Many-body effects are essential in a physically motivated CO2 force fieldKuang Yu and J. R. Schmidt

Citation: The Journal of Chemical Physics 136, 034503 (2012); doi: 10.1063/1.3672810 View online: http://dx.doi.org/10.1063/1.3672810 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/136/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Approximating quantum many-body intermolecular interactions in molecular clusters using classical polarizableforce fields J. Chem. Phys. 130, 164115 (2009); 10.1063/1.3121323 Diffusion Monte Carlo Study Of The Equation Of State Of Solid pH2: Role Of ManyBody Interactions AIP Conf. Proc. 850, 370 (2006); 10.1063/1.2354741 Comparison of low-order multireference many-body perturbation theories J. Chem. Phys. 122, 134105 (2005); 10.1063/1.1863912 Many-body force field models based solely on pairwise Coulomb screening do not simultaneously reproducecorrect gas-phase and condensed-phase polarizability limits J. Chem. Phys. 120, 9903 (2004); 10.1063/1.1756583 A many-body model to study proteins. II. Incidence of many-body polarization effects on the interaction of thecalmodulin protein with four Ca 2+ dications and with a target enzyme peptide J. Chem. Phys. 119, 1874 (2003); 10.1063/1.1579479

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THE JOURNAL OF CHEMICAL PHYSICS 136, 034503 (2012)

Many-body effects are essential in a physically motivated CO2force field

Kuang Yu and J. R. Schmidta)

Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison,Wisconsin 53706, USA

(Received 13 October 2011; accepted 7 December 2011; published online 17 January 2012)

We develop a physically motivated many-body force field for CO2, incorporating explicit three-body interactions parameterized on the basis of two- and three-body symmetry adapted perturbationtheory (SAPT) calculations. The potential is parameterized consistently with, and builds upon, oursuccessful SAPT-based two-body CO2 model (“Schmidt, Yu, and McDaniel” (SYM) model) [K. Yu,J. G. McDaniel, and J. R. Schmidt, J. Phys Chem B 115, 10054 (2011)]. We demonstrate that three-body interactions are essential to achieve an accurate description of bulk properties, and that previoustwo-body models have therefore necessarily exploited large error cancellations to achieve satisfactoryresults. The resulting three-body model exhibits excellent second/third virial coefficients and bulkproperties over the phase diagram, yielding a nearly empirical parameter-free model. We show thatthis explicit three-body model can be converted into a computationally efficient, density/temperature-dependent two-body model that reduces almost exactly to our prior SYM model in the high-densitylimit. © 2012 American Institute of Physics. [doi:10.1063/1.3672810]

INTRODUCTION

Carbon dioxide (CO2) has been widely studied viacomputer simulation techniques for the past few decadesdue to its importance in both industrial applications andenvironmental issues. Recently, there has been increasinginterest in modeling the behavior of CO2 in complex, hetero-geneous environments (e.g., adsorption in nonporous materi-als), or extreme conditions (e.g., characteristic of CO2 seques-tration). Here the accuracy of traditional, non-polarizable,empirical CO2 models, which have been parameterizedtypically to reproduce bulk structural, thermodynamic, and/ordynamic phenomena under moderate conditions, is drawninto question.

As such, our group recently developed an ab initio,physically motivated CO2 model based on density functionaltheory-symmetry adapted perturbation theory (DFT-SAPT)calculations on CO2 dimers, known as the “Schmidt, Yu,and McDaniel” (SYM) model.1 Utilizing the SAPT energydecomposition, each physically meaningful component ofthe SAPT energy (electrostatic, exchange, induction, disper-sion, etc.) was fit individually to a corresponding term inthe force field, employing a physically appropriate functionalform. Since some of these terms (e.g., exchange-inductionand exchange-dispersion) are mathematically well definedbut not physically meaningful and vanish outside regimes ofstrong charge overlap, we grouped these terms with their con-ventional induction/dispersion counterparts. This is in linewith literature recommendations2 and avoids unphysical di-vergences in the individual terms that cancel in their summa-tion (e.g., induction + exchange-induction).3 We believe thatthis physically motivated methodology offers a compelling

a)Electronic mail: schmidt@chem.wisc.edu.

advantage over conventional approaches, where only the to-tal interaction energy is fit.

The resulting force field is polarizable (via a Drudemodel), naturally incorporating many-body electrostatic ef-fects, and yields quantitative agreement with a diverse setof structural, thermodynamic, and dynamic observables (den-sity, heat capacity, diffusion constants, vapor-liquid equilib-rium, etc.). However, this model does not explicitly treatnon-electrostatic many-body effects such as many-body ex-change/dispersion.

Although the SYM model was parameterized almost ex-clusively on the basis of ab initio SAPT calculations (e.g.,all parameters where fit to ab initio data rather than experi-mental results), we were forced to introduce a single empiri-cal parameter to achieve excellent agreement with propertiesacross the phase diagram: a scale factor of 0.935 was appliedto the original fitted dispersion coefficients (i.e., reducing theSAPT-calculated dispersion by ∼6%), a fact we attributed toomission of non-electrostatic three-body effects (whereas in-duction is included explicitly via polarizability). As expected,application of this empirical scale factor decreases the accu-racy of the calculated second virial coefficient (which is sen-sitive to only the two-body potential), especially in the lowtemperature region, although predictions of all bulk proper-ties were dramatically improved.

Although neglected by nearly all empirical potentials,such three-body interactions have been demonstrated to besmall but important in many cases, such as rare gases,4–9

simple diatomic molecules,10, 11 alkali ions,12 large fullerenemolecules,13 metal-ligand complexes,14 and water.15 Many-body effects can be partitioned into simple many-bodyelectrostatic effects (e.g., induction), and complex non-electrostatic effects (e.g., many-body exchange/dispersion).While the former have been incorporated into a host of

0021-9606/2012/136(3)/034503/7/$30.00 © 2012 American Institute of Physics136, 034503-1

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034503-2 K. Yu and J. Schmidt J. Chem. Phys. 136, 034503 (2012)

polarizable empirical force fields, the latter are nearly univer-sally neglected.14, 15

Currently, Persson’s empirical Gaussian charge polariz-able potential for carbon dioxide (GCPCDO) model16 and themodel of Oakley and Wheatley (OW)17 are the only CO2 po-tentials to include explicit non-electrostatic three-body inter-actions, and thus the only potentials that are (in principle) ca-pable of reproducing both experimental second/third virial co-efficients and bulk properties. The GCPCDO model utilizes a“smeared” Gaussian model for electrostatic interactions, in-corporating polarization via a molecular point-dipole approx-imation. Model parameters were empirically refined to repro-duce the experimental second virial coefficient over a range oftemperatures. Persson incorporates many-body dispersion ef-fects (neglecting other many-body effects, such as exchange)via an Axilrod-Teller form, treating the magnitude of these in-teractions as an adjustable empirical parameter to reproduceexperimentally measured third virial coefficients. Unfortu-nately, this model is very computational expensive (no simpleanalytic gradients are available), and yields substantial errorswhen applied to the bulk, particularly in the medium densityregion.16

The OW model employs a different strategy, parameter-izing the CO2 pair potential interaction using Moller-Plessetsecond-order perturbation theory (MP2) calculations on CO2

dimers with a variety of basis sets. This ab inito data arefit to an anisotropic Lennard-Jones plus point charge form,yielding a pairwise-additive, non-polarizable dimer potential(note that the anisotropic form of the OW potential, Eq. (2) inRef. 17, does not obey exchange symmetry with respect to theCO2 molecules due to a sign error in the fourth or fifth term).Three-body dispersion contributions are approximated at theHartree-Fock SAPT level and fit to an Axilrod-Teller form.Three-body exchange contributions are estimated by subtract-ing the fitted three-body dispersion and induction componentsfrom the sum of the non-additive MP2 energies and three-body SAPT dispersion. However, the authors do not validatethe underlying pair potential via second virial calculationsprior to the addition of the three-body terms. Calculations(using the isotropic form of their potential, due to ambigu-ity resulting from the above error) show that the second virialpredicted by this model are, for a wide range of temperatures,approximately 10% lower than the experimental values; seesupplementary material18 (note that calculation on our ownMP2 CO2 dimers shows that anisotropy has only a small ef-fect on the virial). This large virial error is not surprising asMP2 is known to overestimate dispersion energies for largebasis sets.19 As such, the resulting bulk properties predictedby this model are not particularly accurate (e.g., 5%–10% er-ror in enthalpy of vaporization and ∼20% in the gas-liquidcoexistence densities and pressures).

Nonetheless, the basic approach employed by Oakleyet al. is promising–using explicit electronic structure datato parameterize many-body interactions. As such, we reporthere an extension of our recent SAPT-based “SYM” CO2

model to include explicit three-body exchange and dispersionterms based on three-body DFT-SAPT calculations; many-body induction is already present in the underlying polariz-able “SYM” model. Consistent with the physically motivated

philosophy of the SYM model, we utilize the inherent SAPTenergy decomposition to add corresponding three-body ex-change and dispersion terms to our SYM model, fitting theenergy decomposition component-by-component to mitigatespurious cancellation of errors both between the various three-body terms (e.g., between three-body dispersion, exchange,and induction) and across terms of different order (e.g., be-tween two- and three-body terms). We show that althoughthe energetic contribution of the non-electrostatic three-bodyterms is quite small, these terms make an extremely large con-tribution to the internal pressure, and are thus indispensable ina truly physically motivated model (as opposed to includingthem as an ad hoc correction to the two-body potential). Withthis modification, we obtain highly accurate results for boththe second/third virial coefficients and bulk properties, thusmaking significant progress towards a robust, transferable,physically motivated, and truly parameter-free CO2 model—achieving the “right answer, for the right reason.”

Finally, we discuss how this explicit three-body modelcan be reduced, using rigorous statistical-mechanical con-siderations, to a density- and temperature-dependent effec-tive two-body potential that can be efficiently implementedin existing molecular simulation packages. Intriguingly, weshow that in the limit of liquid-like density this new effectivepotential reduces almost exactly to our prior extremely suc-cessful SYM model, thus essentially “deriving” this modelfrom its more rigorous many-body counterpart. As such,this density- and temperature-dependent effective SYM[ρ, T]model should retain the extremely high accuracy (for a vast ar-ray of properties across the phase diagram) and computationalefficiency of the SYM model, while also gaining asymptoticcorrectness via accurate second/third virial coefficients.

METHODS

Our methodology requires a set of representative CO2

trimers to parameterize a physically motivated three-bodypotential. We employ the Transferable Potentials for PhaseEquilibria (TraPPE) force field20 to run a bulk CO2 molec-ular dynamics (MD) simulation with a liquid-like den-sity and 3000 K temperature, such that both repulsiveand attractive regions are sampled. Approximately 200trimer geometries are randomly selected from the trajec-tory and three-body DFT-SAPT calculations are conductedutilizing the SAPT2008 suite of codes.21 We use the Perdew-Burke-Ernzerhof exchange-correlation functional22, 23 in con-junction with Dunning-style aug-cc-pVDZ and aug-cc-pVTZbasis sets; a complete basis set extrapolation is conducted forthe dispersion term (which converges slowly),24

ECBS = 3β

3β − 2βEtz − 2β

3β − 2βEdz.

Based on test calculations for a selection of Ar trimerswith corresponding DZ/TZ/QZ bases, a value of 2.01 isadopted for the exponent β in the equation above, compa-rable to values of 2.2 and 2.4 that have been reported forMP2 and coupled-cluster methods including singles, doubles,and perturbative triples (CCSD(T)) correlation calculations,respectively.24

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034503-3 Three-body effects are essential for CO2 J. Chem. Phys. 136, 034503 (2012)

-0.04 -0.02 0 0.02 0.04

SAPT E3b

ind (mH)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08D

rude

Osc

illa

tor

thre

e-bo

dy in

duct

ion

(mH

)

FIG. 1. Comparison of the SAPT calculated three-body induction energy,compared to the corresponding quantity from the Drude model. Data areshown for 208 CO2 trimers with a total interaction energy of less than+15 kJ/mol. The rms error between the Drude model and SAPT is 5.4 μH.

The resulting total three-body interaction energy can bewritten as a summation of physically meaningful terms,21

ESAPT(DFT)int [3, 3] = E3b

exch + E3bdisp + E3b

ind,

E3bexch = E(1)

exch[3, 3](KS) + E(2)exch-disp[3, 3](CKS),

E3bdisp = E(3)

disp[3, 3](CKS),

E3bind = E(2)

ind[3, 3](CKS) + E(2)exch-ind[3, 3](CKS).

Here, E3bind = E(2)

ind[3, 3](CKS) + E(2)exch-ind[3, 3](CKS) repre-

sents the induction contribution, and has already been in-cluded (implicitly) in the polarizable Drude oscillator ap-proach of the existing SYM model. As the Drude modelclosely reproduces the CO2 polarizability tensor, it must cer-tainly reproduce the induction energy outside regions of sig-nificant charge penetration. As a more strenuous test, we com-pare the results of our Drude model with the three-body in-duction calculated from 3B SAPT calculations in Figure 1,yielding overall very good agreement for almost all sampledtrimers; larger deviations are seen only for trimers with veryhigh interaction energies that are unlikely to be sampled un-der ambient conditions. Note that the results are not a “fit,”in that we are simply taking the results of our prior Drudemodel (which was parameterized for dimer induction energy)and applying it to many-body induction.

The three-body dispersion contribution, E3bdisp

= E(3)disp[3, 3](CKS), was fit with a generalized Axilrod-Teller

functional form.25 The remaining contributions are dominatedby exchange, E3b

exch = E(1)exch[3, 3](KS) + E(2)

exch-disp[3, 3](CKS),and are represented by an isotropic exponential function in amean-field way. The details of the fitting procedure will bediscussed later.

We utilize these three-body terms in conjunction with anaccurate dimer potential. Here we employ our former SYMmodel,1 which was parameterized in a consistent, physicallymotivated manner to SAPT calculations on CO2 dimers. TheSYM model was fit to the SAPT energy decomposition sep-arately for each interaction energy component to mitigate

200 300 400 500 600 700 800 900 1000 1100 1200 1300Temperature (K)

-0.25

-0.2

-0.15

-0.1

-0.05

0

Sec

ond

Vir

ial C

oeff

icie

nt (

L/m

ol)

FIG. 2. The second virial coefficients. Experimental data1 (black solid line)are shown alongside simulation data from the SYM-1.0 (red cross), SYM-0.935 (blue plus), and SYM-0.970 (green dot, present work).

error cancellation, with explicit and physically appropriatefunctional forms for exchange, electrostatics, induction, anddispersion.

The SYM model contains a single empirical scale fac-tor, λ = 0.935, applied to the two-body dispersion term toachieve accurate bulk properties. We will use the generic no-tation SYM-λ, where λ is the corresponding scale factor, todenote such scaled models. SYM-0.935 thus accounts in anaverage sense for the neglected many-body terms but some-what compromises the accuracy of the dimer potential. Sincein this case we seek a true dimer potential, we initially ex-amine the unscaled SYM-1.0, and benchmark the dimer po-tential via the second virial coefficients, B2(T) (see Figure 2).The unscaled potential slightly underestimates B2 in the lowtemperature region, corresponding to a small error in the un-derlying two-body potential that we attribute to the inherentlimitations in underlying SAPT calculations (note that trivialerror was present in our prior reported calculations of B2; noother results or conclusions are affected, and the corrected re-sults are shown above). Our prior SYM-0.935 slightly over-estimates B2 in low temperature region, thus yielding an“incorrect” two-body potential (although bulk properties aredramatically improved over the unscaled model). As such,prior to introducing three-body terms, we compensate for this(small) error by substituting a scale factor of 0.97 for disper-sion (SYM-0.970), yielding essentially perfect B2 coefficientsin the range 200 K–1300 K. Note that this scale factor is muchsmaller than in our prior work (0.935), as the latter includedeffective three-body interactions to yield the correct experi-mental bulk density. In contrast, here we correct only for er-rors in the two-body potential, guaranteeing a clean separationof two- and three-body portions of the total potential. There-fore, SYM-0.970 is used throughout all the calculations laterin this work, in conjunction with the newly introduced three-body terms.

The SAPT three-body dispersion energies are fit with ageneralized Axilrod-Teller form in the basis of atomic sites:

E3bdisp

∼=∑

{A,B,C}CtAtBtC

1 (1 + 3 cos θA cos θB cos θC)/R3ABR3

BCR3CA

+ CtAtBtC2 /R3

ABR3BCR3

CA,

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034503-4 K. Yu and J. Schmidt J. Chem. Phys. 136, 034503 (2012)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

SAPT E3b

disp (mH)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2F

itte

d 3b

dis

pers

ion

ener

gy (

mH

)

FIG. 3. Three-body dispersion fitting. The rms error in the fit is 3.7μH.

where A, B, and C runs over all intermolecular atom trimers,and tA, tB, and tC stand for the atom types (C and O) of A, B,and C. The traditional Axilrod-Teller functional form, whichis derived for isotropic atoms, is inadequate to fit the SAPTenergies for anisotropic molecular species such as CO2. Wetherefore incorporate an additional isotropic term (as givenabove) to increase the flexibility of the fitting function, corre-sponding to a relaxation of several of the inapplicable assump-tions made during the derivation of the Axilrod-Teller form.25

The final fitting is shown in Figure 3 and fitted parameters areshown in Table I.

We also examine the effect of the three-body exchangeinteraction, E3b

exch. Although this term is smaller in magni-tude (at typical intermolecular separations) and faster decay-ing (exponential vs. algebraic), we find its inclusion to be im-portant for quantitative accuracy. We note that the GCPCDOmodel does not account for this effect, although it was likelyincluded implicitly in the empirical fitting procedure. Unfor-tunately, a physically appropriate functional form for three-body exchange has been derived only for atomic (and notmolecular) species and is extremely complex; generalizationto molecular species is nontrivial and would yield many freeparameters, which would need to be parameterized based onour SAPT data. Furthermore, current three-body DFT-SAPTmethods do not explicitly calculate three-body exchange-dispersion under the coupled Kohn-Sham (CKS) approxi-mation, E(2)

exch-disp[3, 3](CKS), but rather E(2)exch-disp[3, 3](KS),

the far less accurate uncoupled counterpart. Following therecommendations of Podeszwa and Szalewicz,21 we approx-

TABLE I. Three-body dispersion parameters.

Trimer C1/100 C2/100type (kJ/mol · Å9) (kJ/mol · Å9)

OOO 18.978 4.9638COO − 8.1498 10.261CCO 9.6611 1.1445CCC − 50.381 − 44.709

imated E(2)exch-disp[3, 3](CKS) via its uncoupled Kohn-Sham

(KS) counterpart by scaling based on the ratio of the CKSand KS dispersion,

E(2)exch-disp(CKS) = E(2)

exch-disp(KS)E(3)

disp(CKS)

E(3)disp(KS)

.

The associated scale factor can be quite large, up to afactor of 5–10, with associated uncertainty. Given the uncer-tainty in both the functional form for three-body exchangeand the magnitude of the exchange-dispersion, we there-fore treat the total remaining many-body exchange poten-tial E3b

exch= E(1)exch[3, 3](KS) + E

(2)exch-disp[3, 3](CKS) in a sim-

plified, mean-field way via a completely isotropic form,

E3bexch

∼=∑

OAOBOC

α exp(−β(3)(RAB+RBC+RCA)),

approximating the three-body exchange energy as propor-tional to the mutual overlap of all three electron densities.Although the above form neglects the complex angular de-pendence, it will still allow us to capture the average (at-tractive) effect of three-body exchange. In particular, thethree-body exchange is dominated by trimers in near equi-lateral triangular configurations where all three monomersexhibit strong overlap. The above approximate form workswell in this regime, and the magnitude of the three-body ex-change itself rapidly decreases as the mutual trimer overlapdecreases. For simplicity, we account only for interactions be-tween intermolecular oxygen trimers (since the carbon atomsare “buried”). We approximate the three-body exchange expo-nent, β(3), in terms of the corresponding two-body exponent,β(3) ≈ β(2)/2 = 1.9644 Å−1. This approximation has been ver-ified by Ne, Ar, and Kr trimer calculations in an equilateral ge-ometry, and should be fairly robust in quasi-equilateral casesthat dominate the three-body exchange energy.

Given the uncertainty in both the magnitude of the to-tal three-body exchange and the functional form, we do notattempt to fit the remaining pre-factor, α, to the explicit three-body SAPT exchange energies. Rather, we treat α as a quasi-empirical parameter fit to experimental third virial/density,and obtain a final value of −3.0 × 107 kJ/mol. Note, however,that the average calculated and fitted three-body exchange en-ergies agree within about 20% (−2.1 μH vs. −2.6 μH), sug-gesting that we are in fact including the correct “physics” intothe model, at least in an average sense.

Overall, the three-body parameterization procedure cap-tures the correct balance between the various three-bodyterms: three-body induction (via our prior Drude model, asvalidated by explicit three-body SAPT), dispersion (via fit-ting to explicit three-body SAPT), and exchange (in a mean-field sense, to within approximately 20% of the average SAPTthree-body exchange). Nonetheless, the inclusion of a quasi-empirical exchange term means the resulting many-body po-tential cannot be considered a strictly ab initio potential, asthis term was fit to experimental data. Advances in the imple-mentation of three-body SAPT, yielding accurate three-bodyexchange at the CKS level, would presumably facilitate thedevelopment of a truly ab initio many-body model.

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034503-5 Three-body effects are essential for CO2 J. Chem. Phys. 136, 034503 (2012)

TABLE II. Third virial coefficients at selected temperatures. All numbersare given in (ml/mol2); numbers given in parenthesis omit the three-bodyportion of the potential, i.e., the SYM-0.970 model.

T (K) EPM2 TraPPE SYM-3B GCPCDO Exp

280 2910 3080 4700 (2814) 5140 5636,a 5165b

300 2922 3060 4538 (2956) 4790 4927,a 4753b

320 2700 2870 4263 (2908) 4460 4423,a 4360b

340 2560 2680 3963 (2850) 4046 3996b

aReference 26.bReference 27.

RESULTS AND DISCUSSION

Many-body SYM-3B model

Third virial coefficients at selected temperatures werecalculated using these parameters and the results are givenin Table II. Compared to non-polarizable Elementary Phys-ical Model 2 (EPM2) and TraPPE models, our underlyingpolarizable SYM-0.970 model (which omits explicit non-electrostatic three-body effects) does not yield improved thirdvirial coefficients. However, addition of the new three-bodyterms (SYM-3B model) yields a dramatic improvement. Thusin contrast to other systems like water,15 the non-additivity ofCO2 potential mainly comes from dispersion and exchangeeffects rather than induction (a result which is unsurprisinggiven the vanishing dipole of CO2).

The bulk density of the SYM-3B model was examined atthree phase points, representing gas, liquid, and supercriticalstates, respectively. All explicit three-body simulations werecarried out using a hybrid Monte Carlo (MC) algorithm28 witha cutoff of 12 Å for both two-body and three-body interac-tions. Long range dispersion corrections for two-body parts29

are included in conjunction with a particle mesh Ewald treat-ment for electrostatic interactions. The three-body energy ofa certain trimer ABC is explicitly calculated only when allthree involving distances (rAB, rBC, and rCA) are smaller thanthe cutoff value. The remaining (long range) part of three-body energy is treated using Kirkwood approximation, as dis-cussed below. The system size is set to 400 molecules. AllMD simulations were carried out in an NVT ensemble witha cutoff of 14 Å for two-body dispersion interaction. HybridMC calculations are conducted using a modified version ofTowhee,30 and all MD simulations are conducted using GRO-MACS 4.31 The percentage errors compared to experimentaldata are quite small and are shown in Table III. It is worthnoting that although GCPCDO exhibits excellent third virialcoefficients (by construction, as these were explicitly fit), itdisplays typically 8% error in the same density region (below20 mol/l). The structure of bulk CO2 was also tested via radialdistribution functions and are nearly identical to our previousSYM-0.935 potential (see Figure 4), which in turn are in ex-cellent agreement with the experimental bulk structure.

Effective ρ/T-dependent SYM[ρ,T] model

Inclusion of an explicit three-body term is computa-tionally expensive. However, in a homogeneous system thetotal three-body energy can be calculated under Kirkwood

TABLE III. Comparison of calculated and experimental bulk densities inthe gas (300 K, 60 bar), liquid (300 K, 100 bar), and supercritical (320 K,140 bar) regions.

T (K) P (bar) Density (mol/l) Exp.a Error %

300 60 3.99 4.14 − 3.6300 100 17.64 18.21 − 3.1320 140 15.73 15.98 − 1.6

aReference 32.

approximation,

E3bABC =Natom

ρ2atom

6

∫drABdrACgtAtB (rAB)gtBtC (rBC)gtCtA (rCA)

×V 3b(rAB, rAC)

E3b = 8

27E3b

OOO + 1

27E3b

CCC + 4

27

(E3b

COO + E3bOCO + E3b

OOC

)

+ 2

27

(E3b

CCO + E3bCOC + E3b

OCC

),

where the necessary radial distribution functions, g(r), are es-timated from our previous SYM-0.935 model at a variety of(fixed) densities and temperatures using MC simulations inan NVT ensemble. The validity of Kirkwood approximationat 320 K is illustrated in the supplementary material.18 Wefind this approximation to be excellent in other regions ofthe phase diagram as well, except at high temperature/high-density zones (e.g., 800 K and 24 mol/l). We also employthe Kirkwood approximation via the above equations to es-timate the long range correction to the three-body dispersionenergy in our explicit three-body simulations, excluding theshort range part of the integration (rAB < rc, rBC < rc, andrCA < rc).

Using the three-body energy, E3b(ρ, T), we then derivea density/temperature-dependent two-body polarizable effec-tive potential (the SYM[ρ, T] model) such that,⟨

E2bSAPT−disp

⟩ + λ(ρ, T)⟨E2b

disp

⟩= ⟨

E2bSAPT−disp

⟩ + 0.97⟨E2b

disp

⟩ + 〈E3b〉,

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Distance (Å)

0

0.5

1

1.5

2

Rad

ial D

istr

ibut

ion

Fun

ctio

n

C-C

O-O

C-O

FIG. 4. Radial distribution functions for bulk CO2 (320 K, 18mol/l) calcu-lated with explicit SYM-3B model (black), plotted alongside results fromSYM-0.935 model (red).

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034503-6 K. Yu and J. Schmidt J. Chem. Phys. 136, 034503 (2012)

where the angle brackets denote an equilibrium average at agiven density and temperature, evaluated from explicit MCsimulations, and 〈E2b

SAPT−disp〉 is the average calculated two-body energy at a given density and temperature (excludingdispersion contributions), 〈E2b

disp〉 is the average two-body dis-persion contribution, and 0.97 is the scale factor (determinedabove) which yields the correct B2 (and thus the correct two-body potential). In other words, we develop an effective two-body potential such that the average potential energy of thehomogeneous fluid (at a given density and temperature) forthe effective two-body model and the explicit many-bodymodel are equal,⟨

ESYM[ρ,T]⟩ρ,T = ⟨

ESYM−3B⟩ρ,T .

This constraint is enforced by choosing an appropri-ate temperature/density-dependent scale factor, λ(ρ, T), ap-plied to the SAPT two-body dispersion, as in the prior SYMmodel. Results for λ(ρ, T) are tabulated in the supplementarymaterial.18 Note that although the resulting SYM[ρ, T] modelcontains no explicit many-body terms, it is polarizable andthus implicitly retains many-body electrostatic contributions.

In contrast to the single empirical scale factor (0.935)we introduced in our previous SYM-0.935 model, the pa-rameter λ(ρ, T) in the SYM[ρ, T] model is rigorously de-termined from the three-body terms in the SYM-3B modeland is density- and (weakly) temperature-dependent. How-ever, note that at liquid-like densities and moderate temper-atures, we find thatλ(ρ, T) ≈ 0.94, very close to the empiri-cal value utilized previously, thus motivating our prior SYMmodel on rigorous statistical mechanical grounds.

We calculate density profiles at two different temper-atures just above the critical temperature, as shown inFigure 5; liquid phase data are also examined, yielding ac-curacy within ∼3% (see supplementary material18). Note thatsince our potential is density dependent, and small correctionis required when evaluating the pressure in an NVT ensem-ble, P3b

corr = ρ2

N〈E2b

disp〉 ∂λ(ρ,T)∂ρ

. In general, most of the densityerrors are between 1% and 4%, slightly worse than our priorSYM model; this is not surprising, as the latter was empiri-cally tuned to reproduce the experimental density. The maxi-

0 200 400 600 800 1000

Pressure (bar)

0

5

10

15

20

25

30

Den

sity

(m

ol/L

)

320K

360K

FIG. 5. Density profiles in 320 K and 360 K. Black solid lines are experi-mental data32 and red crosses are from the SYM[ρ, T] model simulations.

mum error, at 320 K and 110 bar, is due to the near divergenceof the compressibility near the critical point (thus, the corre-sponding error in pressure is much smaller).

In the high-density (liquid-like) regions of the phase di-agram, the present SYM-3B/SYM[ρ, T] models yields verysimilar (and even slightly better) densities than our prior SYMmodel (i.e., SYM-0.935 model), which is not unexpected asthey employ very similar scale factors in this region. Giventhat the structure and density are nearly identical to the SYM-0.935 model, other bulk properties should be similarly un-changed (and thus in excellent agreement with experimen-tal values). Furthermore, in the low density region λ(ρ, T)asymptotically approaches 0.97 (as the three-body potentialvanishes in the low density limit), yielding the correct two-body potential and thus correct B2(T). While correct in thesetwo limits, the present SYM-3B/SYM[ρ, T] is slightly lessaccurate in the medium density region, likely due to the ap-proximate treatment of three-body exchange.

We contrast the present approach for converting an ex-plicit many-body potential to an effective two-body potentialwith that of Oakley et al.17 In the latter case, the authors foundthat the non-additive (many-body) component of the potentialscaled approximately as ρ2.5, with a proportionality constantthat was fit by comparing with an explicit many-body sim-ulation. They used this density-dependent fit in conjunctionwith the pair potential to efficiently approximate the many-body energy in their underlying MC simulations. Note thatthe procedure advocated by Oakley et al. is equivalent to in-troducing an external density-dependent pressure (which maybe either positive or negative) on the system. While this pres-sure can account for the mean-field effect on the system, e.g.,enforcing the correct density due to average many-body ef-fects, it cannot reproduce the alterations in the structure ofthe liquid due to these many-body effects. In contrast, weoutline a methodology to arrive at an explicitly density- andtemperature-dependent pair potential, which accounts for theaverage three-body effects at the level of effective intermolec-ular interactions. This approach does allow for such structuralalterations via a modified, effective pair potential. In the limitof weak, perturbative three-body interactions, these method-ologies will be equivalent but may yield significantly differentanswers when applied to systems with significant many-bodyeffects.

CONLUSIONS

Overall, the explicit SYM-3B CO2 model we presenthere is particularly exciting as it is physically motivated inall aspects, with two- and three-body terms parameterizedon the basis of the corresponding SAPT energy decomposi-tion, and extremely minimal empirical input. The model re-produces not only bulk properties but also second/third virialcoefficients, demonstrating correct two- and three-body in-teractions. The latter are critical for bulk CO2, accountingfor the pressure corrections as large as 100 bar and densitychanges of up to 10% in condensed phase. Although it is cer-tainly possible to account for such three-body effects via a(density-independent) two-body potential, as in our previousSYM model, this necessarily leads to at least some physically

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034503-7 Three-body effects are essential for CO2 J. Chem. Phys. 136, 034503 (2012)

incorrect behavior in some regimes, as manifested by, for ex-ample, incorrect third virial coefficients, and somewhat vio-lates the tenants of a “physically motivated” model.

We show that we can accurately approximate this three-body SYM-3B model in terms of a computationally efficientdensity-/temperature-dependent two-body effective potential,yielding the SYM[ρ, T] model. This model essentially re-duces to our prior SYM model in the limit of liquid-like den-sities, thus providing a rigorous “derivation” of our previousmodel. Since the SYM model offers excellent agreement witha wide range of bulk structural, thermodynamic, and dynamicproperties, we thus also expect (by construction) similar ex-cellent agreement for the SYM-3B and SYM[ρ, T] models.We thus anticipate that the present SYM-3B and SYM[ρ, T]models will be extremely robust and transferable, from clus-ters to the bulk, across the phase diagram, and with possibleapplications to heterogeneous environments and/or extremeconditions.

ACKNOWLEDGMENTS

Computational resources for this work were providedby the National Science Foundation via Grant No. CHE-0840494. Additional computing resources were providedvia the Center for High Throughput Computing at theUniversity of Wisconsin.33 This research was supported byDOE via Grant No. DE-FG02–09ER16059.

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