Reversible integrators for basic extended system molecular dynamics. ALESSANDRO ... gator for the Newtonian dynamics of an N-particle atomic system originates ..... mine the constraints forces, in such a way that both the atomic coordinates ...
MOLECULAR PHYSICS, 1999, VOL. 97, NO. 7, 825 ± 832
Reversible integrators for basic extended system molecular dynamics ALESSANDRO SERGI1 , MAURO FERRARIO1 * and DINO COSTA2 1 Istituto Nazionale per la Fisica della Materia (INFM) and UniversitaÁ degli Studi di Modena e Reggio Emila, Dipartimento di Fisica, Modena, Italy 2 Istituto Nazionale per la Fisica della Materia (INFM) and UniversitaÁ degli Studi di Messina, Dipartimento di Fisica, Messina, Italy (Received 4 January 1999; revised version accepted 20 May 1999) Starting from a family of equations of motion for the dynamics of extended systems whose trajectories sample constant pressure and temperature ensemble distributions (Ferrario, M., 1993, in Computer Simulation in Chemical Physics, edited by M. P. Allen and D. J. Tildesley (Dordrecht: Kluwer)), explicit time reversible integration schemes are derived through a straightforward Trotter factorization of the dynamic Liouville propagator, along the lines ® rst described by Tuckerman, M., Martyna, G. J., and Berne, B. J., 1992, J. chem. Phys., 97 , 1990. The original Andersen’s constant-pressure dynamics are recovered in the limit of zero coupling with the Nose thermostat. Reversible integration schemes are derived as a generalization of the velocity Verlet algorithm, with direct handling of the velocity dependent forces in such a way that both predictions and relative iterative corrections are not required. For the sake of clarity both the equations of motion and the Trotter factorization are kept to the basic level. The proposed structure can accommodate easily, when needed, complications such as multiple timesteps and more e ective thermostats (Nose± Hoover-chain). Finally, an application is made to a model molecular system subjected to holonomic constraints by means of the shake algorithm. In the constant pressure case it is no longer possible to avoid using a prediction for the constraint contribution to the volume acceleration; however, recourse to a minimal iteration scheme still achieves excellent overall behaviour for the proposed integration algorithm, with no perceptible di erence from the unconstrained case.
1.
Introduction
The coupling of a physical system with a thermal or pressure reservoir can be regarded as a thermodynamic constraint. The extended system approach in molecular dynamics simulation, pioneered by Andersen in his 1980 seminal paper for the constant pressure case [1], treats the thermodynamic constraints in a dynamic way, where the baths are represented solely by a few additional degrees of freedom. In this context, one introduces an extended Hamiltonian, in which the physical and the extra degrees of freedom are treated at the same level, and the equations of motion are derived by standard methods [2]. These equations are then cast in a non-canonical form by means of a coordinate transformation in order to sample the properties of the physical system from the appropriate statistical ensemble by numerical integration over uniform time separations [3]. Alternatively,
* Author for correspondence.
following Hoover’s approach [4, 5], the extended equations of motion are not derived from a Hamiltonian and are postulated directly in non-canonical form. Within this general scheme, several di erent equations of motion have been proposed, all leading to an ensemble equilibrium probability density for the physical system, after integration over the extra variables [4, 6± 12]. In fact, the form of the kinetic term for the extended variables is not ® xed and there is freedom for di erent choices that give rise to di erent dynamic trajectories, which equally sample in a correct way the phase space of the desired ensemble [13, 14]. In this paper we refer to a particular set of equations of motion, previously introduced by one of the authors [11], which combines constant pressure with constant temperature thermodynamic constraints. This set ful® ls the requirement that, following Andersen [1], reference can be made to a Hamiltonian system via a noncanonical transformation. In general the extended system procedure requires the introduction of velocity dependent forces, which are dif® cult to integrate numerically in the context of a Verlet
Molecular Physics ISSN 0026± 8976 print/ISSN 1362± 3028 online # 1999 Taylor & Francis Ltd http://www.tandf.co.uk/JNLS/mph.htm http://www.taylorandfrancis.com/JNLS/mph.htm
826
A. Sergi et al.
scheme without spoiling the time reversal properties of the algorithm. The Trotter formalism [15± 17] provides for an elegant solution to these problems. Indeed, it has been shown that an appropriate breakup of the propagator for the Newtonian dynamics of an N-particle atomic system originates the well known velocity Verlet algorithm, providing at the same time a theoretical foundation for such a simple and commonly used integration scheme [17]. Along this route the recourse to an iterative procedure (which requires a guess ® rst and a correction later of the atomic velocities) is de® nitely avoided. Moreover, this approach leads to a procedure that allows one to derive in a systematic way multiple timestep strategies in molecular simulations, highly suitable for systems with strong separations in timescales [17, 18]. The aim of this paper is to show how a rather simple, basic, molecular dynamics reversible integrator can be deduced from the family of equations mentioned above [11], by means of a Trotter symmetric factorization of the Liouville propagator. Recent applications of the Trotter formalism to extended system dynamics have appeared in the literature. Procacci and Berne [18] and Marchi and Procacci [19] proposed an algorithm for constant pressure dynamics, specialized for application to ¯ exible molecules, that propagates scaled coordinates rather than real ones. Martyna et al. [20] proposed a number of explicit reversible integrators starting from a particular set of equations of motion [12]. This set di ers from that of [11] mainly in the dynamic behaviour of the extra volume variable, and a non-canonical transformation that casts the equations of [12] in canonical form has not been de® ned; in particular, the latter cannot be reduced to the Andersen’s equations of motion for the constant pressure, constant enthalpy case. Now it is known that, starting from a given set of equations of motion, the integrators that can be developed are not unique, but depend on the speci® c way the propagator is factorized. As a matter of fact, the use of either complicated equations of motion or sophisticated factorization can result in somewhat di cult-to-handle algorithms. The integrators we develop here share a common, simple structure, ultimately based on a unique choice for the factorization of the propagator. Such integrators can be extended readily, with minor modi® cations, to deal with systems subjected to holonomic constraints and, as we will show in subsequent sections, they are all stable and exhibit fairly good ¯ uctuation properties. The integrators are tested both on a Lennard-Jones ¯ uid and on a simple molecular model, comprising of a linear chain of seven centres of force. This latter, previously introduced elsewhere, has been proposed as a candidate for a multi-site, soft-core
approach for the simulation of model liquid crystalline systems [21].
2.
Reversible algorithms for extended system dynamics
In [11]the equations of motion proposed by Andersen [1] for constant pressure molecular dynamics simulation were combined with the Nose± Hoover approach for constant temperature [3]. In particular, by introducing ® rst an extended Lagrangian in scaled coordinates, a system of Hamiltonian ® rst-order equations of motion was derived for the canonical scaled coordinates and momenta and rearranged in terms of real variables by means of a non-canonical transformation. Then it was proved that the molecular dynamics trajectory samples the isobaric± isothermal ensemble. Consider a system composed of N particles where N N f r g ;f p g are the 3N particle coordinates and 3N momenta. We introduce the extra degree of freedom described by the coordinate ² and momentum P ² to play the role of the heat bath at a given temperature T , with M ² the related inertial parameter. The volume V and the momentum P V represent the reservoir at the pressure Pext , with M V the corresponding inertial parameter. The Hamiltonian, when expressed using real non-canonical variables, transforms into the conserved energy of the extended system: H
ˆ
X
N
iˆ 1
2
pi
2m i
‡
2
N
F… r ; V †
PV 2M V
‡
P ext V
‡
2
P²
‡
2M ²
‡
gk B T ²:
…
1†
The coupling of the system with the bath variables is achieved through the forces F ² and F V that act on the extended variables ² and V , respectively [11]: N
F² … p ; PV †
N
N
FV … r ; p ; V †
ˆ
ˆ
X
N
iˆ 1
1 3V
2
pi
mi X
2
‡
PV MV
¡
g kB T ;
pi
mi
iˆ 1
2†
…
3†
!
2
N
…
r i Fi
‡
¡
P ext
N
¡
@F… r ; V † @V
;
where F i ˆ ¡ @ F =@ r i is the force on particle i. The choice g ˆ 3N ‡ 1 stems from the fact that, including the barostat, the number of thermostated degrees of freedom is 3N ‡ 1 and the fact that the equations of motion for the NPT dynamics to be solved numerically are written using real variables [11]:
827
Reversible integrators for extended molecular dynamics pi
r_ i
ˆ
p_ i
ˆ
Fi
V_
ˆ
PV ; MV
P_ V
ˆ
²_
ˆ
P_ ²
ˆ
‡
ri
¡
pi
mi
FV P²
P² M²
¡
i
pi
1; N ;
ˆ
1 PV 3V M V
i
ˆ
1; N ;
P² ; M²
PV
¡
M²
1 PV 3V M V
;
F² :
…
4†
It is known that, for a system without severe timescale separations, these equations are robust with respect to the choice of the inertial parameters and that their integration does yield the correct sampling of the statistical mechanical ensemble [11]. The original Andersen’s equations of motion for the NPH ensemble [1]can be obtained from the above set in the limit … ²; P ² ; F ² † ! 0. By denoting G ˆ f r N ; p N ; V ; P V ; ²; P ² g the 6N ‡ 4 variables in the phase space of the system and using the relation C _ ˆ iL^ C , the Liouville operator iL^ corresponding to expression (4) can be obtained in a straightforward way: iL^
ˆ
iL^ 0 ‡ i L^ 1 ‡ i L^ 2 ‡ iL^ 3 ‡ iL^ 4 ‡ iL^ 5 ‡ iL^ 6 ‡ iL^ 7 ‡
ˆ
X
iL^ 8 N
‡
‡
‡
X
mi @ ri N
Fi
iˆ 1
‡
@ pi
¡
X
t= 2†
ˆ
P² … 0†
²…
t= 2†
ˆ
² … 0†
PV …
t= 2†
ˆ
PV … 0† exp
pi…
t= 2†
ˆ
p i … 0† exp
t= 2†
ˆ
t†
ˆ
2
V … 0†
ri…
N
pi
P²
P² @ ¡ M² @ pi @
iˆ 1
F²
‡
@
:
@ P²
…
5†
V…
t†
ˆ
p i…
t†
ˆ
V…
ˆ
^9 U
½
^8 U
2
^3 U ^3 U ^9 U ^ ¬… ½ † where U
½
2 ½
2 ½
2 ˆ
½
2
^2 U ^4 U
^7 U ½
2 ½
2
½
2
^1 U ^5 U
½
^6 U ½
2 ½
2
2
^5 U
^6 U
½
2
^4 U
2
^ 0… ½ † U ^1 U ^7 U
;
exp… iL^ ¬ ½ † ;
½ ½
2 ½
2
½
2
^2 U ^8 U
ˆ
0; 9.
ˆ
t P V … t= 2† 3V … 0†
t†
ˆ
²…
P² …
t†
ˆ
P² …
‡
¡
t
2 t
2M ²
FV … 0† ; p² …
t= 2†
t= 2† ;
PV …
t= 2† t= 2†
t PV … 2M V 3V …
‡
t mi
p i…
t= 2†
t PV … t= 2† ; 2M V 3V … t= 2† t
‡
2 t
¡
2M ²
t
2
2M ²
6† t= 2†
t= 2† FV …
P² …
t P² … 2M ²
‡
‡
t
2
t†
Fi …
p² …
‡
t
¡
t= 2†
t
‡
t= 2† ;
PV …
2MV
½ ²…
t= 2†
2M V
t
t= 2†
exp
2
t P² … 2M ²
¡
¡
t= 2†
PV …
2
t= 2† ;
P² …
2M V
t= 2†
pi …
½
…
¬
t†
t
2M²
‡
exp PV …
F² … 0† ;
Fi … 0† ;
exp
1 PV @ 3V M V @ p i
pi
P² @ M ² @²
‡
M ² @ PV
X
N
2
r i … 0† exp
Starting from this ten-term decomposition of the Liouville operator the approximate discrete time propagator can now be derived by means of a Trotter factorization along the route shown in [17]. ^…½† G
‡
‡
t
‡
V…
PV @ 1 PV @ ‡ ri 3V M V @ r i M V @ V
iˆ 1
PV
¡
@ PV
N
iˆ 1
@
@
FV
X
t
P² …
iL^ 9
pi @
iˆ 1
This is our basic choice for the factorization of the ^… ½† G ^ … ¡ ½ † ˆ 1) orig propagator. The time reversibility ( G inates from the symmetric factorization in equation (6). 3 The error implied by this factorization is of order ½ . To obtain the explicit form of the integration scheme ^ … ½ † in equation (6) to the phase we apply the operator G space point G . In particular, the direct translation technique [20], which consists of the sequential application ^ ¬ … ½ † to the point G , yields a version of the of each U algorithm that can be mapped directly into the lines of a computer code. Setting ½ ˆ t the algorithm for the case of NPT dynamics has the following structure:
F² …
¡
t PV … t= 2† ; 2MV 3V … t†
t†
t= 2† ; t= 2† ; t† :
…
7†
828
A. Sergi et al.
In the above expressions no physical meaning can be associated with the half timestep values at t= 2 which result from a partial application of the evolution ^. operator G Starting from the more general case of the NPT ensemble, the derivation of the reversible algorithms for the NV T and NPH dynamics follows in a straight^8 ˆ U ^7 ˆ U ^6 ˆ U ^3 ˆ U ^ 2 ˆ 1 in forward way. Setting U equation (6) one recovers the discrete propagator for the NV T dynamics. The corresponding reversible algorithm has been presented elsewhere [17, 22]although in slightly di erent forms. In order to derive explicitly the constant pressure algorithm, given by the original Andersen’s equations ^ 10 ˆ U ^9 ˆ U ^8 ˆ U ^ 5 ˆ 1 in [1], it is su cient to set U the full discrete time propagator of equation (6). The explicit form of the algorithm, which resembles the original velocity Verlet algorithm, reads ri…
t†
ˆ
t= 2† t= 2†
t PV … MV 3V …
r i … 0† exp 2
p i…
t†
ˆ
p i … 0† exp
t
‡
V…
t†
ˆ
2
V … 0†
‰
Fi … 0†
‡
t= 2†
t† Š
exp
Fi …
t†
ˆ
t= 2†
ˆ
P V … 0†
V…
t= 2†
ˆ
V … 0†
H
ˆ
iˆ 1
2
pi
2m i
‡
N
t= 2†
ˆ
PV … 0† exp
t†
ˆ
r i … 0† exp
ri …
‡
1 3V 0 …
¡
†
‡
1 3V
…
pi…
t†
t†
ˆ
‡
2
t PV … t= 2† ; 2MV 3V … t†
…
‡
8†
t= 2† ;
…
9†
F… r ; V †
2
‡
PV 2M V
‡
F ² … 0† ; t P² … 2M s
¡
t PV … MV
p i … 0† exp
t= 2†
t
‡
2
F V … 0† ;
t= 2† t
¡
…
2
¬
1†
¡
P V … t= 2† MV
t= 2†
P² …
M²
Pext V :
t t Fi … 0† exp PV … 2m i 2M V
…
10†
It is worthwhile to make a direct comparison with the alternative NPT equations of motion of Martyna et al. [12]. In [12]temperature control was achieved by means of a Nose± Hoover chain [23]; here, for a more direct comparison, we derive a reversible algorithm for a single Nose thermostat variable through a factorization
MV
t= 2† ; t = 2†
¬ PV …
t= 2†
P² …
M² t
2
‰
Fi … 0†
t P² … M²
²…
t†
ˆ
² … 0†
P² …
t†
ˆ
P² … 0†
V…
t†
ˆ
V … 0† exp
PV …
t†
ˆ
PV … 0† exp
‡
t
2
‡
‰
exp
t
t
¡
M²
‡
F² …
MV t M²
‡
t
2M ²
t
2 ;
t= 2† ;
3PV … t= 2†
¡
¡
t= 2†
P² …
‡
F ² … 0†
2
FV … 0†
t† Š exp
Fi …
‡
t= 2†
¬ PV …
‡
1
t
¡
MV
FV … 0† ;
t PV … 2M V
mi
2
p i … 0† exp
‡
t
‡
t
‡
2
and FV is given by equation (3). The NPH Hamiltonian, expressed using the real non-canonical variables, transforms into the conserved energy of the system N
PV …
t= 2† ;
PV …
X
P² … 0†
‡
t
‡
ˆ
‡
‰ F V … 0† ‡ F V … t† Š ; 2 where we have de® ned the half time step values
PV …
t= 2†
t= 2† ; t= 2†
t PV … 2MV
t PV … MV
‡
PV … 0†
¡
t
P² …
t p i … 0† mi
‡
t t PV … Fi … 0† exp 2mi 2MV 3V …
‡
similar to that in equation (5). The inclusion of a Nose± Hoover chain thermostat would require substitution of the terms U 8 U 9 and U 9 U 8 with the appropriate propagator and, in turn, adoption of either a smaller timestep within a multiple-step approach or a higher order scheme to avoid numerical problems [20]. The algorithm then reads:
P² …
FV … P² …
t† ; ;
t= 2† t† Š
t= 2† ; …
11†
829
Reversible integrators for extended molecular dynamics where ¬ and FV
ˆ
ˆ
…
1
X
N
N ‡ 1† = N , for F² is valid the expression (2), 2
N
iˆ 1 "
‡
pi mi
1 3V 3V
X
N
iˆ 1
!
2
pi
mi
‡
Fi r i
¡
@ F … r; V † @V
#
¡
P ext : …
12†
The energy H of the extended system is again a constant of the motion and formally retains the same expressions given in equation (1). 3.
S ystems with holonom ic constraints
The algorithms shown in the previous section can be extended to treat systems subjected to holonomic constraints. Starting from the algorithm written in explicit form (equation (8)), this task can be achieved by applying the constraints exactly in the same way as in the case of the velocity Verlet algorithm. This amounts to a two-step a posteriori procedure that solves for the unknown Lagrange multipliers … ¶ j ; j ˆ 1; K † , that determine the constraints forces, in such a way that both the atomic coordinates and momenta satisfy exactly (i. e. with any desired accuracy) at all times the holonomic N constraints imposed on the system, ¼ j … f r g † ˆ 0 … j ˆ 1; K † . In order to advance coordinates and momenta from time 0 to time t one has to calculate a ® rst set of Lagrange multipliers … ¶ j … 0† ; j ˆ 1; K † to correct the atomic positions by requiring N K ¼ j … f r … t; f ¶ … 0† g † g † ˆ 0 … j ˆ 1; K † and successively a second set … ¶^j … t† ; j ˆ 1; K † to correct the atomic N K momenta by requiring ¼_ j … f p … t; f ¶^ … t† g † g † ˆ 0 … j ˆ 1; K † [24, 25]. Now it turns out that, within the approach for molecular systems known as the atomic stress formulation [26, 27], the constraints give a non-trivial contribution to the virial pressure corresponding to j ¶ j @¼ j =@ V . F Obviously the correction a ects the force V … 0† which, in turn, determines the advance in time of the atomic coordinates through the momentum P V … t= 2† . However the constraint forces at time zero can be calculated through the `shake’ procedure only after the free ¯ ight atomic positions are known. A ® rst prediction of F V … 0† can be made using the Lagrange multipliers f ¶^K g computed in the previous step. An iterative procedure is then required in order to solve consistently for the Lagrange multipliers and all the correction terms depending upon them. Since the correction to the virial is of higher order in t, reliable results can be obtained without iterating on the values of f r N … t† g . N Finally, in our scheme the momenta f p … t† g do not depend on F V … t† , and it is su cient to know the value P
of F V … 0† , including the constraint term depending upon K f ¶ … 0† g calculated previously. For this reason there is no need to iterate self-consistently the constraint correcN tions for the momenta f p … t† g and stress calculation. This is at variance with other schemes (e.g., [20], equation (11)) where additional iterations are required due to the fact that f p N … t† g depends on FV … t† at the same time. In order to completely avoid any iterative treatment, that can be very expensive since constraint forces must be recalculated at every step, a di erent approach should be taken. When studying molecular entities a convenient solution could be the molecular virial approach where the barostat variable is coupled solely to the molecular centres of mass [18, 27]. In the particular case of macromolecules the e ective strategy proposed recently by Marchi and Procacci [19] suggests replacing the sti est bonds with rigid constraints while the others are modelled explicitly in terms of intramolecular potentials. Within this approach the constraints concern only small molecular entities, and the di erent timescales in the dynamics are tackled with a multiple timestep breakup of the relevant terms in the propagator. 4.
Results for simple model systems
In order to assess the reliability of the integration scheme in equation (7) we have performed extensive MD simulations both for an atomic system and for a simple molecular model. A further comparative integration via the algorithm in equation (11) has concerned the atomic model. The ® rst system is composed of N ˆ 1000 LennardJones particles enclosed in a cubic box with periodic boundary conditions. The potential is truncated and shifted at the minimum of the well depth. In the usual reduced units, m i ˆ 1 for the mass of each particle and " ˆ ¼ ˆ 1 for the Lennard-Jones parameters, the external temperature and pressure are held ® xed at T ˆ 1:5 and P ˆ 5:0, respectively, corresponding to a dense liquid phase for the sample. All calculations are made with a ® xed timestep t ˆ 2: 5 10 3 . In the same units the mass of the Nose thermostat is M ² ˆ 200. During the simulations we have tested several values of the inertial masses for the dynamics of the volume 5 M V ranging from 10 to 10. The results we report below refer to M V ˆ 1 10 4 and M V ˆ 100 for the algorithms in expressions (7) and (11), respectively; these values for the inertial parameters yield similar timescales for volume ¯ uctuations. As a general strategy, starting from the lattice con® guration, the system is allowed to melt through typically 50 000 MD NV E steps. Then, upon sequential compression by means of MD runs at constant pressure, ¡
¡
¡
830
A. Sergi et al. Table 1. Equilibrium properties of the Lennard-Jones ¯ uid and of the system subjected to holonomic constraints, as obtained through the reversible integrators presented here. The average values of the conserved energy H , the total E, potential F , and kinetic K energies and of the volume V and of the pressure P are displayed, along with the respective ¯ uctuations (indicated by the ¯ s ) corresponding to one standard deviation. The ® rst two columns refer to the behaviour of the atomic system, as generated by the algorithms of equations (7) and (11), respectively, while the last refers to the dynamics of the molecular model obtained via equation (7). Eqn (7) H
dH
E dE F dF
Figure 1. NPT molecular dynamics behaviour of the conserved quantities for the Lennard± Jones system: … a † conserved energy H (equation (1)) relative to the reversible integrator (equation (7)) based on the Nose± Andersen equations of motion (4); and … b † conserved energy H (equation (1)) relative to the algorithm (11) based on the Martyna et al. [12] equations. Note that the vertical scale is the same for both panels.
the system is taken to the prescribed thermodynamic conditions. From this equilibrated con® guration, long NPT trajectories are generated up to 5 105 timesteps. Reported data refer to these cumulation runs. Our results for the atomic system are shown in ® gures 1 and 2. Detailed numerical values of averages and thermodynamic ¯ uctuations for all relevant observables are reported in table 1. Figure 1 shows the behaviour of the conserved energy H , equation (1), for the two algorithms of equation (7) and equation (11). As seen in ® gure 1, both algorithms yield a stable MD evolution and the conserved quantities do not exhibit any tendency to drift away from the average values. Generally we have obtained a conservation as good as 2± 3 parts over 105 . The behaviour of the two integrators can be regarded as essentially identical over the entire timespan corresponding to about half a million steps. When looking at the trajectories generated with the algorithm in equation (7) for the volume, the kinetic energy and the trace of the internal pressure tensor (i.e., the ® rst term of F V in equation (3)) one ® nds
K dK P dP V dV
8.996 75 0.000 12 2.985 0.065 0.736 0.035 2.23 0.06 4.99 0.18 1505.0 13.0
Eqn (11) 8.996 59 0.000 13 2.986 0.067 0.736 0.035 2.25 0.05 4.99 0.18 1505.0 13.0
Eqn (7) plus constraints 1163.912 0.008 136.7 3.5 21.9 1.3 68.9 1.9 14.93 0.86 6240.0 84.0
behaviour which is identical to that obtained from equation (11). The ¯ uctuations exhibit no apparent regularities and, as can be seen from table 1, the algorithms reproduce the same averages correctly. In general, the averages obtained through equation (7) largely are independent of the exact value of the inertial parameter for the dynamics of the volume variable. Conversely, we have observed that, by using a mass M V ˆ 10 in the algorithm of equation (11), the reduced pressure exhibits long period ¯ uctuations, while smaller values can even give rise to some instability problems. Possibly such problems would not have been observed had we used Nose± Hoover chains to thermostat the system. In order to give further evidence of the equivalence among the ensemble averages calculated through the dynamics from equations (7) and (11), ® gure 2 shows the distributions of the kinetic energy, the volume and the momentum obtained via the di erent approaches. All curves compare exactly with each other, and for clarity sake the vertical scales have been shifted. The distribution of the momenta is also in excellent agreement with the theoretical (Maxwell± Boltzmann) expectation, also shown in ® gure 2. The molecular model is a system of 500 linear rigid molecules in an orthogonal box with separate periodic boundary conditions in the three Cartesian directions.
831
Reversible integrators for extended molecular dynamics
core, multi-site approach liquid crystalline behaviour in model mesogens. A purely mechanical analysis of the system has shown evidence of a rich phase diagram, including solid, smectic, nematic and isotropic phases. In this context we are not concerned with a similar analysis. Instead, we have assessed the reliability of our approach in the presence of holonomic constraints by generating long trajectories at ® xed external temperature and pressure. In order to detect any tendency to numerical drift of the conserved energy H , the simulations are performed at rather high reduced temperature and pressure, T ˆ 46 and P ˆ 14: 8. If we use the values of the molecular parameters of [21] ( m ˆ 1: 99 23 22 10 g, ¼ ˆ 3:9 AÊ, and " ˆ 6: 0 10 J) we obtain the corresponding real values of T ˆ 2000 K and P ˆ 1:5 kbar. Under these thermodynamic conditions, the system samples a completely disordered, isotropic liquid phase. The timestep chosen is t ˆ 0: 001 (corresponding to 1 fs) in order to treat properly the fast librational motions in the system. The stability and the ¯ uctuation properties of the integration scheme in equation (7) are assessed for different values of the inertial masses for the thermostat and barostat variables, ranging from 1 to 100 for M ² and from 0.0001 to 0.01 for M V . As for the pure Lennard-Jones ¯ uid, the NPT simulations start from a well equilibrated con® guration and the analysis is made over 5 quite a long time, usually ranging from 5 10 to 6 2 10 MD steps. The results for the molecular system are shown in ® gure 3. Averages and thermodynamic ¯ uctuations of several observables are reported in table 1 along with the atomic system values. Figure 3 shows the behaviour of the constant of motion, equation (1), as obtained by setting M ² ˆ 100 and M V ˆ 0:001. We have detected no signi® cant di erences with the atomic case: no drift a ects the constant of motion on a time interval as large as two million steps, and the conservation is of 2 parts over 105 , similar to that for the other system. The distribution of the volume variable, shown in ® gure 3, also is comparable with those obtained for the atomic system. From the data so far collected we conclude that the inclusion of constraints does not alter dramatically the overall behaviour of the proposed integration schemes. Moreover, a careful choice of the inertial parameters yields reliable results, although the iterative procedure is limited to only one correction step, as detailed in the previous section. ¡
Figure 2. Distribution of … a† the kinetic energy, … b† the volume, and … c† the linear momentum for the NPT dynamics of the Lennard± Jones system. The curves are shifted for visualization. On each panel, the lower and upper curves refer to algorithms (7) and (11), respectively. The theoretical Maxwell± Boltzmann distribution of the momenta is also displayed on top in the panel … c† .
Each molecule is composed of seven equivalent centres of force uniformly distributed from end to end, each being a Lennard-Jones site. As in the previous case, the potential is truncated and shifted at the bottom of the well depth. Two equal masses are positioned at the molecule ends, while the other ® ve sites are massless centres of force. Adjacent sites are constrained at a distance 0: 6¼ , where ¼ is the Lennard-Jones length parameter. This class of models was introduced by Paolini et al. [21] in order to characterize by means of a soft-
¡
5.
Conclusion
We have derived basic NPT reversible algorithms using the Trotter symmetric factorization of the time propagator for the relevant equations of motion. In particular we have shown that the reversible algorithm
832
Reversible integrators for extended molecular dynamics the June 98 Erice School on Computer Simulation of Liquid Crystals. Some preliminary calculations performed by Dr A. Fiorino during the early stage of this work are gratefully acknowledged. References
Figure 3. NPT molecular dynamics for the system with holonomic constraints: … a† behaviour of the conserved energy H (equation (1)) for the Nose± Andersen equation of motion (4); and … b † distribution of the volume.
based on the equations of motion of [11] are robust and easy to implement. We have derived as a special case the reversible integrators for Andersen NPH dynamics, and we have proved that a simple factorization of the relevant Liouville propagator is able to give stable dynamics in all cases under study. All the integration schemes so far introduced are suited to the study of atomic systems. Their extension, within the atomic stress formulation, to the treatment of systems subjected to holonomic constraints is feasible with good results on a practical level, even if the development of a truly reversible scheme would deserve further investigation. A.S. and D.C. would like to thank Dr Piero Procacci for very stimulating discussions and comments during
[1] Andersen, H. C., 1980, J. chem. Phys., 72, 2384. [2] Goldstein, H., 1980, Classical Mechanics, 2nd Edn (Reading, MA: Addison-Wesley). [3] Noseí, S., 1986, Molec. Phys., 57, 187. [4] Noseí, S., 1991, Progr. theoret. Phys. Suppl., 103, 1. [5] Hoover, W. G., 1986, Phys. Rev. A, 34, 2499. [6] Hoover, W. G., 1985, Phys. Rev A, 31, 1695. [7] Parrinello, M., and Rahman, A., 1980, Phys. Rev. L ett., 45, 1196. [8] Noseí, S., and Klein, M. L., 1983, Molec. Phys., 50, 1055. [9] Ferrario, M., and Ryckaert, J. P., 1985, Molec. Phys., 54, 587. [10] Melchionna, S., Ciccotti, G., and Holian, B. L., 1993, Molec. Phys., 78, 533. [11] Ferrario, M., 1993, Computer Simulation in Chemical Physics, edited by M. P. Allen and D. J. Tildesley (Dordrect: Kluwer) p. 153. [12] Martyna, G., Tobias, D. J., and Klein, M. L., 1994, J. chem. Phys., 101, 4177. [13] Jellinek, J., 1988, J. phys. Chem., 92, 3163. [14] Jellinek, J., and Berry, R. S., 1988, Phys. Rev A, 38, 3069. [15] Suzuki, M., 1985, J. math. Phys., 26, 601. [16] Forest, E., and Ruth, R. D., 1990, Physica D, 43, 105. [17] Tuckerman, M., Martyna, G. J., and Berne, B. J., 1992, J. chem. Phys., 97, 1990. [18] Procacci, P., and Berne, B. J., 1994, Molec. Phys., 83, 255. [19] Marchi, M., and Procacci, P., 1998, J. chem. Phys., 109, 5194. [20] Martyna, G. J., Tuckerman, M., Tobias, D. J., and Klein, M. L., 1996, Molec. Phys., 87, 1117. [21] Paolini, G. V., Ciccotti, G., and Ferrario, M., 1993, Molec. Phys., 80, 297. [22] Ferrario, M., Fiorino, A., and Ciccotti, G., 1997, Physica A, 240, 268. [23] Martyna, G. J., Klein, M. L., and Tuckerman, M., 1992, J. chem. Phys., 97, 2635. [24] Ryckaert, J. P., Ciccotti, G., and Berendsen, H. J. C., 1997, J. comput. Phys., 23, 327. [25] Andersen, H. C., 1983, J. comput. Phys., 52, 24. [26] Melchionna, S., and Ciccotti, G., 1997, J. chem. Phys., 106, 195. [27] Ciccotti, G., and Ryckaert, J. P., 1986, Comput. Phys. Rep., 4, 345.
