Generating function density functional theory: free-energy functionals and direct correlation functions for hard-spheres

Physica A 296 (2001) 347–363www.elsevier.com/locate/physa

Generating function density functional theory:free-energy functionals and direct correlation

functions for hard-spheresA. Gonz(alez∗, J.A. White

Departamento de F� sica Aplicada, Facultad de Ciencias, Universidad de Salamanca,37008 Salamanca, Spain

Received 22 January 2001

Abstract

A recent density functional theory for the inhomogeneous hard-sphere -uid expresses thefree-energy in terms of a set of derivatives, with respect to the particle radius, of a simplegenerating function. In the present paper this generating function approach (GFA) is reformulatedin terms of general conditions on the coe1cients of the di2erential operator involved in thetheory. This gives more -exibility to the theory and allows for ready derivation of new functionalsthat are tested in the calculation of density pro5les of hard spheres near a hard wall. Comparisonwith the results of the previous formulation of the GFA shows a better performance of the newfunctionals and also gives a much deeper understanding of the structure of the theory andits connections with the fundamental measures theory of Rosenfeld. A simple modi5cation ofthe GFA pair direct correlation functions yields explicit analytic expressions with very goodagreement with simulation results. c© 2001 Elsevier Science B.V. All rights reserved.

PACS: 61.20.Gy; 68.45.−v

Keywords: Density functional theory; Hard spheres; Inhomogeneous -uids;Direct correlation function

1. Introduction

Traditionally, most of the e2ort dedicated to the density functional theory (DFT) ofclassical -uids has focused on the search for theories [1] able to give a reasonable(and sometimes very good!) description of a number of inhomogeneous situations thatappear under di2erent circumstances. Adsorption, wetting and freezing are examples of

∗ Corresponding author. Tel.: +34-923-29-44-36; fax: +34-923-29-45-84.E-mail address: [email protected] (A. Gonz(alez).

0378-4371/01/$ - see front matter c© 2001 Elsevier Science B.V. All rights reserved.PII: S 0378 -4371(01)00175 -3

348 A. Gonz�alez, J.A. White / Physica A 296 (2001) 347–363

problems well studied by means of these theories [1–3]. The free-energy functionalsemployed for these applications are usually built upon a semiempirical basis. As anexample, for the hard-sphere -uid it is usual to consider a known expression for theexcess (over ideal) free-energy density of the homogeneous -uid (e.g., the Percus–Yevick (PY) or the Carnahan–Starling (CS) results), written in terms of a weighted orsmoothed inhomogeneous density. Second-order functional di2erentiation of the result-ing free-energy functional gives rise in the homogeneous limit to the direct correlationfunction (DCF) of the bulk -uid. Experience shows that a functional that yields agood bulk DCF (e.g., the PY DCF for the hard-sphere -uid) has a good performancein highly inhomogeneous situations. Consequently one imposes the functional to yieldthis good bulk DCF—usually through a density-dependent weighting function. Thismeans that in this semiempirical approach the homogeneous -uid structure and ther-modynamics are needed as an input.A di2erent approach, more close to 5rst principles, is the so-called fundamental-

measures theory (FMT) [4–7] in which one assumes that the excess free-energy densityof the inhomogeneous -uid is a function of a set of six weighted densities. The densityindependent weight functions considered have a geometric interpretation since they arethe fundamental measures of a hard sphere. For a uniform hard-sphere -uid the FMTyields as a result the Scaled Particle Theory (or PY) excess free-energy density andthe PY DCF instead of requiring it as an input. When applied to one dimension itgives the exact free-energy functional for the inhomogeneous -uid of hard rods whichalso has the FMT structure.Recently, a new density functional theory for the inhomogeneous hard-sphere -uid

has been proposed which is close to the FMT [8]. The generating function approach(GFA) assumes that the excess free-energy density can be written as a set of derivativesw.r.t. the particle radius of a simple generating function G. The choice of G is basedon free-volume considerations and the theory only requires one density-independentweighting function. In addition, G is essentially the exact excess free-energy of azero-dimensional (0D) cavity which cannot hold more than one particle. This estab-lishes a link between the GFA and the FMT in the light of recent work that presentsnew FMT functionals based on the exact 0D free-energy [9–15]. Moreover, a funda-mental measure theory for mixtures of parallel hard cubes (parallelepipeds) has beenrecently formulated in terms of derivatives of the exact 0D free-energy w.r.t. the edgelengths of the parallelepiped (which, of course, reduces to a cube when all the lengthsare set equal) [16,17]. Due to its simple structure in terms of derivatives, the GFAis remarkably successful at describing dimensional crossover situations in which thee2ective dimensionality D of the -uid is reduced below D=3. Also, like the FMT, theone-dimensional version of the GFA yields the exact free-energy functional for hardrods. Furthermore, the 2D GFA functional is equivalent to the one obtained from theFMT [8].The GFA was designed so that the coe1cients of the derivatives were obtained from

5tting to the free-energy of the homogeneous -uid. In the present paper we propose amore -exible formulation of the GFA in terms of a set of conditions for the coe1cients.

A. Gonz�alez, J.A. White / Physica A 296 (2001) 347–363 349

These conditions are of very general nature, including, e.g., the constraints imposed bythe 2nd and the 3rd coe1cients of the virial expansion of the free-energy density in theuniform limit. Since we obtain more than three independent conditions and only threecoe1cients are required, di2erent functionals are derived. The main weakness of theselies in their failure to yield an accurate pair DCF for the uniform -uid. For instance,in some cases, the DCF fails in reproducing the correct 0th term in its diagrammaticexpansion, i.e., the pair exclusion function. A simple modi5cation avoids this problem.In one case we obtain a result which is identical to the PY pair DCF. In another case(CS) the modi5ed DCF has a remarkable agreement with the results of simulations athigh packing.The paper is organized as follows. For completeness, in Section 2, we present a

brief account of the FMT, with special emphasis on the aspects that are relevant tothe present work. In Section 3 the GFA is formulated in terms of general conditionson the coe1cients of the di2erential operator involved in the theory. These conditionsare determined from a detailed investigation of the behavior of the GFA pair DCF inthe uniform limit. In Section 4 several GFA functionals are derived and an applicationto the canonical problem of adsorption at a wall is considered in order to examine theperformance of the functionals. In Section 5 the misbehavior at low densities of theGFA pair DCF in the uniform limit is corrected by means of a simple modi5cation.From these modi5ed results we obtain the structure factor and the radial distributionfunction for the uniform hard-sphere -uid. A summary of the results is presented inSection 6.

2. The Fundamental measure theory of Rosenfeld

The excess contribution Fex[�] to the free-energy functional of a one componenthard-sphere -uid can be expressed as

�Fex[�] = �F[�]− �Fid[�] =∫

dr�({n}) ; (2.1)

where � = 1=kBT is the inverse temperature, �(r) is the inhomogeneous density andFid[�] is the ideal-gas contribution. In this equation �−1� represents an excess free-energy density which in the theory of Rosenfeld (denoted with the label ROS) iswritten as [4–7]:

�ROS = �1 + �2 + �3 ; (2.2)

�1 =−n0 log(1− n3); �2 =n1n2 − n1 · n2

1− n3; �3 =

(1=3)n32 − n2(n2 · n2)8�(1− n3)2

:

(2.3)

Note that this equation includes both scalar and vector quantities. The n’s are weighteddensities

n(r) =∫

�(r′)!()(r− r′) dr′ ; (2.4)


where the four scalar and two vector density-independent weight functions !() are thefundamental geometric measures of a sphere of radius R [4–7]:

!(3)(r) = �(R− r) ; (2.5)

!(2)(r) = �(R− r); !(2)(r) =rr�(R− r) ; (2.6)

!(1)(r) =!(2)(r)4�R

; !(1)(r) =!(2)(r)4�R

; (2.7)

!(0)(r) =!(2)(r)4�R2 ; (2.8)

where � and � are, respectively, the Heaviside step and the Dirac delta function.The pair DCF can be obtained from the second functional derivative of the excess

free-energy functional

−c(2)(r1; r2) = ��2Fex

��(r1)��(r2)=∫

dr∑;�

��!()(r− r1)!(�)(r− r2) ; (2.9)

where �� ≡ @2�=@n@n�. In the uniform limit the derivatives �� are independent ofr and thus

−c(2)(r) =∑;�

��!() ⊗ !(�) ; (2.10)

where ⊗ denotes the usual convolution product. Taking into account the explicit formof �ROS and the weights !(), from Eq. (2.10) one obtains a pair DCF that coincideswith the PY result for a three-dimensional hard-sphere -uid [4–7]. In the low-densitylimit (�→ 0) this DCF becomes the (minus) pair exclusion function �(2R − r), i.e.,the Mayer function of the system.Finally, we note that Kierlik and Rosinberg [18,19] derived a version of the theory

of Rosenfeld involving only scalar weighted densities:

�K–R =−n̂0 log(1− n̂3) +n̂1n̂21− n̂3

+n̂32

24�(1− n̂3)2; (2.11)

where n̂3 = n3, n̂2 = n2, and n̂1, n̂0 di2er from n1, n0. This expression for the ex-cess free-energy density �−1� leads to an excess free-energy functional Fex which iscompletely equivalent to the Rosenfeld functional [20].

3. The Generating function approach

In the GFA [8] the excess free-energy density �−1� is written in the formof a di2erential operator DR acting on a generating function G = (1 − n3(r; R))(log(1− n3(r; R))− 1) where n3 is the weighted density that appears in the Rosenfeldtheory and its dependence on R is made explicit. For hard spheres DR reads

DR =3∑i=1

aiR3−i

@i

@Ri; (3.1)


where the ai’s are dimensionless constants that can be determined by 5tting �=DRGin the uniform limit to a known expression of the excess free-energy density of thehomogeneous hard-sphere -uid. In Ref. [8] two 5ts were considered: the PY 5t, andthe Carnahan–Starling (CS) 5t, which lead to the sets

a1 =16�

; a2 = 0; a3 =1

24�(3.2)

and

a1 =− 718�

; a2 =16�

; a3 =1

36�; (3.3)

respectively. Substitution of these sets into (note that the prime denotes derivativew.r.t. R):

�= DRG =−( a1R2 n

′3 +

a2Rn′′3 + a3n′′′3

)log(1− n3)

+(a2R(n′3)

2 + 3a3n′3n′′3

) 11− n3

+ a3(n′3)3 1(1− n3)2

; (3.4)

yields two di2erent excess free-energy densities for the inhomogeneous hard-sphere-uid:

�PY =−(

16�R2 n

′3 +

124�

n′′′3

)log(1− n3) +

n′3n′′3

8�(1− n3)+

(n′3)3

24�(1− n3)2(3.5)

and

�CS =−( −718�R2 n

′3 +

16�R

n′′3 +1

36�n′′′3

)log(1− n3)

+(

16�R

(n′3)2 +

112�

n′3n′′3

)1

1− n3+

136�

(n′3)3

(1− n3)2: (3.6)

When applied to the canonical problem of adsorption at a wall both 5ts gave sensibleresults (with a better value of � at contact in the CS case) but far from the accuracyof the results of the theory of Rosenfeld [8,18,19]. In particular, the main di2erencewith Rosenfeld’s results was a kink located at a distance of one particle diameter fromthe wall. This kink was ascribed to the presence of a Dirac delta function in the pairDCF obtained from both 5ts for the homogeneous hard-sphere -uid.The pair DCF of the GFA is easily obtained from (2.9) and (3.4). It can be expressed

as

−c(2)(r1; r2) = DR

∫1

1− n3(r; R)!(3)(r− r1)!(3)(r− r2) dr : (3.7)

In the uniform limit, where �(r)→ �=constant, n3 =(4=3)�R3� is the packing fractionand does not depend on r so that (3.7) reduces to

−c(2)(r) = DR

(!(3) ⊗ !(3)

1− n3

): (3.8)


Considering that DR is a linear di2erential operator we obtain

−c(2)(r) = 11− n3

DR(!(3) ⊗ !(3))+ !(3) ⊗ !(3)DR

(1

1− n3

)+ cross terms ;

(3.9)

where the cross terms only depend on products of the 5rst (or second) derivative of1=(1− n3) w.r.t. R times the 5rst (or second) derivative of !(3) ⊗ !(3). From (3.9) itis easy to realize that the term DR(!(3) ⊗!(3))=(1− n3) gives the leading contributionto the pair DCF at low densities. In general DR(!(3) ⊗ !(3)) di2ers from the pairexclusion function and consequently (3.8) does not exhibit the correct low densitybehavior. Moreover, the third derivative w.r.t. R of the convolution !(3) ⊗ !(3) yieldsa Dirac �-function at r = 2R = ", as mentioned above. This allows us to identify thecondition

(NO DELTA) a3 = 0 ;

as the one that leads to a pair DCF without the unphysical �-term.More information can be obtained from the GFA pair DCF of the homogeneous

hard-sphere -uid (3.8) if one considers its Fourier transform

−c(2)(k) = DR

((!(3)(k; R))2

1− n3

); (3.10)

because the compressibility equation of state is derived from

−c(2)(k = 0) = DR

((V3(R))2

1− n3

); (3.11)

where V3(R)=!(3)(k=0)=(4=3)�R3 is the volume of a sphere of radius R. A densityexpansion of (3.11) allows us to write

−c(2)(k = 0) = DR((V3(R))2) + �DR((V3(R))3) + �2DR((V3(R))4) + · · · ;(3.12)

where we have used the identity n3 = �V3(R) which applies in the uniform limit. Thenext step is to identify the coe1cients of expansion (3.12) with those coming fromthe virial expansion of the equation of state. In particular, the exact second and thirdvirial coe1cients lead to the conditions

DR((V3(R))2) = 8V3(R) (3.13)

and

DR((V3(R))3) = 30(V3(R))2 ; (3.14)

respectively. Considering the general expression for DR (3.1), conditions (3.13) and(3.14) require

(2nd VIR) �(a1 + 5a2 + 20a3) = 1 ;

(3rd VIR) 4�(a1 + 8a2 + 56a3) = 10 :


A further condition can be obtained from the 0D limit in which one considers avery inhomogeneous situation with cavities that cannot hold more than one particle[9–15]. In the strict 0D limit the density is �(r)=$�(r) where the 0D packing fraction $represents the average occupation of the cavity (06 $6 1). In this case n3 = $�(R−r)and one obtains [8]

�Fex($) = (1 + G($))DR(V3(R)) : (3.15)

Taking into account that the exact excess free-energy in this situation is given by[9–15]

�F0Dex ($) = $+ (1− $) log(1− $) = 1 + G($) ; (3.16)

one recovers the exact 0D limit provided that

DR(V3(R)) = 1 ; (3.17)

which from (3.1) leads to the condition

(0D) 4�(a1 + 2a2 + 2a3) = 1 :

Considering relation (3.16) one can express the GFA excess free-energy density in theform

�= DR(�F0Dex (n3)) ; (3.18)

which shows the dimensional crossover structure of the GFA. One can check that theset (3.2) that originates the PY 5tted excess free-energy density �PY ful5lls condition(0D) and thus, in the strict 0D limit, �PY yields the exact result (3.16). It can beshown [8] that its 1D limit does not have nonintegrable singularities as it happenedin the original theory of Rosenfeld, and the result of its 2D limit is close to thatof Rosenfeld. Furthermore, the GFA can be extended to one and two-dimensions bysimply replacing n3 in G (or in F0D

ex ) by the corresponding one or two-dimensionalcharacteristic weighted density. Also, the operator DR should be modi5ed accordingly.In one dimension one obtains the exact free-energy functional for the one-dimensionalinhomogeneous hard-rod -uid. In two dimensions a 5t to the SPT free-energy densityyields a functional that is completely equivalent [8] to results of the theory of Rosenfeldfor the inhomogeneous hard-disk -uid [21].

4. Free-energy functionals

In the GFA the excess free-energy density for the inhomogeneous hard-sphere -uid� (3.4) depends on three coe1cients ai. In previous work [8] two di2erent sets of coef-5cients were obtained from 5tting to known results in the uniform limit. In the presentpaper we propose a more -exible alternative method for obtaining GFA functionals.The method consists in using the conditions derived in Section 3 for calculating thecoe1cients ai. Since the four conditions (NO DELTA, 2nd VIR, 3rd VIR, and 0D) are


linearly independent and they involve only three unknowns ai, only three of them canbe considered at the same time. In what follows we present the results for some of thepossible systems of equations that one can formulate. In all cases, we have consideredthe (2nd VIR) condition in order to obtain a reasonable equation of state.The solution to the system (NO DELTA, 2nd VIR, 3rd VIR) is

a1 =− 32�

; a2 =12�

; a3 = 0 : (4.1)

Substitution of the set (4.1) into the general expression for � (3.4) leads to

�VIR =−(− 32�R2 n

′3 +

12�R

n′′3

)log(1− n3) +

(n′3)2

2�R(1− n3); (4.2)

where the label (VIR) is due to the striking property of �VIR in (4.2) that in the uniformlimit it reduces to the excess free-energy density of a hard-sphere -uid with the PYvirial equation of state [22]. Because we did not use the (independent) condition (0D),it is not ful5lled by (4.1). As a matter of fact, using the set (4.1) we obtain DR(V3(R))=−2. Indeed, this indicates that �VIR does not achieve the correct dimensional crossoverto 0D but it also leads to a more serious problem since the pair DCF obtained fromfunctional di2erentiation of �VIR in the uniform limit becomes positive at low densities.This is easily seen if one considers the relation [8]

−c(2)(r = 0) + �c(2)(k = 0) = DR(V3(R)) ; (4.3)

which is valid for the GFA pair DCF in the uniform limit.A di2erent set of coe1cients can be derived from the conditions (NO DELTA, 2nd

VIR, 0D). We obtain

a1 =− 14�

; a2 =14�

; a3 = 0 ; (4.4)

which substituted into (3.4) yields

�1+2 =−(− 14�R2 n

′3 +

14�R

n′′3

)log(1− n3) +

(n′3)2

4�R(1− n3): (4.5)

The label (1 + 2) in (4.5) comes from the fact that �1+2 gives rise to a free-energyfunctional that is completely equivalent to the functional originated from the sum ofthe contributions �1 and �2 of the Rosenfeld theory (2.3). This is readily shown byconsidering the di2erence∫

dr (�1 + �2 − �1+2)

=∫

dr(−(n0 +

14�R2 n2 −

14�R

n′2

)log(1− n3)− n1 · n2

1− n3

); (4.6)

in which the identities n2 = n′3 and n1 = n′2=4�R have been used. Also, since [20]∫dr

n1 · n21− n3

=−∫

dr∇ · n1 log(1− n3) ; (4.7)


we obtain∫dr (�1 + �2 − �1+2) =

∫dr

(∇ · n1 − n0 − 1

4�R2 n2 +1

4�Rn′2

)log(1− n3) :

(4.8)

Furthermore, from (2.4)–(2.8) and using n1 =−∇n3=4�R one has

∇ · n1 − n0 − 14�R2 n2 +

14�R

n′2

=�(r)⊗[− (∇ · ∇)!(3)(r; R)

4�R− 2

!(2)(r; R)4�R2 +

!(2)′(r; R)4�R

]; (4.9)

where the dependence of !(2) and !(3) on R has been made explicit. Considering theidentity (∇ · ∇)!(3)(r; R) = !(2)′(r; R)− 2!(2)(r; R)=r, we obtain

∇ · n1 − n0 − 14�R2 n2 +

14�R

n′2 = �(r)⊗[!(2)(r; R)2�Rr

− !(2)(r; R)2�R2

]= 0 ;

(4.10)

where the last equality is easily established by Fourier transforming the convolu-tion product and verifying that the second term of the product vanishes. From (4.8)and (4.10) we demonstrate the equivalence between the functionals �

∫dr�1+2 and

�∫dr(�1 + �2).

If one considers the conditions (0D, 2nd VIR, 3rd VIR) the set (3.2) is obtained.Of course this is not an unexpected result since this set yields exactly the PY com-pressibility equation of state and the correct crossover to 0D. What about the CS 5t(3.3)? In this case, one needs to consider the (4th virial) �2 term in the expansion(3.12) which for the CS equation of state leads to

DR((V3(R))4) = 72(V3(R))3 (4.11)

or equivalently to

4�(a1 + 11a2 + 110a3) = 18 : (4.12)

Condition (4.12) with (2nd VIR) and (3rd VIR) gives the CS set (3.3) as a result.This procedure could also have been used in the PY compressibility case, substitutingthe (0D) condition by

DR((V3(R))4) = 76(V3(R))3 ; (4.13)

i.e.,

4�(a1 + 11a2 + 110a3) = 189 : (4.14)

So far we have obtained di2erent free-energy functionals for the inhomogeneoushard-sphere -uid. How do they fare in an inhomogeneous situation? To give an an-swer to this question we have investigated the density pro5le of a one-componenthard-sphere -uid in contact with a planar hard wall. This is a canonical test for anytheory of inhomogeneous -uids. Details of the calculation are given in Ref. [8] where


Fig. 1. Density pro5les for a hard-sphere -uid near a planar hard wall; (a) bulk density �B = 0:2 and(b) �B = 0:9.

the results of the GFA with �PY were compared with the accurate results of thetheory of Rosenfeld (ROS). Here we also present the results of �CS , �VIR, and �1+2.In Fig. 1a we plot the density pro5les obtained from the di2erent theories at low pack-ing (�B = 0:2). Perhaps the best result at low densities is obtained from �1+2 becauseit ful5lls both the (NO DELTA) and the (0D) conditions. For this functional, the maindi2erences with ROS arise at contact and are due to its failure to reproduce the ex-act 3rd virial coe1cient of the equation of state. �PY and �CS perform almost equal,with a similar kink at z = ", and a better value at contact for �CS due to the contact


sum rule �P = �(contact). 1 The pro5le obtained from �VIR shows a poor agreementwith the accurate results of Rosenfeld due to the above-mentioned poor behavior ofits pair DCF at low densities. At high densities (Fig. 1b, �B = 0:9) things are ratherdi2erent. The pro5le �1+2 is dominated by its bad value at contact and the term �3,absent in �1+2, is clearly needed. The overall best agreement with the pro5le of theFMT is achieved by �VIR with a satisfactory value at contact (given by the PY virialequation of state) and a signi5cantly better behavior than �PY or �CS , specially nearthe 5rst peak where the absence of a kink in �VIR (due to the (NO DELTA) con-dition) is determinant. We 5nally note that given the good performance of �VIR athigh densities it could be tempting to consider a “Carnahan–Starling” functional inthe form (2=3)�ROS + (1=3)�VIR. The results are discouraging. Using this functionalin the present application of adsorption at a wall one obtains pro5les almost equalto (2=3)�ROS(r) + (1=3)�VIR(r), with the accurate CS value at contact but inferior to�ROS(r) for z¿ 0.

5. Direct correlation functions and structure factors for hard-spheres

As argued in Section 3, the GFA pair DCF of the homogeneous hard-sphere -uidat low densities behaves as DR(!(3) ⊗ !(3)). In general this result does not match theexact 0th order term in the diagrammatic expansion of c(2)(r), given by the Mayerfunction of the system. Di2erent sets of coe1cients thus lead to di2erent (wrong)low-density behaviors. In particular, the CS and PY sets yield a low-density pair DCFthat includes a delta function contribution while the VIR set leads to a positive DCFat low densities. The (1+2) set yields the correct low-density behavior as one wouldexpect from considering that (i) �

∫dr�1+2 and �

∫dr(�1 + �2) are equivalent, and

(ii) �1 + �2 yields the leading contribution of �ROS at low-density.In Fig. 2a we plot the pair DCF’s of the di2erent theories at low density (�B =0:2)

comparing with Monte Carlo data [23]. Since the Rosenfeld’s DCF coincides with theaccurate PY pair DCF, it agrees very well with simulation while the other theoriesbehave as commented above. We note that the di2erence between the GFA(1+2)pair DCF and simulation comes from 3rd virial contributions that become increasinglyimportant as the density grows, as one can observe in Fig. 2b where we plot thepair DCF’s at �B = 0:9. By comparing Figs. 2a and b one can see how the relativedi2erences between the GFA(PY, CS, and VIR) pair DCF’s decrease as density growsand are mainly due to the di2erent isothermal compressibilities (T =1=(1−�c(2)(k=0))and to the delta terms in the GFA(PY and CS) cases.An important point in the present work is that we have obtained a pair DCF (3.8)

with (3.3) that yields the CS compressibility equation of state although it has a poorlow density behavior and a delta function at r = ". Both problems can be solvedby using an idea previously applied for hard disks [10]. The procedure consists in

1 This sum rule applies for a -uid near a planar hard wall and is satis5ed by any non local DFT.


Fig. 2. Pair direct correlation function c(2)(r) of the homogeneous hard-sphere -uid; (a) bulk density �B=0:2and (b) �B = 0:9. The results of the GFA are the unmodi5ed ones (see text).

Fig. 3. Pair direct correlation function c(2)(r) of the homogeneous hard-sphere -uid. The solid lines showthe PY DCF, the dashed lines represent the modi5ed GFA(CS) pair DCF and the symbols are simulationdata (see text).

replacing DR(!(3) ⊗ !(3)) in (3.9) by the pair exclusion �(" − r). This leads to theappropriate low density behavior and to the suppression of the unphysical delta termin the DCF. The results are very good as one can see in Fig. 3 where we present themodi5ed GFA(CS) pair DCF at three densities (�B=0:2; 0:6, and 0:9). For the sake ofcomparison, in Fig. 3 we also present the PY pair DCF and the Monte Carlo resultsof Ref. [23]. We note that, although substitution of DR(!(3) ⊗ !(3)) by �(" − r) was


Fig. 4. Same as Fig. 2 but now we present the modi5ed results of the GFA (see text).

done in order to improve the low-density behavior of the DCF, the best performanceis obtained at high density where the agreement with simulation is remarkable (ofcourse our theory cannot reproduce the positive tail in Monte Carlo data that appearbeyond the range of the pair exclusion function). We also note, however, that thePY pair DCF achieves better agreement with simulation at low densities. It would beappealing if on the basis of this new DCF one could rebuild a FMT-like free-energyfunctional that yield this DCF. However, this is not possible since the obtained DCFdoes not ful5lls the Schwartz identity for second derivatives and thus there is not anyfunctional whose second functional derivative yields this DCF as a result. In any case,the availability of a new analytical (both in real and in Fourier spaces) pair DCF forthe homogeneous hard-sphere -uid can be relevant for those density functional theoriesthat require the structure of the -uid as an input, as it happens, e.g., for the smootheddensity approximation of Tarazona [24].The same scheme can be applied to the other GFA pair DCF’s. In Fig. 4 we plot the

modi5ed DCF’s. In all cases (except for (1+2) in which the DCF remains unchanged)there is a signi5cant improvement with respect to the DCF’s plotted in Fig. 2. It isworth to mention that applying the present scheme to the GFA(PY) pair DCF leadsexactly to the PY pair DCF. Furthermore, a close inspection of the GFA �PY allowsone to identify the modi5cations that should be done in �PY in order to yield the PYpair DCF. These modi5cations transform �PY into �K–R and therefore we obtain an


Fig. 5. The structure factor S(k) of the homogeneous hard-sphere -uid; (a) bulk density �B = 0:2 and(b) �B = 0:9. The solid lines show the PY structure factor, the dashed lines are the results of the modi5edGFA(CS) and the dots are the results of the unmodi5ed one (see text).

alternative derivation of the scalar version of the FMT of Rosenfeld. We note that thecondition (2nd VIR) [or, more speci5cally, (3.13)] is equivalent to consider that theintegral over DR(!(3) ⊗ !(3)) is equal to the volume enclosed by the pair exclusion�(" − r) and thus the present modi5cation preserves the compressibility equation ofstate in all the sets of coe1cients considered. On the other hand, DR(!(3) ⊗!(3))(r =0)=DRV3(R) but �(")=1 and therefore the value taken by the DCF at r=0 is shifted(1−DRV3(R))=(1−n3) as one can observe by comparing Fig. 4 with Fig. 2 (Note thatthe (0D) condition of �1+2 and of �PY implies no displacement in these cases).

The structure factor S(k)=1=(1−�c(2)(k)) obtained from both the modi5ed and theunmodi5ed GFA(CS) pair DCF’s is compared in Fig. 5 with the accurate PY result.As it can be observed, the unmodi5ed theory yields a poor structure factor (as it alsohappens for the unmodi5ed GFA(PY), see Fig. 2 in [8]); a dephasing occurs and the5rst peak is grossly underestimated at high densities. The modi5ed theory yields amuch better structure factor for all densities, no noticeable dephasing occurs and the5rst peak is described more adequately.Finally, in Fig. 6 we present the radial distribution function (RDF) g(r) obtained

from the (modi5ed) GFA(CS) structure factor by inverting the well known identity


Fig. 6. Radial distribution functions g(r) of the homogeneous hard-sphere -uid obtained from the structurefactor; (a) bulk density �B = 0:2 and (b) �B = 0:9. The solid lines show the PY g(r), the dashed lines arethe results of the modi5ed GFA(CS) and the symbols represent simulation data (see text).

S(k) = 1 + �∫dr eik·r[g(r) − 1]). This function is compared with the PY RDF and

our results for g(r) from Monte Carlo simulation. The most relevant feature of themodi5ed GFA(CS) RDF is its failing to ful5lls the core condition g(r ¡") = 0. Thisis due to the fact that the corresponding DCF vanishes o2 the Mayer function range.Since the only theory that simultaneously veri5es g(r ¡") = 0 and c(2)(r ¿") = 0 isthat of PY we 5nd that the modi5ed GFA(CS) RDF should violate the core condition.Outside the exclusion zone the agreement with simulation is fairly good even at highpackings. The slight dephasing and less pronounced oscillations are probably a sideconsequence of the non-vanishing core.

6. Summary

We have presented a new formulation of the generating function approach densityfunctional theory in terms of quite general conditions instead of using a 5t to theuniform -uid free-energy density as we did in previous work. The -exibility of thepresent approach allowed us to obtain new GFA functionals. In particular, a functionalequivalent to one previously considered [9–15] in the context of dimensional crossoverin the FMT was obtained from considering the conditions that arise from a correct 0D


crossover, the 2nd virial coe1cient, and a DCF without an unphysical delta functionterm at r=". If instead of the 0D condition one imposes the exact 3rd virial coe1cientone obtains a functional whose uniform limits yields the PY virial equation of state.Finally, considering 0D, 2nd virial, and 3rd virial one recovers the PY 5t of the originalderivation of the GFA; in the same line, 2nd virial, 3rd virial, and CS 4th virial allowsus to recover the CS 5t.The performance of the new functionals in the prototypical problem of adsorption

of hard spheres at a planar hard wall depends on the density regime considered. Atlow densities the best results are obtained from �1+2 whereas at high densities �VIR

fares much better, without the kink that arises at z = " in the density pro5le obtainedfrom the PY or the CS 5ts of the earlier GFA. The failing of �VIR at low densities(and of the other functionals in the same or lesser degree) has driven us to consider amodi5cation in the GFA pair DCF of the uniform hard-sphere -uid in order to achievethe correct low-density behavior given by the pair exclusion function of the system.With this modi5cation we have obtained a GFA(PY) pair DCF that is identical to thatof PY. In the CS case the DCF derived following this scheme exhibits an excellentagreement with simulation at high densities. At low densities the worse agreement canbe ascribed to the missing positive tail in the proposed DCF.The modi5cation of the DCF that we have considered also represents a signi5cant

improvement in the structure factor which leads to a good radial distribution functionof the homogeneous hard-sphere -uid. In this RDF the violation of the core conditiong(r ¡") = 0 is also traced back to the lack in the DCF of the positive tail that isobserved in simulations.

Acknowledgements

We thank S. Velasco and R. Evans for reading and commenting on the manuscript.This work was supported by the Comisi(on Interministerial de Ciencia y Tecnolog(Va ofSpain under Grant PB 98-0261 and by Junta de Castilla y L(eon and FSE under grantSA097/01.

References

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Generating function density functional theory: free-energy functionals and direct correlation functions for hard-spheres