Molecular dynamics study of pressure in molecular systemsTetsuo Tominaga and Sidney Yip Citation: The Journal of Chemical Physics 100, 3747 (1994); doi: 10.1063/1.466362 View online: http://dx.doi.org/10.1063/1.466362 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/100/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Molecular dynamics study on entrainment phenomenon in model molecular systems AIP Conf. Proc. 1518, 729 (2013); 10.1063/1.4794669 A molecular dynamics study of the role of pressure on the response of reactive materials to thermal initiation J. Appl. Phys. 107, 093517 (2010); 10.1063/1.3340965 Effects of pressure on structure and dynamics of model elastomers: A molecular dynamics study J. Chem. Phys. 129, 154905 (2008); 10.1063/1.2996009 Tracer surface diffusion at high pressures: Molecular-dynamics study J. Chem. Phys. 113, 3868 (2000); 10.1063/1.1287716 Glass formation in simple ionic systems via constant pressure molecular dynamics J. Chem. Phys. 90, 7384 (1989); 10.1063/1.456218

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Molecular dynamics study of pressure in molecular systems Tetsuo Tominagaa) and Sidney Yip Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

(Received 19 July 1993; accepted 18 November 1993)

In molecular dynamics simulation of molecular systems, an atomistic model is needed to describe the intramolecular effects on system properties such as pressure. An expression for computing the pressure is derived based on the virial theorem, with explicit kinetic, intra-, and intermolecular contributions. It is shown that the virial terms arising from three- or four-body forces that depend only on internal angles are zero by using the Wilson S vector technique, and that only the two-body forces appear in the pressure expression. Molecular dynamics simulations are carried out for an atomistic model of benzene with intramolecular interactions based on ab initio harmonic potentials and intermolecular interactions given by semiempirical atom-atom potentials. Calculated total pressure depends on parameters of intermolecular potentials. An intramolecular contribution to the pressure is studied for both solid and liquid phases. In solids where the molecules are compressed, the intramolecular pressure contributions are appreciable and positive, while in liquids where molecular deformations are relatively small, the contributions are small and negative. A possible further improvement of the atomistic model of benzene is discussed.

I. INTRODUCTION

In molecular dynamics simulation the system pressure P is often a useful quantity for monitoring the simulation condition, even though it is not as readily available as the other common state variables, volume V and temperature T. To calculate pressure one can apply the thermodynamic definition of p= -aElaV, where E is the internal energy, valid at low temperature, or the virial expression valid at arbitrary T. While such calculations are straight forward for atomistic systems, 1 an ambiguity arises when one is dealing with molecular systems. Because a molecular sys­tem can be described in terms of either a rigid-molecule model2 or an atomistic model,3,4,5 different expressions ex­ist for determining the pressure.6

,7 The first derivation of an expression for pressure in an atomistic model of a mac­romolecular system was made by Weiner.3 The same result was obtained by Honnell et al. 6 for highly idealized models of chain molecules, and used to analyze the intermolecular and intramolecular contributions separately.5

In this work we first present a derivation of the expres­sion for pressure in a molecular system in which intermo­lecular interaction is represented by a sum of two-body interactions, and intramolecular potential is made up of two-, three-, and four-body contributions. In contrast to the previous derivations3,6 which rely on a scaling argu­ment, we show explicitly, using the Wilson S vector repre­sentation,8 that the three- and four-body potentials give no contributions to the pressure. We next apply the pressure expression to a specific atomistic model of benzene9

,10 and calculate equation of state properties for both solid and liquid phases. By comparing the results with experiment we determine the accuracy of this particular potential de-

a)Pennanent address: Japan Synthetic Rubber Co., Ltd. 25, Miyukigaoka, Tsukuba, Ibaraki 305, Japan.

scription. We find that the use of ab initio force constants for intramolecular part of the potential and semiempirical atom-atom potential for the intermolecular interactions leads to a potential model which gives pressures of several kilobars at the experimental densities and temperatures of liquid and solid benzene (at one bar pressure).

We have modified the parameter values in the Buck­ingham potential used to represent intermolecular part of the interaction in an attempt to reduce the calculated pres­sures. Specifically we allow the exponential prefactor and the coefficient of the dispersion (inverse power) term to vary. We find that while such modification gives a consid­erable improvement in the calculated pressure, agreement in the heat of vaporization is now worse. Thus, it appears that fundamental improvement in the potential function description will require consideration of electrostatic ef­fects which are ignored in the present atomistic model.

The paper is organized as follows. In Sec. II we derive the virial pressure expression for a molecular system whose potential is expressed as the sum of intramolecular and intermolecular contributions, with detailed analysis of the individual terms in the intramolecular contribution dis­cussed in an appendix. In Sec. III we describe the details of the atomistic model potential function for benzene and cal­culate by means of molecular dynamics simulation radial distribution functions, molecular structure, internal en­ergy, and system pressure. Then we discuss the results of modifying the potential function. By analysis of the in­tramolecular and intermolecular components of the pres­sure through the concept of bond force,I1,4,5 we gain some insight into the different nature of the molecular deforma­tions that exist in the solid and the liquid. Finally, in Sec. IV we summarize our main results and comment on pos­sible further improvement of the potential function model.

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3748 T. Tominaga and S. Yip: Molecular dynamics study of pressure

II. THEORETICAL ANALYSIS OF PRESSURE

Let there be v molecules consisting of n atoms in a volume V. The total number of atoms in the system N is equal to vn. The Lagrangian of an atomistic model of the system is given by

N 1 L= ;~1 2m,i-;-U(.-N), (1)

where rU ... ,rN are positions of atoms, and U(.-N) = U(rl> ... ,rN) is a potential energy of the system.

Using the virial theorem, the pressure of the system is given byl

I N _ 1 N aU(.-N) P=3V .2: m,i-;-3V .2: riO ar.'

,=1 1=1 , (2)

where the bar means the time average. In a molecular sys­tem, the potential energy can be divided into inter- and intramolecular parts

(3)

Correspondingly, the pressure can be rewritten as the sum of kinetic, intermolecular, and intramolecular contribu­tions,

P=Pkin +Pinter+Pintra' (4)

When the system is in equilibrium, Pkin is given by

(5)

where p is a number density of the system, k B is Boltz­mann's constant, and T is the temperature of the system. For an intermolecular potential which is assumed to be a sum of two-body interactions,

(6)

where rapj denotes the distance between atoms a; on mol­ecule i and b j on molecule j, Pinter becomes

(7)

In general, for a molecule the intramolecular potential is given by sums of two-body, three-body, and four-body in­teractions,8,12

Uintra(.-N) = 2: <1>2 (Ri ) + 2: <1>3 (Ri ) + 2: <l>4(R;), (8) i i j

where R; denotes the internal coordinate of molecule i. The two-body potential includes the bond stretch contribution and the nonbonded pair potential,

(9)

where A; denotes a pair of atoms in molecule i with sepa­ration distance r).. .• The three-body potential describes

I

bond angle bending, whereas out-of-plane wag and the tor-sions associated with dihedral bending are described by

four-body potentials. In the Appendix a proof is given us­ing the Wilson S vector technique8 that the three- and four-body potentials do not contribute to the instantaneous pressure, i.e., the virial terms for such potentials are zero, therefore P intra becomes

(10)

Combining Eqs. (4) with (5), (7), and (10), one obtains an expression for calculating the pressure of an atomistic model of molecular systems.

III. APPLICATION TO BENZENE

A. Potential model

We use the same atomistic model of benzene as in Refs. 9 and 10, where the vibrational properties of benzene in the condensed phases have been well reproduced. In this model, the potential energy function is specified as a com­bination of ab initio intramolecular interactions and semi­empirical atom-atom intermolecular interactions. The former consists of quadratic functions of the internal coor­dinates developed for benzene by Pulay et al. 13 The latter is expressed by atom-atom functions of the Buckingham form, which are extensively used for aromatic hydrocar­bons. 14

The intramolecular force field for benzene is given by a sum of functions of internal coordinates, 13

tPintra(R) =tPs(R) +tP B(R) +tPw(R) +tPT(R) +tPdR), (11)

where tPs is the atom-atom bond stretching contribution, tP B and tP ware the in-plane and out-of-plane (wag) bond angle bending contributions, respectively, tPT is the tor­sional potential associated with dihedral angle bending, and tPc represents cross terms. The vector R has as its components 30 internal coordinates.

The internal coordinates chosen for the benzene force field are as followS:9,1O r; (i= 1, ... ,6) corresponds to the carbon(C)-hydrogen(H) bond length, R; (i= 1, ... ,6) is the CC bond length, P; (i= 1, ... ,6) denotes the CH in-plane deformation, q19 and Q20; (i=a, b), which are linear com­binations of CCC in-plane bond angles a; U= 1, ... ,6), describe B lu and E2g deformations, respectively, y; (i= 1, ... ,6) refers to the CH wagging, and Q28 and q29; (i=a, b), which are linear combinations of CCCC dihedral angles S; (i= 1, ... ,6), describe B2g and E2u defor­mations, respectively.

The intramolecular potential components in Eq. (11) are written in terms of the internal coordinates defined above. The bond stretching contribution is

1 6 1 6

tPs(R) =2 ;~1 kr(r;-ro)2+2 ;~1 kR (R;-Ro)2, (12)

where ro and Ro are the equilibrium C-H and C-C bond lengths, which are determined to be ro= 1.077 A and Ro = 1.395 A, respectively.13 Note that these lengths are slightly different from their accepted experimental values,

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T. Tominaga and S. Yip: Molecular dynamics study of pressure 3749

TABLE I. Intermolecular potential parameters for benzene.

Parameters" Parameter

seth Ace aCC Cce ACH aCH CCH AHH aHH CHH

RW 290.5 3.60 0.069 107.5 3.67 2.293 8.78 3.74 -0.184 Ml 290.5 3.60 0.069 126.0 3.67 2.990 8.78 3.74 -0.184 M2 290.5 3.60 0.069 139.8 3.67 3.440 8.78 3.74 -0.184

'Units for A.}, a'j' and Cjj are 10- 18 J, A -I, and 10- 18 J A 6, respectively. bRW, MI, and M2 represent the refined Williams parameters, modified parameter sets 1 and 2, respectively.

rcc= 1.397 A, rCH= 1.084 A.IS The in-plane bond angle bending contribution, which is a three-body potential, is written as

The out-of-plane wagging contribution and the torsional potential, which are four-body potentials, are expressed as

1 6

tPw(R) =2 ,tl k r (Yi)2, (14)

and

1 2 1 2 1 2 tPr(R) =2 k qzs (Q2S) +2 kqz9a (Q29a) +2 k qZ9/Q29b) ,

(15)

respectively. The last term in Eq. (11) consists of cross terms including bond length-bond length, bond length­angle, and angle-angle types. The force constants for these potentials have been determined from ab initio Hartree­Fock calculations,13 and the adopted values for the model are listed in Table II in Ref. 10.

The intermolecular potentials are taken to be semi­empirical atom-atom pair functions of the Buckingham form,14

tPinter(rij) =Aij exp( -aijrij) -Ci/i/, (16)

where rij is the distance between atoms i andj in different molecules, and Aij , and aij' and Cij are parameters which depend on the type of i-j interaction, C-C, C-H, or H-H. Two sets of parameters are known for the Buckingham potential; one is the Williams parameters l4,16 derived from the crystal structure and properties of several aromatic hy­drocarbons and the other is the refined Williams parame­ters l7 for benzene using lattice vibrational frequency data obtained from coherent inelastic neutron scattering mea­surements on deuterated benzene. The refined Williams parameter set was adopted 10 for the present simulation and then modified in order to obtain improved results for pres­sure. These parameter sets are listed in Table I.

Using this atomistic potential model, Anderson et al. 9

,10 have carried out molecular dynamics simulations of liquid (305 K) and solid (10 K) benzene to investigate the vibrational properties of benzene in the condensed

phase. Vibrational frequencies are analyzed in terms of the frequency spectrum of the velocity autocorrelation func­tion and the results are found to match well with infrared and Raman data. The dynamic structure factor is calcu­lated and the agreement with inelastic neutron scattering measurements is good. It is also shown that the model describes the structural and dynamical properties well.

B. Simulation procedures

Simulations have been carried out for a 32 molecule system in the solid and liquid states. For solid phase sim­ulations, temperatures were chosen to be 78 and 138 K, where measurements of the lattice parameters of the ben­zene crystal have been reported. ls,19 The benzene crystal is an orthorhombic system with space group Pbca and four molecules per unit cell. The lattice parameters and densi­ties at 78 and 138 K are a=7.29 A, b=9.47 A, c=6.74 A, and 1.114 g/cm3 and a=7.39 A, b=9.42 A, c=6.81 A, and 1.094 g/cm3, respectively. For liquid phase simula­tions, temperatures were set to be 283 and 353 K where their densities at ambient pressure have been published.2o

They are 0.890 and 0.813 g/cm3, respectively. The molecular dynamics (MD) simulation technique

employed in this work is basically the same as that in Refs. 9 and 10. The simulations were carried out in the micro­canonical ensemble with periodic boundary conditions. A fifth-order Nordsieck predictor-corrector algorithm was used to integrate the equations of motion with the integra­tion time step set at 3.1 X 10-4 ps. In the force evaluation, the intermolecular forces were calculated in the usual man­ner using atom-atom pair potentials which are truncated and shifted with the cutoff distance being half the simula­tion cell side length. On the other hand, since the intramo­lecular potential is described in terms of internal coordi­nates, a coordinate transformation from internal to Cartesian coordinates should be done to calculate the in­tramolecular forces. Such a transformation was carried out using techniques well known in molecular vibrational anal­ysisS (see Ref. 10 for details).

The simulation cell dimensions and the initial config­uration of the molecules were determined from experimen­tal data. For solid runs, the cell was chosen such that its side lengths were equal to twice their respective lattice parameters, and the initial configuration was specified by the reported benzene crystalline structure. IS,19 For liquid simulations, the cubic cell with the desired density was used, and the initial configuration was determined by using the benzene crystalline structure at 270 K.21 The simula­tion cell dimensions for the solid and liquid runs are listed in Table II.

In the present simulations, each system was equilibri­ated over a period of 3.1 ps (10 000 time steps). A simpli­fied equilibrium method was used where the center-of-mass kinetic energy components and a coupled rotational­vibrational component were maintained at their expected values for a specified temperature. Following equilibria­tion, each simulation was continued for 3.1 ps. In calcu­lating the pressure, the expressions derived in the previous section were applied for the benzene model. Long-range

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3750 T. Tominaga and S. Yip: Molecular dynamics study of pressure

TABLE II. Simulation cell dimensions for solid and liquid benzene.

Temperature Density X y Z State (K) (glcm3 ) (A) (A) (A)

Solid 78.0 1.114 14.58 18.94 13.48 Solid 138.0 1.094 14.78 18.84 13.62 Liquid 283.2 0.890 16.71 16.71 16.71 Liquid 353.2 0.813 17.22 17.22 17.22

corrections due to a force cutoff were considered in evalu­ating the intermolecular parts of pressure and energy.

C. Results

The structure of the benzene system was analyzed by intermolecular radial distribution functions and time­averaged molecular internal coordinates. In Fig. 1 the center-of-mass radial distribution functions gCM(r) for the solid and liquid phases are shown. For the solids at 78 and 138 K, there are peaks at about 5.0 and 5.9 A. The first peaks correspond to the four symmetry-equivalent nearest neighbors, and the second peaks come from the four sec­ond and four third nearest neighbors. Compared with gCM(r) at 10 K,1O it is found that these peaks are thermally broadened. For liquids at 283 and 353 K, there are single broad peaks with their maximum at about 5.5 A and first minimum at 7.5 A.. Note that gCM(r) at 283 and 353 K are quite similar and they are also similar to gCM(r) at 305 K.1O

In Fig. 2 the atom-atom intermolecular radial distri­bution functions, gccC r) , gCH (r) , and gHH (r) are pre­sented. For the solid cases, these functions exhibit a large amount of structure arising from the various neighbors. Comparing these functions at 78 K with 138 K, we find that there are shifts of the second or third peak as well as the temperature broadening of peaks. Such temperature changes in the atom-atom intermolecular radial pair cor­relation functions become obvious if we compare our re­sults with corresponding functions at 10 K.1O For the liq­uid cases, these functions show only a minor structure and the temperature dependence is not significant.

4.0r---~-~-~-~-~---'

3.5 3.0

~ 2.5 ... ~ 2.0 0)

1.5 1.0 0.5 r;l

I,·r; O~~,~~,~·~~----~--~--~--~ 3.0 4.0 5.0 6.0

riA 7.0 8.0 9.0

FIG. 1. Molecular center-of-mass radial distribution functions gCM(r) at 78 K (solid line), 138 K (dashed line), 283 K (dotted line), and 353 K (dash-<lot line).

1.8r-~---:r--~-~~-~---' 1.6 1.4 1.2

:s 1.0 o & 0.8

0.6 0.4 0.2 O~~=-~-~-~ ___ ~_~ 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0

(a) riA

1.4r-~-~-_-~-~-~--,

1.2

1.0

:s 0.8 13 0) 0.6

0.4

0.2

o~--~--~--~--~--~--~~ 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0

(b) riA

-c-

1.2r-_-~~-~~-~~---,

1.0

0.8

'10.6 0)

0.4

0.2 ji Ii OL-~_~~ __ ~~_~~~~

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 (c) riA

FIG. 2. Intermolecular radial distribution functions (a) gcc(r); (b) gCH(r); and (c) gCH(r) at 78 K (solid line), 138 K (dashed line), 238 K (dotted line), and 353 K (dash-<lot line).

Time-averaged molecular internal coordinates for solid and liquid phases are listed in Table III. The equilibrium values of the internal coordinates used in the model and experimentally accepted values for gaseous benzene15 are also included in this table for reference. It is observed that in the solid and liquid simulations benzene molecules are slightly compressed. A molecular deformation will make a contribution to the pressure. The intramolecular contribu­tion to the pressure is analyzed later by using the covalent bond force ll ,4,5 rather than the bond lengths.

The pressure of the flexible benzene model can be di­vided into the kinetic, intermolecular, and intramolecular contributions. Calculated values of these contributions and total pressure for solid and liquid phases are presented in Table IV. For the original model where the refined

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T. Tominaga and S. Yip: Molecular dynamics study of pressure 3751

TABLE III. Time-averaged molecular internal coordinates.

Temperature Parameter r R f3 a r [)

CK) set CA) CA) (deg) (deg) (deg) (deg)

78.0 RW 1.0759 1.3939 0.01 120.0 -0.01 0.0 MI 1.0764 1.3942 -0.01 120.0 0.0 0.0

138.0 RW 1.0759 1.3935 -0.01 119.9 0.0 0.0 MI 1.0763 1.3940 -0.01 119.9 om 0.0 M2 1.0767 1.3943 -0.01 119.9 -0.01 0.0

283.2 RW 1.0766 1.3942 0.0 119.9 om 0.0 MI 1.0768 1.3944 0.0 119.9 -0.04 0.0 M2 1.0770 1.3946 0.0 119.8 -0.06 0.0

353.2 RW 1.0770 1.3946 0.0 119.8 0.02 0.0 M2 1.0772 1.3947 0.0 119.8 om 0.0

Model' 1.077 1.395 0.0 120.0 0.0 0.0 Experimentb 1.084 1.397 0.0 120.0 0.0 0.0

'Reference 12. bReference 14.

Williams parameters (RW) are used in the intermolecular potential, calculated total pressure at 78, 138, 283, and 35 K are 4.36, 5.81, 4.45, and 3.65 kbar, respectively; the agreement with experimental value (I bar) is therefore not very good. This is due to overestimation of the virial pres­sure. In general, in a gas phase the kinetic pressure pre­dominates over the virial pressure including intermolecular and intramolecular contributions while in liquid and solid phases the kinetic and virial contributions are essentially equal in magnitude. In the present simulations, however, calculated virial contributions at 78, 138, 283, and 353 K are, respectively, 3.22, 3.89, 1.26, and -0.04 kbar, which are expected to be -1.14, -1.85, -3.19, and -3.69 kbar. The reason for this seems to come from the intermolecular pressure rather than the intramolecular pressure. The in­termolecular pressure should be negative in liquid and solid phases while the intramolecular pressure could be either positive or negative, then positive values of the in­termolecular pressure at 138, 283, and 353 K are not ad­equate. Therefore in order to obtain proper pressure val­ues, the intermolecular potential should be modified such that the intermolecular pressure becomes more negative.

TABLE IV. Calculated and extrapolated pressure for benzene.

Temperature (K)

78.0

138.0

283.2

353.2

Parameter set

RW MI

extrapolation RW MI M2

extrapolation RW MI M2

extrapolation RW M2

extrapolation

Ptot Plin Pintra (kbar) (kbar) (kbar)

4.36 1.14 3.38 -1.97 1.06 1.74

0.0 1.11 2.24 5.81 1.92 3.35 0.23 1.89 2.05

-3.22 1.85 1.20 0.0 1.93 1.99 4.45 3.19 0.87 1.81 3.05 0.49

-0.33 3.25 -0.12 0.0 3.22 -0.05 3.65 3.69 -0.20 0.95 3.82 -0.54 0.0 3.67 -0.67

Pinter (kbar)

-0.16 -4.76 -3.35

0.54 -3.71 -6.26 -3.92

0.39 - 1.72 -3.46 -3.17

0.16 -2.34 -3.04

TABLE V. Calculated energy for benzene.

Temperature Parameter Etot Ekin Eintra Einter (K) set Ow/mol) Ow/mol) (lw/mol) (lw/mol)

78.0 RW -23.7 12.0 6.87 -42.6 MI -47.8 11.1 6.32 -65.2

138.0 RW -6.13 20.6 12.6 -39.3 Ml -28.3 20.2 12.4 -60.9 M2 -42.9 19.8 12.1 -74.7

283.2 RW 42.4 42.0 25.6 -25.2 Ml 25.0 40.2 24.5 -39.7 M2 19.5 42.8 26.7 -49.9

353.2 RW 64.4 53.2 33.2 -21.9 M2 46.6 55.0 34.3 -42.7

Calculated total energies along with kinetic, intra-, and intermolecular contributions for solid and liquid phases are listed in Table V. If we assume the internal energy is inde­pendent of phase and benzene obeys the ideal gas law, the heat of vaporization per mol can be written as

!:Jivap = -Einter+RT, (17)

where R stands for the gas constant. Then calculated in­termolecular energies will be compared with experimental heats of vaporization. Calculated and experimental heats of vaporization for liquids are shown in Table VI. It can be seen that R W underestimate experimental heats of vapor­ization. This implies that the intermolecular potential should be modified such that the intermolecular energy becomes more negative.

As a modification of the intermolecular potential, we will simply change parameters CCH and ACH in the Buck­ingham potential to make the carbon-hydrogen attractive force stronger. Two sets of modified parameters MI and M2 listed in Table I are considered. In Fig. 3 the potential energy, force, and virial for RW, Ml, and M2 are shown. We can expect that Ml and M2 give more negative inter­molecular energy and virial pressure than RW.

In Fig. 4 the center-of-mass radial distribution func­tions and the atom-atom intermolecular radial distribution functions at 138 K for RW, MI, and M2 are presented. Changes of these functions by the modifications are found to be small, and no significant structure change is expected. Similarly, changes of these functions at 78, 283, and 353 K are also small. Calculated values of the pressure using the modified parameters are listed in Table IV. More negative values of the intermolecular pressure are obtained as was expected, and there are also decreases in the intramolecular

TABLE VI. Calculated and experimental heats of vaporization.

Temperature Parameter ~~ .UJ;~~" ~u.:~ - ~H;~~ " (K) set Ow/mol) Ow/mol) (lW/mol)

283.2 RW 27.6 34.6 -7.0 Ml 42.1 7.5 M2 52.3 17.7

353.2 RW 24.8 30.7 -5.9 M2 45.6 14.9

"Reference 29.

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3752 T. Tominaga and S. Yip: Molecular dynamics study of pressure

5.0,----r.-____,r---~--~-____,

4.0

3.0 (5 .€ 2.0

~ 1.0

O~----~~=-~~~ __ ~ -1.0 C-H \:>::;::::/.,.. .. "" -2.0,-:---::-,"-::---:-~-~-_~_--'

1.0 2.0 3.0 4.0 5.0 6.0 (a) rIA

1.0,..---~....,-..,.............-r-~--....,_-__.

0.8

0.6

~ 0.4 b ~ 0.2

0r-----~~==~~~--~ -0.2 C-H(\':::;--'"

-O.4L..--~--~-~-~~---' 1.0 2.0 3.0 4.0 5.0 6.0

(b) rIA

1 O.O.---~-.--r-~-r-~--_-----'

8.0

6.0 (5 4.0 .€

~ 2.0

°r-------tt-~~~~~9 ... ,.<:-"", .. ,,;''"0"" -2.0

-4.0

1.0 (c)

2.0 3.0 4.0 rIA

5.0 6.0

FIG. 3. Ca) Intermolecular potentials; (b) forces; and (c) virial functions for C-C, C-H, and H-H pairs on the refined Williams parameters (solid line), modified parameter sets Ml (dashed line), and M2 (dotted line).

pressure. These effects give improvements in the total pres­sure. In solids Ml provides -1.97 kbar at 78 K and 0.23 kbar at 138 K, and in liquids M2 provides -0.33 kbar at 283 K and 0.95 kbar at 353 K. Calculated energies for solids and liquids and heats of vaporization for liquids us­ing modified parameter sets are presented in Tables V and VI, respectively. Parameter sets Ml and M2 give more negative intermolecular energies but they are too negative, so that calculated heats of vaporization are larger than the experimental values.

It is interesting to estimate values of the intra-, and intermolecular pressure which provide the zero pressure and look into their roles in the experimental condition. They are extrapolated by assuming a linear dependence of the pressure on the parameter eCHo Values of the ex-

3.5r--~-~-~-~~-~--,

3.0

2.5

:s 2.0 :::!

8. 1.5

1.0

0.5

O~~~~~~~_~ __ ~~ 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0

(a) rIA

1.6.--~---~--~--~----~

1.4

1.2 _ 1.0 ... 1l 0.8 0)

0.6 0.4

0.2

O~--~~--~----~--~----~ 2.0 3.0

(b)

4.0 5.0 riA

6.0 7.0

1.4r--~-~_-~-__ -~

1.2

1.0

:s 0.8 13

Cl 0.6

0.4

0.2

O~~~~-~--~ __ ~_---, 2.0 3.0 4.0 5.0 6.0 7.0

(c) rIA

'C"

1.2.---~--~--~----~--_---,

1.0

0.8

'10.6 C»

0.4

0.2

OL-~~--~--~----~--~--~ 1.0 2.0 3.0 4.0 5.0 6.0 7.0

(d) rIA

FIG. 4. (a) Molecular center-of-mass radial pair correlation functions gCM(r), intermolecular radial pair correlation functions (b) gedr); (c) gCH(r); and (d) gHH(r) at 138 K on the refined Williams parameters (solid line), modified parameter sets Ml (dashed line), and M2 (dotted line).

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T. Tominaga and S. Yip: Molecular dynamics study of pressure 3753

4

2 0 0

0 Q

¢

-2 /),.

-4 /),.

¢ 0 Pint,alPkin -6 /),. fCHrolksT -8 <> <> fccRoIks T

-10 50 100 150 200 250 300 350 400

(a) T/K

4

2 0

0

o ~ Q

<> -2 /),.

-4 /),.

0 PintraiPkin <> -6 /),. fCHrolksT -8 ¢ fcc RoIksT <>

-10~~--~--~--~--~ __ ~~ 0.8 0.85 0.9 0.95 1.0 1.05 1.1 1.15

(b) p/g/cm3

FIG. 5. (a) Temperature and (b) density dependence of Pintra/Pkin,

fCHre/kaT, and fccRe/kaT.

trapolated pressure are presented in Table IV, where data ofRW and Ml are used for solids and data ofRW and M2 are used for liquids. It is observed that in solids Pintra makes significant contributions to the pressure while in liquids it makes small negative contributions. Ratios of Pintra to Pkin at 78, 138,283, and 353 K are 2.0, 1.0, -0.02, and -0.18, respectively.

Since the intramolecular pressure is due to the bond stretching forces, the concept of the covalent bond force I I,4,S will give insight to this contribution. For benzene there are two kinds of covalent bond forces corresponding to C-H and C-C bonds. The average covalent bond forces of C-H and C-C bonds are written by

I fCH= - 6N M ~ kr(rj-ro), (18)

and

I fcc = - 6N

M ~ kR(Rj-Ro), (19)

respectively, where N M is the number of benzene molecules and i runs over all C-H or C-C bonds within the system. It can be seen from these expressions that the average co­valent bond forces represent how and how much benzene molecules are deformed. In Fig. 5, fCHrolkBT, fccRoi k BT, and Pinlral Pkin are plotted against temperature and density. In solids with low temperature and high density, fCHrolkBT and fccRoIkBT take negative values, which

indicates benzene molecules are compressed. In liquids with high temperature and low density, absolute values of fCHrolkBT and fccRoIkBT are small, which means defor­mations of benzene molecules are small. The temperature and density dependence of Pintral Pkin correspond with those of fCHrolkBT and fccRoIkBT. Then we can see that the intramolecular pressure is determined by molecular de­formations.

Molecular deformations should be determined by tem­perature, density, and the intra- and intermolecular inter­actions. Effects of the intermolecular interaction on C-H and C-C bond lengths can be seen in Table III. As the C-H intermolecular attractive forces are increased, C-H and C-C bond lengths become longer. The change of C-H and C-C bond lengths will result in the change of the intramolecular pressure. This change, although not as large as the change of the intermolecular pressure, is actu­ally observed in Table IV. Therefore intermolecular inter­actions make a contribution to the intramolecular pressure as well as intermolecular pressure.

IV. DISCUSSION AND CONCLUSION

We have derived a pressure expression for an atomistic model of molecular systems based on the virial theorem. The pressure expression consists of the kinetic, intra-, and intermolecular contributions. In deriving the intramolecu­lar contribution, it is shown that the virial terms arising from the three-body and four-body forces are zero by using the Wilson S vector technique and that only the two-body forces appear in the pressure expression. The same conclu­sion has been reached by Weiner3 and Honnell et al. 6 using a scaling argument due to Green.22 While they proved that forces arising from angular potentials do not contribute to the pressure on the basis of the physical argument that angular potentials should remain unchanged under a uni­form volume change, we showed explicitly that the virial terms associated with angular potentials, describing in­plane and out-of-plane bond angle bending and dihedral angle bending, all vanish. Angular variables considered in our proof are generally used in describing atomistic models of molecular and polymeric systems. Since functional forms of the angular potentials are not specified, our proof is therefore general and can be applied to any molecular and polymeric system.

Molecular dynamics simulations of liquid and solid benzene have been carried out here using an atomistic ben­zene model with intramolecular interaction based on ab initio harmonic potentials and intermolecular interaction given by semiempirical atom-atom pair potentials of the Buckingham form. The refined Williams parameters are adopted for the Buckingham potentia1.9

,10 We find the cal­culated pressures fall in the range of 3.7-5.8 kbar for the experimental conditions of density and temperature which correspond to atmospheric pressure. At the same time the heats of vaporization are underestimated by -20%. We have attempted to modify the Buckingham potential pa­rameters to increase carbon-hydrogen attraction, which makes the intermolecular pressure and energy more nega­tive. This led to a general improvement in the pressure

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3754 T. Tominaga and S. Yip: Molecular dynamics study of pressure

results (by a factor of 2 or better); however, the calculated heats of vaporization are greater than the experimental values by about 50%.

By considering the deformations of benzene molecules, we are able to understand the different nature of the con­tributions of intramolecular pressure to the total pressure. In the crystal where molecules are compressed, the in­tramolecular pressure contribution is large and positive, whereas in the liquid where molecular deformations are relatively small this contribution is small and negative. The intermolecular interaction plays an important role in de­termining the pressure. It makes not only a direct contri­bution to the pressure as the i'ntermolecular pressure, but also an indirect contribution through molecular deforma­tions.

The adequacy of a potential model is always an issue in doing molecular simulation. In calculating thermodynamic properties like the pressure using an atomistic model of molecular systems, the intermolecular potential effects are relatively more important than the intramolecular effects. While the atomistic potential model used in this work was successful in studies of vibrational properties of benzene9,10

and also gave the correct molecular packing of the mole­cules,1O we find that it gives an overestimate of the pres­sure. This suggests that the thermodynamic properties can be quite sensitive to the intermolecular interactions. Our attempts at refining the parameters for the Buckingham potentials show that it is not easy to change the parameter values to obtain improvements in both the pressure and heat of vaporization.

In the case of the six-site rigid model,2,23-27 it is also difficult to obtain reasonable results in the pressure, energy, and structure for both liquid and solid phases. Adan et al.2

modified the Lennard-Jones parameters of the original model proposed by Evans and Watts,23 and predicted with sufficient accuracy the experimental values of internal en­ergy and virial pressure of the real liquid at 298 K. By including quadrupole-quadrupole interactions in an inter­molecular potential model, Claessens et al. 25 have obtained good results in the pressure, heat of vaporization, and structure for a liquid phase. The quadrupolar interaction appears to suppress the occurrence of the stacked (paral­lel) packing configuration of the benzene molecules. Such a configuration was predicted by the model of Evans and Watts and of Adan et al. but is not observed experimen­tally. The same model, however, overestimated the pres­sure in the solid phase and was not able to predict very well the structure.26 Yashonath et al. 27 used a six -site aniso­tropic atom-atom potential model with a point quadrupole located on each carbon atom to study the structures of liquid and solid phases, but did not calculate the pressure. Linse and collaborators28,29 have used a twelve-site rigid model with potential parameters fitted to ab initio data and electrostatic interactions represented by fractional charges placed on the carbon and hydrogen atoms. In this case, reasonable results have been obtained for heats of vapor­ization, while the pressure also was not calculated. In sum­mary, existing studies have shown that a rigid molecule model of six-site Lennard-Jones interaction plus

quadrupole-quadrupole interaction25 can properly describe the liquid state properties of pressure, heat of vaporization, and intermolecular structure of benzene, however, the same model does not predict very well either the pressure or the crystal structure of the solid.26

Thus far we have not considered electrostatic effects which are found to be important especially in rigid benzene models. Therefore, in order to achieve further improve­ments in the pressure, energy, and structure for both liquid and solid phases, it seems worthwhile to incorporate elec­trostatic interactions in our atomistic model. Concerning the intramolecular potential functions adopted in this work, little can be said about whether they are adequate from our pressure calculations. Since they give acceptable results for internal vibrations, we expect they also give reasonable intramolecular contributions to the pressure.

ACKNOWLEDGMENT

We would like to thank J. Anderson for helpful dis­cussions on the simulation code we have used in this work.

APPENDIX

In order to prove that three-body and four-body in­tramolecular potentials do not contribute to the pressure, we show the virial terms related to these interactions are equal to zero.

(1) Valence angle bend

k

An angle aj formed by two bonds rji and rjk is given by

(Al)

where eji is an unit vector along bond rji' If an intramo­lecular potential u is a function of a j' its virial term is expressed by the Wilson S vectors as

~ au(aj) au ~ra' ~=(rj'Sj+ri'Si+rk'Sk) -a,' (A2) a u~ a,

where a runs all atoms in a molecule. S vectors for atoms i, k, and j are given by

(A3)

(A4)

(AS)

respectively. Vectors Si' Sk' and Sj' are parallel to forces acting on atoms i, k, and j, respectively. Equation (AS)

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T. Tominaga and S. Yip: Molecular dynamics study of pressure 3755

implies that a force acting on atom j consists of forces caused by atoms i and k. Putting Eq. (A5) into Eq. (A2), we get

From Eq. (A6), we can say that the virial for the valence angle potential is decomposed into contributions from two valence bonds which form the valence angle. Such a de­composition can be generalized to any potential which can be expressed in terms of bond vectors only. Since vectors Si and Sk are perpendicular to bond vectors fji and fjk, re­spectively, the virial is zero

(A7)

(2) Out-of-plane wag

I

j~ ------'. k

k An angle rj between a bond fji and a plane defined by two bonds fjk and fj' is given by

. _t(eji'ejkxej1) rj=sm . ,

smaj (AS)

where aj is an angle between two bonds fjk and fj" If an intramolecular potential u is a function of rj' its virial term can be written by

S vectors for atoms i, k, /, and j are given by

tan rjeji]' (AlO)

Sk=- . J JI I [ e ·,Xe·· rjk sm aj cos rj

tan rj ] ::::-r-:::- (e jk- cos a je j') , sm aj (All)

(A13)

respectively. Using Eqs. (AlO)-(A13) and following the same argument as the valence angle potential, we can show

(3) Torsional bend

k

An angle Tj between the planes determined by atoms i, j, k and j, k, /, respectively, is given by

_ (eij' ejkXek,) _I [(eijXejk) . (ejkXek1)] Tj-Sgn. cos .. ,

S10 ak S10 a j S10 ak (AI5)

where a j is an angle between bonds f ji and f jk, and ak is an angle between bonds fjk and fk" If an intramolecular po­tential u is a function of Tj, its virial term can be written by

~ au(~) au £., fa' ~= (fj' Sj+fi' S;+fk' Sk+f,' Sf) -a .' a UCa ~

(AI6)

S vectors for atoms i, j, k, and / are given by

(A17)

S (rjk-rij cos aj)(eij X ejk)

j= rijrjk sin2 aj (A1S)

ejkXek' S, . 2

rk,sm ak (A20)

respectively. These S vectors also satisfy similar relations as Eqs. (A5) and (A13),

Si+Sj+Sk+S,=O.

Putting Eq. (A21) into Eq. (AI6), we get

~ aU(Tj) £., f . --= [ -f··· S'-f'k' (S+S·) a a ar

a IJ I J I J

au +fk"S,] -.

aTj

(A21)

(A22)

Vectors Si' SI+Sj' and Si are parallel to forces between atoms i and j, atoms j and k, and atoms k and / and are perpendicUlar to bond vectors fij' fjk' and fk', respec­tively. Then the virial is equal to zero,

(A23)

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3756 T. Tominaga and S. Yip: Molecular dynamics study of pressure

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