms that ameliorate the computational burden of these calculations. The standard path integral approach is based on a second order Trotter decomposition of the Boltzmann operator,27 e−β H /P = e−β V /2P e−β T /P e−β V /2P + O (β/P )2 . ˆ

ˆ

ˆ

ˆ

(1)

where Hˆ is the system Hamiltonian, Tˆ and Vˆ are the kinetic and potential energy operators, respectively, and P is the so-called Trotter number. However, improved efficiency of path integral calculations can be achieved via higher order factorization schemes of the Boltzmann operator, which are designed to reduce the Trotter number.28 Takahashi and Imada (TI) developed a fourth-order path integral scheme

135, 064104-1

© 2011 American Institute of Physics

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064104-2

A. Perez and M. E. Tuckerman

valid for the trace of exp(−β Hˆ ).29 The TI scheme was subsequently given a more practical formulation by Li and Broughton.30 It is noteworthy that convergence of the TI estimator for the internal energy can be achieved with a factor of four fewer beads over the second-order approach.31 For example, the TI method has been recently applied in ab initio path integral hybrid Monte Carlo to study fluoride ionwater clusters.32 However, when using the TI algorithm, the derivation of estimators for physical observables must proceed via the trace of exp(−β Hˆ ). As a consequence, estimators for certain quantities of interest, such as spatial probability distributions, require spatial derivatives of the forces, which are often undesirable and generally impractical in AIPIMD calculations. For force-field based path integral calculations, however, these estimators could be easily evaluated via path integral Monte Carlo or via PIMD if the Hessian is available. A more direct and systematic approach was formulated by Suzuki,33 and later applied to a variety of problems, including time propagation in classical dynamics,34 and more recently to imaginary time path integrals, mostly within Monte Carlo schemes.35–38 Although the error in the Suzuki approach lies “between” the TI and second order methods, it simplifies the derivation of estimators for both positionand momentum-dependent observables and, therefore, allows their accuracy to be improved in a systematic manner. In this paper, we introduce efficient higher order path integral algorithms that permit the evaluation of spatial and momentum distribution functions in bulk systems using closed and open path integral molecular dynamics (OPIMD),42 respectively. Since the classical forces must be computed in PIMD calculations in order to propagate the equations of motion, it is surprising that they have not been used more routinely to improve the convergence of estimators. We demonstrate the performance of our approach on various solvable model systems and molecular applications, including one that employs AIPIMD, using both closed and open path integrals to compute energies as well as spatial and momentum distributions. The present paper is organized as follows: In the first part (Sec. II), we review PIMD, the Takahashi-Imada, and the Suzuki factorization methods. We describe the implementation of these higher-order factorization schemes within PIMD. The implementation of the Suzuki algorithm in OPIMD is also presented. Estimators for spatial and momentum dependent observables, with special emphasis on structural properties such as probability distributions, are presented. In the second part, we present and discuss the numerical results of higher-order PIMD algorithms to various example systems, including the ab initio hydrogen molecule. Finally, conclusions and the future directions are presented. II. THEORY A. Review of path integral molecular dynamics

In this section, we briefly review the methodology of path integral molecular dynamics. Quantum equilibrium and thermodynamic properties at temperature T can be com-

J. Chem. Phys. 135, 064104 (2011)

puted using the Feynman path-integral formulation of quantum statistical mechanics. Let Hˆ denote the Hamiltonian of an N-particle system, rˆ 1 , . . . , rˆ N denote the N Cartesian position operators of the particles, m1 , . . . , mN denote their masses, and V (ˆr1 , . . . , rˆ N ) denote the N -particle potential. For notational convenience, let r denote the full set of N coordinates. The quantum canonical partition ˆ function Z(N, , T ) = Tr[e−β H ], where β = 1/kB T and  is the volume, can be represented as a functional integral over N distinct cyclic imaginary-time thermal paths rI (τ ), τ ∈ [0, β¯]    1 β¯ Dr1 . . . DrN exp − dτ Z(N, , T ) = ¯ 0 D()  ×

1 mi r˙ 2i (τ ) + V (r(τ )) 2 i=1 N

 ,

(2)

where the trace requires that rI (0) = rI (β¯), and D() is the spatial domain defined by the volume. The τ integral in the exponential is denoted S[r(τ )], which is called the imaginarytime or Euclidean action. In Eq. (2), it is assumed that exchange statistics can be safely neglected and that the particles, therefore, obey Boltzmann statistics. Practical implementation of Eq. (2) requires that the continuous paths be discretized, and this is where the order of the decomposition of the operator exp(−β Hˆ ) becomes important. For example, using the second order factorization given by Eq. (1), we can write the ˆ P , where the operator Boltzmann operator as exp(−β Hˆ ) ≈  ˆ = exp(−β Vˆ /2P ) exp(−β Tˆ /P ) exp(−β Vˆ /2P ) was intro duced. If a complete set of coordinate eigenstates is inserted ˆ according to between each factor of  Tr[e−β H ] = ˆ

 P

ˆ s+1 rP +1 =r1 , d N rs rs ||r

(3)

s=1

where rs = r1,s , . . . , rN,s , and the matrix elements ˆ s+1  are evaluated, the resulting finite-P partition rs ||r function becomes N

 mi P 3P /2  ZP (N, , T ) = d NP r 2 2πβ¯ i=1 

 N

 P   1 2 2 × exp −β mi ωP (ri,s − ri,s+1 ) 2 s=1 i=1 +

1 V (rs ) P

,

(4)

ri,P +1 =ri,1

√  P P /(β¯), and s inwhere d NP r = N i=1 s=1 dri,s , ωP = dexes the imaginary time slices. In Eq. (4), the discretized path kinetic energy mi r˙ 2i (τ )/2 term becomes a harmonic nearest-neighbor interaction mi ωP2 (ri,s − ri,s+1 )2 /2. As noted by Chandler and Wolynes,4 if a set of NP uncoupled Gaussian integrals is introduced into Eq. (4) according to

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064104-3

Improving path integral convergence

 ZP (N, , T ) = N d NP p d NP r   N   P   p2i,s 1 2 2 + mi ωP (ri,s − ri,s+1 ) × exp −β 2mi,s 2 s=1 i=1 1 + V (rs ) , (5) P ri,P +1 =ri,1 then the N -particle quantum partition function becomes isomorphic to that of a “classical” system of N cyclic polymer chains of P “pseudoparticles” or beads, each with an arbitrary mass mi,s . In the last two and subsequent expressions, {r} denotes the full set of NP pseudoparticle coordinates for N cyclic polymers and ri,s is the coordinate of atom i corresponding to the imaginary time slice s. In Eq. (5), the normalization constant N was introduced to cancel out the uncoupled Gaussian integrals. Note that, unlike  with real polymer chains, the potential energy term (1/P ) s V (r1,s , . . . , rN,s ) acts only among pseudoparticles with the same imaginary time index s. The true quantum partition function is recovered in the limit P → ∞. In practice, the Trotter number P must be chosen large enough so that the discretization error becomes negligible. The argument of the exponential in Eq. (5) can be concisely written as −βH({r}, {p}), where H({r}, {p}) is the Hamiltonian for this effective classical system. Because Eq. (5) is in the form of a phase space integral, it can be evaluated using an MD approach. Although MD is employed only as a sampling scheme, PIMD has the advantage over path-integral Monte Carlo of scalable and efficient parallelization. PIMD also allows for seamless transition to approximate quantum dynamical schemes such as centroid8, 43–45 or ring-polymer10, 46–48 molecular dynamics. Finally, PIMD is preferable to Monte Carlo in ab initio path integral calculations, as it allows for full system moves, which are important for efficient sampling, as movement of even a single particle requires recalculation of the electronic structure. Unfortunately, as was shown by Hall and Berne,49 straightforward application of MD to the Hamiltonian H({r}, {p}) in Eq. (4) suffers from slow convergence due to severe ergodicity problems arising from the stiff nearest-neighbor harmonic coupling. Tuckerman et al. proposed a solution to this problem7 that involves the following steps: (1) The pseudoparticle coordinates , . . . , ri,P are first transformed to a set of variables yi,s ri,1 = β Tsβ ri,β in terms of which the harmonic coupling is   diagonalized: mi ωP2 s (ri,s − ri,s+1 )2 /2 = mi s ωs2 y2i,s /2. Tsβ can either generate the normal modes of the cyclic chain, so that {ωs } are proportional to the normal-mode frequencies, or the so-called “staging” modes,7 yi,1 = ri,1 , yi,s = ri,s − [(s √ − 1)ri,s+1 + ri,1 ]/s, in which case ω1 = 0 and ωs = ωP s/(s − 1). (2) The fictitious mass parameters mi,s are adjusted so that in the free-particle limit, the uncoupled modes yi,s , s = 2, . . . , P all have the same associated frequency and therefore move on the same time scale. This ensures that all modes are sampled with equal efficiency and are not limited by the fast modes. (3) Each Cartesian component of each mode variable is coupled to a separate thermostat (e.g., a Nosé-Hoover chain thermostat50 ), for a total of 3NP thermostats, in order to increase ergodicity and efficiency

J. Chem. Phys. 135, 064104 (2011)

of equilibration. It was shown7 that this algorithm is as efficient as the staging path-integral Monte Carlo approach of Ceperley and Pollock51 (see Figs. 1 and 3 of Ref. 7). This PIMD algorithm is also sufficiently flexible that it can be easily adapted for path integrals at constant pressure, i.e., in the NPT ensemble,52 grand-canonical path integrals53 and ab initio path-integrals,17 in which the potential energy surface and forces are obtained from electronic structure calculations performed on the fly as the PIMD calculation is carried out. Observable properties obtained from the thermal expecˆ = Tr[Oˆ exp(−β Hˆ )]/Z of an operator Oˆ can tation value O be computed directly from path configurations generated via PIMD if Oˆ depends only on position operators. In this case, the thermal expectation value is computed from an ensemble average over the effective classical system of cyclic poly ˆ ≈ (1/P ) P O(rs )P , where O(r) is the eigenmers, O s=1 value of Oˆ and · · ·P indicates an average over a classical canonical ensemble distribution exp[−βH({r}, {p})], which becomes exact in the limit P → ∞. In this way, the equilibrium configurational properties (radial and orientational distributions,. . . ) of a quantum system can be calculated straightforwardly. In addition, the derivatives of the partition function generate estimators for thermodynamic properties that can be computed in any ensemble. Thus, PIMD allows average energies, free energies, molar volumes, heat capacities, and average enthalpies (if the isothermal-isobaric ensemble is employed52 ), and as shown by Johnson and co-workers, adsorption isotherms (if the grand-canonical ensemble is employed) (Refs. 53, 55–57) to be computed. B. The Takahashi-Imada scheme

The discretized partition function in Eq. (4) takes the form of an NP -particle classical partition function with an effective potential given by the sum of the harmonic nearest neighbor coupling and the attenuated external potential, N P ω2   (2) ({r}) = P mi (ri,s − ri,s+1 )2 Veff 2 i=1 s=1 +

P 1  V (r1,s , . . . , rN,s ). P s=1

(6)

TI (Ref. 29) extended the accuracy of this second-order expression to fourth order in the trace using the following decomposition of Tr[exp(−β Hˆ )]: Tr[e−β H ] = Tr[e−β T /P e−β C/P ]P , ˆ

ˆ

ˆ

(7)

where the operator Cˆ is given by β2 ˆ ˆ ˆ [V , [T , V ]]. Cˆ = Vˆ + 24P 2

 N 1 β¯ 2  1 ˆ =V + (∇i Vˆ )2 . 24 P m i i=1

(8)

When Eq. (8) is used to evaluate the discretized path integral, the result is a modification to the effective potential in Eq. (6). Specifically, Eq. (6) acquires an additional term involving the square of the classical forces. The extended TI effective potential is

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064104-4

(4) Veff

A. Perez and M. E. Tuckerman

({r}) =

(2) Veff

≡

(2) Veff

({r}) +

N  P  i=1 s=1

1 24mi P 2 ωP2

J. Chem. Phys. 135, 064104 (2011)



∂V ∂ri,s

2

({r}) + VTI ({r}) .

In a similar manner, a primitive TI estimator for the heat capacity at constant volume is derived in Appendix B. As noted previously, the ensemble average in Eq. (12) is taken with respect to the second-order action. For the remainder of the paper, it will be assumed that all averages are taken with respect to the second-order action, and we will omit the “(2)” subscript from all subsequent expressions for statistical averages. The expectation value of a general observable Oˆ can be derived using a generating functional approach29

(9)

The application of the improved action is feasible (although possibly involved) in PIMD based on force fields but very expensive in ab initio PIMD as it requires calculation of second derivatives of the potential. This problem does not occur, of course, in path integral Monte Carlo. However, we can still use the improved action in PIMD if we regard the TI term VTI ({r}) as a factor that weights configurations sampled from the second-order action. That is, if VTI (2) constitutes a mild perturbation to Veff , which is not always ˆ guaranteed, then for an observable O, a fourth order estimator can be written as ˆ (4) = O

OTI ({r})e−βVTI ({r}) (2) , e−βVTI ({r}) (2)

ˆ =− O

1 βTr[e−β Hˆ ]

d ˆ ˆ Tr[e−β(H +λO) ]|λ=0 , dλ

where λ is a real variable. In applying Eq. (13) to the derivation of estimators, the TI approach is used to evaluate the trace before the λ derivative is applied. The higher order commutator term in Eq. (8) is evaluated using Vˆ + λOˆ as the potential energy term if Oˆ is position dependent or using Tˆ + λOˆ as the kinetic energy if Oˆ is momentum dependent. Applying this procedure to an observable that depends on the positions of N particles, that is, Oˆ = Oˆ (ˆr1 , . . . , rˆ N ), leads to a general expression of its expectation value

(10)

where · · ·(2) denotes an ensemble average with respect to the second-order action, and OTI ({r}) is an estimator for the observable Oˆ that is consistent with the use of the fourthorder decomposition in Eq. (7). Thus, each path configuration generated by standard PIMD is weighted according to the function wTI ({r}) = e−βVTI ({r}) . This weighting factor has the property of wTI → 1 in the limit of large Trotter numbers P , so that the averages from second and higher order schemes become identical. Such a weighting scheme was suggested in Ref. 35 by Voth and co-workers although its implementation for spatial and momentum distribution functions (the latter requiring an OPIMD formulation) and its performance on the calculation of such distribution functions was not tested by these authors.

ˆ ≈ O

   N P 1 s=1 O (rs ) + i=1 P

1 12mi P ωP2

  (∇i,s V ) · (∇i,s O) wTI .

wTI 

(14)

As an example, the expectation value of the potential energy operator is      N P 1 1 2 s=1 V (rs ) + i=1 12mi P ωP2 (∇i,s V (rs )) wTI P . Vˆ  ≈ wTI  (15)

C. Estimators in the Takahashi-Imada scheme

Thus, the TI estimator for the potential energy reads

In general, the improved estimators must be worked out individually using standard relations of statistical mechanics.78 As an example, the primitive estimator for the internal energy is obtained from the relation E = −∂ln Z/∂β and is given by

PETI ({r}) =

P 1  V (rs ) + 2VTI ({r}) . P s=1

From this potential energy estimator and Eq. (11), we can easily derive the primitive TI estimator for the quantum kinetic energy

N P ω2   3N P − P mi (ri,s − ri,s+1 )2 εTI ({r}) = 2β 2 i=1 s=1 P 1  + V (r1,s , . . . , rN,s ) + 3VTI ({r}) , P s=1

KETI ({r}) =

(11)

P N 3NP ω2   mi (ri,s − ri,s+1 )2 + VTI ({r}) . − P 2β 2 i=1 s=1

(16)

so that the average internal energy is given to fourth order by εTI ({r})wTI ({r})(2) . E≈ wTI ({r})(2) 

 2

rˆij ≈

   P 1 2 s=1 (ri,s − rj,s ) + P

(13)

The expectation value for the distance square between two atoms, i and j , is

(12)

1 6P ωP2



1 ∇ V mi i,s

wTI 

−

1 mj

rˆij2

= (ˆri − rˆ j )2

   ∇j,s V · (ri,s − rj,s ) wTI .

(17)

Since the distance square is a two-body operator, the particle sum in Eq. (14) reduces to the i and j terms in Eq. (17). Similarly, the expectation value of the distance rˆij = |ˆri − rˆ j | between two particles is      P 1 1 |r w − r | + F i,s j,s s TI 2   s=1 P 12P ωP , (18) rˆij ≈ wTI  Downloaded 28 Dec 2011 to 128.122.251.131. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions

064104-5

Improving path integral convergence

where Fs is given by the virial expression Fs =

J. Chem. Phys. 135, 064104 (2011)



1 1 ∇i,s V − ∇j,s V mi mj

 ·

(ri,s − rj,s ) , |ri,s − rj,s |

(19)

and the identity d|x|/dx = sign (x) = x/|x| has been used. The TI expectation value for the radial distribution function between a pair of particles with indices i and j has a more complicated expression       P 1 1 s=1 δ(|ri,s − rj,s | − r) + k=i,j 12mk P ωP2 (∇k,s V ) · ∇k,s δ(|ri,s − rj,s | − r) wTI P g (r) ≈ , (20) 4π r 2 ρwTI 

where ρ is the number density. Using the identity    f (x)δ (x − xo ) dx = − f  (x)δ (x − xo ) dx

pression Eq. (20) are worth investigating given the impressive convergence properties of TI estimators (see Appendix D). In Sec. II E, we will use the generalized Suzuki factorization scheme to devise a simpler higher order estimator for the radial distribution function. The TI expectation value of the static structure factor at wave vector q for a many-body system is

in Eq. (20) leads in principle to an expression that involves the derivative of the forces, which is generally undesirable and impractical in AIPIMD. Approximations to the TI ex-

eiq·(ˆri −ˆrj )  ≈

   P 1 s=1 1 + P

iq 12P ωP2

·

q · 12P ωP2

−

1 mj

∇j,s V



  eiq·(ri,s −rj,s ) wTI

(22)

where fi,s = −∇i,s V . It can be easily seen that the real part of the structure factor can be computed efficiently using fast Fourier transforms. We note, in passing, that TI estimators do not reduce to the classical limit when P = 1. Therefore, this limit must be handled separately when computing the averages. D. The symmetrized Suzuki factorization

(21)

.

using the same notation as Ref. 35, the Boltzmann operator can be split as follows: e−2β H /P = e−β Ve /3P e−β T /P e−4β Vm /3P e−β T /P e−β Ve /3P

5 β +O , (23) P ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

where Vˆe = Vˆ +



1 1 fi,s − fj,s mi mj  × sin [q · (ri,s − rj,s )] wTI , +

1 ∇ V mi i,s

wTI 

We see that the TI correction term is expected to become relevant at large values of the momentum transfer q. In the limit of |q| → 0, the performance of both estimators should be identical regardless the number of beads. More explicitly, the real part of Eq. (21), which can be measured in scattering experiments, is given by  P  1  1 Re {S (q)} ≈ cos [q · (ri,s − rj,s )] wTI  P s=1



Vˆm = Vˆ +

 N  β¯ ˆ 2 α ∇i V , 6mi P i=1

N  (1 − α) β¯ i=1

12mi

P

2 ∇i Vˆ

,

(24)

and α ∈ [0, 1] is a free parameter. In our numerical calculations, we found that the choice α = 0 is optimal to converge observables, in agreement with Ref. 35. Consider writing the discretized canonical quantum partition function for a system of N particles in three dimensions as ZP (β) =

 P /2

d N r2s−1 r2s−1 |e−2β H /P |r2s+1 , ˆ

(25)

s=1

In this section, we derive a discretized path integral expression for the canonical partition function based on the generalized Suzuki factorization (GSF). Following Chin34 and

where, again, the cyclic condition ri,P +1 = ri,1 is obeyed by each particle’s thermal path.

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064104-6

A. Perez and M. E. Tuckerman

J. Chem. Phys. 135, 064104 (2011)

Each matrix element r2s−1 |e−2β H /P |r2s+1  must be computed separately using the factorization in Eq. (23)

 β ˆ r2s−1 |e−2β H /P |r2s+1  ≈ exp − [Ve (r2s−1 ) + Ve (r2s+1 )] 3P  ˆ ˆ × d N r2s r2s−1 |e−β T /P |r2s e−βVm (r2s )/3P r2s |e−β T /P |r2s+1 . ˆ

In Eq. (26) the potential part was evaluated explicitly and the integral over r2s was introduced to resolve the intermediate Vˆm operator. To proceed, if we now insert the matrix element of the free particle propagator r|e

−β Tˆ /P



|r  = r|e

−β pˆ 2 /2mP





|r  =

mP 2πβ¯2

3/2

 mP  2 × exp − (r − r ) 2β¯2

(27)

N

 mi P 3P /2  ZP = d NP r 2 2πβ¯ i=1 × exp −β

N  P  mi i=1 s=1

2

 ωP2 (ri,s − ri,s+1 )2

 P /2   2 4 × exp −β Ve (r2s−1 ) + Vm (r2s ) . (28) 3P 3P s=1 Unlike the second-order algorithm where all beads are equivalent, in the GSF scheme, even and odd values of replica index s are treated differently (but still equivalent within each subset). Voth and co-workers35 proposed a scheme to compute observables via PIMD by redistributing factors in Eq. (28) as follows: N

 mi P 3P /2  ZP = d NP r wJJV (r1,1 , . . . , rN,P ) 2 2πβ¯ i=1 × exp −β

 N P   mi s=1

i=1

 wJJV (r1,1 , . . . , rN,P ) = exp −β

P /2 N   i=1 s=1

1 9mi ωP2



  

∇i V (r2s−1 ) 2 ∇i V (r2s ) 2 × α + (1 − α) . P P

Note again that in Eq. (29), odd and even values of s have different force and potential contributions based on the value of α, which is somewhat inconvenient for practical implementations. Thus, as an alternative, we propose to write the partition function in Eq. (28) as N

 mi P 3P /2  ZP = d NP r wGSF (r1,1 , . . . , rN,P ) 2 2πβ¯ i=1   N  P   mi V (rs ) 2 2 × exp −β ω (ri,s − ri,s+1 ) + , 2 P P s=1 i=1 (31)





weighted with a factor

(30)

into Eq. (26), and multiply the resulting P /2 matrix elements, we obtain a higher-order path integral expression for the canonical partition function



(26)

2

 ωP2 (ri,s

− ri,s+1 )

 P /2   2 4 −β V (r2s−1 ) + V (r2s ) . 3P 3P s=1

2

(29)

Eq. (29) suggests that each configuration obtained from a modified PIMD consistent with this expression must be

where now each configuration generated by the standard PIMD of Subsection A is weighted according to 

 P /2   V (r2s ) − V (r2s−1 ) wGSF (r1,1 , . . . , rN,P ) = exp −β 3P s=1 × wJJV (r1,1 , . . . , rN,P ),

(32)

which renders its implementation very easy in existing PIMD codes. As stated in Ref. 35, fluctuations in wJJV (and of course in our wGSF ) may become very large, especially at low Trotter numbers or for strongly singular potentials, which may degrade the efficiency of the higher order factorization scheme. In fact, this might be expected from the cancellation of potential terms in the sampling function of Eq. (32). In our numerical tests, however, we did not find significant differences between the weighting functions Eqs. (30) and (32). E. Estimators within the GSF scheme

In general, GSF estimators can be derived from the thermodynamic relations in the same way as for TI estimators. For example, using the relation T  = 1/ (βZ) N i=1 mi ∂Z/∂mi we obtain a primitive GSF estimator for the quantum kinetic

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064104-7

Improving path integral convergence

J. Chem. Phys. 135, 064104 (2011)

energy KEGSF =

N P ωP2  

3N P − 2β 2

E = T  + V  ≈ mi (ri,s − ri,s+1 )2 ×

i=1 s=1



 1 ∇i V (r2s−1 ) 2 + α P 9mi ωP2 i=1 s=1  

∇i V (r2s ) 2 + (1 − α) , P P /2 N  

s=1

ˆ ≈ O (r1 ) wGSF (r1 , . . . , rP ) O wGSF (r1 , . . . , rP )

P /2 2  V (r2s−1 ) wGSF  . P s=1 wGSF 

(33)

(34)

The expectation value of the internal energy of a bound system of N particles is

e

iq·ˆri

e

P /2 2  eiq·ri,2s−1 wGSF  , ≈ P s=1 wGSF 

i=1

2

ri,2s−1 · ∇i V (r2s−1 )

 (35)

where the path integral virial theorem58 was used to reduce the variance in the quantum kinetic part. For an unbound system, the latter expression must be modified according to  N P /2  1 2 3N E ≈ + (ri,2s−1 − r¯i ) 2β P wGSF  s=1 2 i=1 

(39)

which exhibits the expected convergence behavior. Finally, the imaginary time position correlation function of particle i is given by !

  P /2 β¯ 2  [ri,2s−1 · ri,2(s+l)−1 ]wGSF  · rˆ i (0) ≈ rˆ i 2l P P s=1 wGSF  (40) with l = 0, 1, . . . , P /2, for which no expression exists within the TI algorithm.

(36)

 where r¯i = (1/P ) Ps=1 ri,s is the ring polymer centroid of particle i. More interesting is the radial probability distribution g (r) ≈

P /2 1 2  P s=1 N (N − 1)

×

N  δ(|ri,2s−1 − rj,2s−1 | − r)wGSF  , (37) 4π r 2 ρwGSF  i,j =i

which contrasts with the complicated TI expression given by Eq. (20). The GSF expectation value of the static structure factor is in principle given by

[wGSF cos (q · r¯i ) + iwGSF sin (q · r¯i )]e−β¯ ≈ wGSF 

where the propagator of a particle in a linear potential was used in the derivation.54 Numerical tests show, however, that this expression does not converge to the expected result. Therefore, a different estimator is introduced iq·ˆri

N  1

· ∇i V (r2s−1 ) + V (r2s−1 ) wGSF ,

In the GSF scheme, the most statistically significant beads for the calculation of observables are those with odd s values. In the last equality of Eq. (33), we have used the fact that all beads with odd indices are topologically equivalent. The expectation value of the potential energy operator Vˆ is V  ≈



+ V (r2s−1 ) wGSF ,

which reduces to the corresponding TI expression, Eq. (16), after the factors α/9 and (1 − α) /9 are replaced by 1/24. Following Ref. 35, the advantage of the GSF method is that estimators for position dependent observables Oˆ (ˆr) can be easily derived via quantum operator identities

P /2 2  O (r2s−1 ) wGSF  . = P s=1 wGSF 

P /2 

2 P wGSF 

2 2

q

/(24mi P 2 )

,

(38)

We remark that all GSF estimators are computed using half the number of beads of the ring polymer. This means less sampling efficiency than ordinary second order or TI estimators, which use the full chain. We will later show that this trade-off (fewer beads versus longer sampling times) is not really an issue in our calculations of model systems provided one uses a sufficient number of beads. The optimal choice of Trotter number will, of course, depend on the system and the physical conditions. F. GSF approach for open path integrals

The sampling of the off-diagonal elements of the density matrix can be achieved by opening the ring polymer.51, 59, 60 OPIMD can be used to calculate the momentum distribution, which can be compared directly with neutron Compton scattering experiments.39–41 Normal mode OPIMD calculations were used to compute the proton momentum distribution in ice Ih.61 Recently, Morrone et al.42 developed staging OPIMD and applied to various hydrogen-bonded

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064104-8

A. Perez and M. E. Tuckerman

J. Chem. Phys. 135, 064104 (2011)

systems.19, 42, 62 From these studies, it has become clear that OPIMD calculations are difficult to converge and it would be desirable to devise a more efficient algorithm (see Ref. 63). Since the Takahashi-Imada scheme is only valid for the trace, it cannot be used in OPIMD. In this section, we investigate whether a GSF scheme can actually improve the convergence of OPIMD. A review of the OPIMD staging algorithm is presented in Appendix C. Since the momentum distribution is a single-particle property, we restrict our discussion to a single particle in three dimensions. For an homogeneous many body system, in principle all particles except one are represented by closed ring polymers. An improvement to this limitation in sampling was introduced by Morrone et al.42

The effective potential for ordinary OPIMD is Veff =

P  m s=1

2

ωP2 (rs − rs+1 )2

[V (r1 ) + V (rP +1 )]  V (rs ) + . 2P P s=2 P

+

(41)

In the OPIMD formalism, the open polymer of a single particle is represented by P + 1 beads with coordinates r1 , . . . , rP +1 . Following a derivation similar to that leading to Eq. (31), we attempt to improve OPIMD averages by using the GSF weighting factor

2 2 2 2 2 wOPIMD (r1 . . . , rP +1 ) = e−β [2V (r2 )−V (r1 )−V (rP +1 )]/6P e−β(1−α)[∇V (r2 )/P ] /(9mωP ) e−βα[(∇V (r1 )/P ) +(∇V (rP +1 )/P ) ]/(18mωP )

× e−β

P /2

2 2 2 k=2 {[V (r2s )−V (r2s−1 )]/3P + [α(∇V (r2s−1 )/P ) +(1−α)(∇V (r2s )/P ) ]/(9mωP )}

which is applied to the configurations generated by the standard OPIMD effective potential, Eq. (41). In particular, the open path end-to-end distance distribution is a one-body property whose GSF estimator reads g (r) ≈

δ (|r1 − rP +1 | − r) wOPIMD  . 4π r 2 wOPIMD 

(43)

The momentum distribution (per unit volume) is then calculated via42   ∞ sin (qr) 2 r dr. n (q) = 4π g (r) (44) qr 0 We remark that OPIMD typically suffers from undersampling at small end-to-end distances because of the larger configuration space associated with large end-to-end distances compared to small ones. III. RESULTS AND DISCUSSION A. Quantum harmonic oscillator

Numerical results for the internal energy are presented in Table I for a harmonic potential, V (x) = mω2 x 2 /2, at β¯ω = 10. Natural units (m = ω = ¯ = kB = 1) are used. Eight million PIMD steps were simulated using a time step of 0.05 with samples written out every 100 steps. Staging mode variables were used for the discrete path integral. The higher order schemes are practically converged at P = 8, whereas the second order estimators (virial and primitive) require at least 32 beads. The convergence behavior of the internal energy with respect to the Trotter number is plotted in Fig. 1 (top). The TI approach exhibits the expected fourth order convergence. At low values of P , the GSF algorithm seems to give results between the TI (at the same order of accuracy) and the standard second order schemes. Table II shows the expectation value of x 2 for the harmonic oscillator at β¯ω = 10. The analytical expression is

,

(42)

given by x 2  = (¯/(2mω)) coth (β¯ω/2) and is equal to 0.5 at these conditions. The 4th-order methods  are superior than the usual second order estimator (1/P ) Ps=1 xs2 . Similar to the internal energy, the former is practically converged at P = 8, while the latter requires 32 beads. Also shown in Table II is a comparison between different GSF weighting functions with α = 0. The GSF scheme advocated here (Eq. (31)) performs as well as that suggested by Voth and coworkers (Eq. (29)). Table III shows the expectation value for |x| and the efficiency is found to be similar to that of the observables already considered. As before, the performance of different GSF weighting factors (with α = 0) is similar. The numerical results for the position probability distribution using the GSF scheme with α = 0 and the second-order algorithm are displayed in Fig. 1 (bottom), along with the analytic ground state expression P (x) = (mω/π ¯)1/2 exp(−mωx 2 /¯). Convergence is achieved for the GSF algorithm using only 8 beads, whereas the standard second order scheme requires more than 16 beads. The numerical results for the imaginary time correlation function using our GSF scheme, Eq. (31) with α = 0, and the second-order algorithm are displayed in Fig. 2. Convergence is achieved for the GSF algorithm again using only 8 beads, whereas the standard second order scheme needs more than 16 beads. The numerical results are compared with the exact expression: C(τ ) = (¯/mω)[coth (β¯ω/2) − cosh (β¯ω/2 − τ ¯ω)/ sinh (β¯ω/2)]. The static structure factor was computed for the quantum harmonic oscillator at β¯ω = 10. At these physical conditions the system is ground-state dominated and the analytic form is known S(q) = exp[(−¯q 2 )/(4mω)]. Fig. 3 shows the comparison between the usual second-order estimator

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064104-9

Improving path integral convergence

J. Chem. Phys. 135, 064104 (2011)

TABLE I. Numerical expectation values for internal energy of the harmonic potential mω2 x 2 /2 at β¯ω = 10. All values are in natural units (m = ω = ¯ = kB = 1). The first column is the Trotter number P . Second and third column correspond to the second-order primitive and virial estimators, respectively. Fourth column is the fourth-order Takahashi-Imada primitive value. Fifth column is the GSF virial estimator sampled according to Eqs. (31) and (32), with α = 0. The exact value is 0.5, the classical value (P = 1) is kB T = 0.1 (numerical: 0.098680). P

Primitive 2nd

Virial 2nd

TI 4th-order

GSF, Eq. (31)

0.1857 0.3128 0.4245 0.4790 0.4973 0.4973 0.5057

0.1849 0.3134 0.4288 0.4753 0.4939 0.4987 0.5007

0.3259 0.4526 0.4952 0.4988 0.4991 0.5020 0.5033

0.2657 0.4150 0.5006 0.4945 0.4969 0.5014 0.4982

2 4 8 16 32 64 128

and the fourth-order estimator based on the TI relation in Eq. (22). The convergence is less spectacular than for the internal energy but there is still an improvement. In particular, the TI estimator is converged at P = 8, while the secondorder estimator at least P =16. Similarly, Fig. 4 shows a comparison between GSF, Eq. (39), and TI estimators of the static structure factor for P = 4 and P = 8. We implemented the GSF scheme within OPIMD following the staging algorithm proposed by Morrone et al.,42 which is briefly described in Appendix C. As shown by Fig. 5, the

-1

log 2 (Energy)

-1.1 -1.2 Prim Vir TI GSF

-1.3 -1.4 -1.5 -1.6 -1.7 2

3

4

5

6

7

TABLE II. Numerical values for x 2  for the quantum harmonic potential (V (x) = mω2 x 2 /2) at β¯ω = 10. First column is the Trotter number P . Second column corresponds to the usual second order primitive estimator. Third column is the Takahashi-Imada fourth order value. Fourth column corresponds to Eqs. (31) and (32) with α = 0. Fifth column corresponds to Eqs. (29) and (30), also with α = 0. Natural units (m = ω = ¯ = kB = 1) are used. The exact value is 0.5. The classical value (P =1) is kB T /mω2 = 0.1 (numerical: 0.099991). The typical errors are 2 × 10−3 for the primitive, 5 × 10−4 for the TI, 5 × 10−4 GSF, Eq. (31), and 2 × 10−4 GSF, Eq. (29) estimators. P 2 4 8 16 32 64 128

Primitive 2nd

TI 4th

GSF, Eq. (31)

GSF, Eq. (29)

0.1884 0.3129 0.4248 0.4753 0.4957 0.4978 0.5019

0.3311 0.4512 0.4934 0.4962 0.5031 0.4969 0.5022

0.2718 0.3945 0.4857 0.4951 0.5040 0.4976 0.5017

0.2717 0.4066 0.4773 0.4932 0.4990 0.4960 0.5002

GSF scheme given by Eqs. (31) and (32) with α = 0 helps to converge the end-to-end distance distribution. The end-to-end distance distribution is a quantity that is very often difficult to converge in OPIMD. Calculating the end-to-end distance distribution of the open polymer requires extensive sampling (see Fig. 3 in Ref. 42). B. Hydrogen molecule: Force field approach

The hydrogen molecule was simulated using a Morse potential, V (r) = D[1 − e−a(r−req ) ]2 , where req is the internuclear equilibrium distance, D is the dissociation energy measured from the potential minimum, and a is related to the curvature of the potential at its minimum. The parameters used in our calculations are listed in Table IV. At these physical conditions, the system is ground-state dominated, and the exact value of the lowest energy is E0 = ¯ω/2 − (¯ω/2)2 /(4D) = 0.009816 a.u. (6.1596 Kcal/mol). In total, 108 PIMD steps were simulated with a time step of 0.1 a.u. Samples were written out every 50 steps. Staging mode variables were used for the discrete path integral.

log 2 P 0.7

0.5 0.4 0.3

0.8

C (τ)

0.6

P (x)

1.0

P=4 2nd P=4 GSF P=8 2nd P=8 GSF P=16 2nd P=16 GSF P=64 2nd P=64 GSF EXACT

P=8 2nd P=8 GSF P=16 2nd P=16 GSF P=64 2nd P=64 GSF EXACT

0.6 0.4

0.2

0.2

0.1 0 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x FIG. 1. Harmonic potential V (x) = mω2 x 2 /2 with β¯ω = 10. Top: Convergence of the internal energy with respect to the Trotter number P. Bottom: Normalized position probability distribution.

0.0 0

2

4

6

8

10

τ FIG. 2. Imaginary time correlation function for the harmonic potential V (x) = mω2 x 2 /2 with β¯ω = 10. Solid line: our GSF algorithm with α = 0. Dotted line: Standard second-order scheme.

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064104-10

A. Perez and M. E. Tuckerman

J. Chem. Phys. 135, 064104 (2011)

0.35

P=4 2nd P=4 GSF P=8 2nd P=8 GSF P=16 2nd P=16 GSF EXACT

0.3

N (r)

0.25 0.2 0.15 0.1 0.05 0

-4

-2

0

2

4

r

FIG. 3. Static structure factor S(q) = eiqx  for the harmonic potential V (x) = mω2 x 2 /2 with β¯ω = 10. The TI curves were computed numerically with estimator given by Eq. (22).

Numerical values of internal energy are shown in Table V. The TI and GSF estimators are converged using between 16 and 32 beads, while second order estimators appear to require in excess of 100 beads. However, from Fig. 6 (top), we see that the there is some degradation in the performance of both TI and GSF compared to the harmonic oscillator, Fig. 1 (top). Figure 6 (bottom) shows the normalized position probability distribution for the internuclear distance. The GSF result seems to be converged at around P = 16 beads, whereas primitive algorithm requires almost 32 beads. Also shown is a fit to the quantum harmonic oscillator (QHO, in "legend) using the ground state expression PQHO (r) 2 = (μω)/(π ¯)e−μω(r−1.43) /¯ , where ω and μ are given by Table IV. Figure 7 shows the end-to-end distance distribution for the hydrogen molecule at 300 K after 8×106 OPIMD steps. GSF seems to be converged at around 16 beads, whereas primitive algorithm requires almost 32 beads. The numerical results are compared with the analytic expression for the 1.0 P=4 2nd P=4 TI P=4 GSF P=8 2nd P=8 TI P=8 GSF EXACT

iqx

S(q) = Re

0.8 0.6 0.4

FIG. 5. The open path end-to-end distance distribution for the harmonic potential V (x) = mω2 x 2 /2 potential with β¯ω = 10 as obtained from OPIMD. Comparison between GSF approach (Eqs. (31) and (32) with α = 0) and the primitive algorithm.

ground state density matrix in the harmonic limit

 # μω μω exp − r2 ρ (r) = 4π ¯ tanh(β¯ω/2) 4¯ tanh(β¯ω/2) #   μω 2 μω ≈ exp − r . (45) 4π ¯ 4¯ C. Lennard-Jones cluster

In order to test the TI and GSF schemes for the case of a strongly interacting singular potential, we studied a LennardJones cluster of N = 13 particles at 50 K. The potential is of the standard form V (r) = 4[(σ/r)12 − (σ/r)6 ], where σ = 2.93 Å and  = 370 K. Each particle was assigned the mass of a proton, and the temperature of the cluster was set to 50 K. Path integral simulations were carried out for 4×106 steps using a time step of 2.0 a.u. The convergence of the energy estimators for this system is shown in Table VI, and the convergence of the spatial probability distribution is shown in Fig. 8. Again, even for this many body case, it can be seen that the TI and GSF schemes improve convergence with no additional overhead. As before, it is found that the GSF method performs “in between” the 2nd-order and TI schemes. As was TABLE III. Numerical values for |x| for the harmonic oscillator potential (V (x) = mω2 x 2 /2) at β¯ω = 10. Same column labels as Table II. Natural units (m = ω = ¯ = " kB = 1) are used. The exact ground state value √ for a harmonic system"is ¯/(π mω) = 1/ π = 0.564189584. The classi√ 2 cal value (P = 1) is 2kB T /(π mω ) = 1/( 5π) = 0.25231325 (numeri−3 cal: 0.256). The typical errors are 2 × 10 for the primitive, 4 × 10−4 for the TI, 5 × 10−4 GSF, Eq. (31), and 2 × 10−4 GSF, Eq. (29) estimators.

0.2 0.0

P

0

1

2

3

4

5

q [1/length] FIG. 4. Real part of the static structure factor eiqx  for the quantum harmonic potential V (x) = mω2 x 2 /2 with β¯ω = 10 for P =4 and P =8. The GSF curves were computed numerically with the (real part) estimator given by Eq. (39).

2 4 8 16 32 64 128

primitive 2nd

TI 4th

GSF, Eq. (31)

GSF, Eq. (29)

0.3463 0.4471 0.5202 0.5500 0.5617 0.5630 0.5653

0.6233 0.5710 0.5641 0.5623 0.5659 0.5627 0.5655

0.4167 0.5037 0.5560 0.5617 0.5666 0.5631 0.5652

0.4160 0.5109 0.5501 0.5596 0.5629 0.5620 0.5649

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064104-11

Improving path integral convergence

J. Chem. Phys. 135, 064104 (2011)

TABLE IV. Morse potential V (r) = D[1 − e−a(r−req ) ]2 parameters for the H2 molecule. Also shown is the reduced mass μ and the harmonic frequency √ ω = a (2D/μ). All values are in atomic units.

1.401413 0.174553 1.021317 918.0584 0.019917

req D a μ ω

2.4

log 2 (Energy)

H2 Morse parameters

2.6

2.2 Prim Vir TI GSF

2 1.8 1.6 1.4 1.2 2

D. Bulk water at room temperature with a quantum force field

The flexible quantum water model q-SPC/Fw of Voth and co-workers79 was employed to show the utility of the TI and GSF algorithms in the PIMD simulation of bulk liquids. This force field, which derives from the well-known SPC model, was explicitly re-parametrized for its use in path integral simulations to avoid the over-counting of nuclear quantum effects.

4

5

6

7

log 2 P 3.0 2nd P=8 GSF P=8 2nd P=16 GSF P=16 2nd P=64 GSF P=64 QHO

2.5 2.0

P(r)

observed for the example of the Morse potential, there is some degradation in the convergence of all estimators as a function of P compared to the harmonic oscillator. For the run lengths performed, we fit the P dependence and found that the TI and GSF methods converge roughly quadratically with exponents 2.0 and 1.84, respectively, rather than the ideal quartic dependence. For the same run length, the second-order algorithms, however, show sub-quadratic convergence with an exponent of 1.31 (virial). It is also worth noting, however, that at very low P , the distribution obtained with GSF is rather noisy, which is a clear sign that the calculations are not converged with respect to the Trotter number. This problem is exacerbated at lower temperatures, where the exponential weighting factor wGSF exhibits more severe fluctuations. Nonetheless, the GSF scheme still shows some improvement at intermediate Trotter numbers over the second order algorithm provided the system is not “too quantum.”

3

1.5 1.0 0.5 0.0 1.0

1.2

1.4

1.6

1.8

2.0

r (a.u.) FIG. 6. Hydrogen molecule at T =300 K: the interaction between the two hydrogen atoms was described by a Morse potential (see Table IV). Top: Convergence of the internal energy with respect to the Trotter number P. Bottom: Normalized position probability distribution of the internuclear distance. Also shown is a fit to the quantum harmonic oscillator (QHO, in legend).

We carried out PIMD simulations of length 100–150 ps with this model for Trotter number values of P = 2, 4, 8, 16, 32, 64, and 128 beads, all at a temperature of 300 K using the PINY _ MD code.64 All simulations were carried out using 64 water molecules in a periodic box of length 12.416 Å.

1.4

P 4 8 16 32 64 128

Primitive 2nd 2.243 3.789 5.165 5.849 6.089 6.134

Virial 2nd 2.272 3.787 5.138 5.877 6.067 6.133

TI 4th-order 4.016 5.496 6.036 6.143 6.159 6.176

GSF, α = 0 3.123 4.346 5.760 6.106 6.104 6.166

1.2 1.0

N (r)

TABLE V. Numerical values for internal energy (in Kcal/mol) for the hydrogen molecule at T =300 K. The interaction between two hydrogen atoms was described by a Morse potential, see Table IV. First column is the Trotter number P . Second and third columns are the second-order primitive and virial estimator, respectively. Fourth column corresponds to the primitive estimator in the TI approach. Fifth column corresponds to the GSF virial according to Eqs. (31) and (32), with α = 0. The exact ground state value (at 0 K) is E0 = ¯ω/2 − (¯ω/2)2 / (4D) = 6.1596 Kcal/mol. The typical error bars (in Kcal/mol) are 0.012 for the primitive, 0.027 for the virial, 0.002 for TI, and 0.03 for GSF estimators.

0.8 2nd P=9 GSF P=9 2nd P=17 GSF P=17 2nd P=65 GSF P=65 QHO

0.6 0.4 0.2 0.0 -1.0

-0.5

0.0

0.5

1.0

r (a.u.) FIG. 7. The open path end-to-end distance distribution for the hydrogen molecule at T = 300 K. The hydrogen-hydrogen interaction was modeled by a Morse potential, see Table IV. Comparison between GSF approach (Eqs. (31) and (32) with α = 0) and the primitive algorithm for various Trotter numbers. Also shown is a fit to the quantum harmonic oscillator (QHO, in legend).

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064104-12

A. Perez and M. E. Tuckerman

0.35

3.0

2nd P=8 GSF P=8 2nd P=16 GSF P=16 2nd P=64 GSF P=64

0.30 0.25

2nd P=8 GSF P=8 2nd P=16 GSF P=16 2nd P=32 GSF P=32

2.5 2.0

0.20

P(r)

P(r)

J. Chem. Phys. 135, 064104 (2011)

0.15

1.5 1.0

0.10

0.5

0.05 6.0

7.0

8.0

9.0

10.0

0.0 1.0

11.0

1.2

r (a.u.)

1.4

1.6

1.8

2.0

r (a.u.)

FIG. 8. Spatial probability distribution of the interatomic distance in a cluster of 13 Lennard-Jones particles at 50 K.

FIG. 9. Normalized position probability distributions of the internuclear distance of the hydrogen molecule at T = 300 K as obtained from ab initio path integral molecular dynamics.

Real-space interactions are cut off at a distance of half of the box length, and reciprocal-space interactions are computed using the smooth particle-mesh Ewald approach. The path integral equations of motion were integrated using a time step of 0.25 fs within the staging algorithm.7 In Table VII, we show the convergence of the energy estimators as a function of P for the second-order, TI, and GSF schemes. The importance of the energy, particularly the proton kinetic energy, is highlighted by deep inelastic neutron scattering measurements,65–67 from which this energy can be extracted. From the table, it is clear that the energy is converged to within 0.2 kcal/mol per molecule using the TI scheme with just P = 8 beads, while the GSF virial estimator is converged at P = 32 beads. The second order energies require at least P = 64 or even 128 beads to reach convergence. These results clearly demonstrate the power of the TI and GSF schemes for energy prediction. Interestingly, we also computed radial distribution functions, but it was found that these converge so quickly with P (all methods converged by with P = 8 beads at this temperature) that the TI and GSF schemes did not offer a significant advantage for this distribution function. The exception is the intramolecular peak of the OH radial distribution function. At second order the height of the first peak of the OH radial distribution function is 20.14 (P = 4), 16.15 (P = 8), 14.28 (P = 16), 13.67 (P = 32), 13.50 (P = 32, 64, and

A fully ab initio PIMD calculation was performed to illustrate the utility of our approach in more advanced methods. The GSF approach was implemented in the plane wave density functional theory68, 69 code CPMD (version 3.13.2).70 A kinetic energy cutoff of 75 Ry was employed to expand the electronic orbitals with core regions represented by TroullierMartins pseudopotentials.71 Exchange and correlation were treated via the BLYP functional.72–74 The hydrogen molecule was placed in a cubic box of side 9 Å and isolated molecule boundary conditions were imposed using the method of Martyna and Tuckerman.75 Normal mode variables were used for the discrete path integral, together with massive Nosé-Hoover chain thermostats50 to ensure adequate canonical sampling.76 The Car-Parrinello algorithm11 was employed with a time step of 2 a.u., and a fictitious electronic mass of 340 a.u. These parameters were carefully chosen to ensure stability of the overall simulations. The number of beads was varied to monitor the convergence of the position probability distribution with respect to the Trotter number. The results of the AIPIMD calculations on the hydrogen molecule at 300 K are shown in Fig. 9, which

TABLE VI. Numerical values for internal energy (in Kcal/mol) for a cluster of 13 Lennard-Jones particles at T = 50 K. The first column is the Trotter number P , the second and third columns are the second-order primitive and virial estimator, respectively. The fourth column gives the primitive estimator in the TI approach, and the fifth column gives the GSF virial according to Eqs. (31) and (32), with α = 0. Typical errors are (in Kcal/mol) 0.0012 (primitive), 0.0006 (virial), 0.0006 (TI), and 0.0313 (GSF).

TABLE VII. Numerical values for the internal energy (in Kcal/mol) per molecule for bulk water simulated at 300 K using the q-SPC/Fw model (Ref. 79). Simulation details are given in the text. The first column is the Trotter number P , the second and third columns are the second-order primitive and virial estimator, respectively. The fourth column gives the primitive estimator in the TI approach, and the fifth and sixth column give the GSF primitive and virial energies with α = 0.

P 4 8 16 32 64 128

Primitive 2nd

virial 2nd

TI 4th-order

GSF, α = 0

−1.7147 −1.5021 −1.3853 −1.3313 −1.3136 −1.3121

−1.7154 −1.5036 −1.3840 −1.3299 −1.3131 −1.3065

−1.4687 −1.3675 −1.3238 −1.3083 −1.3069 −1.3103

−1.5183 −1.3753 −1.3252 −1.3045 −1.3064 −1.3043

128). Within the GSF scheme, the peak heights are 18.08 (P = 4), 15.97 (P = 8), 13.26 (P ≥ 16). E. Hydrogen molecule revisited: Ab initio approach

P 4 8 16 32 64 128

Primitive 2nd

Virial 2nd

TI 4th-order

GSF (prim)

GSF (vir)

−2.724095 0.392339 2.646778 3.616124 3.890687 4.012118

−2.722710 0.401126 2.652240 3.569497 3.763296 4.019538

5.872565 3.834473 3.815622 3.97602 3.987196 4.036441

−2.523691 0.917932 2.360696 3.757817 3.902920 4.038041

−2.283750 0.432549 3.314681 4.020392 3.974610 4.036345

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064104-13

Improving path integral convergence

displays the probability distribution of the internuclear distance. In total, 600 000 steps were accumulated. The trend as a function of P agrees quite well with the results from the force field based-PIMD calculation (Fig. 6). The small differences are attributed to the differences in the underlying models. From Fig. 9, it is clear that the GSF approach requires half the number of beads to reach the same degree of convergence compared to the second-order approach. It will be interesting to apply our method to investigate the structure of bulk hydrogen-bonded liquids such as water or solvated ions with AIPIMD.

IV. CONCLUSIONS

We have presented higher-order methods to improve the convergence of averages in path integral molecular dynamics with respect to the Trotter number. The scheme uses the forces on the path-integral beads within the Takahashi-Imada and generalized Suzuki factorization approaches to construct an a posteriori weighting of the configurations sampled from standard PIMD. As estimators must be modified for use within this approach, we have also introduced appropriate estimators for the spatial and momentum distribution functions, the latter within the open path-integral MD scheme of Ref. 42. The new algorithms were tested in various model systems. These new schemes are superior to the traditional second-order approach and yield a factor of 2–4 or more savings in computational time and/or resources. It is found that higher-order schemes (TI and GSF) improve convergence of the internal energies for the problems considered, with GSF performing in between the second-order and TI schemes (the best). We found that the expected fourth-order convergence is attained for soft harmonic potentials but that there is some degradation in the convergence behavior of both the secondorder and the higher-order algorithms for singular potentials. The performance of the GSF method in the calculation of radial distribution functions is affected at low Trotter numbers (and/or extremely low temperatures) but rapidly improves as the latter increases. The methods are very easy to implement in existing path integral codes and carry no significant overhead. Moreover, they can be applied a posteriori after a conventional path integral calculation (that is, using the second-order action) is performed. While improvement via reweighting is not always guaranteed, as some of the numerical examples demonstrate, it is expected to be advantageous if there is good overlap between the second and higher order TI or Suzuki probability distributions. The generalized Suzuki factorization was implemented in the ab initio PIMD scheme and confirmed the behavior predicted in the model systems. The Suzuki higher order factorization method was found useful (although to a lesser extend) to improve the convergence of open-path integral molecular dynamics as well. In the future, we will apply these methods to estimate nuclear quantum effects in bulk systems. For example, it will be interesting to compute the GSF radial distribution functions using the q-SPC/Fw water model79 at temperatures lower than 300 K.

J. Chem. Phys. 135, 064104 (2011)

ACKNOWLEDGMENTS

We gratefully acknowledge computer resources from City University of New York (CUNY) High Performance Computing Center (HPCC) at College of Staten Island. A.P. is grateful for support from the Horizon fellowship at New York University. M.E.T. acknowledges (National Science Foundation) NSF CHE-1012545. APPENDIX A: RING POLYMER MOLECULAR DYNAMICS (RPMD) WITH TI CORRECTIONS

In this appendix, we explore the interesting question of whether higher-order schemes can improve imaginary-time methods for approximate quantum dynamics.77 Voth and coworkers have already established35 that the GSF scheme is useful in centroid molecular dynamics.8, 9 Recently, Craig and Manolopoulos extended the applicability of the primitive Hamiltonian to approximate Kubotransformed correlations functions;10 they termed the new scheme as RPMD. In this section, we investigate whether TI or GSF correction factors can improve RPMD. The RPMD Hamiltonian for a system of N interacting particles with the TI correction reads  N  P   p2i,s mi + (r − ri,s+1 )2 H = 2 i,s 2m 2(β¯/P ) i s=1 i=1 

  β 2 ¯2 ∇i,s V 2 (A1) + + V (rs ) 24mi P (the cyclic condition ri,P +1 = ri,1 is understood). The equations of motion generated by this Hamiltonian are pi,s , r˙ i,s = mi mi (2ri,s − ri,s+1 − ri,s−1 ) − ∇i,s V (rs ) p˙ i,s = − (β¯/P )2 

 2 N  ∇ij,s V (β¯)2 ∇j,s V − . 12mj P P j =1 (A2) In principle, the calculation of the bead force requires the knowledge of the local Hessian matrix. If we specialize to the case of a non-interacting system of particles, then the momentum equation simplifies to mi (2ri,s − ri,s+1 − ri,s−1 ) − ∇i,s V (rs ) p˙ i,s = − (β¯/P )2  (β¯)2 2 (r ) ∇ V . (A3) × 1+ s 12mi P 2 i,s Next, we introduce the momentum centroid p¯ i = (1/P ) Ps=1 pi,s and the centroid position  r¯ i = (1/P ) Ps=1 ri,s . These new variables are used in the calculation of the Kubo-transformed velocity and position autocorrelation functions, respectively. For the particular case of a single particle in a harmonic oscillator potential, V (x) = mω2 x 2 /2, we have the following equations of

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064104-14

A. Perez and M. E. Tuckerman

J. Chem. Phys. 135, 064104 (2011)

motion for the centroid coordinates: x˙¯ =

p¯ , m

p˙¯ = −

 P  1 (β¯)2 2 2 ¯ 1+ (mω ) mω xs = −mT I ω2 x, P 12mP 2 s=1 (A4)

where the frequency-dependent mass mT I (ω) = m[1 + (β 2 ¯2 /12P 2 )ω2 ] related to the TI corrections has been introduced. We see that the TI factors break the classical and harmonic limits in RPMD. A similar conclusion can be drawn for the GSF case (not shown explicitly). Thus, this extended RPMD does not have the desirable limits to approximate quantum dynamics. However, the harmonic limit may be recovered if we choose the dynamical or kinetic masses to be m → mT I . Unfortunately, there is no obvious way to proceed for more general potentials. APPENDIX B: A PRIMITIVE ESTIMATOR FOR THE HEAT CAPACITY AT CONSTANT VOLUME

The heat capacity at constant volume is given by the fluctuation of thermodynamic derivatives 



  1 ∂ 2Z 1 ∂Z 2 2 − 2 Cv = k B β Z ∂β 2 Z ∂β   (B1) = kB β 2 [ E 2 − E2 ]. The first term in Eq. (B1) is 1 ∂Z 3NP =− Z ∂β 2β +

 ! B − 2 + 3β 2 C + D wTI/GSF , (B2) wTI/GSF  β 1

where B=−

P N  mi P  i=1

2¯2

(ri,s − ri,s+1 )2 ,

(B3)

s=1

P P ¯2  1  2 (∇ V ) , D = − V (rs ) i,s 24mi P 3 s=1 P s=1 i=1 (B4) for the TI method and

C=−

N 

C=−

N  i=1

P /2 ¯2  [α(∇i V (r2s−1 ))2 9mi P 3 s=1

+ (1 − α) (∇i V (r2s ))2 ], D=−

P /2 2  [V (r2s−1 ) + 2V (r2s )] 3P s=1

for the GSF algorithm. The second term in Eq. (B1) is



3N P 3N P 2 1 1 ∂ 2Z = + + 2 2 Z ∂β 2β 2β wTI/GSF   !

3N P + 2 B × (6 − 9N P ) βC + β3  3N P 2 (B6) − D + A wTI/GSF , β where

 B 2 A = − 2 + 3β C + D . β

(B7)

This higher-order expression for the heat capacity is expected to inherit the same convergence problems as its second-order counterpart. APPENDIX C: STAGING ALGORITHM FOR OPEN PATH INTEGRAL MOLECULAR DYNAMICS

In this appendix, we present the main equations of the OPIMD staging algorithm.42 The momentum distribution is a single-particle property. Therefore, we limit our discussion to a single particle of mass m in three dimensions. The effective potential for an open path integral in primitive variables reads Veff =

P  m s=1

2

ωP2 (rs − rs+1 )2

[V (r1 ) + V (rP +1 )]  V (rs ) + . 2P P s=2 P

+

(C1)

Note that the effective potential for ordinary PIMD can be recovered from Eq. (C1) by closing the open chain (r1 = rP +1 ). A transformation to staging variables is then used to diagonalize the free-particle contribution. The staging variables y = {y1 , y2 , . . . , yP +1 } are defined by 1 (r1 + rP +1 ) , 2 1 ys = rs − [(s − 1) rs+1 + r1 ]; s = 2, . . . , P , s (C2) yP +1 = (r1 − rP +1 ) , y1 =

where r = {r1 , r2 , . . . , rP +1 } denotes the primitive variables describing the open polymer. Note that the mode variable |yP +1 | is the end-to-end distance, which determines the momentum distribution. The inverse transformation from staging to primitive variables is 1 r1 = y1 + yP +1 , 2  P

 (P /2 − s + 1) s−1 rs = y1 + yt ; yP +1 + P t −1 t=s s = 2, . . . , P ,

(B5)

1 rP +1 = y1 − yP +1 . (C3) 2 Forces are usually evaluated in Cartesian coordinates and then transformed back to staging forces fys . To this end, one uses

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064104-15

Improving path integral convergence

J. Chem. Phys. 135, 064104 (2011)

the chain rule

0.3

fys = −

∂V ∂ys

 =

P +1

 t=1

∂rt ∂ys

 frt ,

where frt = −∂V /∂rt is the primitive force on the t-th bead. The partial derivatives ∂rt /∂ys can be evaluated using Eq. (C3) to yield fy1 =

P +1  t=1



0.2 0.15

0.05 0



This recursion is initiated from the second mode force (fy2 ) and terminates at the mode force fyP . Finally, the effective potential in staging coordinates reads  ms m 2 2 ωP yP +1 + ωP2 y2s = 2P 2 s=2 P

[V (r1 (y)) + V (rP +1 (y))]  V (rs (y)) + , 2P P s=2 P

+

P=32 EXACT

0.1

frt ,

s−2 fys−1 ; s = 2, . . . , P , fys = frs + s−1  P

 P /2 − t + 1 1 frt . (C5) fyP +1 = (fr1 − frP +1 ) + 2 P t=2

Veff

0.25

(C4) N (yP+1)



(C6) where ms = [s/(s − 1)]m. In order to carry out the dynamics, the OPIMD Hamiltonian is formed by adding a kinetic energy term P +1 2   s=1 ps /2ms (the ms are the arbitrary dynamical masses) to the effective potential, Eq. (C6). The harmonic contribution to 2 harm = 0 and fharm mode forces are fharm yP +1 = −mωP yP +1 /P , fy1 ys 2 = −ms ωP yP for s = 2, . . . , P . This free particle analysis suggests the choice mP +1 = m/P , m1 = m, and ms = ms for s = 2, . . . , P for the dynamical masses. This choice for mP +1 is particularly important for efficient sampling of the momentum distribution since it controls the dynamics of the relevant variable yP +1 in OPIMD. As an illustrative example, the open path end-to-end distance distribution estimator ρ (x) = δ (x1 − xP +1 − x) was computed for the harmonic potential, V (x) = mω2 x 2 /2, at β¯ω = 10. Under these physical conditions, the system is ground-state dominated and the exact expression for the density matrix is

 # mω mω exp − x2 ρ (x) = 4π ¯ tanh(β¯ω/2) 4¯ tanh(β¯ω/2) #   mω mω 2 (C7) ≈ exp − x . 4π ¯ 4¯ The numerical results are shown in Fig. 10 and compared to the analytical expression. The agreement between the numerical and analytical curves confirms the correctness of our implementation. The analytical momentum distribution is ulti$ 2 mately obtained using n (q) = ρ (x) e−iqx/¯ dx ≈ e−¯q /(mω) (not shown).

-4.0

-2.0

0.0

2.0

4.0

yP+1 FIG. 10. The open path end-to-end distance distribution for the harmonic potential V (x) = mω2 x 2 /2 with β¯ω = 10. Four million staging open PIMD steps were accumulated using a time step of 0.01 and 32 beads. Dots: Numerical (P = 32). Dotted-dashed line: Exact.

APPENDIX D: THE TAKAHASHI-IMADA ESTIMATOR FOR THE RADIAL DISTRIBUTION FUNCTION

It is interesting to study approximations to the TI estimator for the radial distribution function, Eq. (20), given the excellent convergence properties of the TI estimator for internal energy. To simplify the notation, let us consider a particle in one dimension. Equation (20) then reads  1 g (x) ≈ dx1 . . . dxP ZP  

 P  1  1 ∂s V 1+ ∂s δ(xs − x) × P s=1 P 12mωP2 (2)

× e−β(Veff +VTI ) ,  (2) where Veff = Ps=1 [ m2 ωP2 (xs − xs+1 )2 + V P(xs ) ] and VTI  = (1/24mωP2 ) Ps=1 [ ∂Ps V ]2 . We now perform a partial integration of this expression to eliminate the partial derivative of the delta funcion to get  P wTI  1 g (x) ≈ [1 + C (xs , xs−1 , xs+1 ; β)] wTI  P s=1 × δ(xs − x) ,

(D1)

where 1 C (xs , xs−1 , xs+1 ; β) = − 12mωP2



∂ss V P



−β

∂s V P





 ∂s V × mωP2 (2xs − xs−1 − xs+1 ) + P

× 1+

1 12mωP2



∂ss V P

  (D2)

is a non-local TI correction factor that involves the Hessian matrix of the potential. In the derivation, the boundary terms

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064104-16

A. Perez and M. E. Tuckerman

J. Chem. Phys. 135, 064104 (2011)

3.0 Hess P=8 wout P=8 Hess P=16 wout P=16 Hess P=32 wout P=32 2nd P=128 2nd P=16

2.5

P(r)

2.0 1.5 1.0 0.5 0.0

1.0

1.2

1.4

1.6

1.8

2.0

r (a.u.) FIG. 12. Normalized position probability distribution versus the Trotter number for the Morse potential at 300 K according to TI expression Eq. (D1). Equation (D2), which includes Hessian terms (Hess, in legend), and Eq. (D3), which neglects them (wout, in legend), are shown.

Figure 12 shows that TI results without Hessian are practically converged at 16 beads, confirming our expectations. The above expressions can be easily generalized for the many-body case in three dimensions. The expectation value of the radial distribution function of a pair of particles i and j is  P wTI  1 g(r) ≈ [1 + C(rs , rs−1 , rs+1 ; β)] (4π r 2 ρwTI ) P s=1

FIG. 11. Normalized position probability distribution for the harmonic potential V (x) = mω2 x 2 /2 with β¯ω = 10. Including (top) and neglecting (bottom) Hessian terms in TI expression. Second-order (2nd, in legend) curves are shown for comparison.

do not contribute because each term is Boltzmann-suppressed. If we neglect the Hessian in Eq. (D2), then we obtain an approximate correction factor, 

 ∂s V 1 ˜ −β C (xs , xs−1 , xs+1 ; β) = − P 12mωP2  

∂s V . (D3) × mωP2 (2xs − xs−1 − xs+1 ) + P Fig. 11 shows the convergence of the TI position probability distribution with the Trotter number for the harmonic oscillator at β¯ω = 10 with (top) and without (bottom) Hessian terms (second derivative of the potential for 1D systems). Curiously, the strange double hump in the TI curve at P = 4 when the Hessian is included disappears without the Hessian terms, and the curve is generally better. Interestingly, for Trotter numbers greater than 8 beads, the TI curves with and without Hessian are almost identical, which suggests that the approximate expression Eq. (D3) may be of some use in ab initio PIMD calculations of bulk systems, where evaluation of the local Hessian is prohibitive. We tested our conjecture for the Morse potential, which has a positiondependent Hessian, using the same conditions as in Sec. III B.

× δ(|ri,s − rj,s | − r) ,

(D4)

where the full TI correction factor reads

 1  −1 ∇kk V C(rs , rs−1 , rs+1 ; β) = − m P 12ωP2 k=i,j k

 β  −1 ∇k V + m P 12ωP2 k=i,j k 

 ∇k V × mk ωP2 (2rk,s − rk,s−1 − rk,s+1 ) + P  



N  ∇ik V ∇i V 1 + , (D5) P P 12mi ωP2 i=1 with V = V ({rs }). The approximate TI correction is obtained by neglecting all Hessian terms in Eq. (D5), and can be implemented easily in ab initio PIMD calculations. 1 R.

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064104-17

Improving path integral convergence

J. Chem. Phys. 135, 064104 (2011)

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