statistical thermodynamics of fluids with both dipole and quadrupole moments

Statistical thermodynamics of fluids with both dipole and quadrupolemomentsAna L Benavides Francisco J Garciacutea Delgado Francisco Gaacutemez Santiago Lago and Benito Garzoacuten Citation J Chem Phys 134 234507 (2011) doi 10106313599465 View online httpdxdoiorg10106313599465 View Table of Contents httpjcpaiporgresource1JCPSA6v134i23 Published by the AIP Publishing LLC Additional information on J Chem PhysJournal Homepage httpjcpaiporg Journal Information httpjcpaiporgaboutabout_the_journal Top downloads httpjcpaiporgfeaturesmost_downloaded Information for Authors httpjcpaiporgauthors

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THE JOURNAL OF CHEMICAL PHYSICS 134 234507 (2011)

Statistical thermodynamics of fluids with both dipoleand quadrupole moments

Ana L Benavides1a) Francisco J Garciacutea Delgado1 Francisco Gaacutemez2

Santiago Lago2 and Benito Garzoacuten3

1Departamento Ingenieriacutea Fiacutesica Divisioacuten de Ciencias e Ingenieriacuteas Campus LeoacutenUniversidad de Guanajuato Apdo E-413 Leoacuten Guanajuato 37150 Meacutexico2 Departamento Sistemas Fiacutesicos Quiacutemicos y Naturales Universidad Pablo de OlavideCtra de Utrera Km 1 Seville 41013 Spain3Facultad de Farmacia Universidad San Pablo CEU Madrid Spain

(Received 2 March 2011 accepted 22 May 2011 published online 17 June 2011)

New Gibbs ensemble simulation data for a polar fluid modeled by a square-well potential plus dipole-dipole dipole-quadrupole and quadrupole-quadrupole interactions are presented This simulationdata is used in order to assess the applicability of the multipolar square-well perturbation theory[A L Benavides Y Guevara and F del Riacuteo Physica A 202 420 (1994)] to systems where morethan one term in the multipole expansion is relevant It is found that this theory is able to reproducequalitatively well the vapor-liquid phase diagram for different multipolar moment strengths corre-sponding to typical values of real molecules except in the critical region Hence this theory is usedto model the behavior of substances with multiple chemical bonds such as carbon monoxide and ni-trous oxide and we found that with a suitable choice of the values of the intermolecular parametersthe vapor-liquid equilibrium of these species is adequately estimated copy 2011 American Institute of

Physics [doi10106313599465]

I INTRODUCTION

One of the more fruitful approaches within the statisticalmechanics of fluids has been to model thermodynamicproperties with the simplest intermolecular potential func-tions able to reproduce the features of interest Given theimportance of polar fluids in process engineering and otherbranches of applied science1 2 various interaction modelshave been developed in the last few decades3ndash22 Most ofthese models include overlap andor dispersion forces throughhard-sphere (HS) Lennard-Jones or Yukawa (Y) potentialsbesides the electrostatic interactions Perhaps the most com-monly used is the Stockmayer potential3 which consists in aLennard-Jones potential plus dipolar interactions The multi-polar square-well model is one of the simplest model manifestoverlap and dispersion forces through a square-well (SW) po-tential together with dipole-dipole quadrupole-quadrupoledipole-quadrupole octopole-octopole or hexadecapole-hexadecapole interactions The multipolar square-wellperturbation theory provides an analytic equation of state forpolar fluids (MSWEOS) based on the statistical-mechanicsperturbation theory that has been developed and used to an-alyze on a consistent basis the effects of the range of the SWpotential as well as the strength of polar moments9 10 ThisMSWEOS has the advantage that several analytic equationsof state for SW potentials are available in the literature whichis an important ingredient in this theory This advantage canalso be found in the polar fluids works of Alavi and Feyzi19

a)Author to whom correspondence should be addressed Electronic mailalbfisicaugtomx Tel +55 477 7885100 ext 8422 Fax +55 4777885100 ext 8410

and of Henderson et al12 that considered for the overlap anddispersion terms the SW and the Y potential respectively

More recently and taking a step further the MSWEOShas been successfully applied to model thermodynamic prop-erties of real single component polar substances which firstnonzero multipole moments were the dipole moment (wa-ter and ammonia) the quadrupole moment (carbon diox-ide and nitrogen) the octopole moment (methane and car-bon tetrafluoride) and the hexadecapole moment (sulphurhexafluoride)23ndash27 Thus this theory can be applied to modelthe behavior of single component real substances of quasi-spherical shape Besides this MSWEOS could be used as animportant ingredient to study more complex fluids in moreelaborated theories such as the Statistical Associating FluidTheories as for example SAFT-VR-D (Ref 15) or SAFT-VR-Q (Ref 18) to model the polar monomer interactions orin coarse-graining modeling theories28

Briefly the MSWEOS is made up of separate terms rep-resenting the effects of overlap and dispersion forces on onehandmodeled by a SW term and of point multipolar interac-tions on the other In spite of its success the MSWEOS hasbeen rigorously tested against simulation results either for SW+ dipole-dipole and for SW + quadrupole-quadrupolepotentials29 In this work we present the results of Gibbsensemble Monte Carlo (GEMC) simulations for SW+ dipole-dipole + dipole-quadrupole + quadrupole-quadrupole interactions We included the crossed dipole-quadrupole interaction in order to assess the applicability ofthe MSWEOS to systems where more than one term in themultipole expansion is relevant Although the importance ofcrossed interactions have been suggested11 13 20 this workpresents together with the work of Vrabec and Gross20 one

0021-96062011134(23)2345078$3000 copy 2011 American Institute of Physics134 234507-1

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234507-2 Benavides et al J Chem Phys 134 234507 (2011)

of the few contributions where a simple dispersion term andmore than one multipole moment are explicitly consideredand applied to models as well as to real substances

The systems considered in this work cover a widerange of reduced dipole and quadrupole moments μ= (μ2(εσ 3))

12 and Q = (Q2(εσ 5))

12 respectively where ε

and σ are the energy parameter and hardcore diameter of theSW interaction The dipoles and quadrupoles are taken to belocated at the center of the molecules besides the quadrupoleis axial Since the effect of the polar interactions on the ther-modynamics has been found to be approximately additiveover the contribution of the bare SW potential9 10 (result alsofound by Henderson et al12 in the context of dipolar Yukawafluids via simulation studies) in this work simulation modelsconcentrate on systems with various values of μ and Q butkeeping a fixed SW range λ = 15 For the selected SW rangevalue the multipolar effects have been found to be relativelyhigh whereas for λ gt 15 those effects ndashat constant μ andQndash decrease in importance10 Furthermore the values of λ

that have been found suitable to model real substances are notfar from this value

Up to now the MSWEOS considered in this work hasnot been applied to real systems that have both permanentdipole and quadrupole moments We will apply it to two sim-ple but very important substances carbon monoxide (CO) andnitrous oxide (N2O) The first of these substances is knownfor its toxic effects in human beings and it is a fundamen-tal substance in chemical synthesis The latter one becauseit is a greenhouse gas and its presence in the air favors theglobal warming phenomena Moreover N2O is an importantsubstance in supercritical extraction From a molecular pointof view the atoms in both substances are linked by σ andπ chemical bonds In previous work we have shown that inthese cases dipole and quadrupole should be simultaneouslyconsidered11 to give quantitative agreement with experimentIt has been recently shown that we can also fit this kind of sub-stances by means of shifting the dipole out of the molecularcenter21 but this option is not considered here

So this paper is scheduled as follows the description ofthe potential model and the MSWEOS derived from pertur-bation theory are shortly reviewed in Sec II as well as thesimulation details The comparison between simulation dataand the theoretical approach will be considered in Sec III InSec IV the MSWEOS will be applied to the cases of CO andN2O The main conclusions of this work are given in Sec V

II MSW PERTURBATION THEORYAND SIMULATION DETAILS

A Perturbation theory

Assuming a point dipole μ and an axial quadrupole Q ina multipolar square-well fluid with any pair of particles withtheir centers a distance r apart interact with the potential

u(r 1 2μ Q) = uSW (r ) + uD(r 12μ)

+ uQ(r 12Q)

+ uDQ(r 12μ Q) (1)

The first term is the square-well potential described bythree parameters a diameter σ a depth ε and a range λ andis given by

uSW (r ) =

⎧

⎪

⎨

⎪

⎩

infin r le σ

minusε σ lt r le λσ

0 r gt λσ

(2)

The last three terms are the dipole-dipole dipole-quadrupole and quadrupole-quadrupole interaction terms1 2

The orientations of each charge distribution are described by1 = (θ1 φ1) and 2= (θ2 φ2)where θ i and φi are the polarand azimuthal angles for the linear charge distribution withrespect to the axis defined by the intermolecular distance rFor simplicity we define the following variables related tomultipolar terms

c1 = cosθ1 s1 = sinθ1 c2 = cosθ2 s2 = sinθ2

c = cos(ϕ1 minus ϕ2)

The dipole-dipole term is then expressed as

uD =

(

minusμ2

r3

)

(2c1c2 minus s1s2c) (3)

The quadrupole-quadrupole term is given by

uQ =

(

3Q2

4r5

)

(

1 minus 5(

c21 + c2

2

)

minus 15c21c2

2

+2(s1s2c minus 4c1c2)2)

(4)

The sum of the dipole-quadrupole plus quadrupole-dipole interactions is represented by the term

uDQ = minus

(

3μQ

2r4

)

(

c1(

3c22 minus 1

)

minus c2(

3c21 minus 1

)

minus2(c2 minus c1)s1s2c)

(5)

In order to apply the high temperature perturbation ex-pansion for the potential given by Eq (1) we have selectedthe hard-sphere potential (uHS) as reference potential and asperturbation potential the attractive part of the square-wellpotential (uSWprime ) plus the electrostatic interactions (uD + uQ

+ uDQ)On the basis of the Barker and Henderson high-

temperature perturbation expansion the reduced excessHelmholtz free energy a = ANkT for a MSW fluid of N par-ticles contained within a volume V and at a temperature T canbe written as the sum of four terms9

aM SW = aH S +

sum

i=1

aSW prime

i

T lowasti+

sum

i=2

aDQDQi

T lowasti

+

sum

i=3

aDQDQSW prime

i

T lowasti (6)

In this expression T = kTε is a reduced temperaturewith k being the Boltzmannacutes constant and ε the depth ofthe SW potential The first and second terms correspond tothe complete SW perturbation expression for the free energywhere each aSW prime

i term is a function of the reduced density

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234507-3 Dipole and quadrupole moments J Chem Phys 134 234507 (2011)

ρ = (NV)σ 3 and of the SW range λ The third term whoseleading term is of second-order in 1T (the first-order term iszero) gives the complete polar series that appears in the pertur-bation theory for multipolar hard-spheres4 The ai

DQDQ termsin this polar series depend on ρ and on the reduced multipo-lar moments μ and Q

The last series includes terms with integrals that involveproducts of multipolar and square-well potentials but noticethat this series starts up to third-order in 1T that is the first-and second-order terms of this type are zero For simplicity inthe MSWEOS this last term has been neglected Expressionsfor all these terms are given in Ref 9 So in a more compactform the MSW reduced free-energy can be written as

aM SW (T lowast ρlowast λ μlowast Qlowast)

= aSW (T lowast ρlowast λ) + aM (T lowast ρlowast μlowast Qlowast) (7)

and only requires square-well and multipolar equations ofstate For the aSW contribution several equations of state areavailable in the literature and in this work we have selectedtwo analytical equations of state (EOS) Gil-Villegas et al30

(SW1) and Espiacutendola et al31 (SW2) SW1 gives an equationthat gives an estimation of the complete SW high-temperatureperturbation expansion and SW2 is a fourth-order SW hightemperature perturbation expansion In Figure 1 it can beseen that both equations when compared with available simu-lation data32ndash37 describe well the vapor liquid phase diagramof a SW fluid of range λ = 15 It is important to remarkthat better expressions for the SW equation are available asfor instance the one obtained by the self consistent Ornstein-Zernike (SCOZA) integral equation method38 and that theMSWEOS is not restricted to the use of equations obtainedby the perturbation theory methodology but we selected SW1and SW2 because they give analytic expressions that permit tocalculate all the thermodynamic properties in an easier way

FIG 1 Phase diagram of a SW potential of range λ = 15 The continuousline represents the equations of state predictions of MSWEOS with SW1 andthe discontinuous line those with SW2 Simulation data is presented as openupward triangles for del Riacuteo et al (Ref 32) filled upward triangles for Elliotand Hu (Ref 33) open circles for Patel et al (Ref 34) filled circles for Vegaet al (Ref 35) open squares for Kim et al (Ref 36) and diamonds for Kimet al (Ref 37)

The multipolar contribution aM =sum

i=2 (aDQDQi )

(T lowasti ) is considered by means of the following Padeacute approxi-mant expression as obtained by Larsen et al4 in the contextof multipolar hard-spheres

aM =aM

2

T lowast2

[

1 minus

(

aM3

aM2 T lowast

)]minus1

(8)

with aM2 and aM

2 given in the Appendix

B Simulation details

In our simulations we have used the canonical versionof the GEMC method39 that gives simultaneously the coexis-tence densities of vapor and liquid of a fluid under its criticaltemperature In order to check the accuracy of the MSWEOSwe have carried out simulations corresponding to a SW fluidof range λ = 15 with a point dipole and a point quadrupoleWe have performed simulations for these systems using a to-tal number of N = 512 particles (256 particles in each box)In the GEMC method simulations are performed in cycleseach cycle consisting in three steps The first step is a con-ventional NVT MC cycle N attempts to move a particle andwhere the rate of move acceptation is about 50 In the sec-ond step an attempt to change the volume is made with anacceptation of about 30 The third step consists of a cer-tain number of particle insertiondeletion attempts This num-ber is variable and is chosen in order to obtain an acceptationrate of 1ndash3 We used 4000 cycles to reach the equilibriumplus 4000 additional cycles to obtain ensemble averages Esti-mated errors were obtained from the standard deviations fromaverages taken every 100 cycles The long-range dipolar con-tributions have been estimated using a reaction field schemewhich has been shown to be very accurate for Stockmayer andmolecular systems40ndash43 The critical properties were obtainedfrom the results of the GEMC simulations using the rectilin-ear diameter law and the critical exponent law with exponent13 as described by MacDowell et al41 Betancourt et al44

have used the finite size scaling method to obtain dipolar SWfluids critical data and found that their results were in reason-able agreement with previous results29 within the techniqueemployed here

III COMPARISON BETWEEN MSWEOS ANDSIMULATION DATA

The properties studied were the orthobaric reduced den-sities ρlowast

L and ρlowast

G as functions of the reduced temperature T

for different values of the reduced dipole and quadrupole mo-ments μ and Q A complete list of simulated thermody-namic states is shown in Table I Some of these cases are pre-sented in Figure 2 The comparison immediately shows thatMSWEOS is a valuable tool except in the vicinity of the crit-ical point We can notice that using SW1 or SW2 does notmake an important difference but SW2 provides a better esti-mation of critical properties as can be seen in Figure 1

In Table II critical data are presented for the casesconsidered in this work The critical densities are wellpredicted with the MSWEOS using both SW1 and SW2

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234507-4 Benavides et al J Chem Phys 134 234507 (2011)

TABLE I Coexistence densities for dipole-quadrupole SW fluids (λ = 15)All the symbols correspond to reduced values as defined in the main textSubscripts indicate the error bar in the last digit

μ2 Q2 T ρL ρG

025 025 075 07714 00234

08 07594 002414

09 07337 00253

095 0711 00222

1 0681 00272

11 0611 0055112

025 10 075 08913 002 2550

08 08953 002338

085 08845 0023145

09 08468 00232614

095 08353 001994

10 08216 002064

11 0792 001798

115 0761 00192

05 2 14 08425 001575

15 08126 001866

16 076613 002551

175 0663 00383

17 0721 00321

18 0583 00443

10 2 145 08215 001795

155 07876 002136

15 08154 002037

165 07497 002516

16 07887 002126

175 0681 00321

17 0721 0026813

18 0662 00352

12 05 095 07707 00061

1 074613 00071

105 07298 00121

11 0721 00221

115 067312 00221

12 065712 00341

2 1 125 08645 001365

135 07984 002024

13 08214 001795

145 07178 00271014

14 0781 00221

155 0612 00392

15 0701 003269

16 0593 00483

The critical temperatures are better predicted by using SW2as expected We noticed that as the multipolar moment in-creases this theory stops behaving properly specially whenQ is increased This trend was previously observed for puredipolar and quadrupolar square-well potentials29 Neverthe-less these high multipolar models range out of the multipolarmoments usually found for real substances This fact guar-antees the feasibility of the theory on the prediction of both

FIG 2 Simulation data of this work and MSWEOS with SW1 (continuousline) and with SW2 (discontinuous line)

vapor-liquid equilibrium and saturation pressure of the polarcompounds of interest This is precisely the aim of the nextsection

IV APPLICATION TO REAL SUBSTANCES

The MSWEOS has been applied to the cases of CO andN2O In Table III we show the best suitable reduced inter-molecular parameter values λ μ and Q by using this equa-tion with SW1 and with SW2 The parameter fit has been donein order that the theoretical predictions give simultaneously

TABLE II Critical data for the cases considered in this work with theMSWEOS using SW1 and SW2 and GEMC data Error bars indicated asin Table I

MSWEOS MSWEOSSystem (SW1) (SW2) GEMC

μ2 Q2 ρc Tc ρc Tc ρc Tc

025 025 0325 1410 0291 1365 0283 13815

025 10 0336 1694 0313 1651 02755 171

050 2 0350 2265 0334 2229 0316 212

1 2 0345 2355 0331 2320 0295 212

12 05 0323 1664 0299 1626 0292 1494

2 1 0323 2048 0305 2017 0296 171

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234507-5 Dipole and quadrupole moments J Chem Phys 134 234507 (2011)

TABLE III Reduced parameters range λ dipole μ and quadrupole Q and the corresponding MSWEOS predictions for the real parameters σ εk μ andQ with SW1 and SW2 Experimental dipole and quadrupole moments for each substance (Refs 51ndash55) are also included (Expt)

Substance λ μ Q σ Aring (εk)K μ middot 1030Cm Q middot 1040Cm2

CO SW1 16 02 minus 06 350 8807 048 minus50SW2 15 015 minus 05 353 10415 040 minus47Exp 037a

minus95b minus877c

N2O SW1 16 01 minus 10 363 18410 037 minus133SW2 14 01 minus 05 369 29010 047 minus87Exp 055a

minus122d minus1103e

aReference 51bReference 52cReference 53dReference 54eReference 55

well the orthobaric and saturated pressures of the real sub-stance For this purpose the fitting process was done by se-quential choice of the best λ μ and Q in order to improvethe overall agreement The intermolecular potentials proposedin this work are effective potentials and we do not expectthat the theoretical intermolecular parameters obtained by theMSWEOS are exactly those corresponding to real substancesbut still we give the corresponding values for σ εk μand Q

In Figures 3 and 4 the phase diagrams (orthobaric and sat-uration pressure curves) for CO and N2O are shown togetherwith NIST experimental data45 As can be seen the MSWEOSwith both SW1 and SW2 reproduces the experimental data forvapor-liquid equilibrium quite well and the agreement for thevapor pressure is good in a broad range of temperatures fromthe triple point temperatures45 (CO Tt = 6795 K and N2O Tt

= 18233 K) to close to the critical temperatures In bothcases the critical points are overestimated by the theory (seeTable IV)

In Table IV we present some vapor-liquid experimentaldata46ndash49 together with the corresponding MSWEOS predic-tions for CO and N2O For both substances the critical proper-

FIG 3 Phase diagrams (orthobaric and saturation pressure curves) for car-bon monoxide Circles represent the NIST data (Ref 45) and lines have thesame meaning as in Figure 2

ties temperature Tc density ρc and pressure pc are betterpredicted with MSWEOS with SW2 except for the criticaldensity for N2O The normal boiling temperature Tb and thevaporization enthalpies Hvap and entropies Svap for COare better predicted with MSWEOS with SW1 For N2O al-though MSWEOS with both SW EOS are in fair agreementwith experimental values MSWEOS with SW2 gives betterestimations for the vaporization properties In both the casesstandard vaporization entropies are of the same order than85 J Kminus1 molminus1 that it is the empirical figure given by Trou-tonrsquos rule for non-associated liquids50 Thus we can concludethat CO and N2O show no clear indication of association inspite of their multipole moments

We compare the MSWEOS predictions for the multipolarmoments with experimental data just to show that it gives rea-sonable predictions It is important to remark that the reportedexperimental multipolar moments are frequently calculated inthe gas phase and these values could be different in the liquidphase Besides for molecules with both dipole and quadrupolemoments the quadrupole moment is origin-dependent Thenthe multipolar moments predicted by this theory could onlybe considered as effective ones As can be seen in Table III

FIG 4 Phase diagrams (orthobaric and saturation pressure curves) for ni-trous oxide Circles represent the NIST data (Ref 45) and lines have thesame meaning as in Figure 2

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234507-6 Benavides et al J Chem Phys 134 234507 (2011)

TABLE IV Experimental (Expt) and MSWEOS predictions for some thermodynamic properties for CO and N2O Critical properties temperature Tcdensityρc and pressure pc normal boiling temperature Tb and vaporization enthalpies Hvap and entropies Svap

Substance Tc (K) ρc (kgm3) pc (Mpa) Tb (K) Hvap (Jmol) Svap (JmolK)

CO Expt 1344540 3101 3503 81635 5908 7231MSWEOS 14285 3209 506 814 6137 754SW1MSWEOS 13966 3060 456 821 6337 772SW2

N2O Expt 3095615 4532 7242 184675 17815 9647MSWEOS 33763 4670 1164 1848 16286 881SW1MSWEOS 33158 4821 1096 1896 17947 946SW2

the CO dipole moment predicted by MSWEOS with bothSW EOS are nearly equal than the experimental value51 Thequadrupole moment MSWEOS predictions are of the same or-der of magnitude than the experimental reported values52 53

For N2O the MSWEOS predictions for the effective dipolemoment are close to the experimental value51 The predictedquadrupole moment is again of the same order of magni-tude than the experimental reported values54 55 For both sub-stances all the predicted moments are contained in the in-terval of rather the scattered of quantum chemistry reportedvalues45 It is important to remark that the MSWEOS givesthe same results for negative or positive quadrupole momentssince the Padeacute expression is built with the second-order a2

DQ

and third-order a3DQ perturbation terms that are expressed

in terms of squared quadrupole moments (see Eqs (A6) and(A7) in the Appendix) so we decided to use the negative val-ues of the quadrupole moment in the parameter fit

V CONCLUSIONS

In this paper we have favorably compared new Gibbsensemble simulation data for multipolar square-well poten-tials with the MSW theory Once tested against simulationdata we have shown how this very simple potential modelthe square-well potential plus multipole contributions is ableto give quantitative agreement for orthobaric densities andvapor pressures of substances containing multiple chemicalbonds This is a paradox because a correct molecular shapeis considered a key element for any accurate perturbationtheory However we found that a simple spherical modelplus one or two multipole terms gives good results not onlyfor short molecules as CO but also for more elongatedmolecules as N2O giving results comparable to more compli-cated models28 56 The issue of anisotropy seems to be moresubtle that we thought and we rather think that we should talkabout the electron cloud anisotropy (ECA) This ECA canbe well described either by a reliable molecular shape withthe position of nuclei giving the molecular shape or by ananisotropy given by multipole charge distribution as it wassuggested in classical texts of angular momentum in quantummechanics57 Our results here are closer to this second vision

Recently Abascal and Vega58 have found by molecularsimulation studies with several intermolecular potentials for

solid water that a suitable effective potential for this substancein the liquid state able to reproduce its different solids shouldinclude besides the dipolar interactions the quadrupolar onesThe MSWEOS could be an appropriate tool to consider thesetypes of polar interactions

ACKNOWLEDGMENTS

This work was funded by grants CTQ2007ndash60910 of theSpanish MICINN 61418 CONACYT (Meacutexico) PROMEP(SEP) Meacutexico and CONCYTEG (Meacutexico)

APPENDIX POLAR TERMS IN MULTIPOLARSQUARE-WELL EQUATION OF STATE

The polar terms for the MSWEOS can be estimated bythe following Padeacute expression4

aM =aM

2

T lowast2

[

1 minus

(

aM3

aM2 T lowast

)]minus1

(A1)

with

aM2 = aD

2 + aQ

2 + aDQ

2 (A2)

aM3 = aD

3 + aQ

3 + aDQ

3 (A3)

The terms aD2 and aD

3 in Eqs (A2) and (A3) are thedipole-dipole second- and third-order terms that can be ex-pressed as fifth-order polynomials in density

aD2 =

(

minus1

6

)

(ρlowastμlowast4)5

sum

i=0

aiρlowasti (A4)

aD3 =

(

1

54

)

(ρlowast2μlowast6)5

sum

i=0

biρlowasti (A5)

The terms aDQ

2 and aDQ

3 in Eqs (A2) and (A3) arethe second- and third- order dipole-quadrupole perturba-tions terms that are also expressed as fifth-order density

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234507-7 Dipole and quadrupole moments J Chem Phys 134 234507 (2011)

TABLE V Coefficients for the polar terms

i 0 1 2 3 4 5

ai 41888 28287 08331 00317 00858 minus 00846bi 164493 198096 74085 minus 10792 -09901 minus 10249ci 25133 21795 10423 02596 1097 minus 00573di 1394906 2419354 1782237 558448 124629 minus 344187ei 1394906 2989225 3023091 1718428 902115 minus 702773fi 15708 15957 10079 04237 01740 00215gi 12566 13470 09326 04562 01906 00865hi 17952 17551 10376 03890 01561 minus 00082ri 10472 11631 08552 04506 01913 01465Si 5329686 12873491 15339542 11105044 7201578 minus 2488879

polynomials

aDQ

2 =

(

minus1

2

)

(ρlowastμlowast2 Qlowast2)5

sum

i=0

ciρlowasti (A6)

aDQ

3 =

(

2

5

)

(ρlowastμlowast4 Qlowast2)5

sum

i=0

diρlowasti

+

(

12

35

)

(ρlowastμlowast2 Qlowast4)

times

5sum

i=0

eiρlowasti

+

(

1

480

)

(ρlowast2μlowast4 Qlowast2)5

sum

i=0

fiρlowasti

+

(

1

640

)

(ρlowast2μlowast2 Qlowast4)5

sum

i=0

giρlowasti (A7)

The terms aQ

2 and aQ

3 in Eqs (A2) and (A3) are thesecond- and third-order quadrupole-quadrupole perturbationterms and can be expressed also as fifth-order density polyno-mials

aQ

2 =

(

minus7

10

)

(ρlowast Qlowast4)5

sum

i=0

hiρlowasti (A8)

aQ

3 =

(

36

245

)

(ρlowast Qlowast6)5

sum

i=0

riρlowasti

+

(

1

6400

)

(ρlowast2 Qlowast6)5

sum

i=0

siρlowasti (A9)

The coefficients ai bi ci di ei fi gi hi ri si are includedin Table V

1C G Gray and K E Gubbins Theory of Molecular Fluids I (ClarendonOxford 1984)

2K Lucas Applied Statistical Thermodynamics (Springer-VerlagHeidelberg Berlin 1991)

3W H Stockmayer J Chem Phys 9 398 (1941)4B Larsen J C Rasaiah and G Stell Mol Phys 4 987 (1977)5M S Wertheim Mol Phys 37 83 (1977)6C H Twu and K E Gubbins Chem Eng Sci 7 879 (1978)7B Saager and J Fischer Fluid Phase Equilib 72 67 (1992)8M E van Leeuwen B Smit and E M Hendriks Mol Phys 78(2) 271(1993)

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