On the determination of orientational temperature from computer ...

JOURNAL OF CHEMICAL PHYSICS

VOLUME 114, NUMBER 15

15 APRIL 2001

On the determination of orientational configurational temperature from computer simulation A. A. Chialvoa) Chemical and Analytical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6110 and Department of Chemical Engineering, University of Tennessee, Knoxville, Tennessee 37996-2200

J. M. Simonson Chemical and Analytical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6110

P. T. Cummings Departments of Chemical Engineering, Chemistry, and Computer Science, University of Tennessee, Knoxville, Tennessee 37996-2200 and Chemical Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6268

P. G. Kusalik Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, B3H 4J3, Canada

共Received 23 October 2000; accepted 31 January 2001兲 A straightforward derivation for the configurational temperature associated with the orientational ⍀ N portion of the configurational phase space of the molecules in an open system is presented. Explicit relationships are given for determining the configurational temperatures in classical simulations of molecular liquids, such as water, and their forms and their evaluation discussed. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1357792兴

INTRODUCTION

quently we need tools to test the validity of strategies used not only for the random translational,5 but also for orientational moves8 regarding the principles of microscopic reversibility9 and the most general infinitesimal rotation.10 A subtler situation may arise when dealing with metastable systems such as supercooled liquids or glasses, for which the configurational/orientational relaxation times become large.11 For these 共or any兲 systems, the most general thermodynamic definition of the temperature will involve the derivative ( ⳵ E/ ⳵ S) V,N , where both the entropy, S, and energy, E, will depend on all the microscopic degrees of freedom 共kinetic and configurational兲 of the system. While the motion of individual particles within these systems might be consistent with the desired temperature 共the measure of the time-average kinetic energy兲, the corresponding averages over configurational 共translational and orientational兲 degrees of freedom may not. This scenario might become more complex at low temperature and high density because of the possible coupling between translational and orientational degrees of freedom in supercooled liquids,12 which might hinder the expected energy equipartition. Yet, there are other compelling reasons behind our desire to assess the temperature of a system by analyzing average configurational measures rather than averages over the kinetic degrees of freedom. For instance, in the validation of microstructural information from x ray, electron, and neutron scattering spectra by means of reverse Monte Carlo techniques we require ways to diagnose the correctness of the procedure used in the processing and test the internal consistency of the raw data.3 Moreover, many researchers would like to be able to estimate the temperature of a system based

The temperature of an equilibrium system has been almost exclusively studied through the average kinetic energy associated with the translational and rotational degrees of freedom of their molecules.1 However, the recent publication of Rugh2 on the dynamical approach to thermodynamics highlighted alternative ways to define the temperature of a closed system, in terms of configurational properties in the microcanonical ensemble, and opened the door to the development of new tools for the validation and testing of new simulation strategies, and for the interpretation and analysis of systems away from equilibrium. The distinction between kinetic and configurational temperatures might appear as a mere academic curiosity. However, there are numerous simulation scenarios where either the control or evaluation of the kinetic temperature is not possible,3 or the systems under study are far from equilibrium.4 Moreover, there is always the need for new tools to check the validity or self-consistency of simulation algorithms5 and the configurational temperature provides one of the few consistency tests available for Monte Carlo simulation. Typically, the constants of motion are the targets for testing the validity and accuracy of the integration approach in the simulation of dynamical systems. Yet, the conservation of these quantities is not always a necessary or sufficient condition for either accuracy6 or correctness7 of the algorithm. Nondynamical simulations, such as Monte Carlo techniques, do not involve conserved quantities, and consea兲

Author to whom correspondence should be addressed.

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J. Chem. Phys., Vol. 114, No. 15, 15 April 2001

Orientational configurational temperature

on configurational information gathered from those spectra.13 As a first step toward the achievement of these goals, in a recent paper we revived an earlier theoretical development, the so-called hypervirial theorem developed by Hirschfelder,14 as a well-suited tool to make possible the molecular simulation route to positional and orientational configurational-temperatures.8 We implemented their calculation and reported results for a simple quadrupolar LennardJones system at normal conditions and for the TIP4P water model at low temperature. In the present communication we present an alternative derivation of an expression to determine the orientational configurational temperature for open systems, and discuss its evaluation in a standard computer 共MD or MC兲 simulation of a molecular liquid. ORIENTATIONAL CONFIGURATIONAL TEMPERATURE

The second term on the right-hand side of Eq. 共3兲 can be written explicitly as

␻i • ␶˙ i ⫽ ␻i • ⫹

兺j

冉

兺

␣ ⫽x,y,z

⫽ ␻i •

冉兺 j

兺

␣ ⫽⍀ x ,⍀ y ,⍀ z

T rot⬅

N 兺 i⫽1 ␻i •Ii • ␻i

gk

⫽

N 兺 i⫽1 L␫ •I⫺1 i •L␫

gk

,

共1兲

where we use the conventional notation k, Ii , Li , and ␻i to denote Boltzmann constant, the tensor of inertia, the angular momentum, and the angular velocity of molecule i, respectively. By assuming isokinetic conditions, the first and second time derivatives of the instantaneous temperature are T˙ rot⫽

N ␻i •L˙i 2 兺 i⫽1

gk

共 ⳵ ␶i / ⳵ ␣ j 兲 • 共 ⳵ ␣ j / ⳵ t 兲

共 ⳵ ␶i / ⳵ ␣ j 兲 • 共 ⳵ ␣ j / ⳵ t 兲

␻ j •ⵜ ⍀ j ␶i ⫹p j •

冊

兺j ⵜ r ␶i j

冊

共4兲

.

For systems in equilibrium, the components along the three positional (r ␣ ) and orientational (⍀ ␣ ) axes are equivalent, the linear and angular momenta are not correlated, while the positions and orientations are not correlated with their corresponding gradients 共forces and torques兲. Therefore, from Eqs. 共3兲–共4兲 we have that

冓兺 N

As recently indicated by Baranyai,15 the expression derived by Butler et al.5 for the configurational 共translational兲 temperature does not apply to open systems 共i.e., systems in phase equilibria, etc.兲, and consequently this author proposed a simpler derivation that fulfilled that condition. In this section we take Baranyai’s approach, and derive the corresponding expression for the configurational orientational 共as opposed to configurational translational兲 temperature of an open system, an analysis that parallels the corresponding hypervirial derivation given by Gray and Gubbins.16 We start with the expression of the instantaneous rotational kinetic temperature of a d-dimensional system, composed of N rigid molecules with g⬅dN degrees of freedom, i.e.,

0⫽

i⫽1

冔 冓兺 N

␶i •I⫺1 i • ␶i

⫹

i⫽1

N

␻i •

兺

j⫽1

which can also be written as

冓兺 N

0⫽

i⫽1

冔 冓兺 兺 N

I⫺1 i : ␶i ␶i

⫹

N

i⫽1 j⫽1

冔

共5兲

冔

共6兲

␻ j •ⵜ ⍀ j ␶i ,

Li L j :I⫺2 i ⵜ ⍀ j ␶i .

Note that, for a particular configuration, the only contribution to the average in the second term of Eq. 共6兲 is for i ⫽ j, because there is no correlation between the angular momenta of different molecules, i.e., ⫺2 0⫽ 具 I⫺1 i : ␶i ␶i 典 ⫹kT 具 Ii :Ii ⵜ ⍀ i ␶i 典

⫽ 具 ␶i • ␶i 典 ⫹kT 具 ⵜ ⍀ i ␶i 典 .

共7兲

In Eq. 共7兲 we have invoked the fact that the inertia tensor is the same for all molecules, and that Li Li ⫽kTIi , where the bar indicates an average over all particles for a particular configuration. Finally, from Eq. 共7兲 and the fact that ␶ ⫽⫺ⵜ ⍀ ␾ we obtain the final expression for the orientational configurational temperature, 2 ␾典, kT rot⫽⫺ 具 ␶ 2i 典 / 具 ⵜ ⍀ i ␶ i 典 ⫽ 具 共 ⵜ ⍀ i ␾ 兲 2 典 / 具 ⵜ ⍀ i

⫽0,

共2兲

6515

共8兲

which is precisely the same expression as that derived from Hirchfelder’s hypervirial theorem.

and T¨ rot⫽ ⫽

N ˙ N ˙ ¨ Li •I⫺1 2 兺 i⫽1 i •Li ⫹2 兺 i⫽1 ␻i •Li

SIMULATION IMPLEMENTATION AND FINAL REMARKS

gk N N ␶i •I⫺1 ˙i 2 兺 i⫽1 i • ␶i ⫹2 兺 i⫽1 ␻i • ␶

gk

⫽0,

共3兲

˙ i ⬅ ␶i is the rotational equation of morespectively, where L tion, and the upper dot 共dots兲 over any quantity represents a first 共higher兲 time derivative. The first time-derivative of the instantaneous temperature gives no additional information beyond the well-known fact that the angular velocity and corresponding torque are perpendicular to each other, i.e., not only the summation, but also each term in the summation of Eq. 共2兲 must be identically zero.

Starting from expression 共8兲, where ␾ (r N ,⍀ N ) is a potential depending on distance 共r兲 and orientation 共⍀兲, we have the angular gradient operator ⵜ ⍀ (¯)⬅l⫻ⵜ r (¯), which defines the torque ␶, ⵜ ⍀ ␾ ⬅l⫻ⵜ r ␾ ⫽⫺ ␶.

共9兲

In Eq. 共9兲 the operator ‘‘⫻’’ denotes cross-product of two vectors, and l⫽l x ˆi ⫹l y ˆj ⫹l z kˆ is the vector position of molecular sites with respect to the corresponding center of mass. Thus, the divergence of the torque becomes

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J. Chem. Phys., Vol. 114, No. 15, 15 April 2001

ⵜ ⍀ • ␶⫽⫺ⵜ ⍀ •ⵜ ⍀ ␾ ⫽⫺ⵜ 2⍀ ␾ ,

共10兲

l⫽l x ˆi ⫹l y ˆj ⫹l z kˆ

ⵜ 2⍀ (¯)

where is the angular Laplacian. Consequently, the orientational temperature, Eq. 共8兲, can be represented as

⫽ 共 l y ⳵ / ⳵ z⫺l z ⳵ / ⳵ y 兲 ˆi ⫹ 共 l z ⳵ / ⳵ x⫺l x ⳵ / ⳵ z 兲 ˆj ⫹ 共 l x ⳵ / ⳵ y⫺l y ⳵ / ⳵ x 兲 kˆ ,

kT rot⫽ 具 ␶2 典 / 具 ⵜ 2⍀ ␾ 典 and

⫽⫺ 具 ␶2 典 / 具 ⵜ ⍀ • ␶典 ⫽ 具 ␶2 典 / 具 共 l⫻ⵜ r 兲 • 共 l⫻ⵜ r ␾ 兲 典 ,

共11兲

where the angular Laplacian of the intermolecular potential, ⵜ 2⍀ ␾ ⫽(l⫻ⵜ r )•(l⫻ⵜ r ␾ ), can be written more explicitly by solving the two cross-products, as follows:

冏

冏

ˆi

ˆj

kˆ

lx

ly

lz

⳵/⳵x

⳵/⳵y

⳵/⳵z

共 l⫻ⵜ r 兲 ⫽

共12兲

,

冏 冏 ˆi

ˆj

kˆ

共 l⫻ⵜ r ␾ 兲 ⫽⫺ l x

ly

lz ,

fx

fy

fz

ⵜ r ␾ ⫽⫺ f x ˆi ⫺ f y ˆj ⫺ f z kˆ ,

⫽⫺ 共 l y f z ⫺l z f y 兲 ˆi ⫺ 共 l z f x ⫺l x f z 兲 ˆj ⫺ 共 l x f y ⫺l y f x 兲 kˆ , 共13兲 where f x ⫽⫺( ⳵ ␾ / ⳵ x) is the x-component of the force f. Therefore, the denominator of Eq. 共11兲 becomes

共 l⫻ⵜ r 兲 • 共 l⫻ⵜ r ␾ 兲 ⫽⫺ 共 l y ⳵ / ⳵ z⫺l z ⳵ / ⳵ y 兲共 l y f z ⫺l z f y 兲 ⫺ 共 l z ⳵ / ⳵ x⫺l x ⳵ / ⳵ z 兲共 l z f x ⫺l x f z 兲 ⫺ 共 l x ⳵ / ⳵ y⫺l y ⳵ / ⳵ x 兲共 l x f y ⫺l y f x 兲

⫽⫺l 2y 共 ⳵ f z / ⳵ z⫹ ⳵ f x / ⳵ x 兲 ⫺l z2 共 ⳵ f y / ⳵ y⫹ ⳵ f x / ⳵ x 兲 ⫺l 2x 共 ⳵ f z / ⳵ z⫹ ⳵ f y / ⳵ y 兲 ⫹l x l z 共 ⳵ f x / ⳵ z⫹ ⳵ f z / ⳵ x 兲 ⫹l y l z 共 ⳵ f y / ⳵ z⫹ ⳵ f z / ⳵ y 兲 ⫹l y l x 共 ⳵ f y / ⳵ x⫹ ⳵ f x / ⳵ y 兲 ⫹2 共 l x f x ⫹l y f y ⫹l z f z 兲 .

For the sake of completeness, we introduce the corresponding positional 共translational兲 configurational temperature, T trans , whose expression has been derived either from the hypervirial theorem8 or from Baranyai’s approach,15 i.e.,

␾ 共 r 兲 ⫽q i q j 共 1/r⫺ 共 ␧ r f ⫺1 兲 r 2 / 共 2␧ r f ⫹1 兲 r 3c 兲 , then f x ⫽q i q j x 共 1/r 3 ⫹2 共 ␧ r f ⫺1 兲 / 共 2␧ r f ⫹1 兲 r 3c 兲

kT trans⫽ 具 共 ⵜ r ␾ 兲 2 典 / 具 ⵜ r2 ␾ 典

→ ⳵ f x / ⳵ x⫽⫺q i q j 共 3x 2 /r 5 兲 ⫹ f x /x

⫽⫺ 具 f2 典 / 具 ⵜ r •f典 ,

共15兲

and ⵜ r •f⫽⫺ 关共 ⳵ f x / ⳵ x 兲 ⫹ 共 ⳵ f y / ⳵ x 兲 ⫹ 共 ⳵ f z / ⳵ x 兲兴 ⫽⫺2 共 ⳵ ␾ / ⳵ r 兲 /r⫺ 共 ⳵ 2 ␾ / ⳵ r 2 兲 .

共16兲

To make these working expressions more explicit, let us consider the two types of interactions that we might usually find in a typical water model 关such as the SPC 共Ref. 17兲 or TIP4P 共Ref. 18兲 model for water兴, i.e., involving electrostatic and Lennard-Jones potentials. For ␾ (r)⫽q i q j /r i j , where q i denotes a point charge on the site i and r i j ⫽r is the distance between the pair of sites i,j, f x ⫽q i q j x/r → ⳵ f x / ⳵ x⫽⫺q i q j 共 3x /r 兲 ⫹ f x /x 3

2

5

→ ⳵ f x / ⳵ y⫽⫺q i q j 共 3xy/r 5 兲 ,

共19兲

and Eq. 共18兲 becomes automatically satisfied if we recall that the molecule’s electroneutrality is implicitly fulfilled, i.e., 兺 ␣ , ␤ q ␣1 q ␤2 ⫽0. Obviously, the corresponding Ewald sum expressions might be determined in a straightforward manner. Otherwise, for the Lennard-Jones potential ␾ (r) ⫽4␧ 关 ( ␴ /r) 12⫺( ␴ /r) 6 兴 we have that, f x ⫽␧ 关 48共 ␴ 12/r 14兲 ⫺24共 ␴ 6 /r 8 兲兴 x → ⳵ f x / ⳵ x⫽⫺␧ 关 672共 ␴ 12/r 16兲 ⫺192共 ␴ 6 /r 10兲兴 x 2 ⫹ f x /x

⳵ f x / ⳵ y⫽⫺␧ 关 672共 ␴ 12/r 16兲 ⫺192共 ␴ 6 /r 10兲兴 xy⫽ ⳵ f y / ⳵ x,

共20兲

and

→ ⳵ f x / ⳵ y⫽⫺q i q j 共 3xy/r 兲 ⫽ ⳵ f y / ⳵ x, 共17兲 5

and analogously, for y and z-components. Finally, from Eq. 共16兲, ⵜ r2 共 q i q j /r 兲 ⫽0,

共14兲

共18兲

and hence this Laplacian will be similarly equal to zero for any multipolar interaction. Note that, if we use a reaction field approach for the electrostatic interactions, with a reaction field dielectric ␧ r f and a cut-off radius r c , i.e.,

ⵜ r2 ␾ 共 r 兲 ⫽␧ 关 528共 ␴ 12/r 14兲 ⫺120共 ␴ 6 /r 8 兲兴 .

共21兲

Note that according to Eqs. 共17兲–共21兲, only short-ranged 共nonharmonic兲 potentials contribute to the positional Laplacian, and consequently, to the definition of the positional configurational temperature 关Eq. 共15兲兴. In contrast, both shortand long-ranged potentials contribute to the angular Laplacian in the definition of the orientational configurational temperature 关Eq. 共8兲兴. This feature, in principle, suggests that the translational and orientational configurational temperatures would exhibit different dependence on the range of the inter-

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J. Chem. Phys., Vol. 114, No. 15, 15 April 2001

Orientational configurational temperature

molecular potential acting on the molecules of the system. Consequently, we might also expect they would exhibit different relaxation behavior.8 According to the previous analysis 关Eqs. 共8兲 and 共15兲兴 both configurational temperatures are defined in terms of the ratio between two independent averages. These averages are taken over the number of particles and over the trajectory in the phase space. Yet, we can define an instantaneous temperature, average only over the N-particles constituting the system.16 Finally we can average that instantaneous temperature as usual, over the phase space trajectory. This means that we can replace the ratio of two averages by the average of the ratio, i.e., T⫽

冓冔

␰ 具 ␰ 典 N,t →T⫽ ␨ 具 ␨ 典 N,t

共22兲 t

by invoking the following identity in the thermodynamic limit;19

冓 冔 冓冔

1 1 1⫹ ␦ ␨ 具 1⫺ ␦ ␨ ⫹ 共 ␦ ␨ 兲 2 ¯ 典 ⫽ ⬇ ␨ 具␨典 具␨典

共23兲

with ␦ ␨ ⫽( ␨ ⫺ 具 ␨ 典 )/ 具 ␨ 典 , where 具 ␰ 典 N,t and 具 ␨ 典 N,t are the numerator and k-times the denominator of the right-hand side of either Eq. 共8兲 or 共15兲 for a finite system of size N. Therefore, Eq. 共23兲 reduces to

冓冔 冓冔

␰ 1 ⫽ 具␰典 ␨ ␨

共24兲

by assuming that the numerator and denominator are not correlated 共independent兲. Thus, according to Eqs. 共23兲–共24兲,

冓冔

␰ 具 1⫺ ␦ ␨ ⫹ 共 ␦ ␨ 兲 2 ¯ 典具 ␰ 典 具 1⫹ 共 ␦ ␨ 兲 2 ¯ 典具 ␰ 典 ⫽ ⫽ 共25兲 ␨ 具␨典 具␨典

since 具 ␦ ␨ 典 ⫽0, and finally, T⫽

冉

冊

具 ␰ 典 具 ␨ 2典 具␰典 ⫹O 共 N ⫺1 兲 . 2 ⫹¯ ⫽ 具␨典 具␨典 具␨典

共26兲

In the thermodynamic limit, lim N→⬁

冓冔

␰ 具 ␰ 典 N,t ⫽ ␨ 具 ␨ 典 N,t

. t

共27兲

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ACKNOWLEDGMENTS

This research was sponsored by the Division of Chemical Sciences, Geosciences, and Biosciences, Office of Basic Energy Sciences, and by the Environmental Management Science Program 共TTP OR17-SP22兲, under Contract No. DE-AC05-00OR22725 with Oak Ridge National Laboratory, managed and operated by UT-Battelle, LLC. P.T.C. was supported by the Division of Chemical Sciences, Geosciences, and Biosciences, Office of Basic Energy Sciences, U.S. Department of Energy. P.G.K. acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada. A.A.C. acknowledges fruitful discussions with Dr. Andras Baranyai during the visit to the Department of Theoretical Chemistry at Eo¨tvo¨s University 共Budapest, September 2000兲. D. A. McQuarrie, Statistical Mechanics 共Harper and Row, New York, 1976兲. 2 H. H. Rugh, J. Phys. A 31, 7761 共1998兲. 3 G. To´th and A. Baranyai, Mol. Phys. 97, 339 共1999兲. 4 G. Ayton, O. G. Jepps, and D. J. Evans, Mol. Phys. 96, 915 共1999兲. 5 B. D. Butler, G. Ayton, O. G. Jepps, and D. J. Evans, J. Chem. Phys. 109, 6519 共1998兲. 6 D. Greenspan, J. Comput. Phys. 91, 490 共1990兲. 7 A. R. Tindell, D. J. Tildesley, and J. Walton, CCP5 Newsletter 4, 26 共1982兲. 8 A. A. Chialvo, J. M. Simonson, P. G. Kusalik, and P. T. Cummings, in Proceedings of Foundations of Molecular Modeling and Simulation (FOMMS 2000), AICHE Symposium Series 共to be published兲. 9 M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids 共Oxford University Press, Oxford, 1987兲. 10 V. J. Menon, Eur. J. Phys. 21, 91 共2000兲. 11 S. Sastry, P. G. Debenedetti, F. H. Stillinger, T. B. Schro” der, J. C. Dyre, and S. C. Glotzer, Physica A 270, 301 共1999兲. 12 C. Theis and R. Schilling, Phys. Rev. E 60, 740 共1999兲. 13 H. Heinze, P. Borrmann, H. Stamerjohanns, and E. R. Hilf, Z. Phys. D: At., Mol. Clusters 40, 190 共1997兲. 14 J. O. Hirschfelder, J. Chem. Phys. 33, 1462 共1960兲. 15 A. Baranyai, J. Chem. Phys. 112, 3964 共2000兲. 16 C. G. Gray and K. E. Gubbins, Theory of Molecular Fluids 共Oxford University Press, Oxford, 1985兲. 17 H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, and J. Hermans, in Intermolecular Forces: Proceedings of the Fourteenth Jerusalem Symposium on Quantum Chemistry and Biochemistry, edited by B. Pullman 共Reidel, Dordrecht, 1981兲, pp. 331–342. 18 W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey, and M. L. Klein, J. Chem. Phys. 79, 926 共1983兲. 19 E. M. Pearson, T. Halicioglu, and W. A. Tiller, Phys. Rev. A 32, 3030 共1985兲. 1

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