Chemical Physics Letters 591 (2014) 237–242

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Chemical Physics Letters

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Bridge function of the repulsive Weeks–Chandler–Andersen (WCA) fluid

0009-2614/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cplett.2013.11.025

⇑ Corresponding author. Fax: +49 231 755 3748.E-mail address: stefan.kast@tu-dortmund.de (S.M. Kast).

Daniel Tomazic, Franziska Hoffgaard, Stefan M. Kast ⇑Physikalische Chemie III, Technische Universität Dortmund, Otto-Hahn-Straße 6, 44227 Dortmund, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 September 2013In final form 8 November 2013Available online 22 November 2013

The bridge function of a simple liquid is calculated for the repulsive part of the Weeks–Chandler–Ander-sen (WCA) separation of the Lennard–Jones potential. We employ explicit molecular dynamics simula-tions of the potential of mean force between constrained dimers in order to extract bridge data nearzero separation and illustrate the difference to full Lennard–Jones results. We compare direct, reciprocalspace and iterative, real space inversions of the Ornstein–Zernike equation. Bridge functions for variousthermodynamic states are analyzed as to their parametric dependence on the renormalized indirect cor-relation function, which has consequences for the analytic representation of the free energy functional.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

For a simple homogeneous liquid, the Ornstein–Zernike (OZ)integral equation (IE),

ðqh�cÞðrÞ ¼ qZ

hðr0 � rÞcðr0Þdr0 ¼ hðrÞ � cðrÞ � tðrÞ ð1Þ

connects the total correlation function (TCF) h with the direct corre-lation function (DCF) c (t = h � c is the indirect correlation function,ICF) at a spatial coordinate r. Together with the closure relation

hðrÞ þ 1 ¼ e�buðrÞþhðrÞ�cðrÞþBðrÞ ¼ e�buðrÞþtðrÞþBðrÞ ð2Þ

where u is the interaction potential and b = (kBT)�1 with tempera-ture T and Boltzmann constant kB, the OZ equation forms a com-plete theory of the liquid structure at equilibrium (characterizedby the pair distribution function, PDF, g = h + 1). Knowledge of theso-called bridge function B is an essential ingredient; neglecting Bcorresponds to the hypernetted chain (HNC) approximation [1]. Be-yond improving structural predictions by IE theories, knowledge ofB is also essential for calculating the free energy of a liquid [2–6]. Inthis respect, it is remarkable that the short range region of B, whichis completely superseded by the repulsive interaction in the closurerelation (2), is important for the free energy (most notably via thezero separation theorem [7]) and satisfies a fundamental inequalitywith respect to the DCF [8], while B close to zero distance does notinfluence any of the correlation functions noticeably. This numericalfact is responsible for the considerable body of literature devoted tothe extraction of bridge data from standard molecular simulationsbeyond the repulsive core (with recent applications to molecularinteraction site fluids, see e.g. [9,10]), whereas the determination

of inner-core data requires special simulation techniques due tothe insensitivity of the correlation functions. Such special tech-niques, that are basically free energy simulation methods, havebeen used for constructing the bridge function of the hard sphere[11] and the Lennard–Jones (LJ) fluid [12]. Accurate bridge functionmodels are available that reproduce the known LJ inner-core behav-ior [13,14].

The general goal of this Letter is the calculation of the bridgefunction of an intermediate case, the repulsive part of theWeeks–Chandler–Andersen (WCA) separation of the LJ potential,

uLJðrÞ ¼ 4e ððr=rÞ12 � ðr=rÞ6Þ ð3Þ

constructed by truncating the latter at the LJ minimum, rmin = 21/6 r,and shifting by the minimum energy, e, i.e.

uWCA-repðrÞ ¼4e ððr=rÞ12 � ðr=rÞ6Þ þ e () r < 21=6r0 () r P 21=6r

(ð4Þ

Our interest in this context is fourfold. (i) Besides tabulatingaccurate data including the inner-core region in comparison withthe full LJ case for possible use by the IE community to test approx-imate theories, we want to establish, (ii) direct molecular dynamics(MD) simulations of the potential of mean force (PMF) betweenconstrained atom pairs [15] as a means for treating the difficult re-gion, (iii) we compare the established direct (reciprocal space)inversion of the OZ equation as e.g. used in [9,10] with an iterativereal space variant [16–18], and (iv) we analyze the parametricdependence of the thus obtained bridge function on a renormal-ized ICF t⁄ = t � bu which has been shown earlier by us [5,6] tobe a possible candidate for a thermodynamically consistent bridgefunction parametrization. In particular, such a functional form sat-isfies the state function condition of the free energy to be indepen-dent of the coupling parameter integration path.

238 D. Tomazic et al. / Chemical Physics Letters 591 (2014) 237–242

2. Theory and methods

2.1. Bridge functions close to zero separation from PMF simulations

Since the PMF W is related to the PDF via

bWðrÞ ¼ � ln gðrÞ ¼ �ðtðrÞ � buðrÞ þ BðrÞÞ ð5Þ

(for simple liquids the radial distance coordinate r is sufficient),a specially crafted MD simulation technique such as the con-strained pair approach by Ciccotti et al. [15] for computing themean force between the particle pair can be applied to extractbridge values within the core region. The PMF can be subdividedinto a solvent-mediated component, Wv, and the interaction poten-tial, u,

WðrÞ ¼WvðrÞ þ uðrÞ ð6Þ

such that

�bWvðrÞ ¼ tðrÞ þ BðrÞ ð7Þ

The l.h.s. results from integration of the solvent-mediated meanforce (which is zero for infinite distance),

WvðrÞ ¼ �Z r

1dr0 �Fvðr0Þ ð8Þ

where

�Fvðr12Þ ¼12

r12

r12ðFvðr1Þ � Fvðr2ÞÞ

� �r12

ð9Þ

is the average solvent force exerted on a constrained site pair atpositions r1 and r2 with distance r12 = |r2 � r1|. B is therefore ob-tained from Wv by supplementing (7) with an ICF from invertingthe OZ equation. Note that the upper integration limit is problem-atic since we can only treat a limited range of constraint distancesby MD simulations. By splitting Wv into

WvðrÞ ¼ �Z r

rx

dr0 �Fvðr0Þ �Z rx

1dr0 �Fvðr0Þ ð10Þ

with variable rx we can circumvent this issue by explicit PMF sim-ulation of the first term on the r.h.s. and taking the second term di-rectly from the simulated PDF, gMD, of the pure liquid by

�Z rx

1dr0 �Fvðr0Þ ¼ �b�1 ln gMDðrxÞ � uðrxÞ ð11Þ

2.2. Inversion of the OZ equation

In contrast to molecular site IEs and in the absence of long rangeinteractions the direct inversion of the OZ equation in reciprocalspace is unproblematic [9,10]. Starting with the Fourier transform(FT) of the TCF (assumed to be known over a sufficiently long rangewhere it has practically decayed to zero),

hðkÞ ¼ FT gMDðrÞ � 1� �

; ð12Þ

the ICF is given by

tðrÞ ¼ gMDðrÞ � 1� cðrÞ ¼ gMDðrÞ � 1� FT�1 hðkÞ1þ qhðkÞ

" #: ð13Þ

The resulting ICF enters (7) together with simulated PMF data inorder to obtain B including the inner-core region.

The alternative is an iterative, real space inversion [16–18]based on

BðrÞ ¼ f ðrÞ lnðgMDðrÞÞ þ buðrÞ � tðrÞ� �

ð14Þ

which, upon insertion in (2), basically leads to a constraint on thesolution of the OZ/closure system in that g is set to the MD valuesover a specified range. Here, f(r) defines a switching function (cubicpolynomial) varying between 1 and 0 in a transition region betweendistances rin and rout, beyond which the HNC approximation is as-sumed to be valid. In practice this means that the OZ equation issolved together with (14) as effective closure relation, implying agrand canonical long range correction to the PDF. If B would be ex-tracted in this way, the core region would be undefined due to thereasons discussed above. Here, we use this approach solely in orderto obtain the ICF t which is then inserted similar to the direct ap-proach together with simulated PMF data for extracting B via (7).

2.3. Computational details

MD simulations were performed with LAMMPS [19] in thecanonical ensemble using the Nosé–Hoover thermostat [20,21]and a time step of 5 fs for a range of states corresponding to [12].LJ parameters were r = 2.79 Å and e = 0.0709 kcal mol�1 resem-bling a liquid neon model [22] (note that all results are specifiedin reduced units, i.e. for density q⁄ = qr3 and temperatureT⁄ = kBT/e). Initially, the atoms were placed randomly inside thesimulation volume and then preequilibrated in the microcanonicalensemble starting with zero velocities by limiting the maximumdisplacement per time step to 0.1 Å for 2.5 ps. Similar to thesimulation setup in [12] the LJ potential was smoothly switchedto zero using a cubic polynomial between 8 and 10 Å for produc-tion purposes unless specified otherwise. For reference PDF data(used for direct and iterative inversion as well as for PMF longrange offset determination) and for explicit PMF simulations twodifferent setups were used. The former were calculated from pro-duction runs based on 15000 atoms; for the latter, the mean forceswere recorded for two constrained atoms out of a total of 5002 (tocheck convergence, for a single state in addition 15000 solventatoms were simulated) employing the SHAKE algorithm [23].

PDF data was computed after 5 ps microcanonical and 250 pscanonical ensemble equilibration periods from simulations over5 ns using 10000 snapshots up to a distance of 30 Å with a bin sizeof 0.06 Å. Since the tail of the PDF oscillates around one for dis-tances between 20 and 30 Å no long range correction of the PDFwas applied. To estimate the error we calculated for a single state(T⁄ = 1.0 q⁄ = 0.8) the PDF standard error for every 100 snapshots,finding 0.004 in the area of the first peak and around 0.0005 be-tween 20 and 30 Å. The resulting data were therefore used withoutfurther smoothing. For PMF calculations, 195 distances between0.01 and 19.9 Å were evaluated in steps of 0.03 Å for (0.01–0.1) Å, 0.05 Å for (0.1–6) Å, 0.1 Å for (6–10) Å, 0.3 Å for (10–19.9) Å. 25000 snapshots of 12.5 ns production simulation wereevaluated after a 117.5 ps equilibration phase in the canonicalensemble. The mean statistical error of the PMF for all states wasfound to be below 0.0005 kcal mol�1. For the long range PMF offset,the forces were integrated for 15 different values of rx between 4.4and 10 Å followed by averaging over the results. To check the accu-racy of the calculations we also derived the PDF from the PMF anddetermined the maximum deviation from the directly simulatedPDF. The largest value for all state points was 0.06; the maximummean deviation was about 0.004.

The IE calculations were performed on a logarithmic grid of 512points ranging from 5.98 � 10�3 Å to a maximum distance of164.02 Å for the same densities and temperatures as in the MDsimulations, employing the Talman method for the fast Fouriertransform [24] including zero padding over twice as many points.The IE solutions were converged by the ‘modified direct inversionof iterative subspace’ (MDIIS) method [25] to a threshold ofmax(Dc) < 10�5 between two successive iteration steps. For theiterative inversion of the OZ/closure equation the switch function

-5

0Aρ* = 0.4 T * = 1.5ρ* = 0.6 T * = 1.5

D. Tomazic et al. / Chemical Physics Letters 591 (2014) 237–242 239

in (14) turned off the reference PDF between 17 and 21 Å. The di-rect inversion of the simulated PDF was performed on a linear gridfrom 0 to 19 Å with a spacing of 0.1 Å. The simulated PDF wasinterpolated to the target grids using cubic spline interpolationwhereby it was set to 1 for r > 30 Å. The iterative inversion requiresa potential energy which was chosen as identical to the one appliedin the MD simulations. For testing the extracted bridge functionsthey were truncated at 7 Å before reinserting into the OZ/closureequations. Here, the full LJ potential was applied in order to allowfor a comparison with reference LJ data.

-30

-25

-20

-15

-10

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

BLJ

ρ* = 0.7 T * = 1.5ρ* = 0.8 T * = 1.5ρ* = 0.9 T * = 1.5ρ* = 0.8 T * = 1.0ρ* = 0.8 T * = 0.81ρ* = 0.85 T * = 0.72

-30

-25

-20

-15

-10

-5

0

0.0 0.2 0.4 0.6 0.8 1.0 1.2B

LJ

B

ρ* = 0.8 T * = 1.5ρ* = 0.8 T * = 0.81ρ* = 0.85 T * = 0.72

B

iterative inversiondirect inversionreference

-0.3

-0.2

-0.1

0.0

1.0 1.5 2.0 2.5 3.0 3.5

BLJ

C

ρ* = 0.8 T * = 1.5ρ* = 0.8 T * = 0.81ρ* = 0.85 T * = 0.72

C

iterative inversiondirect inversionreference

3. Results and discussion

3.1. Analysis of the methodology

For all analyses presented below, raw data is provided as Sup-plemental information (in the case of bridge functions from itera-tive inversion the results were interpolated to a regular grid bycubic polynomials).

Figure 1 illustrates the solvent-mediated component of the PMFfor both, the full LJ and the repulsive part of the WCA potential(WCA-rep). The statistical noise is apparently negligible as ex-pected from the size of the simulation systems. Overall, the shapeof the PMF curves in the LJ and the WCA-rep case are similar whilelarger discrepancies are found for the core region; the long rangepart of Wv is apparently only marginally influenced by the pres-ence of a long range attractive interaction tail. For each state theclose contact value of Wv is slightly more attractive for WCA-repthan for LJ while the intermediate range at slightly larger distancesthan the potential minimum (at around 1.12 r) is more repulsivefor LJ than for WCA-rep.

The two variants for bridge extraction, direct reciprocal spaceand iterative real space inversion were tested for the LJ fluid in

-14-12-10

-8-6-4-202

0.0 0.5 1.0 1.5 2.0 2.5

βWv LJ

Aρ* = 0.4 T * = 1.5ρ* = 0.6 T * = 1.5ρ* = 0.7 T * = 1.5ρ* = 0.8 T * = 1.5ρ* = 0.9 T * = 1.5ρ* = 0.8 T * = 1.0ρ* = 0.8 T * = 0.81ρ* = 0.85 T * = 0.72

-14-12-10

-8-6-4-202

0.0 0.5 1.0 1.5 2.0 2.5

βWv W

CA-

rep

Bρ* = 0.4 T * = 1.5ρ* = 0.6 T * = 1.5ρ* = 0.7 T * = 1.5ρ* = 0.8 T * = 1.5ρ* = 0.9 T * = 1.5ρ* = 0.8 T * = 1.0ρ* = 0.8 T * = 0.81ρ* = 0.85 T * = 0.72

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5

βWv

r / σ

C

LJWCA-rep

Figure 1. Solvation contribution to the PMF as a function of pair distance forvarious states represented by different line styles as specified in the panels; (A) LJfluid, (B) repulsive WCA fluid, (C) enlarged comparison of LJ and repulsive WCAresults for q⁄ = 0.85 and T⁄ = 0.72.

direct comparison with reference data by Llano-Restrepo andChapman [12] as illustrated in Figure 2. Overall we find closeagreement between the various approaches with the largest dis-crepancies in the depletion zone between the first and the secondRDF maximum at around r/r = 1.5. For certain thermodynamicstates, larger discrepancies between our and reference data are

r / σ

Figure 2. Bridge data for the LJ fluid as a function of pair distance for various statesrepresented by different line styles as specified in the panels; (A) from iterativeinversion, (B and C) comparison of iterative (real space) and direct (reciprocalspace) inversion, and with reference data by Llano-Restrepo and Chapman [12].

−0.10−0.08−0.06−0.04−0.02

0.000.020.04

1.0 1.5 2.0 2.5 3.0 3.5

BLJ

A

N = 15000, rin = 8 Å, rout = 10 ÅN = 5000, rin = 8 Å, rout = 10 ÅN = 5000, rin = 10 Å, rout = 12 ÅN = 5000, rin = 12 Å, rout = 15 Å

−20

−15

−10

−5

0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

BLJ

r / σ

B

N = 15000, rin = 8 Å, rout = 10 ÅN = 5000, rin = 8 Å, rout = 10 ÅN = 5000, rin = 10 Å, rout = 12 ÅN = 5000, rin = 12 Å, rout = 15 Å

−20.1

−19.9

−19.7

Figure 3. Bridge data for the LJ fluid from direct inversion under various simulationconditions (different particle numbers N and inner/outer ranges rin/rout of theswitching function that turns off interactions) as a function of pair distance forq⁄ = 0.9 and T⁄ = 1.5; (A and B) represent different plot ranges. An 8/10/12/15 Ådistance in the real simulated system corresponds to reduced distances r/r of about2.87/3.58/4.30/5.38. In the inset, the orange line corresponding to the larger systemis indistinguishable from the lowest curve.

-30

-25

-20

-15

-10

-5

0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

BW

CA-

rep

Aρ* = 0.4 T * = 1.5ρ* = 0.6 T * = 1.5ρ* = 0.7 T * = 1.5ρ* = 0.8 T * = 1.5ρ* = 0.9 T * = 1.5ρ* = 0.8 T * = 1.0ρ* = 0.8 T * = 0.81ρ* = 0.85 T * = 0.72

-30

-25

-20

-15

-10

-5

0

0.0 0.2 0.4 0.6 0.8 1.0 1.2

B

B

ρ* = 0.8 T * = 1.5ρ* = 0.8 T * = 0.81ρ* = 0.85 T * = 0.72

B LJB WCA-rep

-0.3

-0.2

-0.1

0.0

1.0 1.5 2.0 2.5 3.0 3.5

B

r / σ

C

ρ* = 0.8 T * = 1.5ρ* = 0.8 T * = 0.81ρ* = 0.85 T * = 0.72

B LJB WCA-rep

Figure 4. Bridge data from iterative inversion as a function of pair distance forvarious states represented by different line styles as specified in the panels; (A) forthe repulsive WCA fluid, (B and C) comparison of repulsive WCA and full LJ fluids.

-3.0-2.5-2.0-1.5-1.0-0.50.00.5

-4.0 -2.0 0.0 2.0 4.0 6.0

BLJ

Aρ* = 0.4 T * = 1.5ρ* = 0.6 T * = 1.5ρ* = 0.7 T * = 1.5ρ* = 0.8 T * = 1.5ρ* = 0.9 T * = 1.5ρ* = 0.8 T * = 1.0ρ* = 0.8 T * = 0.81ρ* = 0.85 T * = 0.72

-3.0

-2.0

-1.0

0.0

BLJ

ρ* = 0.8 T * = 1.5

Biterative inversiondirect inversion

240 D. Tomazic et al. / Chemical Physics Letters 591 (2014) 237–242

found in the second depletion region. These differences are of asimilar order of magnitude as those found by varying the simula-tion particle number and cutoff distances, as shown in Figure 3.In general, the cutoff effect is more important than the finite sizeeffect for our simulations; 5000 solvent particles are apparentlysufficient for production purposes. The reference data [12] wasgenerated based on a much smaller simulation system whichmight explain why our bridge data are systematically slightly high-er in the core region. The increase of statistical error with largerpair distance is apparently responsible for the increasing discrep-ancies at longer range.

The small local differences do not significantly influence theinternal energy as derived from IE-computed PDFs obtained byinserting bridge data directly into the closure. Results are summa-rized in Table 1 where we also compare with data from an analyt-ical equation of state of the LJ fluid [26]. In summary, the iterativeand the direct inversion methods are practically equivalent andreliable. In practice, the iterative variant is simpler to apply tomore complex systems described by site–site IEs since it avoidsproblems of ill-conditioned matrices [10].

3.2. Analysis of bridge functions

Figure 4 shows bridge data from iterative inversion for theWCA-rep in comparison with the LJ case. Overall, the shapes arequite similar, more so for smaller densities and higher tempera-tures as expected. The limiting value of B for vanishing distanceis systematically smaller for LJ compared to WCA-rep. Most signif-icant differences are again found in the first depletion region.

The key property of bridge functions that we want to examinein this Letter is related to its parametrization as a function of cor-relation functions. Since the seminal work by Duh and Haymet [27]it is well known that no universal bridge parametrization by theICF alone exists independent of the thermodynamic state. To thisend they established a parametric representation of B as a functionof t for corresponding distances (‘Duh–Haymet plot’). On the otherhand, we have shown earlier [5,6] that a bridge function that de-pends on a renormalized ICF t⁄ = t � bu belongs to a class of bridgemodels that satisfies the criterion of path independence of the cou-pling parameter integration leading to the free energy. We havedemonstrated this property numerically by explicit integration ofa modified Verlet (VM) fit to LJ reference bridge data [5] wherethe ICF was renormalized by the long range WCA component only.Such an approach is equivalent to scaling the interaction betweenthe WCA-rep and LJ cases studied in this Letter. It turned out thatgood results are obtained in this way which raises the question asto why this is the case. We are now in the position to providenumerical evidence since we expect a – at least state-specific –universal bridge parametrization between the WCA-rep and theLJ potentials for the latter approach to work.

Table 1Reduced excess internal energy Ured = Uex/(Ne) of the LJ fluid for various statesobtained from solving the OZ/closure equation with bridge data from differentsources, iterative (‘iter’) and direct (‘direct’) inversion, reference data by Llano-Restrepo and Chapman [12] (‘ref’) in comparison with results from the analytical LJequation of state (‘EoS’) [26].

q⁄ T⁄ UEoSred Uiter

red Udirectred Uref

red

0.4 1.5 �2.70892 �2.70534 �2.70356 �2.690720.6 1.5 �3.96024 �3.96508 �3.96128 �3.973640.7 1.5 �4.58747 �4.57796 �4.57650 �4.572590.8 1.5 �5.13638 �5.11805 �5.11622 �5.135680.9 1.5 �5.52627 �5.50659 �5.50820 �5.489750.8 1.0 �5.52345 �5.53666 �5.53153 �5.524630.8 0.81 �5.71954 �5.70218 �5.70468 �5.708380.85 0.72 �6.12489 �6.11456 �6.11788 �6.11703

-4.0

-8.0 -6.0 -4.0 -2.0 0.0 2.0

ρ* = 0.85 T * = 0.72ρ* = 0.8 T * = 0.81

-0.30-0.25-0.20-0.15-0.10-0.050.000.050.10

-0.5 0.0 0.5 1.0 1.5

BLJ

t *

ρ* = 0.85 T * = 0.72ρ* = 0.8 T * = 0.81ρ* = 0.8 T * = 1.5

C

iterative inversiondirect inversion

Figure 5. Bridge data for the LJ fluid from different inversion methods as a functionof the renormalized ICF t⁄ = t � bu for various states represented by different linestyles as specified in the panels; (A–C) represent different plot ranges.

D. Tomazic et al. / Chemical Physics Letters 591 (2014) 237–242 241

In Figure 5 we depict the modified (i.e. renormalized) Duh–Hay-met plot for the LJ case. In contrast to the non-renormalized casewe find a characteristic turn on the positive side which correlatesroughly with the first maximum of the PDF, i.e. with the onset ofthe repulsive core. The curves continue on the left towards the lim-iting value at close contact; the attractive region is located aroundthe origin. The bridge function shows apparently a bifurcation intotwo different functional dependencies. However, even in theattractive region (panel C of Figure 5) the bridge function can obvi-ously not be fully described by a single function of t⁄, although amodified Verlet fit (which means a single function ansatz) appearsto work well [5]. The most remarkable observation is, however,that all curves almost coincide in the attractive regions, indepen-dent of the thermodynamic state.

We can shed more light on this situation by analyzing the repul-sive WCA case in comparison. The numerical inversion results areshown in Figure 6. It turns out that all curves again almost coincidein the attractive potential region. The situation of Ref. [5] here cor-responds to the transition between the solid and the dashed linesin the upper branch in panel B. The underlying assumption for avalid coupling parameter integration over a specific functionalform of B(t⁄) is that this form is independent of the potential de-tails, in this case for both the WCA-rep and the full LJ cases. Figure 6shows that this is indeed the case in the attractive, upper branchregion. This is supported by the superimposed curve (dotted lines)of the VM closure parametrization presented in Ref. [5] which ledto a successful free energy prediction. The VM bridge model prac-tically coincides with both, the WCA-rep and the LJ bridges as afunction of the respective ICFs in the attractive region. The WCA-rep bridge data obtained in this Letter therefore in retrospect val-

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

-4.0 -2.0 0.0 2.0 4.0 6.0

BW

CA-

rep

Aρ* = 0.4 T * = 1.5ρ* = 0.6 T * = 1.5ρ* = 0.7 T * = 1.5ρ* = 0.8 T * = 1.5ρ* = 0.9 T * = 1.5ρ* = 0.8 T * = 1.0ρ* = 0.8 T * = 0.81ρ* = 0.85 T * = 0.72

-3.5-3.0-2.5-2.0-1.5-1.0-0.50.0

-6.0 -4.0 -2.0 0.0 2.0 4.0

B

ρ* = 0.8 T * = 1.5ρ* = 0.8 T * = 0.81ρ* = 0.85 T * = 0.72

BB LJ(t *LJ)B VM(t *WCA-attr)B WCA-rep(t *WCA-rep)

-0.25-0.20-0.15-0.10-0.050.000.050.10

-0.5 0.0 0.5 1.0 1.5

B

t *

ρ* = 0.8 T * = 1.5ρ* = 0.8 T * = 0.81ρ* = 0.85 T * = 0.72

C

B LJ(t *LJ)B VM(t *WCA-attr)B WCA-rep(t *WCA-rep)

Figure 6. Bridge data for the repulsive WCA and for the LJ fluids from iterativeinversion as a function of the renormalized ICF t⁄ = t � bu for various statesrepresented by different line styles as specified in the panels; (A) pure WCA-represults for all states, (B and C) comparison with LJ data and with a partially (longrange) renormalized bridge model described by the modified Verlet closure (VM),parametrized corresponding to the free energy extraction approach of Ref. [5].

idate the approach chosen earlier. The correspondence and theassociated partial universality, however, break down in the repul-sive core region. Implications for free energy functionals includingbridge functions for the general case covering the full potential arethe topic of ongoing work.

4. Concluding remarks

We have shown that explicit PMF simulations of constrainedpairs yield reliable bridge function data inside the repulsive poten-tial regions of the LJ and the repulsive WCA fluids, irrespective ofthe inversion approach (based on real or reciprocal space formal-isms). Due to the nonlocality and nonlinearity of the OZ equationin conjunction with the closure relation it is very difficult to controlthe impact of statistical noise and other simulation features on theresulting bridge data. Even very accurate simulation data as ob-tained in this Letter result in small differences in the final targetvalues due to practically uncontrollable propagation of error. Forsimple first order thermodynamic quantities such as the internalenergy this is apparently unproblematic, whereas it remains tobe seen if this is also the case for the free energy.

Improvement could possibly be achieved by smoothly interpo-lating the PMF approach to B in the core region with the directinversion method applicable in regions where B is not supersededby the potential. We have not attempted such a method, nor havewe played with smoothing techniques which all will yield furtherdifferent results without a means to objectively measure the ‘cor-rectness’. Another source of concern is the use of a directly invertedICF (which would not be sufficient to obtain B in the core directly)in the context of a PMF-based bridge definition, an inconsistencywhich has also been noted earlier [27]. While there is reason totrust such an approach due to the insensitivity of the ICF to thecore bridge function, many important questions arise. It has for in-stance to be clarified why correlation functions obtained by solvingOZ/closure equations with adequate bridge data do not respond tothe latter in the core region while at the same time this very coreregion has substantial impact on the free energy [3,7]. This prob-lem together with attempts to understand the partial universalityof the bridge functions in the attractive potential region and thediscrepancies in the repulsive part are important areas for futureresearch.

Acknowledgements

We thank the German Federal Ministry of Education and Re-search (BMBF) for financial and the IT and Media Center (ITMC)of the TU Dortmund for computational support.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.cplett.2013.11.025.

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