Generalized-Ensemble Algorithms for the Isobaric–Isothermal Ensemble

Yoshiharu MORI1 and Yuko OKAMOTO1;2

1Department of Physics, Nagoya University, Nagoya 464-8602, Japan2Structural Biology Research Center, Nagoya University, Nagoya 464-8602, Japan

(Received April 13, 2010; accepted June 3, 2010; published July 12, 2010)

We present generalized-ensemble algorithms for isobaric–isothermal molecular simulations. Inaddition to the multibaric–multithermal algorithm and replica-exchange method for the isobaric–isothermal ensemble, which have already been proposed, we propose a simulated tempering method forthis ensemble. We performed molecular dynamics simulations with these algorithms for an alaninedipeptide system in explicit water molecules to test the effectiveness of the algorithms. We found thatthese generalized-ensemble algorithms are all useful for conformational sampling of biomolecularsystems in the isobaric–isothermal ensemble.

KEYWORDS: generalized-ensemble algorithm, multibaric–multithermal algorithm, simulated tempering, replica-exchange method, isobaric–isothermal ensemble, Monte Carlo simulation, molecular dynamicssimulation

DOI: 10.1143/JPSJ.79.074003

1. Introduction

Monte Carlo (MC) and molecular dynamics (MD)simulations with generalized-ensemble algorithms have beenwidely used for studies of proteins and peptides in order tohave efficient conformational sampling (for a review, see,e.g., ref. 1). Three well-known generalized-ensemble algo-rithms are the multicanonical algorithm (MUCA)2,3) (theMD version was developed in refs. 4 and 5), replica-exchange method (REM)6) (the method is also referred to asparallel tempering7) and the MD version, which is referred toas REMD, was developed in ref. 8), and simulated temper-ing (ST).9,10) These methods can be used for obtainingaccurate physical quantities in the canonical ensemble.Among them, REM (particularly REMD) is often usedbecause the weight factor is a priori known (i.e., theBoltzmann factor), while those for MUCA and ST have tobe determined before the simulations.

Multidimensional (or multivariable) extensions of theoriginal generalized-ensemble algorithms have been devel-oped in many ways (see the references in ref. 1) andrecently the general formulations have been given inrefs. 11–13.

In this article, we present a multidimensional simulatedtempering algorithm in the isobaric–isothermal (NPT)ensemble. Several generalized-ensemble algorithms forthe NPT ensemble have been proposed [for example,the multibaric–multithermal (MUBATH) algorithm14–17)

and REM in temperature and pressure18–21)] and here, wepresent an ST algorithm for the NPT ensemble to completegeneralized-ensemble algorithms for this ensemble.

2. Methods

We first briefly review the generalized-ensemble algo-rithms for isobaric–isothermal molecular simulations. Let usconsider a physical system that consists of N atoms and thatis in a box of a finite volume V . The states of the system arespecified by coordinates r � fr1; r2; . . . ; rNg and momentap � fp1; p2; . . . ; pNg of the atoms and volume V of the box.The potential energy Eðr;VÞ for the system is a function of rand V .

We first describe MC simulation algorithms for MUCA,REM, and ST in the NPT ensemble. In these cases,momenta of atoms do not have to be considered. To ensureconvergence to thermal equilibrium, the detailed balancecondition is imposed and a transition probability wðX! X0Þfrom an old state X to a new state X0 is given by theMetropolis criterion.22)

In MUBATH simulations, we introduce a functionHðE;VÞand use a weight factor WmbtðE;VÞ � exp½��0HðE;VÞ� sothat the distribution function fmbtðE;VÞ of E and V may beuniform:

fmbtðE;VÞ / nðE;VÞWmbtðE;VÞ ¼ constant; ð1Þ

where �0 is an arbitrary inverse reference temperature definedas �0 ¼ 1=kBT0 (kB is the Boltzmann constant) and nðE;VÞ isthe density of states.

To perform MUBATH MC simulations, the trial movesare generated in the same way as in the usual constant NPTMC simulations23) and the transition probability from X �fs;Vg to X0 � fs0;V 0g is given by14,15)

wmbtðX! X0Þ ¼ min½1; expð��mbtÞ�; ð2Þ

where

�mbt ¼ �0fH½Eðs0;V 0Þ;V 0� �H½Eðs;VÞ;V�� NkBT0 lnðV 0=VÞg; ð3Þ

and s ¼ fs1; s2; . . . ; sNg is the scaled coordinates defined bysi ¼ V�1=3ri ði ¼ 1; 2; . . . ;NÞ. Here, we are assuming thebox is a cube of side V�1=3.

In REM simulations, we prepare a system that consists ofMT �MP non-interacting replicas of the original system,where MT and MP are the number of temperature andpressure values used in the simulation, respectively. Thereplicas are specified by labels i (i ¼ 1; 2; . . . ;MT �MP),temperature by mt (mt ¼ 1; 2; . . . ;MT ), and pressure by mp

(mp ¼ 1; 2; . . . ;MP).To perform REM MC (REMC) simulations, we carry out

the following two steps alternately: (1) perform a usualconstant NPT simulation in each replica at assigned temper-ature and pressure and (2) try to exchange the replicas. If thetemperature (specified by mt and nt) and pressure (specified

Journal of the Physical Society of Japan

Vol. 79, No. 7, July, 2010, 074003

#2010 The Physical Society of Japan

074003-1

by mp and np) between the replicas are exchanged, thetransition probability from X � f. . . ; ðs½i�;V ½i�;Tmt

;PmpÞ; . . . ;

ðs½ j�;V ½ j�;Tnt ;Pnp Þ; . . .g to X0 � f. . . ; ðs½i�;V ½i�;Tnt ;Pnp Þ; . . . ;ðs½ j�;V ½ j�;Tmt

;PmpÞ; . . .g at the trial is given by19,20)

wremðX! X0Þ ¼ min½1; expð��remÞ�; ð4Þ

where

�rem ¼ ð�mt� �nt Þ½Eðs

½ j�;V ½ j�Þ � Eðs½i�;V ½i�Þ�þ ð�mt

Pmp� �ntPnp ÞðV

½ j� � V ½i�Þ: ð5Þ

In ST simulations, we introduce a function gðT ;PÞ anduse a weight factor WstðE;V ; T ;PÞ � exp½��ðE þ PVÞ þgðT ;PÞ� so that the distribution function fstðT ;PÞ of T and P

may be uniform:

fstðT ;PÞ /Z 1

0

dV

ZV

dr Wst½Eðr;VÞ;V; T ;P�

¼ constant: ð6ÞFrom eq. (6), it is found that gðT ;PÞ is formally given by

gðT ;PÞ ¼ � ln

Z 10

dV

ZV

dr exp½��ðEðr;VÞ þ PVÞ�� �

; ð7Þ

and the function is the dimensionless Gibbs free energyexcept for a constant.

To perform ST MC simulations, we carry out thefollowing two steps alternately: (1) perform a usual constantNPT simulation and (2) try to update the temperature andpressure. The transition probability from X � fs;V ; T ;Pg toX0 � fs;V; T 0;P0g for this trial is given by

wstðX! X0Þ ¼ min½1; expð��stÞ�; ð8Þ

where

�st ¼ ð�0 � �ÞEðs;VÞ þ ð�0P0 � �PÞV� ½gðT 0;P0Þ � gðT ;PÞ�: ð9Þ

For MD simulations with MUCA, REM, and ST in theNPT ensemble, the actual formulations depend on constanttemperature and pressure algorithms. Here, we employ theMD methods with the Martyna–Tobias–Klein (MTK) algo-rithm,24) whose equations of motion follow Nose25,26) andHoover27) for the thermostat and Andersen28) for the barostat.

To perform MUBATH MD simulations, we solve theusual equations of motion for the MTK algorithm exceptthat it is necessary to modify the equations for pi and p" asfollows:

dpidt¼ �

@H@E

@E

@ri� 1þ

1

N

� �p"

Wpi �

p�

Qpi; ð10aÞ

dp"

dt¼ 3V Pint �

@H@V

� �þ

1

N

XNi¼1

p2i

mi

�p�

Qp"; ð10bÞ

where

Pint ¼1

3V

XNi¼1

p2i

mi

�@H@E

XNi¼1

ri �@E

@riþ 3V

@E

@V

!" #; ð11Þ

p" is the momentum associated with the logarithm of V , p�is the momentum of the thermostat, and W and Q are themasses of barostat and thermostat, respectively.

When we perform MD simulations with REM and ST, themomenta should be rescaled if the replicas are exchanged forthe temperature in REM8) and the temperature is updated inST.12)

In REM MD (REMD) simulations with the MTKalgorithm, let an old state be X � f. . . ; ðr½i�;V ½i�; p½i�; p½i�" ;p½i�� ;Tmt

;PmpÞ; . . . ; ðr½ j�;V ½ j�; p½ j�; p½ j�" ; p

½ j�� ; Tnt ;PnpÞ; . . .g and a

new state X � f. . . ; ðr½i�;V ½i�; p½i�0; p½i�0" ; p½i�0� ; Tnt ;Pnp Þ; . . . ; ðr½ j�;

V ½ j�; p½ j�0; p½ j�0" ; p½ j�0� ; Tmt;PmpÞ; . . .g. When replica exchange

is performed, the momenta should be rescaled as fol-lows:29)

p½i�0 ¼

ffiffiffiffiffiffiffiTnt

Tmt

sp½i�; p½ j�0 ¼

ffiffiffiffiffiffiffiTmt

Tnt

sp½ j� ð12aÞ

p½i�0" ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiTntWn

TmtWm

sp½i�" ; p½ j�0" ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiTmt

Wm

TntWn

sp½ j�" ð12bÞ

p½i�0� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiTntQn

TmtQm

sp½i�� ; p

½ j�0� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiTmt

Qm

TntQn

sp½ j�� ; ð12cÞ

where Wm and Qm are the masses in the simulation at Tmtand

Pmp, and Wn and Qn are the ones at Tnt and Pnp . Then the

transition probability (4) and (5) can also be used in theREMD simulations.

In ST MD simulations with the MTK algorithm, let an oldstate be X � fr;V ; p; p"; p�; T ;Pg and a new state X0 � fr;V ;p0; p0"; p

0�; T0;P0g. If p0; p0", and p0� are written as

p0 ¼ffiffiffiffiffiT 0

T

rp; p0" ¼

ffiffiffiffiffiffiffiffiffiffiffiT 0W 0

TW

rp"; p0� ¼

ffiffiffiffiffiffiffiffiffiffiT 0Q0

TQ

sp�; ð13Þ

where W 0 and Q0 are the masses in the simulation at T 0 andP0, then it can be shown that the transition probability isgiven by eqs. (8) and (9).

While the weight factors in REM simulations are a prioriknown, those in MUBATH and ST simulations have to bedetermined before the simulations. We propose two-dimen-sional extensions of the replica-exchange multicanonicalalgorithm (REMUCA)30) and the replica-exchange simulatedtempering (REST)31) for the isobaric–isothermal ensemble toovercome the difficulty of determining the weight factors.First, we perform a REM simulation for the isobaric–isothermal ensemble and then calculate the density of statesand the dimensionless Gibbs free energy by the multiple-histogram reweighting techniques,32) or the weighted histo-gram analysis method (WHAM),33) using the results ofthe REM simulation. The WHAM equations for isobaric–isothermal simulations are given by

nðE;VÞ

¼

XMT

i¼1

XMP

j¼1

ð1þ 2�ijÞ�1NijðE;VÞ

XMT

i¼1

XMP

j¼1

nijð1þ 2�ijÞ�1 exp½gij � �iðE þ PjVÞ�; ð14aÞ

expð�gijÞ

¼XE

XV

nðE;VÞ exp½��iðE þ PjVÞ�; ð14bÞ

where gij � gðTi;PjÞ, NijðE;VÞ is the histogram of E and V ,nij is the number of the data, and �ij is the integratedautocorrelation time in the simulation at Ti and Pj,respectively. These equations need to be solved self-consistently. We remark that for biomolecular systems, itwas found that �ij may be set to a constant value for all i andj in eq. (14a).33)

J. Phys. Soc. Jpn., Vol. 79, No. 7 Y. MORI and Y. OKAMOTO

074003-2

In the isobaric–isothermal version of REST, the weightfactor can also be determined using the dimensionless Gibbsfree energy obtained from eqs. (14a) and (14b).

3. Results

In order to verify that the generalized-ensemble algo-rithms discussed above can be effective for conformationalsampling and give the same results, we performed MDsimulations with the three generalized-ensemble algorithms.We used a system of an alanine dipeptide in 73 surroundingwater molecules and the system was placed in a cubic cellwith periodic boundary conditions. Both of the backbonedihedral angles � and of the peptide were initially setto 180�. The simulations were performed with the TINKER

program package.34) Several of the programs were modifiedand a few programs were added so that MUBATH, REM,and ST simulations with the MTK algorithm can beperformed. We used the AMBER parm99 force field35) forthe peptide and the TIP3P water model.36) The electrostaticinteractions were calculated by the particle mesh Ewaldmethod.37,38) In the van der Waals interaction calculations,we used the spherical cutoff method and the cutoff distancewas set to 12.0 A. The integration of the equations of motionwas employed by the method proposed by Martyna et al.39)

and the Shake/Rattle/Roll constraint method39–41) was usedso that the water molecules are rigid body molecules. Theunit time step was set to 0.5 fs. The mass parameters W andQ were determined as in ref. 39.

First, we performed the two-dimensional REMD simu-lation. The simulation time was set to 2.0 ns. We used thefollowing six temperature (T1; . . . ;T6) and four pressure(P1; . . . ;P4) values: 280, 305, 332, 362, 395, and 430 K fortemperature and 0.1, 65, 150, and 250 MPa for pressure. Atthe replica-exchange trial, either exchanging temperature (T-exchange) or exchanging pressure (P-exchange) was chosenrandomly and then the pairs fðT1;T2Þ; ðT3;T4Þ; ðT5;T6Þg orfðT2; T3Þ; ðT4; T5Þg for T-exchange and fðP1;P2Þ; ðP3;P4Þg orfðP2;P3Þg for P-exchange were also chosen randomly.

Figure 1(a) shows the time series of T and P in one of thereplicas and Fig. 1(b) shows the time series of the label ofthe replicas in the simulation at 280 K and 250 MPa. Fromthese figures, it is found that random walks in T–P spacewere realized in each of the replicas. The acceptance ratiosin the REM simulation were in the range from 0.152 to 0.205for T-exchange and from 0.296 to 0.446 for P-exchange,respectively, which also implies that sufficient number ofreplica exchange was realized.

We then performed 24 MUBATH simulations of 2.0 ns,where the total simulation time was 48 ns so that it is equalto that in the REMD simulation. In each of the simulations,different initial velocities were given. The data weresampled every 100 fs. The reference temperature was set to430 K. @H=@E and @H=@V were interpolated by the third-order polynomial, following ref. 17.

Figure 2 shows the probability distributions of E and V

from the MUBATH simulations. From Fig. 2(a), it is foundthat the MUBATH simulations gave a uniform distributionin the range where the density of states was obtainedaccurately in the REMD simulation and that the distributionof the MUBATH simulations was much wider than the onesfor the NPT ensemble [see Fig. 2(b)].

Twenty-four ST simulations of 2.0 ns were carried out,which gave us the same number of sampled data as inthe REMD and MUBATH simulations. In each simulation,different initial velocities were given. We used the sametemperature and pressure values as in the REMD simulation.We tried to update the parameter (T or P) every 100 fs andthe data were sampled just before the trials. The choicebetween updating T (T-update) and updating P (P-update)was made randomly and then either Tmt�1 or Tmtþ1 for T-update and Pmp�1 or Pmpþ1 for P-update was also chosenrandomly, where mt and mp stand for the labels correspond-ing to the temperature and the pressure before the trial,respectively.

Figure 3(a) shows the time series of T and P and Fig. 3(b)shows the probability distribution of T and P. These figuresindicate that random walks in T–P space were successfullyrealized. The acceptance ratios in the ST simulations were inthe range from 0.260 to 0.430 for T-update and from 0.426to 0.642 for P-update, respectively.

Figure 4 shows the probability distributions of the back-bone dihedral angles at 298 K and 0.1 MPa in all thesimulations. Compared with the simulations with REMD,MUBATH, and ST, we also performed 24 conventionalisobaric–isothermal simulations of 2.0 ns at 298 K and0.1 MPa with different initial velocities. The dihedral angledistributions in the simulations with the generalized-ensem-ble algorithms had a small peak in 0 � � � 90� and�90 � � 90�, although there was no peak in the rangein the distribution of the conventional simulations.

(a)

(b)

Fig. 1. Results of the two-dimensional REMD simulation: (a) the time

series of T and P in one of the replicas and (b) the time series of the

replica label at 280 K and 250 MPa.

J. Phys. Soc. Jpn., Vol. 79, No. 7 Y. MORI and Y. OKAMOTO

074003-3

We divided the whole ð�; Þ plane into six states (PII, C5,�R, �P, �L, and Cax

7 ) and calculated the population W of eachstate, following ref. 43. The population WPII

of the PII stateis, for example, defined by

WPII¼Zð�; Þ2PII

d� d f ð�; Þ; ð15Þ

where f ð�; Þ is the distribution of the dihedral angles � and . Table I lists the population of each state at 298 K and0.1 MPa obtained by the three generalized-ensemble simu-lations and the MUBATH simulation in ref. 43. The resultsof REMD, MUBATH, and ST almost agree with each otherand with the previous ones in ref. 43 within the error.Therefore, all the simulations with the three generalized-ensemble algorithms were able to provide the same resultsand reproduce the distribution obtained previously.

4. Conclusions

In this article, we presented the extensions of MUCA,REM, and ST for isobaric–isothermal molecular simulations.These algorithms can be effective methods for conforma-tional sampling and give accurate physical quantities in theisobaric–isothermal ensemble. Therefore, one can use the

(a)

(b)

Fig. 3. Results of the ST simulations: (a) the time series of T and P for

2.0 ns and (b) the logarithm of the probability distribution f of T and P.

(a)

(b)

Fig. 2. Logarithm of probability distributions f of E and V (a) in the

MUBATH simulations and (b) for the NPT ensemble at 280 K and

250 MPa (left) and at 430 K and 0.1 MPa (right). The distributions

corresponding to the NPT ensemble were calculated by the single-

histogram reweighting techniques42) using the results of the MUBATH

simulations.

(a) (b)

(c) (d)

Fig. 4. (Color online) Contour maps of probability distribution of the

backbone dihedral angles � and in the simulations with (a) REMD, (b)

MUBATH, and (c) ST and (d) in the conventional isobaric–isothermal

simulations. In these figures, the probability distributions at 298 K and

0.1 MPa are plotted in logarithmic scale and were calculated by the

single-histogram reweighting techniques for MUBATH and by WHAM

for REMD and ST.

Table I. Population of each state at 298 K and 0.1 MPa.

State REMD MUBATH ST MUBATH43Þ

PII 0.012(2) 0.019(3) 0.017(3) 0.017(3)

C5 0.050(5) 0.057(7) 0.053(9) 0.046(6)

�R 0.265(10) 0.294(8) 0.280(10) 0.281(11)

�P 0.668(8) 0.625(11) 0.641(12) 0.648(11)

�L 0.005(4) 0.005(3) 0.008(5) 0.007(2)

Cax7 0.000(0) 0.000(0) 0.000(0) 0.000(0)

074003-4

J. Phys. Soc. Jpn., Vol. 79, No. 7 Y. MORI and Y. OKAMOTO

generalized-ensemble algorithms to study temperature andpressure effects on complex systems such as biomolecularsystems.

We expect the simulated tempering to be a more usefulmethod for simulations in large systems. Many replicas areneeded in REM simulations of large systems, while only onereplica is used in ST simulations. In addition, ST can beeasily implemented into existing program packages com-pared to MUCA (or MUBATH), in which the forcecalculation part of programs has to be modified for MDsimulations. We only have to change simulation parameterssuch as temperature and pressure during simulations for ST.This allows us to write a short script program such as shelland Perl to implement the ST method without extensivemodifications of program packages. Therefore, molecularsimulations with ST will be more easily applicable to largersystems than with the other two generalized-ensemblealgorithms.

Acknowledgments

Some of the computations were performed on the super-computers at the Information Technology Center, NagoyaUniversity and at the Research Center for ComputationalScience, Institute for Molecular Science. This work wassupported, in part, by Grants-in-Aid for Scientific Researchon Innovative Areas (‘‘Fluctuations and Biological Func-tions’’) and for the Next Generation Super ComputingProject, Nanoscience Program from the Ministry of Educa-tion, Culture, Sports, Science and Technology, Japan(MEXT).

1) A. Mitsutake, Y. Sugita, and Y. Okamoto: Biopolymers 60 (2001) 96.

2) B. A. Berg and T. Neuhaus: Phys. Lett. B 267 (1991) 249.

3) B. A. Berg and T. Neuhaus: Phys. Rev. Lett. 68 (1992) 9.

4) U. H. E. Hansmann, Y. Okamoto, and F. Eisenmenger: Chem. Phys.

Lett. 259 (1996) 321.

5) N. Nakajima, H. Nakamura, and A. Kidera: J. Phys. Chem. B 101

(1997) 817.

6) K. Hukushima and K. Nemoto: J. Phys. Soc. Jpn. 65 (1996) 1604.

7) E. Marinari, G. Parisi, and J. J. Ruiz-Lorenzo: in Spin Glasses and

Random Fields, ed. A. P. Young (World Scientific, Singapore, 1998)

p. 59.

8) Y. Sugita and Y. Okamoto: Chem. Phys. Lett. 314 (1999) 141.

9) A. P. Lyubartsev, A. A. Martsinovski, S. V. Shevkunov, and P. N.

Vorontsov-Velyaminov: J. Chem. Phys. 96 (1992) 1776.

10) E. Marinari and G. Parisi: Europhys. Lett. 19 (1992) 451.

11) A. Mitsutake and Y. Okamoto: Phys. Rev. E 79 (2009) 047701.

12) A. Mitsutake and Y. Okamoto: J. Chem. Phys. 130 (2009) 214105.

13) A. Mitsutake: J. Chem. Phys. 131 (2009) 094105.

14) H. Okumura and Y. Okamoto: Chem. Phys. Lett. 383 (2004) 391.

15) H. Okumura and Y. Okamoto: Phys. Rev. E 70 (2004) 026702.

16) H. Okumura and Y. Okamoto: Chem. Phys. Lett. 391 (2004) 248.

17) H. Okumura and Y. Okamoto: J. Comput. Chem. 27 (2006) 379.

18) T. Nishikawa, H. Ohtsuka, Y. Sugita, M. Mikami, and Y. Okamoto:

Prog. Theor. Phys. Suppl. 138 (2000) 270.

19) T. Okabe, M. Kawata, Y. Okamoto, and M. Mikami: Chem. Phys.

Lett. 335 (2001) 435.

20) Y. Sugita and Y. Okamoto: in Lecture Notes in Computational Science

and Engineering, ed. T. Schlick and H. H. Gan (Springer, Berlin,

2002) p. 303; cond-mat/0102296.

21) D. Paschek and A. E. Garcıa: Phys. Rev. Lett. 93 (2004) 238105.

22) N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and

E. Teller: J. Chem. Phys. 21 (1953) 1087.

23) I. R. McDonald: Mol. Phys. 23 (1972) 41.

24) G. J. Martyna, D. J. Tobias, and M. L. Klein: J. Chem. Phys. 101

(1994) 4177.

25) S. Nose: Mol. Phys. 52 (1984) 255.

26) S. Nose: J. Chem. Phys. 81 (1984) 511.

27) W. G. Hoover: Phys. Rev. A 31 (1985) 1695.

28) H. C. Andersen: J. Chem. Phys. 72 (1980) 2384.

29) Y. Mori and Y. Okamoto: J. Phys. Soc. Jpn. 79 (2010) 074001.

30) Y. Sugita and Y. Okamoto: Chem. Phys. Lett. 329 (2000) 261.

31) A. Mitsutake and Y. Okamoto: Chem. Phys. Lett. 332 (2000) 131.

32) A. M. Ferrenberg and R. H. Swendsen: Phys. Rev. Lett. 63 (1989)

1195.

33) S. Kumar, D. Bouzida, R. H. Swendsen, P. A. Kollman, and J. M.

Rosenberg: J. Comput. Chem. 13 (1992) 1011.

34) J. W. Ponder and F. M. Richards: J. Comput. Chem. 8 (1987) 1016.

35) J. Wang, P. Cieplak, and P. A. Kollman: J. Comput. Chem. 21 (2000)

1049.

36) W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey, and

M. L. Klein: J. Chem. Phys. 79 (1983) 926.

37) T. Darden, D. York, and L. Pedersen: J. Chem. Phys. 98 (1993) 10089.

38) U. Essmann, L. Perera, M. L. Berkowitz, T. Darden, H. Lee, and L. G.

Pedersen: J. Chem. Phys. 103 (1995) 8577.

39) G. J. Martyna, M. E. Tuckerman, D. J. Tobias, and M. L. Klein: Mol.

Phys. 87 (1996) 1117.

40) J.-P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen: J. Comput. Phys.

23 (1977) 327.

41) H. C. Andersen: J. Comput. Phys. 52 (1983) 24.

42) A. M. Ferrenberg and R. H. Swendsen: Phys. Rev. Lett. 61 (1988)

2635 [Errata; 63 (1989) 1658].

43) H. Okumura and Y. Okamoto: J. Phys. Chem. B 112 (2008) 12038.

074003-5

J. Phys. Soc. Jpn., Vol. 79, No. 7 Y. MORI and Y. OKAMOTO