Reference interaction site model and molecular dynamics study of structure and thermodynamics of...

Reference interaction site model and molecular dynamics studyof structure and thermodynamics of methanol

D. Costaa� and G. MunaóDipartimento di Fisica, Università di Messina, Contrada Papardo, 98166 Messina, Italy and CNISM,Contrada Papardo, 98166 Messina, Italy

F. SaijaCNR—Istituto per i Processi Chimico-Fisici, Sede di Messina, Salita Sperone, Contrada Papardo, 98158Messina, Italy

C. CaccamoDipartimento di Fisica, Università di Messina, Contrada Papardo, 98166 Messina, Italy and CNISM,Contrada Papardo, 98166 Messina, Italy

�Received 3 August 2007; accepted 4 October 2007; published online 11 December 2007;publisher error corrected 28 February 2008�

Thermodynamic and structural properties of various models of liquid methanol are investigated inthe framework provided by the reference interaction site model �RISM� theory of molecular fluids.The theoretical predictions are systematically compared with molecular dynamics simulations bothat ambient conditions and along a few supercritical isotherms. RISM results for the liquid-vaporphase separation are also obtained and assessed against available Gibbs ensemble Monte Carlo data.At ambient conditions, the theoretical correlations weakly depend on the specific details of themolecular models and reproduce the simulation results with different degrees of accuracy,depending on the pair of interaction sites considered. The position and the strength of the hydrogenbond are quite satisfactorily predicted. RISM results for the internal energy are almost quantitativewhereas the pressure is generally overestimated. As for the liquid-vapor phase coexistence, RISMpredictions for the vapor branch and for the critical temperature are quite accurate; on the other side,the liquid branch densities, and consequently the critical density, are underestimated. We discuss ourresults in terms of intrinsic limitations, and suitable improvements, of the RISM approach indescribing the physical properties of polar fluids, and in the perspective of a more generalinvestigation of mixtures of methanol with nonpolar fluids of specific interest in the physics ofassociating fluids. © 2007 American Institute of Physics. �DOI: 10.1063/1.2803059�

I. INTRODUCTION

Methanol is the smallest organic compound able to ex-hibit a hydrogen bond and is largely used as organic solventin experimental setups. It constitutes a typical example ofassociating fluid, which displays short-range and strongly di-rectional attractive interactions. In comparison with water,the methanol does not give rise to such a completely well-developed network, due both to only one-donor configura-tion and to a weaker strength of the hydrogen bond.1 Thissimplified aggregation topology makes the methanol appeal-ing for the development of statistical models, relating themacroscopic features of association with the details of mo-lecular interaction, and in the construction of perturbationschemes �see Refs. 2 and 3 and references therein�. The in-fluence of the hydrogen bond on the liquid structure is wellcharacterized in the experimental literature, with NMRtechniques4–6 and x-ray and neutron spectroscopy.7–10 Satu-rated and liquid density data have been also widely produced�see, e.g., Refs. 11 and 12�, and results for the liquid-vaporphase coexistence are reported in Ref. 13. The associationproperties have important consequences on the liquid struc-

ture, which is well characterized by winding chains ofhydrogen-bonded monomers with a rather low fraction ofbranching points and an average number of six/ten moleculesper chain.10,14 The radial distribution functions exhibit well-defined signatures of inter- and intrachain correlations, andtypically display a sharp first peak followed by a rapid decayto the asymptotic values, without pronouncedoscillations.15,16

This work is devoted to an extended theoretical investi-gation of structural and thermodynamic properties of severalmodels of fluid methanol, from highly supercritical statesdown to normal conditions, including an overview of theliquid-vapor phase coexistence. We have adopted for thispurpose the framework constituted by the reference interac-tion site model �RISM� theory, formulated by Chandler andAndersen in the early seventies,17 and later extensively ap-plied to investigate the structural properties of molecular flu-ids �see, e.g., the reviews in Refs. 18–20 and the recent text-book of Ref. 21 that gives a good account of the RISMmethod�. The RISM formalism constitutes a matrix generali-zation of the well-known Ornstein-Zernike equation forsimple fluids,22 and is naturally suited when a molecular sys-tem is viewed as an assembly of interaction centers spatiallydisposed to realize a fixed geometry.a�Electronic mail: [email protected]

THE JOURNAL OF CHEMICAL PHYSICS 127, 224501 �2007�

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http://dx.doi.org/10.1063/1.2803059

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http://dx.doi.org/10.1063/1.2803059

Current models of methanol are mostly based on a“united atom” representation of the molecular structure,where the CH3 group constitutes a single interaction site. Inthese three-site models, which do not show any significantdifferences with respect to the full-atom concept,23 the me-thyl group, the oxygen atom, and the hydroxyl proton bear aLennard-Jones interaction plus a Coulomb partial charge. Wehave explicitly considered in this work the models J1 �Ref.15� and his improvement J2,16 both introduced by Jorgensenin the context of the transferable intermolecular potentialfunctions �TIPS� and optimized potential for liquid alcohols�OPLS� classes of molecular potentials, respectively, to-gether with the H1 model by Haughney et al.14,24 Amongother suggestions, not reviewed in this work, Nezbeda haveintroduced a family of “primitive” models, particularly use-ful to develop perturbation schemes, where the short-rangerepulsion is represented by an effective hard sphere, and theeffects of hydrogen bond are accounted for by an attractivesquare-well interaction �see the recent review in Ref. 2 forreferences�. Van Leeuwen and Smit25 and Kvamme26 haveproposed minor modifications to the J2 model to improve theoverall appearance of the liquid-vapor coexistence curve andthe correlation functions at ambient conditions, respectively.

Classical simulation studies have characterized the fluidphase of different models of methanol �see, e.g., Refs.14–16, 23, and 26–30�, including the liquid-vaporcoexistence.25,31 As for theoretical investigations, Pettitt andRossky32 carried out a RISM study of the J1 model address-ing the effects of the hydrogen bond on the liquid structure.In a series of papers, Kvamme made a wide comparisonbetween RISM theory and computer simulations for severalmixtures of small polar molecules, including methanol �seeRef. 30 and references therein�. An extended assessment ofRISM predictions coupled with a partially linearized versionof the hypernetted chain �HNC� closure �named KH afterKovalenko and Hirata, see Refs. 33 and 34�, concerning thefluid phase equilibria of several molecular fluids, includingmethanol, has been carried out in Ref. 35. The molecularOrnstein-Zernike formalism has been used by Richardiet al.36 to calculate the rotational invariant coefficients whichdefine the liquid structure. Kiselev et al.37 have investigatedthe equation of state of methanol in the context of the self-associating fluid theory. Recently, the dielectric properties ofmethanol in nonpolar solutes have been investigated througha mean-field approach by Pieruccini and Saija.3

Turning back to the subject of this work, we aim todiscriminate the model which gives the best agreement be-tween RISM theoretical predictions and simulation results.At the same time, we ascertain the most appropriate tools,among the various strategies offered by the RISM formalism�different closures and various routes to thermodynamics�, tocharacterize the structure and the thermodynamics of themodel. Our predictions are systematically gauged againstnewly generated molecular dynamics data both at normalconditions and along two supercritical isotherms. The RISMliquid-vapor phase separation is compared with availableGibbs ensemble Monte Carlo data.25 We consider this studyas a preliminary step in order to investigate from a theoreti-cal point of view solutions of methanol with other nonpolar

solvent, over a wide range of concentrations, such as, inparticular, mixtures with carbon tetrachloride, which havebeen the subject of recent experimental determinations in thegroup of one of the authors.38–41 In these studies it has beenobserved that, upon successive dilution of methanol, the di-electric and calorimetric results can be consistently describedin terms of an increasing number of closed chains whichform ring aggregates. Molecular dynamics simulations ofmixtures of J2 methanol and CCl4 have also been carried outby Veldhuizen and de Leeuw.42 We are mostly interested infinding signatures of such properties in the framework pro-vided by the RISM theoretical approach.

The paper is organized as follows: in the next section thevarious models for methanol and the RISM theory are intro-duced; details of the molecular simulations are also given. InSec. III we present and discuss RISM and simulation resultsfor the structure and the thermodynamic properties, alongwith the calculation of the liquid-vapor phase coexistence.Last section is devoted to the conclusions and directions forfuture works.

II. MODELS, THEORETICAL APPROACH, ANDSIMULATION STRATEGIES

A. Models for liquid methanol

The molecular geometry of all models envisaged in thiswork, namely, the J1,15 J2,16 and H1,14,24 is sketched in Fig.1. All models are based on the vapor phase values of theintramolecular bond lengths and angles. The interaction po-tential between a site � on a methanol molecule and a site �on a different molecule at distance r is written as

v��r� � v��LJ �r� + v��

coul�r�

= 4��

r�12

− ��

r�6 +

q�q�

r. �1�

Sites � and � can be equally a methyl group �CH3�, anoxygen atom �O�, or a hydroxyl proton �H�. The first termaccounts for the short-range interaction in the form of aLennard-Jones �LJ� potential �where �� is the attractive welldepth and �� is the “collisional diameter,” i.e., v��

LJ ��=0�, while the second contribution arises from the presenceof a partial charge q� on each site. Table I shows the variousprescriptions for different models. J1 is the first potentialdeveloped for liquid methanol15 and shows some obvious

FIG. 1. Geometry of methanol models investigated in this work. The diam-eters of the methyl group and of the oxygen atom correspond to �CC and�OO of model J2 reported in Table I, respectively. The distances among theinteraction sites are, irrespective of the model, LCH3O=1.4246 Å, LOH

=0.9451 Å, and LCH3H=1.9437 Å; �CH3OH=108.5°.

224501-2 Costa et al. J. Chem. Phys. 127, 224501 �2007�


weaknesses; in particular, it underestimates the strength ofthe hydrogen bond. In order to improve upon the model J1,the model H1 �Refs. 14 and 24� has different values of thepartial charges. In the model J2 �Ref. 16� both the LJ param-eters and the partial charges are slightly modified. The in-tramolecular bond lengths and angles are fixed for all modelsat issue. The LJ parameters for different types of sites arecalculated according to the usual Lorentz-Berthelot rules,22

��=�� and ��= ��+�� /2.Although the O and H sites experience a net attractive

interaction during the simulation runs, they cannot overlapdue to the fact that the hydrogen site is well inside the repul-sive core of the oxygen atom �see Fig. 1�. In the RISM cal-culations, instead, the r-space is sampled uniformly, and weneed to introduce an “auxiliary site,”43,44 which adds an extrarepulsion between the O and H sites. Following Pettitt andRossky32 we have set for all models a purely repulsive LJinteraction with �OH=0.2 kcal /mol and �OH=1.85 Å.

B. The RISM integral equation theory

In the RISM formalism17–19 the pair structure of a fluidof identical molecules, each carrying n distinct interactionsites, is characterized by a set of n�n+1� /2 site-site intermo-lecular pair correlation functions h��r�=g��r�−1, whereg��r� are the site-site radial distribution functions. Theh��r� are related to a set of intermolecular direct correlationfunctions c��r� by a matrix generalization of the Ornstein-Zernike equation for simple fluids,22 which reads in k-spaceas

H�k� = W�k�C�k�W�k� + �W�k�C�k�H�k� . �2�

In Eq. �2�, H��h��k��, C��c��k��, and W are n�n sym-metric matrices and � is the number density of the system.The elements of W��w��k�� are the Fourier transforms ofthe intramolecular correlation functions. Provided that themolecules are rigid, we have explicitly

w��k� =sin�kL��

kL��

, �3�

where L�� is the bond length between sites � and � on thesame molecule.

As appropriate for charged systems,20,45 the RISM equa-tion has been complemented by a HNC closure for the directcorrelations c��r�,

c��r� = exp�− �v��r� + ��r�� − ��r� − 1, �4�

where v��r� is the pair potential introduced in Eq. �1�, � isthe inverse of the temperature T in units of the Boltzmannconstant, kB, and the gamma functions are defined as��r�=h��r�−c��r� �see, e.g., the review in Ref. 46�. TheRISM/HNC approach has been successfully applied to pre-dict the structural and thermodynamics properties of severalsystems,20 but shows severe limitations when charged mod-els are investigated in the dilute regime or near the criticalregion. For such reason we have also studied the partiallylinearized version of HNC named KH, which has been al-luded to in the Introduction,33–35 and assumes for c��r�,

c��r� = �exp�− �v��r� + ��r�� − ��r� − 1, if g��r� � 1

− �v��r� , otherwise.� �5�

The KH closure amounts using the full HNC approximationin the density depletion region, where g��r��1, and to re-place it with a mean spherical closure when g��r�1, com-bining the advantages of both concepts. It has been em-ployed to study the fluid phase equilibria of several polarmolecular fluids, including water, methanol, and hydrogenfluoride �see Ref. 34 for a recent account of the progress ofthis approach�.

The coupled set of Eq. �2� and �4�, or �5�, has beensolved numerically with a standard Picard iterative proce-dure. Calculations are performed on a grid of 4096 points,with a mesh size of 0.02 Å. The Coulomb interaction is split-ted into a short-range part and a long-range contribution ac-cording to the Ng method �see Ref. 19 for details�.

In the next section we shall present results for the site-site radial distribution functions g��r� and for the site-siterunning coordination numbers, defined as

N��r� = 4� 0

r

g��r��r�2dr�, �6�

which give the average number of � �or �� sites inside asphere of radius r centered on the � �or �� site. The internalenergy per particle is given by

U

N=

�

2��,�

g��r�v��r�dr . �7�

As for the pressure, a closed formula can be deduced in thecontext of the HNC approximation,47

�P

�= 1 −

�

2��,�

dr�c��r� − 12h��

2 �r�� +1

2�2�3 dk

� ��−1 ln det�I − �W�k�C�k��

+ Tr�W�k�C�k��I − �W�k�C�k��−1� , �8�

224501-3 Structure and thermodynamics of methanol J. Chem. Phys. 127, 224501 �2007�


with a similar expression for the KH closure.33 This expres-sion has been compared along supercritical isotherms withthe equation of state obtained through the compressibilityroute to thermodynamics,22

�P = 0

�

d��S�k = 0��−1, �9�

where the k→0 limit of the static structure factor, S�k=0�, iscalculated according to the expression for interaction site flu-ids,

S�k� =1

n2��,�

��k� + �h��k�� . �10�

As for the Helmholtz free energy, a closed formula analo-gous to Eq. �8� reads47

−�Aex

N=

�2

2 ��,�

dr�c��r� − 12h��

2 �r��

−1

2�2�3 dk�Tr�W�k�C�k�

+ ln det�I − �W�k�C�k�� . �11�

We have compared the expression �11� with the free energyobtained from standard thermodynamic integrations alongconstant-temperature or constant-density paths,

�A��N

=�A��0�

N+

�0

� �P��

d��

��, �12�

�A�T�N

=�A�T0�

N−

T0

T U�T��NkBT�

dT�

T�. �13�

C. Molecular dynamics simulations

In order to test the accuracy of the RISM calculations,we have carried out extensive molecular dynamics �MD�simulations. We have used for this purpose the programMOLDY �Ref. 48� �see also the manual at the web page ofRef. 49 for a complete description of this package�. Simula-tions have been performed in the canonical ensemble �i.e., atconstant temperature T, volume V, and number N of par-ticles� at room conditions and along two supercritical iso-therms, T=550 and 950 K. We have used a standard samplecomposed of N=512 methanol molecules; further runs withN=256 and N=2916 particles have been occasionally ana-lyzed in order to detect any dependence of our results on thesize of the system. The time step of the numerical integration

has been fixed to �t=1 fs, and the electrostatic interaction istreated by the Ewald method �see, e.g., Ref. 50�. For eachdensity and temperature, after an equilibration run of typi-cally 500 ps, we have computed the relevant structural andthermodynamic averages over trajectories spanning a timeinterval of 500 ps–1 ns.

III. RESULTS AND DISCUSSION

For clarity sake we have divided the presentation anddiscussion of our results into two separate parts. In the firstone we introduce and discuss the structural properties at nor-mal conditions, i.e., T=298 K and �=0.0148 Å−3, the mo-lecular number density experimentally determined at pres-sure P=1 atm. In the second part we report results forthermodynamics properties, both along the supercritical iso-therms T=950 K and T=550 K, and at ambient conditions,and we discuss the overall appearance of the liquid-vaporphase coexistence. In what follows we shall use the notationC to indicate the methyl group CH3, O for the oxygen atom,and H for the hydroxyl proton.

A. Liquid structure

In order to elucidate the differences among the variousmodels, we show in Fig. 2 the MD results and RISM/HNCpredictions for the site-site radial distribution functions�RDF� g��r� at room conditions. It appears that the MDRDF involving the C site are almost indistinguishable fromeach other, independently of the model considered �for suchreason, we show at this stage the CO RDF only�, whereassmall differences emerge when the O and H correlations areconsidered. The high degree of short-range order, due to thepresence of the hydrogen bond is especially visible in thehigh and sharp peak of the OH RDF. These features turn outto be slightly underestimated in the J1 model, characterizedby a weaker electrostatic interaction between the O and Hsites, in comparison with the J2 and H1 models �see Table I�.Although the J1 model has been practically superseded bythe more refined J2 interaction potential, data are reportedhere essentially with reference to previous studies.32 TheRISM correlations show the same trend of the MD resultswith even smoother differences among various models.

Hinging upon this preliminary survey, and by comparingseparately the RISM and MD results for all models at issue,we are led to the conclusion that, as far as the local structureis concerned, the best agreement between theory and simu-lations is obtained when the J2 model is considered. For thisreason, we concentrate in Fig. 3 on a detailed comparison ofall RDFs for the J2 model alone. It appears that the RISM

TABLE I. Intermolecular potentials for liquid methanol. Parameters are taken from Ref. 15 for model J1, Ref.16 for J2, and Refs. 14 and 24 for H1. Partial charges q are given in e units; � in kcal/mol; � in angstroms. Inall models, �HH and �HH were originally set equal to zero; in the RISM calculations an extra repulsion is addedbetween the O and H sites, see text.

Model qCH3qO qH �CH3CH3

�OO �CH3CH3�OO

J1 −0.285 0.685 −0.400 0.181 13 0.174 75 3.860 93 3.082 78J2 −0.265 0.700 −0.435 0.207 01 0.170 01 3.775 00 3.071 00H1 −0.297 0.728 −0.431 0.181 13 0.174 75 3.860 93 3.082 78



scheme reproduces the MD results with different levels ofconfidence, depending on the correlation function involved,with the notable accuracy of the CC RDF. The CC correla-tion is almost insensitive to the presence of the hydrogenbond and displays practically the same features with or inabsence of the Coulomb interaction, as can be seen in thetop-left panel of Fig. 3, where we show also the predictionsobtained if the charges in the J2 potential are set equal tozero. Moreover, as again visible in the same panel, the CCcorrelation is well compatible with the RDF of a simpleLennard-Jones fluid �as resolved through a Ornstein-Zernike/HNC scheme�, with the same � and � parameters of the CCinteraction.

The position and the strength of the hydrogen bond, asindicated by the first peak in the OH RDF, shown in thetop-right panel of Fig. 3, are also satisfactorily reproduced.On the other hand, the intermediate-distance secondary OHpeak appears essentially misplaced in comparison with thesimulation data. When looking for possible improvements ofthe OH interaction parameters, it emerges that this dephasingis not an artifact of the OH potential itself, and survivesalmost unaffected if we switch off the Coulomb interaction,and even when the full OH interaction completely reduces tozero �see Fig. 3�.

A main feature in the CO and HH distribution functionsis that the theoretical disposition of particles around a centralsite is satisfactory, with the positions of the first peaks andwells correctly predicted, although the height of the peaksand the depth of the correlation minima are smoother incomparison with the simulation data. This “on average”agreement is well reflected in the quantitative accurate RISMpredictions of the integrated measure of correlations which

defines the running coordination numbers N��r� �see Eq. �6�and the central panels of Fig. 3�. The loss of resolution in theshort-range structure, especially after the first correlationpeak, reflects a general deficiency of the RISM approach,already documented in the past literature, as, for instance,when dealing with common water models.51,52 Predictionsworsen for the CH and OO correlations. In the CH RDF thefirst peak at r�2.8 Å, signaling a linear disposition of par-ticles along the CO axis, and hence essentially related tointrachain correlations, is almost missed, barely replaced bya tiny shoulder at r�3 Å; the second and higher peak, to beattributed to the CH correlations involving sites of moleculeson different chains,53 is instead fairly well reproduced. TheOO correlation retains a poor resemblance with MD results.

All theoretical predictions have been obtained within theHNC closure. Our results are practically unchanged if we usethe KH relation instead, but for the height of the first corre-lation peaks in all g��r�, which appears invariably lowerthan the HNC counterpart. Since, as documented in Fig. 3,the RISM/HNC scheme already underestimates the impor-tance of correlations at close contact, we conclude that atnormal conditions the HNC closure performs slightly betterthan the KH one. On the other hand, as observed in Ref. 54,the KH scheme satisfactorily reproduces the features of thefirst solvation shell, and indeed, we have observed a com-plete agreement between KH and HNC results over thewhole range of the coordination number function illustratedin Fig. 3.

The general picture emerging from our study testifiesthat the RISM approach produces the most accurate predic-tions when the local structure is mainly determined by theshort-range correlations, even in the presence of long-range

FIG. 2. �Color� MD �top� and RISM/HNC �bottom� site-site radial distribu-tion functions g��r� at ambient condi-tions �T=298 K, �=0.0148 Å−3� forall methanol models envisaged in thiswork. In the left panels, OH RDF aregrouped on the left and CO RDF onthe right; in the right panels, HH RDFare grouped on the left and OO RDFon the right; All distances are given inangstroms. Note the different verticalscale in the bottom-right panel.



electrostatic interactions, such as in the case of the CC RDF.In fact, the hypothesis that the molecular shape provides thedominant mechanism for the local order in the liquid under-lies the original formulation of RISM �where a molecule isessentially described as a collection of overlapping hardspheres17� and its fruitful applications. In this sense RISMcan be thought as a natural extension to molecular fluids ofthe “Van der Waals picture” of simple fluids, successfullyformalized in the same years �see, e.g., the discussion in Ref.55�. Moreover, the comparison of the CC RDF with that of asimple Lennard-Jones fluid reveals that the best theoreticalpredictions are obtained when the site-site correlationsweakly depends on the presence of surrounding sites. On theopposite side, the worsening of RISM predictions seems re-lated to the influential presence of indirect correlationsamong the molecular sites: the OH secondary peak barelydepends on the OH potential itself, the CH short-range struc-ture is mainly mediated by the presence of the central oxy-gen and by the strong attraction between the O and H sites,and the correlation between the innermost O sites eventuallydepends in a complicated way from the global arrangementsof all sites around a central molecule. The close resemblanceof RISM predictions for all models envisaged in this study

makes our discussion coherent with the previous work ofPettitt and Rossky on the J1 model,32 where the focus wasmainly on the influence of the hydrogen bond in determiningthe liquid structure. In that study, much emphasis is devotedto the evidence that, although a hydrogen-bonded structureexists for the model at issue, packing requirements associ-ated with excluded-volume effects eventually dominate thewhole appearance of the local order in the fluid.

Several approaches have been proposed to improve uponsome important deficiency of the original formulation of theRISM theory adopted in this work, notably the “diagram-matically proper” formulation of Chandler et al.56 and theuse of more refined sets of closures52,57–61 with, in particular,the effective inclusion of approximate bridgefunctions.52,57,59 Hinging upon the body of our results, wesuggest that the RISM approach could greatly benefit fromthe inclusion of the concept of “molecular closures,”62 to beconsidered as a remedy to the inherent impossibility ofsimple, atomiclike closures—which deal with a single pair ofsites at once—to account for the observed indirect, site-mediated effects on the short-range structure of the fluid. Theidea of molecular closures has been positively set forward bySchweizer and Yethiraj62 in the context of the reduced RISM

FIG. 3. �Color� MD �circles� andRISM/HNC �full lines� g��r� for theJ2 model at ambient conditions. Alldistances are given in angstroms. Inthe top panels, the dashed lines corre-spond to the neutral model, i.e., withq�=0 in Eq. �1�. In the CC panel, thedotted line is the RDF of a LJ fluid. Inthe OH panel, the dotted line is ob-tained when the OH interaction iscompletely switched off. In the centralpanels, the running coordination num-bers N��r�, see Eq. �6�, are alsoshown �squares: MD; dashed line,RISM�, with the corresponding scaleon the right.



approach known as polymer-RISM,63,64 as a mean to includeexplicitly, also at the level of the closure relation, the “indi-rect” pathways by which two particular sites on differentmolecules can interact. In close connection, Raineri andStell65 have preliminary introduced a family of “nonlocal”closures in the r-space, in an effort to correct the well-knowntrivial, ideal gas predictions of RISM concerning the dielec-tric constant of charged fluids �see, e.g., Ref. 22�. Work inthis direction is currently under way.

B. Thermodynamic properties and the liquid-vaporcoexistence

The RISM equation of state and the free energy alongthe supercritical isotherms T=950 and 550 K are comparedwith the corresponding MD results in Fig. 4. In order tocalculate these properties as functions of the density withinthe KH and HNC schemes, we have either used the com-pressibility route of Eq. �9� and the thermodynamic integra-tion formula �12� to obtain the pressure and the Helmholtzfree energy in turn, or we have directly employed the closedformulae �8� and �11�. In the case T=550 K, we display the

HNC data obtained through the closed formula only, giventhe well-known difficulties of such a closure to converge inthe low-density regime, and close to the critical region. TheKH closure instead, explicitly devised to circumvent theseproblems,33 allows for a full integration all over the densityinterval investigated, ranging from �→0 up to the ambientdensity ��0.015 Å−3. It appears from Fig. 4 that, when fa-vorable conditions for the HNC scheme to work properly aremet, both closures give comparable results, and theoreticalpredictions tend to bracket the MD data with an overall bet-ter agreement for the free energy rather than for the pressure.At the lower temperature �T=550 K, right panels of Fig. 4�,discrepancies among various theoretical schemes definitelyemerge. In particular, the closed formula free energy inter-sects the corresponding MD results, whereas the thermody-namic integration procedure yields generally overestimatedpredictions. In comparison with the MD data, the theoreticalpressure is generally too high, over a broad fluid densityrange. These results reflect well the high sensitivity of thepressure to the approximations introduced by the theory andconfirm the general trend observed in Ref. 33 for the SPC

FIG. 4. �Color online� RISM and MD equations of state �top� and free energy �bottom� for the J2 model at T=950 K �left� and T=550 K �right�. Symbols inthe legends. In the top panels smooth interpolations of the simulation data are also shown. The MD free energies are obtained by thermodynamic integrationthrough Eq. �12� of the corresponding simulation pressures.



model of water: in that paper, the main source of errors wasattributed to a general overestimate of the core repulsioncontribution to the pressure �and the chemical potential�when HNC-like closures are adopted. Moreover, as reportedin Ref. 52 for the same model of water, the inclusion of“illegal”diagrams in the RISM formalism drives equally thepressure to higher values. A similar composition of effects is,hence, likely to manifest in our calculations. On the otherhand, the analysis of g��r� carried out in the previous sec-tion shows that the strength of the hydrogen bond is gener-ally well predicted, with possibly minor influence on the ob-served discrepancy among theoretical and simulation results.In the top-right panel of Fig. 4, we finally observe—perhapsdue to a fortuitous cancellation of errors during the integra-tion procedure—a nice agreement between the compressibil-ity route pressure and the MD results in the density regime��0.015 Å−3 which corresponds, at lower temperatures, tothe liquid pocket of the system.

Summing up all observations emerging at T=550 K, itappears that the closed formulae eventually lead to the bestpredictions in the low-to-intermediate density regime wherethe liquid-vapor phase separation takes place.25,31 This con-clusion has been corroborated by further trial thermodynamicintegrations of Eq. �13� along several constant-density pathsin the liquid phase starting from the calculated values of thefree energy at T=550 K �not reported here�. As for the caseT=550 K, the application of this procedure generally resultsin a degradation of theoretical performances in comparisonwith the closed formula predictions.

In Fig. 5 we report our estimate of the liquid-vapor co-existence in comparison with the Gibbs ensemble MonteCarlo �GEMC� results of van Leeuwen and Smit.25 To avoidovercrowding, we have displayed in the figure results con-cerning models J1 and J2 only. All theoretical predictions areobtained within the KH scheme, which ensures the conver-

gence of the numerical procedure in the critical region andon the low-density side of the coexistence curve. When pos-sible, we have verified that HNC predictions show no appre-ciable differences in comparison with their KH counterpart.Data for the critical point of all models are reported in TableII. In order to obtain the liquid-vapor coexistence curve, wehave adopted a standard procedure: once the pressure and thefree energy are calculated along several isotherms in the liq-uid and vapor sides of the binodal, the requirements of equalpressures and chemical potentials in both phases allow us todetermine the coexisting densities at a given temperature. Wehave located the critical point by fitting the RISM resultswith the scaling law for the densities, �L−�V �T−Tcr��,where �L and �V are the densities of the coexisting liquid andvapor phases, respectively, Tcr is the critical temperature, and� is the critical exponent for which we take the nonclassicalvalue of 0.32; we have then used the law of rectilinear diam-eters to determine the critical density �cr.

The main issue emerging from the analysis of Fig. 5 isthat the RISM approach yields too low coexisting liquid den-sities, in comparison with the simulation results. The vaporbranch is generally correctly estimated, but for the regionclosest to the critical point, especially for the J2 model. As aresult, see Table II, the critical temperatures are reasonablywell predicted, with a 4%–6% discrepancy with respect tothe simulation counterpart, whereas the critical densities aredefinitely underestimated. RISM predictions generally pre-serve the trend of GEMC results in passing from model J1 toJ2, with an increase of both the critical temperature and den-sity. The theoretical approach barely discriminates among thesmall differences observed for the J2 and H1 binodals in theGEMC calculations.25

Our predictions agree well with the theoretical estimateof the methanol binodal given in Ref. 35, where a slightlyhigher critical temperature is reported; we have indeed veri-fied that this small discrepancy depends �i� mainly on the useof a mean-field value for the critical exponent � in the scal-ing law for the densities, in place of the nonclassical choiceadopted here, and �ii� weakly on the different prescription forthe auxiliary hydrogen site proposed in Ref. 35. More gen-erally, the results of our investigation follow the trend re-ported in previous works33,35,52 for the liquid-vapor coexist-ence curve of the SPC model of water and other polar fluids,and clearly reflect the tendency of the RISM approach tooverestimate the pressure, already commented on for thecase T=550 K. It is worth noting—see Ref. 52—that thetheoretical predictions for the binodal do not definitely im-prove when the proper formulation of RISM �Ref. 56� isadopted �the liquid branch is slightly better, the vapor branch

FIG. 5. RISM/KH liquid-vapor coexistence points for the J1 �full squares�and J2 �full circles� models of methanol. Open symbols represent the corre-sponding Gibbs ensemble simulation results from Ref. 25. Interpolatingcurves are best fits of the theoretical predictions according to the scaling lawfor the densities and the law of rectilinear diameters, with the full diamondsindicating the critical points �see the text for more details�.

TABLE II. RISM critical temperatures �in kelvins� and densities �in kg /m3�for all models envisaged in this work, together with the corresponding Gibbsensemble Monte Carlo results of Ref. 25.

Model Tcr �RISM� Tcr �GEMC� �cr �RISM� �cr �GEMC�

J1 425 446 169 269H1 462 489 164 251J2 469 498 176 274



follows the opposite trend�, even if a zeroth-order bridgefunction correction is enforced in the closure. We surmisethat a valuable route to the thermodynamic properties couldbe provided by the use of different free energy functionals,which have been recently demonstrated to improve upon Eq.�11� in the context of the interaction site formalism.66 A moreextended comparison with simulation data than that reportedin Ref. 66, especially involving the phase equilibria, wouldbe at this stage highly desirable. We plan to develop suchcalculations for the methanol soon in the future.

As far as the thermodynamic properties at ambient con-ditions are concerned, we compare in Fig. 6 the theoreticalpredictions for the internal energy of various models accord-ing to Eq. �7�, with present and past14 MD estimates. Thebest accuracy is observed for the J2 model, for which RISMpredictions closely interpolate between simulation results,whereas the internal energy for the J1 and H1 models ap-pears slightly overestimated. As for the pressure, simulationresults tend to overestimate the experimental value, thatranges approximately around 1 bar. Our MD value, P=35 MPa, with 10 MPa error bar—obtained with the largestsample employed in the simulations, composed of2916 molecules—is consistent with the range P=10 /30�10 MPa in the interval T=268 /338 K reported inprevious calculations.14 As can be expected from our discus-sion about the binodal and the equation of state at T=550 K, the closed formula predictions obtained with theuse of Eq. �8� severely overestimate the simulation datum byabout a factor of 6. On the other hand, thermodynamic inte-gration yields the much more reasonable value P�70 MPa,slightly worsening at ambient temperatures the accuracy ob-served at T=550 K in the top-right panel of Fig. 4.

IV. CONCLUSIONS

We have presented extended theoretical RISM predic-tions of thermodynamic and structural properties of severalmodels of liquid methanol. Our study encompasses the in-

vestigation of the liquid structure at ambient conditions, theanalysis of the thermodynamic behavior at supercritical andnormal temperatures and densities, and the calculation of theliquid-vapor phase coexistence. All results are assessedagainst newly generated molecular dynamics data and dis-cussed with reference to previous theoreticalcalculations.32,33,35,52 We have compared our predictions forthe liquid-vapor equilibrium with existing Gibbs ensembleMonte Carlo simulations.25 We have generally coupled theRISM equations with a HNC closure, but for the low-densityregime and around the critical region, where the KH partiallylinearized version of HNC is more fruitful for the determi-nation of phase equilibria.33

Most accurate predictions for the liquid structure are ob-tained when packing requirements constitute the dominantmechanism to organize the local order in the fluid, and ap-parently when the indirect correlations, mediated by the pres-ence of surrounding sites around a given interaction center,play only a minor role. The internal energy at ambient con-ditions is well captured, whereas the pressure is generallyoverestimated. As a result, the liquid branch of the binodal�and in turn the critical density� is misplaced to low densi-ties, whereas predictions for the vapor branch and for thecritical temperature are accurate within a few percents.

In the discussion of the results we have indicated severalputative routes to improve the RISM performances. Withparticular reference to the issues emerging for the presentclass of polar models, we suggest that these directions shouldinclude the use of more elaborated closures, possibly involv-ing the concept of “molecular closure.”62 The adoption ofmore refined free energy functionals66 could also constitute auseful tool for the calculation of the thermodynamic proper-ties and phase equilibria. Investigations along these lines arecurrently underway. As mentioned in the Introduction, thiswork paves the way to a more elaborate theoretical analysisof mixtures of methanol, and possibly other lower alcohols,in solution with nonpolar solvent such as carbon tetrachlo-ride, for which experimental and theoretical studies3,38–42,67

have suggested the onset of different aggregation structures,in comparison with what is observed in the pure fluid.

ACKNOWLEDGMENTS

This work makes use of results partly produced by thePI2S2 Project managed by the Consorzio COMETA, aproject cofunded by the Italian Ministry of University andResearch �MIUR� within the Piano Operativo Nazionale“Ricerca Scientifica, Sviluppo Tecnologico, Alta Formazi-one” �PON 2000-2006�. More information is available athttp://www.pi2s2.it and http://www.consorzio-cometa.it

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