JOURNAL OF CHEMICAL PHYSICS
VOLUME 110, NUMBER 8
22 FEBRUARY 1999
Fluid transport properties by equilibrium molecular dynamics.
I. Methodology at extreme fluid states
D. K. Dysthe,a) A. H. Fuchs, and B. Rousseau
Laboratoire de Chimie Physique des Matériaux Amorphes, Bâtiment 490, Université Paris-Sud,
91405 Orsay Cedex, France
~Received 31 August 1998; accepted 17 November 1998!
The Green-Kubo formalism for evaluating transport coefficients by molecular dynamics has been
applied to flexible, multicenter models of linear and branched alkanes in the gas phase and in the
liquid phase from ambient conditions to close to the triple point. The effects of integration time step,
potential cutoff and system size have been studied and shown to be small compared to the
computational precision except for diffusion in gaseous n-butane. The RATTLE algorithm is shown
to give accurate transport coefficients for time steps up to a limit of 8 fs. The different relaxation
mechanisms in the fluids have been studied and it is shown that the longest relaxation time of the
system governs the statistical precision of the results. By measuring the longest relaxation time of
a system one can obtain a reliable error estimate from a single trajectory. The accuracy of the
Green-Kubo method is shown to be as good as the precision for all states and models used in this
study even when the system relaxation time becomes very long. The efficiency of the method is
shown to be comparable to nonequilibrium methods. The transport coefficients for two recently
proposed potential models are presented, showing deviations from experiment of 0%–66%.
© 1999 American Institute of Physics. @S0021-9606~99!50908-2#
I. INTRODUCTION
try to explore the limits of the GK method, and compare its
efficiency with NEMD.
There has for decades been a serious concern that the
relaxation times of molecular liquids are too long to obtain
representative sampling during a single trajectory by EMD.
A recent study1 of n-butane near its boiling point has shown
excellent agreement between NEMD and GK results. The
typical relaxation times for n-butane at the state point used
are, however, very short compared with the total simulation
time. For larger molecules, higher densities and lower temperatures the relaxation times become much longer. We will
in this work compare a number of relevant relaxation times
for different systems and state points in order to cast some
light on necessary equilibration time and the statistical significance of the GK sampling.
It is well known that the finite system size and periodic
boundary conditions tend to suppress the time correlation
functions of atomic systems making the calculated transport
coefficients system size dependent. Recent studies1 probing
for a system size effect or a long time tail of the time correlation functions in n-butane close to the boiling point have
not observed any significant size effect or deviation from
exponential decay. At this state the time correlation functions
have relaxation times comparable to the sound traversal
times. In the gas phase, the mean time for a molecule to
traverse the simulation cell, is shorter than the relaxation
times of the time correlation functions. It is phenomenologically interesting and methodologically necessary to study the
influence of the system size and periodic boundary conditions.
To our knowledge, no systematic study of the influence
of the integration time step d t and cutoff distance r c of the
Molecular Dynamics ~MD! is becoming ever more widespread as a tool for prediction of thermodynamic and transport properties of molecular liquids. Until now, most simulations and all the methodological work in the field of
transport properties of flexible molecules have been performed for n-alkanes at moderate liquid densities and above
ambient temperature. In order to apply the now well established methods to high density, low temperature liquids and
to gases, there are a number of new precautions to be taken.
We will in this work clarify a number of methodological
issues that either have not been raised before or that need
special attention when applying MD to extreme states.
In the last two decades most progress has been made in
nonequilibrium MD ~NEMD! to calculate transport properties. One obvious reason is to try to increase the signal to
noise ratio in the measurements because the equilibrium
method, the Green-Kubo ~GK! formalism, only measures the
time dependent response of the system to spontaneous fluctuations in the system, another reason is to go beyond the
linear transport regime and study, for example, nonNewtonian rheology. It is believed that NEMD is more effective ~precision versus computer time! than GK when calculating a single transport coefficient, although detailed
comparison is lacking. The advantage of the GK method is
that one can obtain all the phenomenological transport coefficients in one single run with one single method. We are
interested in obtaining a complete picture of the transport
mechanisms in molecular fluids and have therefore chosen to
a!
Present address: Department of Physics, University of Oslo, P.O. Box 1048
Blindern, N-0316 Oslo, Norway; electronic mail: d.k.dysthe@fys.uio.no
0021-9606/99/110(8)/4047/13/$15.00
4047
© 1999 American Institute of Physics
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4048
J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
Dysthe, Fuchs, and Rousseau
TABLE I. Mean field molecular potential models.
Site–site potential
u LJ54 e i j (( s i j /r i j ) 122( s i j /r i j ) 6 )1u c ,
r i j <r c ;
u LJ50,
r i j .r c
Site parameters and „constrained… bond lengths
s ii
~Å!
e ii /k B
~K!
mi
~g mol21 )
d aua
~Å!
d bond
~Å!
rc
~Å!
mixing
shift
RB
RB
CH2
CH3
3.923
3.923
72
72
14.5194
14.5194
0.0
0.0
1.53
1.53
9.8075
9.8075
e i j 5 Ae ii e j j
s i j 51/2( s ii 1 s j j )
u c 50.01632• e i j
OPLS
OPLS
CH2
CH3
3.905
3.905
59.38
88.07
14
15
0.0
0.0
1.53
1.53
9.7625
9.7625
e i j 5 Ae ii e j j
s i j 51/2( s ii 1 s j j )
u c 50
AUA~3!
AUA~3!
CH2
CH3
3.516
3.516
79.87
119.8
14
15
0.40
0.18
1.545a 1.533b
1.545a 1.533b
8.79
8.79
e i j 5 Ae ii e j j
s i j 51/2( s ii 1 s j j )
u c 50
SMMK
SMMK
SMMK
CH
CH2
CH3
4.1
3.93
3.77
12
47
98.1
13
14
15
0.0
0.0
0.0
1.54
1.54
1.54
e i j 5 Ae ii e j j
s i j 5 As ii s j j
u c 50
RB
OPLS
AUA~3!
SMMK
Bending potential
Constrained bending angle u 51.9111
Constrained bending angle u 51.9548
u u /k B 5(62543/2)(cos u2cos u0)2 K, u 0 52.0001a,
u u /k B 5(62500/2)( u 21.9897) 2 K
a7
a8
RB
OPLSc
AUA~3!
SMMK
Torsion
a0
1116.0
1116.0
1001.35
428.70
21736.22
2817.37
potential
a1
1462.0
1462.0
2129.52
895.08
a2
21578.0
21578.0
2303.06
223.7
a3
2368.0
2368.0
23612.27
21765.1
10.25
9.825
9.425
u 0 51.9775b
a4
a5
3156.0 23788.0
3156.0 23788.0
2226.71
1965.93
u t /k B 5 ( i50 a i cosi x K
a6
24489.34
a
Adjusted and used for n-decane.
Adjusted for n-pentane, used for n-butane.
c
The RB torsion parameters were used for OPLS to compare results with Ref. 27.
b
intermolecular potentials on the transport properties of molecular fluids has been published. Different authors use very
different d t and r c without any arguments for their preferences. We therefore find it timely to include such a study in
the present article.
In this study we use two recently proposed potential
models and our results may also be used as an evaluation of
the state and property independence of these models.
II. METHOD
We have developed, tested and used a MD code for flexible, multicenter molecules. The code allows constraints between any interaction site to model linear, branched or cyclic
molecules with flexible or constrained bending angles. We
use the RATTLE algorithm to solve the constrained equations of motion.2 We will first briefly describe the types of
potential models we have used and go on to describe the MD
methods employed.
sites is constrained, the angle between neighboring bonds is
either constrained or is subject to a bending potential and the
dihedral angle ~see Fig. 1! is subject to a torsion potential.
Carbons separated by more than three bonds interact via the
same binary interaction potential as for intermolecular interactions to avoid an unphysical overlapping of sites. The collapsed centers of force are characterized by the pair interaction parameters of size, s ii , and well depth, e ii , the mass,
m, and the position of the center of mass and center of force
relative to the bonds, d aua . In addition to s ii and e ii , one
must specify the mixing rules, the cutoff distance, r c and
long range corrections applied. We will use i, j,k and l as
united atom indices.
A. Potential model description
We have used four different mean potential field models
of alkanes. The potential model parameters are all given in
Table I. All four models are so-called united atom ~UA!
models because they do not use each atom as an interaction
site but they use one single Lennard-Jones ~LJ! potential for
each CHi group. The distance between bonded interaction
FIG. 1. Molecular geometry. Left hand side: skeleton diagram of
2-methylpentane showing the carbon positions ~open circles!, centers of
force ~filled circles! and bonds. Right hand side: illustration of the two
dihedral angle definitions.
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J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
Dysthe, Fuchs, and Rousseau
We use the RB model by Ryckaert and Bellemans3 and
the OPLS ~optimized potentials for liquid simulation! model
by Jorgensen et al.4 to compare to previously published data.
The potential model called SMMK proposed by Siepmann
et al.5 is a readjusted version of the UA potential model proposed by Smit, Karaborni, and Siepman.6 The parameters
have been readjusted by comparing vapor liquid equilibrium
data from experiment and Gibbs ensemble Monte Carlo.5
The potential model named AUA~3! proposed by Toxvaerd7
is a readjustment of his own anisotropic united atom ~AUA!
model.8,9 The SMMK and AUA~3! models have not been
used previously for transport coefficient calculations. We
have chosen to use these to evaluate the property independence of these models. Since they are of the same general
UA type of models, the methodological discussions are not
restricted by this choice.
In the AUA model the center of the UA potential is
displaced relative to the carbon center by a distance d aua ~see
Fig. 1!
rcf,i 5ri 1d aua•
( j ri 2r j
u ( j ri 2r j u
5ri 1d aua ei ,
force F(rk ) on the carbon atom at rk due to the intermolecular forces Fi j (i counts the center of force on carbon k and
those of the n carbons bonded to k and j counts the centers of
force on other molecules! can be expressed as
F~ rk ! 5
(i j
F S
H
D G J
d aua
li
21 Fi j ,
~ Fi j •ei ! ei 1 d ik n111
li
d aua
du ~ r cf,i j ! rcf,i j
,
dr cf,i j r cf,i j
~3!
LJ
where Fi j 52
~4!
l i 5 u ( m ri 2rm u and m counts all sites bonded to i. The center
of force velocity v cf,i may in the same manner be expressed
as
v cf,i 5 v i 1
where v e,i 5
d aua
$ v e,i 2 ~ v e,i •ei ! ei % ,
li
(j v i 2 v j .
~5!
~6!
~1!
B. Integration of the equations of motion
where ri is the carbon position and r j are the positions of the
carbon atoms bonded to ri and ei is the unit vector from the
carbon to the center of force ~cf!. The mass of the CHi group
is placed on the carbon position.
Between two carbons that are separated by three bonds
the interaction is modeled by a torsion potential that depends
on the dihedral angle x ~see Fig. 1!
cos x 52cos f 52
4049
~ ri j 3r jk ! • ~ r jk 3rkl !
A12 ~ ri j •r jk ! A12 ~ r jk •rkl !
2
2
,
~2!
where ri j 5ri 2r j . It should be noted that both definitions x
and f are used in the literature, x 50 and f 5 p in the equilibrium ~trans! conformation of a normal alkane. We have
reformulated the torsion potential used by Siepmann et al.5
~see Table I! in terms of the power expansion in cos x. The
forces on the four centers of mass ~cm! joined by neighboring bonds are given by the spatial derivatives of u t ( x ) and
the potential energy u t ( x ) is equipartitioned on the four centers of mass. The bending forces and energies are treated in
the same manner.
For branched alkanes we will distinguish between two
types of torsion potentials: X–CH2 –H2 –Y and
X–CH–H2 –Y, where X and Y may be any CHi group. In
Fig. 1 the torsion around the bond 2–3 is of the first type and
torsion around bond 3–4 of the second type. For computational convenience the second torsion is split into two separate contributions,5 u t,1 : 2–3–4–5 and u t,2 : 2–3–4–6 with
the total torsion potential around 3–4 regained by adding the
two terms: u t ( x , u )5u t,1( x 2 u /2)1u t,2( x 1 u /2). Figure 2
shows u t ( x , u 0 ) using the SMMK torsion potential where u 0
is the equilibrium angle between neighboring bonds. For the
calculations of the heat flux one should shift the potential to
assure that u t ( x 50,u 0 )50 which is not the case for all published X–CH–H2 –Y potential parameters.
The formal generalization of the AUA model8 to
branched alkanes is straightforward @see also Eq. 1!#. The
The equations of motion for the molecules may be written as10
ṙi 5
pi
mi
~7!
and ṗi 5F~ ri ! 1Gi 1Ki ,
where Gi 5
(j l i j ¹ i~~ ri 2rj ! 2 2d 2i j !
and Ki 5 a NH pk .
~8!
~9!
~10!
Here Gi is the constraint force on center of mass i due to the
bonds to other centers of mass j. The Lagrange multipliers
l i j are solved iteratively for each molecule using the
RATTLE algorithm of Andersen.2 The additional ‘‘forces,’’
Ki , in the equations of motion are due to the thermostat
applied. For microcanonical ~NVE! dynamics Ki [0, for the
atomic and molecular versions of the Nosé–Hover thermostat pk 5pi and pk 5 ( iPmol pi respectively. The Nosé–
Hoover multiplier is10
FIG. 2. Total SMMK torsion potential, u t,tot5u t ( x 2 u 0 /2)1u t ( x 1 u 0 /2),
of 2-methylbutane.
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4050
J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
S
D
1 E k ~ p!
] a NH
5 2
21 ,
]t
t NH f kT
where E k,at~ p! 5
and E k,mol~ p! 5
Dysthe, Fuchs, and Rousseau
~11!
p2i
(i m i
1
~12!
~ ( iPa pi !
.
ma
~13!
We will use a and b as indices for molecular quantities. The
term t NH is a characteristic relaxation time of the thermostat
and f is the number of degrees of freedom of the system
atomically, f 53N cf2N bond21, and molecularly, f 53N mol
21. The Nosé–Hoover multiplier is calculated iteratively for
the whole system ~Frenkel and Smit10 describe this for
atomic systems! before applying the velocity part of the
RATTLE bond constraint. The advantage of the molecular
version is that it does not interfere directly with the internal
motions of the molecule. The application of the thermostat is
therefore not coupled to the bond constraining part of the
algorithm and the internal conformation of the molecules
will not be directly affected by the thermostat.
Jq ~ t ! V5
C. Transport coefficients
We use the Green–Kubo ~GK! formalism to calculate
the self-diffusion coefficients, D, the viscosity, h , and the
thermal conductivity, l 11
E
D5 lim lim D ~ t ! 5
1
3N
h 5 lim lim h ~ t ! 5
V
10kT
t →` N→`
1
t
dt
0
E
(a ^ v a~ t ! • v a~ 0 ! & ,
t
0
dt
F(
a
t →` N→`
~14!
T
T
~ t ! P aa
~ 0 !&
^ P aa
1
^ ~ P ab ~ t ! 1 P ba ~ t !!
2 a , b P $ 12,13,23%
(
G
3 ~ P ab ~ 0 ! 1 P ba ~ 0 !! & ,
l5 lim lim l ~ t ! 5
(a bÞa
( ~ r a, a 2r b, a ! iPa
( jPb
( Fi j, b .
V
3kT 2
E
t
0
dt ^ Jq ~ t ! •Jq ~ 0 ! & ,
~15!
(a
H
e a ~ t !v a ~ t ! 2
~18!
1
2 iPa
J
( bÞa
( jPb
( rab~ t !
3 @ Fi j ~ t ! • v cf,i ~ t !# ,
~19!
where all velocities are barycentric, a and b are molecular
indices, the velocities and positions corresponding to the
center of mass of the molecule, and i and j are atomic indices
where iPa signifies indice i running over all united atoms in
molecule a. Simon et al.12 show the equivalence between the
force – velocity product ~in the second term! using the center
of mass and center of force quantities. This is of practical
importance using the RATTLE scheme because only the
center of force quantities are known at the time the heat flux
is computed. The internal energy e a of molecule a is calculated at a later stage using the expression
e a~ t ! 5
t →` N→`
1
2
The heat flux is defined as
2
(a
1
(a m a p a, a p a, b
and P ab ~ t ! V5
1
m v ~ t !21
u t,k ~ t ! 1
u u ,l ~ t !
2 iPa i i
kPa
lPa
(
1
(
(
1
( u LJi j ~ t ! 1 2 iPa
( (b jPb
( u LJi j ~ t ! .
i, jPa
~20!
The autocorrelation functions are evaluated for a total
time t GK depending on the system and state. At times t
1i t GK /N GK ~i is an integer! we record the pressure tensor,
heat flux and molecular velocities and calculate the autocorrelation functions for all times (N GK2i) t GK /N GK ,
iP @ 1,N GK# . We use N GK54096 or 8192 for the viscosity
and thermal conductivity and N GK51024 for the selfdiffusion. For gaseous and for dense liquid systems t GK may
be of the order 100 ps and one may have to run some shorter
additional runs to assure that the initial fast decorrelation is
properly sampled. It is also clear that each new evaluation of
the autocorrelation functions at times t1i t GK /N GK does not
provide much ‘‘new’’ information with regard to the previous evaluation, but we prefer to ‘‘over evaluate’’ rather than
lose information.
~16!
D. Configurational properties and relaxation
where ^ ¯ & is the mean over a trajectory, k is the Boltzmann
factor, T the mean temperature of the simulation, V is the
system volume, N the number of molecules, Jq the heat flux,
v a the instantaneous velocity of the center of mass of molecule a, Pab the elements of the pressure tensor where a and
b are indices running over the three Cartesian coordinates.
We use the molecular definition of the pressure tensor which
has the components
T
P aa
~ t ! 5 P aa ~ t ! 2
1
3
(b P bb~ t !
~17!
We have calculated a number of configurational properties. In this work we have mainly used them to check the
equilibration of the simulated systems, although these properties have an interest in themselves. There are two main
reasons for studying the relaxation times in addition to calculating the transport coefficients. First, one has to assure
that the system is equilibrated, second, one must try to assure
that the trajectory samples a representative part of phase
space. There is no guarantee that the equilibrium reached
represents a global minimum, or that the sampling is representative. In other words, the ergodicity of these systems is
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J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
Dysthe, Fuchs, and Rousseau
only possible to ‘‘verify’’ empirically. The direct way to, as
far as possible, assure that the trajectory samples representatively is to compare equilibrium properties with Monte Carlo
~MC! simulations. Since one does not always have the possibility of comparing with MC13 we want to, as far as possible, use methods available by MD to survey the equilibration and representativity of the sampling.
We have calculated the end-to-end vector, which for
molecule a in a system of n-alkane of n carbons is re2e,a
5ra,12ra,n . This is used for the end-to-end distance distribution P( u re2eu ) and the reorientational autocorrelation functions of first and second rank14
C or,1~ t ! 5
1
N
(a ^ êa~ 0 ! •êa~ t ! &
1
and C or,2~ t ! 5
N
(a
K
~21!
L
3
1
~ êa ~ 0 ! •êa ~ t !! 2 2 ,
2
2
H
1,
if cos x i ~ t ! .0.5
0,
if cos x i ~ t ! <0.5,
H g1 @ x i ~ t !# 5
H
12H t ,
if ~ ri j 3r jk ! •rkl >0
if ~ ri j 3r jk ! •rkl ,0
0,
and H g2 @ x i ~ t !# 512H t 2H g1 .
~23!
~24!
~25!
~26!
where c,dP $ t,g1,g2 % . The fraction of each configuration
N
for a system is defined as X c 5 ( i t H c @ x i # /N t , where N t is
the number of angles subject to a torsion potential in the
system, so that X t 1X g1 1X g2 51. The correlation functions
have the limiting values
C cd ~ t50 ! 5
H
^ X c& ,
0,
C̄ cd ~ t ! 5
C cd ~ t ! 2 ^ X c &^ X d &
,
C cd ~ 0 ! 2 ^ X c &^ X d &
C̄ cd ~ t cd ! 5e 21
~27!
~28!
and t conf5 ^ t cd & .
In addition to the correlation functions of Brown and
Clarke,14 we define the torsion transition function
Ti ~ t ! 512H t @ x i ~ t !# H t @ x i ~ t1Dt !#
2H g2 @ x i ~ t !# H g2 @ x i ~ t1Dt !# .
~22!
For branched molecules we use one characteristic function
around a bond for each branch in the same manner as the
torsion potential is split in two. The functions H t ,H g1 and
H g2 characterize whether the dihedral angles are such that
the torsion potential is in the attractive parts of the local
minima trans, gauche plus or gauche minus, respectively.
Using these characteristic functions, we can define the correlation functions
C cd ~ t ! 5 ^ H c ~ 0 ! •H d ~ t ! & ,
There are four combinations of these autocorrelation
functions that have different behavior C tt ,C 1 5C g1g1
1C g2g2 , C 2 5C g1g2 1C g2g1 and C tg 5C tg1 1C tg2
1C g1t 1C g2t . The conformational relaxation times, t cd ,
are defined by the normalized autocorrelation functions
2H g1 @ x i ~ t !# H g1 @ x i ~ t1Dt !#
where êa 5re2e,a / u re2e,a u . For a long, flexible molecule that
may curl itself up, this is a slightly ambiguous measure, but
the related relaxation times ~given as the times, t or,i when
C or,i ( t or,i )5e 21 ) give relevant information about one aspect
of the structural decorrelation of a system.
In order to watch the internal configuration of the molecules, we calculate the dihedral angle distribution
P @ cos(x)#. To study the dynamics of the conformations we
follow Brown and Clarke14 in defining the characteristic
functions of each dihedral angle
H t @ x i ~ t !# 5
4051
~29!
This function of torsion bond i has the value 1 if the dihedral
angle has switched from one conformation (t, g1, g2) to
another during the time Dt and is zero if there has been no
transition. We can now define the neighbor torsion transition
correlation function
C trans~ t ! 5
^ Ti ~ 0 ! •Ti11 ~ t ! &
^ Ti ~ 0 ! 2 &
21
~30!
which is a measure of the likelihood that dihedral angle i
undergoes a transition when the neighboring dihedral angle
underwent a transition at a time t earlier. It is thus a measure
of the concertation of the conformational changes in a molecule.
1. System relaxation time t s and statistical
independency
Assuming that a property Q obeys Gaussian statistics,
the autocorrelation function C QQ 5 ^ Q(t)Q(0) & measured
during one trajectory of length t p has a variance15
s 2 ~ ^ Q ~ t ! Q ~ 0 ! & t p ! 52 t Q C QQ ~ 0 ! 2 / t p .
~31!
The assumption of Gaussian statistics is related to the representativity of the sampling in phase space, i.e., whether
^ Q(t)Q(0) & t p 5 ^ Q(t)Q(0) & ensemble . The standard method
presented by Allen and Tildesley15 is to calculate the statistical inefficiency to find a minimum time t b over which subaverages should be blocked to ensure that consecutive subaverages are statistically independent. We consider this
method somewhat cumbersome and it requires a lot of disk
space to save subaverages over small t b . When simulating
different molecules at different states, the minimum t b may
vary by two orders of magnitude and it is preferable to estimate the minimum blocking time for every simulation without saving to disk too often.
We will define the relaxation time of the system, t s , as
the longest relaxation time among t or , t conf , t h ,
where
if c5d
if cÞd
and C cd (t→`)5 ^ X c &^ X d & .
h ~ t h ! 5 ~ 12e 21 ! h ~ t→` !
@see Eq. ~15!# and the mean time for a molecule to move one
~mean! molecular diameter
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4052
t d5
J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
r̄ 2e2e
6D
.
For different molecules at different states, t s will be governed by different relaxation processes.
Mundy et al.16 and Mondello and Grest17 have recently
shown that for long alcanes ~n-decane to squalane! one can
use the Rouse model of polymer dynamics and the reorientational relaxation time t or to estimate the viscosity of the
model fluid to within 20%–30%. For their systems they find
t or to be the slowest and only relevant relaxation time for
viscosity calculations. For shorter molecules one cannot expect this to be true and one will at least have to consider t d
as well. Thermal conductivity is sensitive to internal energy
and flexation of the molecules. The relaxation of the internal
conformations of the molecules should therefore be taken
into account as well. From a purely statistical perspective,
one may argue that all conceivable correlations of the systems should have died out before one can consider the next
sampling of the system independent of the preceding. As a
practical rule we propose that: ~1! In order that 95% of the
molecules should have ‘‘forgotten’’ their previous state, the
length of an equilibration trajectory, t e , should be at least
three times t s . ~2! Consecutive subaverages should be
spaced by 1–3 times t s in order to be considered statistically
independent. A corollary to this principle is that the length of
the trajectory, t p , used to sample the phase space should be
much longer than the system relaxation time: t p @ t s . We
have, in some sense, substituted t s for t Q in Eq. ~31!. These
criteria should be sufficient ~although not always necessary!
to obtain reliable error estimates by standard treatment of
subaverages.
E. General comments
At this point we want to add a comment on the cutoff
distance r c of the LJ interactions. The term r c is normally
treated as a practical parameter to increase the computational
speed and one assumes that the effect on the calculated trajectory is small ~see discussion in Sec. III B 2!. The cutoff
also directly affects measured properties, notably the pressure and internal energy and thereby the viscosity and thermal conductivity. The pressure and internal energy may be
corrected assuming the fluid to be a continuum ~structureless
and without fluctuations! beyond r c . Although the continuum assumption implies that the fluctuations of the fluid
outside r c are negligible, the absolute values of the fluctuating heat flux @both terms in Eq. ~19!# and pressure tensor
@second term in Eq. ~18!# are changed.18 It follows that if one
does not correct P ab (t) and Jq (t) before calculating h and
l, one must consider r c as a model parameter and not
merely a practical cutoff distance to increase the computation
speed. We have not performed any such correction and accordingly include r c in Table I as one of the potential parameters.
In order to obtain pressures comparable to experiment it
is, however, necessary to include a long range correction.
The pressures we report are given by
Dysthe, Fuchs, and Rousseau
1
3
(a
P aa 2
16p N 2
3 V2
(i (j
N iN je i j
S
s 6i j
22
s 12
ij
D
,
r 9c,i j
~32!
where N i is the number of centers of force of type i in the
molecule.
The initial configurations are produced by placing all
molecules with identical internal conformation and random
orientation and velocities on fcc lattice sites at a gaseous
density and compressing or expanding the system quickly
while rescaling the velocities.
The total momentum of the system is kept from drifting
by resetting it to zero at intervals of ;10 time steps. This
technique may also be used for the total energy in the NVE
simulations without perturbing the system measurably, but
we have preferred to let the total energy drift as a check on
the stability of the integration of the equations of motion.
Most of the simulations presented here are performed
using NVE dynamics. The reason for this is that the effect of
different thermostats on transport properties has not been
completely established. Evans and co-workers19 have shown
the formal equivalence of the isokinetic ensemble with the
canonical ensemble and they do indeed obtain transport coefficient results that agree with NVE simulations1 within
their precisions. On the other hand, Frenkel and Smit10 have
shown that the mean square displacement is slightly dependent on the relaxation time of the Nosé-Hoover thermostat.
Even though the displayed difference is smaller than the
typical precision of transport coefficient results in molecular
systems, the thermostating of the different degrees of freedom of molecular systems may cause additional problems. It
is also known that using extremely efficient thermostats like
a Nosé-Hoover chain breaks down the time dependent correlations in the fluid. Mundy et al.20 have shown some evidence that both h (t) and C or,1(t) are affected by the time
constant of the atomic Nosé-Hoover thermostat. Because of a
lack of a systematic study of this influence we have preferred
to either use NVE dynamics or molecular NVT with a relatively long time constant t NH .
p5
r 3c,i j
III. RESULTS AND DISCUSSION
Most of the results discussed are directed at clarifying
methodological issues from a practical viewpoint: we wish to
calculate transport properties of molecular systems with as
high precision as possible and with an accuracy that is similar to the precision. By precision we mean the reproducibility
of results under nominally identical conditions, i.e., for the
same molecular models at the same mean temperature and
density. The accuracy of a result is the deviation from the
~unknown! theoretical result for the model system in the
thermodynamic limit of size and time at the given state point.
The precision can be increased by performing longer runs or
sampling over more independent starting configurations. The
accuracy may be improved by using better integrators,
smaller time steps and larger system size. Since calculation
of transport properties requires very long runs the limited
computation resources set an upper limit on both precision
and accuracy and one should try to match the two. We will
Downloaded 15 Jan 2002 to 129.240.85.171. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
also use the results to study the predictive accuracy of the
models used. Information on system, model, state, and trajectory details for all simulations is listed in Table II.
A. Verification of the code
The code has been tested by comparing with three other
programs that have been developed independently for other
purposes.12,21,22 For pure and binary mixtures of linear alkanes, the energies, pressure tensor and heat fluxes have
been compared to two other molecular dynamics
programs12,22 for several time steps starting from the same
configurations. The deviations started in the sixth digit and
increased slowly at first before the two trajectories diverged
~as expected!. For linear and branched molecules, the energies and scalar pressure of some configurations were compared with a Gibbs Monte Carlo program20 yielding deviations in the sixth digit.
B. Errors due to integration of equations of motion
There are three main sources of errors in the trajectories
calculated by integrating the equations of motion: finite
floating number representation causing roundoff errors, finite
time step size d t causing truncation errors and truncation at
distance r c of the intermolecular potentials. For a thorough
discussion of errors introduced by finite difference integration of the equations of motion see Haile.23
1. The length of the integration time step
We will address two connected questions: how does the
time step affect the short time accuracy of the trajectories
and how does it affect the long time relaxation of fluctuations connected to the transport properties? We have performed simulations using double precision real number representation to minimize the roundoff errors. In order to
compare the effect of roundoff errors to that of truncation
errors, we have compared two programs using RATTLE, one
implemented for workstations, another for the Cray T3E.
For liquid n-decane ~system H!, starting from the same
configuration, the energies calculated by the two programs
diverged after 1.5 ps using d t51 fs and after 1 ps using
d t54 fs. The energies calculated by the same program using
time steps of 1 and 4 fs diverge after about 0.75 ps. Thus,
looking only at short time energies from trajectories, it seems
that using time steps of about 2–3 fs for this system will
balance the truncation and roundoff errors. In other words,
there is no gain in accuracy by using d t,2 fs; on the contrary one spends more time computing a less accurate trajectory. We consider this to be a result representative of molecular models in liquid states where round off errors are
introduced by the extensive calculation of the forces between
many neighboring centers of force.
Comparing the energies calculated by the two different
programs for a gaseous state ~system C! on the other hand,
using d t50.5, 1, 2, 4 and 5 fs, we find no indication that
round off errors are more important than truncation errors for
any of these time steps. Careful inspection of the individual
energy components shows that there are fast harmonic oscillations of the bending angles with periods of 0.04–0.06 ps,
of the dihedral angles with periods 0.12–0.18 ps and a cou-
Dysthe, Fuchs, and Rousseau
4053
pling of bending and torsion with periods 0.07–0.09 ps. The
fast oscillations in intramolecular energy and the inverse oscillations in kinetic energy are better resolved by short time
steps. Fourier transform of the intermolecular potential energy reveals no clear coupling with the intramolecular potential energies. Thus in the gas phase the accuracy of the trajectory is limited by truncation errors due to the
intramolecular motions of molecules in ‘‘free flight.’’
To save computing time, the question is how long time
steps can one use? For thermodynamic properties that are
time averaged along a trajectory it is important that the trajectory visits a representative part of the NVE or NVT hypersurface. As long as the total energy is ‘‘reasonably’’ conserved, the details of the trajectory in time are therefore not
important for time independent properties.15 Increasing the
time step d t usually causes an increased loss of total energy
per unit time and thereby ~in the NVE ensemble! a diminishing temperature along the trajectory.
To compare different systems in a coherent manner, we
use the relative changes of the total energy or temperature
QP $ E,T % along a trajectory: d Q5( ] Q/ ] t)/ s Q , where s 2Q
is the variance of Q when the systematic drift has been subtracted. We have performed this analysis for systems H ~2, 4,
6 and 8 fs!, I ~4 and 5 fs! and B ~1, 2.5 and 5 fs!. For the
liquid systems H and I d T520.01 to 20.02 ns21 and the
total drift in temperature is only 20.02 to 20.08 K/ns except
for the 5 fs time step for system I where d E520.09 ns21
and the total drift is 20.6 K/ns. For system B d T520.02 to
20.04 ns21 and the total drift is 20.4 to 21 K/ns. The drift
is smallest for the 2.5 fs trajectory ~B.2!. Increasing the time
step, d E changes from 20.35 to 21.5 ns21 for system H,
from 0.6 to 22.3 ns21 for system I and from 20.3 to 1.4
ns21 for system B. For system H, d t58 fs is close to the
limit of stability for the integration of motion, the noise in
the total energy ~drift subtracted! is eight times larger than
for the smaller time steps. We consider the energy to be
‘‘reasonably’’ conserved ~for time independent properties at
states far from phase coexistence! when the drift in the mean
temperature along the trajectory is much smaller than the
fluctuation in temperature or less than a few Kelvin, which is
the case for all these systems. Although some of these drifts
in total energy and temperature may seem large, they are
sufficiently small to allow comparison of the mean value of a
given property with experimental data and other simulations
unless this value changes more with the change in temperature than the standard deviation of the mean.
For time dependent properties, on the other hand, one
may expect that the accuracy of the calculated trajectory
~with respect to the theoretical trajectory! is much more important because one studies dynamic events of duration up to
several hundred picoseconds. Erpenbeck and Wood24 have
made an extensive study of the effect of trajectory inaccuracy on the velocity autocorrelation function ~VACF! for
hard spheres and hard disks. They found no significant difference between the VACF of single precision and double
precision calculation of the trajectories ~from identical starting points! even though the single precision trajectory lost
its accuracy at times much shorter than the upper sampling
time of the VACF. This result is encouraging since it
Downloaded 15 Jan 2002 to 129.240.85.171. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
4054
J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
Dysthe, Fuchs, and Rousseau
TABLE II. Simulated systems and trajectory information.
Equilibration
Lbl. Molecule
A
Ba
C
D
E
F
G
H
I
J
K
n-butane
n-butane
n-butane
n-butane
n-butane
n-butane
n-butane
n-decane
n-decane
2-methylbutane
2-methylbutane
Model
r
~kg m23)
T
~K!
RB
AUA~3!
AUA~3!
AUA~3!
AUA~3!
AUA~3!
OPLS
AUA~3!
AUA~3!
SMMK
SMMK
583
31.6
31.6
31.6
640
732.3
720.36
559.9
820.9
610
750
291.6
500
500
500
290.0
150.0
150.0
510.9
286.0
323.2
303.2
dt
~fs!
Ensemble
tp
~ns!
4
1
1-2.5
4
4
4
4
4
4
3
3
NVT
NVT/NVE
NVT
NVT/NVE
NVT
NVT/NVE
NVT
NVT/NVE
NVT/NVE
NVT/NVE
NVT/NVE
8
535
8
4
8
8
6
2 38
6,10 30.6
4.5
4.5
te
N mol ~ns!
108
32
108
864
108
108
108
108
108
108
108
0.8
1.5
4
2
0.8
8
6.6
0.7
2.8
0.8
0.8
Production
dt
~fs! Ensemble
4
1–5
4
4
4
4
4
2,4
4
3
3
NVT
NVE
NVT
NVT
NVT
NVE
NVT
NVE
NVE
NVE
NVE
B.1: d t51, r c 54s i j ; B.2: d t52.5, r c 54s i j ; B.3: d t55, r c 54s i j ; B.4: d t52.5, r c 52.5s i j ; B.5: d t52.5,
r c 55.5s i j .
a
is impossible to calculate an accurate trajectory of our molecular systems for several hundred picoseconds with the
finite difference method employed here. It is necessary,
however, to verify that the properties calculated do not
change significantly with the time step d t up to some limiting d t.
For system H approximately 50% of the contribution to
the shear stress autocorrelation function ~SSACF! integral
comes from the first 2 ps, h (t52ps)'0.5h (t→`). We can
not rule out that the local, short time dynamics calculated
‘‘more correctly’’ using a small time step is important for the
accuracy of the transport coefficients. We have therefore calculated the viscosity and thermal conductivity25 for system H
starting from the same configuration using time steps d t
52, 4, 6 and 8 fs. The results are everywhere within one
standard deviation of each other, even for the largest time
step where the total energy is much noisier than for the other
time steps. At high density and low temperature the first 2 ps
contribute only 1/5 of the total viscosity and possible effects
of poor short time dynamics should be even smaller. We
conclude that as long as the integration of the equations of
motion is stable, i.e., the total energy does not diverge completely, the viscosities and thermal conductivities calculated
by GK are as accurate as the precision ~in this case ,5%!.
This allows for much longer time steps than we had anticipated. We note that in NEMD simulations like SLLOD the
large time steps in multiple time step algorithms are typically
2–3 fs,26,27 only a 1/2 to 1/4 of our upper limit.
For gaseous n-butane we have performed simulations
with d t51, 2.5 and 5 fs ~systems B.1, B.2 and B.3, respectively!. The results displayed in Table III show that the diffusion coefficient calculated with d t51 fs is significantly
larger ~9%! than the two others. We do not yet understand
why better resolution of the intramolecular modes in mostly
free flight should increase the diffusion coefficient and suggest that this should be studied further.
2. Truncation of intermolecular potential
In the gas phase we expected that the molecular trajectories may be much modified by a change in the cutoff length
TABLE III. Transport coefficients.
T
~K!
A
B.1
B.2
B.3
B.4
B.5
C
D
E
F
G
H
I
J
K
a
291.6
508.960.7
506.160.6
505.060.7
502.860.7
505.460.7
500
500
290
150.560.4
150
508.560.4
286.960.2
321.060.2
301.860.2
p
~MPa!
1.860.2
2.0860.01
2.0460.01
2.0260.01
1.9860.01
2.0260.01
2.0560.01
2.0560.01
31.360.1
236.360.3
37.960.2
11.960.4
177.260.5
4.160.2
178.160.2
p MD2p exp
p exp
~%!
1
1
2
4
2
1
1
200
0.6
D
(1029 m2 s21 )
7.0660.04
645615
590615
590615
590615
590615
60968
63463
6.0160.04
1.2160.05
0.6360.01
10.960.1
0.39660.008
7.3860.05
2.3760.02
D MD2D exp
D exp
~%!
0.5
66
170
9
211
60
h
(1023 Pa s!
0.15660.004
0.01460.002
0.01260.002
0.01460.002
0.01260.002
0.01360.002
0.01660.002
0.01560.003
0.17760.007
0.7960.02
1.160.17
0.12660.003
3.560.6
0.13560.008
0.4460.02
h MD2 h exp
h exp
~%!
211
8
28
8
28
0
23
15
239
255
219
220
240
222
235
l
~W m21 K21)
0.09360.004
0.01760.003
0.01860.003
0.01660.003
0.01560.003
0.01460.003
0.01960.003
0.02060.004
0.12360.006
0.16860.005
0.16160.005
a
a
0.08360.005
0.1560.01
l MD2l exp
l exp
~%!
218
213
27
27
Too short initial relaxation time.
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J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
Dysthe, Fuchs, and Rousseau
TABLE IV. Relaxation times and configurational properties.
Lbl.
t or
~ps!
A
C
F
G
H
I
K
1.65
2.0
11.3
16.6
6.33
201
2.3
a
t conf
~ps!
20.9
28.6
2830
570
2.68
43
21; 32; 85a
td
~ps!
th
~ps!
Xt
r̄ e2e
~Å!
3.1
0.03
20
37
12
397
10.2
1.1
5.3
6.6
8.8
2.6
75
3.5
0.615
0.48
0.83
0.836
0.571
0.719
0.440
3.506
3.493
3.82
3.755
8.85
9.71
3.8b
t tt (5 t tg ); t 1 ; t 2
From system F.
b
r c at which the intermolecular potential is truncated. We
have therefore performed three simulations of AUA~3!
n-butane at r 531.6 g/l and T5500 K with cutoff lengths
r c,i j P $ 2.5,4.0,5.5% s i j ~systems B.2, B.4 and B.5! started
from the same configuration. Comparing the values in Table
III we find that there is no significant change in the transport
coefficients upon changing r c .
Figure 5 shows that the the second peak of the molecular
pair distribution function of n-butane near the triple point
~system F! has its maximum at about r c . We note that it is
preferable that the fluid is structureless beyond r c , but due to
the long computational times involved we have not performed the same test as in the gas phase.
C. Relaxation, configuration and precision
1. ‘‘Moderate’’ states
4055
changing the conformation are more effective in the liquid
phase. Since the mean kinetic energy for these systems is
always less than half the trans-gauche energy barrier of
e /k B 51328 K one may use a simple activation energy
model for the trans-gauche relaxation process ~the gauche–
gauche barrier is 2167 K and we will for simplicity neglect
the contribution of this relaxation path!: t conf
}exp@2akBT/e# where a is the activation coefficient of the
relaxation process. Using the two relaxation times for each
molecule we find that the activation coefficients are 17 and
16 for n-butane and n-decane, respectively. The two
n-butane simulations C and F are in the gas and liquid phase,
respectively, whereas both n-decane simulations are in the
liquid phase. Since the two activation coefficients are almost
equal we conclude that the rate of the relaxation process is
not significantly different in the gas phase and the liquid
phase. The tenfold difference in the relaxation time between
system C and H is therefore mainly due to some kind of
concertation of the internal molecular motions in the chain.
This notion of concertation is quantified in Fig. 3 which also
shows the neighbor torsion transitions correlation function
C trans(t) which reaches a maximum of 0.6 at t50.16 ps.
This means that when one dihedral angle x i undergoes a
transition at time t the ‘‘probability’’ that the neighboring
dihedral angle i11 undergoes a transition is increased by
60% at time t10.16 ps relative to the uncorrelated probability ^ Ti (0)•Ti (`) & 5 ^ Ti (0) 2 & .
For systems I ~n-decane! and K ~2-methylbutane! at
about 300 K and high density, Fig. 3 shows the configurational autocorrelation functions C̄ cd (t) and C or,1(t) and the
In Fig. 3 we show the conformational, C cd (t), and orientational, C or,i (t), autocorrelation functions for the systems
A ~n-butane! and H ~n-decane! at moderate densities. As
expected, the relaxation of the internal conformation is
slower for system A at 291.6 K than for system H at 511 K.
We show all four configurational autocorrelation functions
that have different behavior, although for these two systems
they are well modeled by one single relaxation time and the
normalized curves thus superimpose. One also observes that
the short n-butane molecules reorient themselves more
quickly than the n-decane molecules. The corresponding relaxation times are given in Table IV. At the states A and H
that have been used repeatedly by several authors for comparing simulation results, the relaxation times considered are
in the range 1.5–21 ps and t s 521 ps for system A and t s
513 for system H. During the total length of the production
runs of t p 58 ns one may perform approximately 100 statistically independent samplings according to the criterion proposed. The standard errors reported in Table III are computed from ten subaverages.
2. Conformational relaxation mechanisms
The conformational relaxation times t conf of the systems
C, F, H and I deserve some comments. The fact that t conf of
gaseous n-butane at 511 K ~system C! is ten times longer
than that of liquid n-decane at the same temperature shows
that there is an important effect of concertation of conformational changes in the chain and/or that the mechanisms for
FIG. 3. Conformational and orientational autocorrelation functions for systems A and H. Legends left hand side: solid lines–C tt (t), dashed lines–
C 1 (t), long dashed lines–C tg (t), dot dashed lines–C 2 (t) and the horizontal, thin, long dashed lines are the t→` limits of the C ab (t). The noisy
curve in the lower left corner is the neighbor torsion transitions correlation
function, C trans(t) and the inset are the short time behavior of the same
function. Legends right hand side: solid lines–C or,1(t) and long dashed
lines–C or,2(t).
Downloaded 15 Jan 2002 to 129.240.85.171. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
4056
J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
FIG. 4. Conformation and relaxation of systems I and K. Left hand side:
normalized conformational relaxation functions C̄ tt (t) ~solid!, C̄ 1 (t) ~long
dashed! and C̄ 2 (t) ~dot dashed! and C or,1(t) ~solid with circles!. Right hand
side: h (t) ~solid lines! and one standard deviation of the same ~long dashed
thin lines!. For system I ~upper right hand corner! the thick line is a 8 ns
long run with sampling frequency 0.04 ps used to correct the initial values
of the upper and lower lines which are the results of a 6 ns long run and ten
independent 0.6 ns long runs, respectively, both with sampling frequency
0.4 ps.
Dysthe, Fuchs, and Rousseau
FIG. 5. Structure, conformation and relaxation of n-butane, systems E, F
and G. Upper left: C̄ ab (t) of systems F @AUA~3! model# and G ~OPLS
model! which are at almost the same state point. Upper right: molecular
radial distribution function for system F that shows the system is not crystalline. Lower left: dihedral angle distribution for system E using MC
~dashed! and MD ~solid!. Lower right: dihedral angle distribution of system
F using MD ~lowest peak, solid! and MC ~highest peak, dashed! and of
system G ~long dashed!.
laxation time t 2 is much longer because it necessarily involves the tg→gg transition.
integrals of the shear stress autocorrelation function h (t).
With respect to Fig. 3 one observes that the longest relaxation times are an order of magnitude larger. The equilibration times for systems I and K should be at least 0.9 and 0.25
ns, respectively. Such long equilibration times are seldom
used, but these results show that at high densities one has to
be very careful. This concerns both equilibrium and nonequilibrium simulations of transport coefficients.
For n-decane t or and t d are dominant, whereas for
2-methylbutane t conf is the largest. For n-decane all four conformational relaxation times are ~within the uncertainty! governed by a single relaxation time t conf'40 ps. For
2-methylbutane there are three different relaxation times:
t tt 5 t tg 521 ps, t 1 532 ps and t 2 585 ps. In order to explain this behavior that is qualitatively different from that of
n-alkanes, we have in Fig. 2 plotted the total torsion potential
of 2-methylbutane which has two low energy configuration
minima ~tg, one dihedral angle gauche, the other trans! and
one high energy conformation minimum ~gg, both angles
gauche!. The three different relaxation times t tt , t 1 and t 2
are connected to three different transition energy barriers
e tg→tg /k B 51393 K, e gg→tg /k B 51650 K and e tg→gg /k B
52234 K, respectively. The relaxation time t tt has contributions from both the tg→tg and tg→gg transitions, but
the first dominates. The relaxation time t 1 has contributions
from both tg→tg and gg→tg transitions, but since the fraction of molecules in the gg conformation must necessarily
undergo the gg→tg transistion this makes t tt , t 1 . The re-
3. Statistical precision and the system relaxation time
Following our criterion of statistical independency we
can only obtain approximately ten independent samplings for
system I during the long trajectory of 6 ns. We have simulated one 6 ns trajectory with ten subaverages and ten independent trajectories of 0.6 ns. The independent trajectories
were started from ten gas phase configurations with different
random orientations of the molecules that each of them were
compressed and equilibrated for 2 ns before the production
run was started. The time dependent viscosity function h (t)
@Eq. ~15!# calculated for the two sets of subaverages are
within one standard deviation of each other ~see Fig. 4!. This
result confirms that the criterion proposed in Sec. II C 1 is a
sufficient criterion for obtaining reliable error estimates even
for long molecules at high densities and low temperatures.
This can also be compared to the assertion of Mondello and
Grest17 that, in order to obtain an accuracy of 10% for long
alkanes, one needs a trajectory of 100–200 t or . Assuming as
usual the accuracy to be proportional to the square root of
independent samplings: since we obtain 17% accuracy from
a 12 ps trajectory one needs a 35 ps trajectory which equals
175 t or to obtain 10% accuracy.
The fact that the ratio of the trajectory length to the
system relaxation time, t p / t s , is a factor 30 smaller for
system I than for systems A and H implies that for t p,H
516 ns and t p,I 512 ns one expects a precision of system I
that is six times lower. For both systems H and I we have
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J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
Dysthe, Fuchs, and Rousseau
4057
TABLE V. Comparison of transport coefficients of n-butane with other authors.
Lbl.
Reference
Method
tp
~ns!
Viscosity
(1023 Pa s!
A
A
A
A
G
G
Travis et al.e
Daivis et al.d
Daivis et al.b
This work
Allen and Rowleyc
This work
NEMD
NEMD
GK
GK
NEMD
GK
5
3
7.7
8
5–8.4a
6
0.15260.001
0.15460.004
0.15660.004
1.33~60.1a!
1.160.17
Thermal conductivity
(W m21 K21)
Self-diffusion
(1029 m2 s21 )
6.760.4
0.09360.004
0.09760.004
0.09360.004
6.9660.04
7.0660.04
0.16160.005
0.6360.01
a
Our estimate; see text.
Reference 1.
c
Reference 27.
d
Reference 30.
e
Reference 31.
b
two sets of trajectories for which we have saved ten intermediary results and calculated the standard deviations ~see
Table III!. The relative standard deviation of the viscosity for
system I is seven times larger than that of system H. The
criterion proposed is thus not only a rule for obtaining reliable error estimates, but can also be used as a rule of thumb
for the expected precision of a system one wants to study.
4. Low temperature n-butane
System F, n-butane at 150 K and high density, is an
interesting test case because one expects long relaxation
times. We did, however, not expect relaxation times in the
order of several nanoseconds as is the case for the internal
conformation ~see Fig. 5!. In order to check that we had
indeed reached the equilibrium distribution of conformations, we used Monte Carlo to compare the energies and
distributions. Figure 5 shows that MC gives more than twice
as high probability of the gauche conformation than from
MD. Both methods converged back to the distributions displayed when started from configurations from the other
method. The total potential energy of the MD simulation is
in fact 4% lower than that of the MC energy. To check the
two methods we performed MC simulations for system E as
well which is far from the phase coexistence curves. Figure 5
shows that the two methods give the same results within
statistical precision for the dihedral angle distribution. Using
long range corrections @see Eq. ~32!# the pressure of this
system is negative: 236 MPa. We conclude that system F is
probably in the gas–liquid two phase region of the phase
diagram despite adjustment of the model to high density
pressure, volume, temperature ~PVT! data.
We tried another potential model, OPLS, in order to obtain more reliable results at this extremely low temperature
~system G!. This model has a much smaller trans-gauche
energy barrier and the conformational relaxation time is
much shorter ~see Fig. 5!. The system relaxation time is 570
ps and the total run time 6 ns. Following the proposed criterion one may use only 3–10 subaverages as statistically independent. The conservative choice gives a standard deviation of about 15% and the viscosity agrees fairly well with
the NEMD result of Allen and Rowley27 ~see Table V!. It
should be noted that the equilibration time reported by Allen
and Rowley ~40–200 ps! was only a fraction of the conformational relaxation time ~570 ps!.
5. Extremely short relaxation times
At high densities where the system relaxation time increases, the correlation functions D(t) and l(t) go to zero in
4–20 time steps. D(t) also shows a back scattering with a
long relaxation time. The practical problem with the extremely short initial relaxation time is to sample often
enough when one needs to continue sampling for a long time
to measure the slower processes. For system I one needs to
sample every 0.04 ps with a duration 400 ps. Increasing the
sampling frequency by a factor 10 to save memory gives an
error in the integral of 25% for the viscosity. For the intradiffusion one may simply use the mean square displacement
and there exists an Einstein formulation of the GK formula
for the thermal conductivity and viscosity as well28 that we
have not implemented.
Another problem is whether one may trust the integral of
a correlation function when the discretization of the trajectory is on the same time scale as the relaxation time. Thermal
conductivity always has the shortest initial decorrelation time
and we do not report l in the cases where l(t)/l(0),1/6
for t,10d t ~systems H and I!.
D. System size dependence at low density
It is well known that the periodic boundary conditions
~pbc! suppress correlation functions at times comparable or
larger than the sound traversal time. In the gas phase there is
another mechanism that may affect the correlation functions
due to the pbc: traversal of the molecules themselves through
the entire simulation cell. To our knowledge, this has not
been studied before. For systems B, C and D the mean time
for a particle to traverse the simulation cell is 5.9, 13 and 53
ps, respectively. It is clear that some of the molecules that
collide at time t50 and part in opposite directions will collide again within the mean traversal time due to the pbc. It is
difficult to estimate the effect on the transport properties of
this unphysical situation. At this gaseous state the relaxation
times of D(t) and h (t) are 9 and 5 ps, respectively, and it is
therefore quite possible that the transport coefficients calculated in the smaller systems are affected by the pbc. Studying
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4058
J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
h (t) and D(t) for systems B, C and D ~see Table III! we find
that within the precision of our simulations there is no significant effect of the system size on the viscosity. The diffusion coefficient, on the other hand, decreases linearly with
N 21/3, as is the case in liquids, but we believe that the
mechanism causing this is different.
E. Comparison with literature, efficiency of GK
versus NEMD
We have also performed simulations on the most studied
flexible molecule, the Ryckaert and Bellemans3 model of
n-butane ~RB in Table I! at a state point near the boiling
point. Table II indicates the state and the trajectory information ~system A!. We used the molecular version of the NoséHoover thermostat with a relaxation time of t NH51 ps.
The average fraction trans conformation during the 8 ns
trajectory was X t 50.61560.007 which agrees well with the
results of Edberg et al.29 of X t 50.606 obtained by sampling
over a trajectory of 0.33 ns. The transport coefficients,
shown together with the results of Daivis and Evans30,1 and
Travis et al.31 in Table V, show excellent agreement except
for the self-diffusion coefficient where the results differ from
those of Daivis and Evans1 by 1.4% which is only slightly
more than two standard deviations. We conclude that our
implementation of the RATTLE algorithm and molecular
Nosé-Hoover thermostat yields results that agree with other
codes based on different integrators and thermostats.
The comparison of precision and trajectory length in
Table V also casts new light on the relative efficiency of the
equilibrium and nonequilibrium methods for calculating
transport coefficients. Assuming that a trajectory is sufficiently long to produce statistically independent samplings,
then making a trajectory two times longer gives an improvement in the precision of A2/2. For the viscosity of system A
one would have to make the trajectory 20 times longer than
that used by Travis et al.31 to obtain the precision they report
~0.7%!, but in the light of other reported precisions we find
this precision to be somewhat overestimated. For the thermal
conductivity the precision is the same as that of Daivis
et al.30 with a trajectory only 2.7 times longer. At high density and low temperature we have compared our viscosity
with that of Allen and Rowley.27 According to their general
statements on run lengths, they have used a production
length of between 5 and 8.4 ns compared with our 6 ns. We
have estimated the accuracy of their result to be somewhat
less than 10% from Fig. 7 in Ref. 27. Thus the efficiency of
the GK method is only a factor 2 less than their NEMD
technique ~SLLOD! for this system, not accounting for their
need of shorter time steps. Cui et al.26 have reported both
GK and SLLOD calculations of viscosity for n-decane at
ambient conditions where the precision of the GK results
was claimed to be slightly better than the SLLOD precision,
while the SLLOD calculation was based on twice as long a
total trajectory. Ryckaert et al.32 have compared the efficiency of a subtraction NEMD method for the viscosity with
GK concluding that the GK method is more efficient for
small system sizes. Erpenbeck33 has compared a color current NEMD method for self-diffusion calculation with GK
and found the GK method to be less N dependent and more
Dysthe, Fuchs, and Rousseau
efficient than the NEMD method. Taking into account that
we obtain all three transport coefficients in the same run, we
consider that GK may have an overall comparable or better
efficiency than the NEMD methods. For multicomponent
systems one will by GK also obtain the interdiffusion coefficient which would require yet another NEMD method and
more CPU time compared to GK. We conclude that if one is
interested in all linear transport coefficients and thermodynamic data for a molecular system one can obtain this with
better efficiency and one single method by using GK compared to an array of NEMD techniques. This is consistent
with what Mondello and Grest17 found for viscosity of long
alkanes based on arguments about the relaxation times governing the precision of the two techniques.
F. Predictive accuracy of new potential models
The potential models SMMK and AUA~3! have been
fitted to reproduce experimental thermodynamic data and we
consider transport coefficients calculated by simulation to be
predictions. We will call the deviation between the model
and experimental results for the transport coefficients the
predictive accuracy of the given model. The experimental
data used for comparison with the simulations ~see Table III!
is taken from Lee34 ~viscosity of gaseous n-butane! and Collings and McLaughlin35 ~viscosity of 2-methylbutane!. The
other transport coefficients are calculated by the correlation
of Assael et al.36 that is as accurate ~or better! as the experimental data for these systems.
One observes from Table III that for both the SMMK
and AUA~3! models and for all molecules the viscosity is
underestimated ~222% to 240%! at high densities. The diffusion coefficients are correspondingly too high ~up to 66%!.
System H is a curious exception where both diffusion and
viscosity are underpredicted ~211% and 220%, respectively!. The low temperature n-butane results are not considered here ~see discussion in Sec. III C 4!. The AUA~3! model
predicts a viscosity that is 15% too high for gaseous n-butane
~10% precision!. The deviations for the thermal conductivity
are not significant. The overall tendency of low predictive
accuracy at high density will be studied more closely in another communication.
IV. CONCLUSION
We have studied the precision, accuracy and efficiency
of the Green-Kubo method for evaluating transport coefficients by molecular dynamics applied to flexible, multicenter
models of linear and branched alkanes in the gas phase and
in the liquid phase from ambient conditions to close to the
triple point.
We find that the cutoff distance, r c , does not have any
significant effect on the accuracy of the transport properties
in the gas phase. The self-diffusion in the gaseous state depends linearly on N 21/3 as is the case in the liquid state,
although the mechanism causing this is believed to be different. Time steps of 1–5 fs are all acceptable for the calculation of equilibrium properties in the gas phase, but the most
accurate trajectory ~using d t51 fs! has a significantly higher
~9%! diffusion coefficient. We have no good explanation for
Downloaded 15 Jan 2002 to 129.240.85.171. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
this result. In the liquid phase time steps of 2–8 fs may be
used without affecting the accuracy of any property and
shorter time steps only make round off errors more important
than truncation errors. As long as the total energy does not
diverge completely, the viscosities calculated by GK are as
accurate as the precision, allowing time steps much larger
than normally used.
We have proposed the use of the system relaxation time
t s as a time unit for equilibration and subaveraging and find
that this gives a sufficient criterion for obtaining reliable estimates of the precision of the transport coefficients. Comparison between single run and independent run calculations
and comparison with NEMD results shows that the accuracy
of the method is as good as the precision. We also find that
the efficiency of the GK method is comparable to or better
than NEMD methods when one wants to calculate all the
linear transport coefficients, even at very high densities, low
temperatures and for large molecules. There is an added advantage of using one single simulation and one method to
calculate all transport properties. We have also shown that
one at the same time obtains valuable insight into the intramolecular relaxation processes of branched and linear
molecules.
The recently proposed mean potential field models
SMMK5 and AUA~3!7 underpredict the viscosity by 22%–
40% and overpredict the diffusion coefficient by up to 66%
for n-butane, n-decane and 2-methylbutane at high densities.
For gaseous n-butane the viscosity is slightly overpredicted
and for all states the thermal conductivity is predicted within
precision.
ACKNOWLEDGMENTS
We thank Jerome Delhommelle and Anne Boutin for
performing the Monte Carlo simulations and for fruitful discussions. We are grateful to Marc Durandeau for having initiated this work and for useful comments and to Total Exploration Production for a grant for one of us ~D.K.D.!. We
acknowledge the Institut du Dévelopement et des Ressources
en Informatique Scientifique ~IDRIS! for a generous allocation of Cray T3E computer time.
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