Variational Principle of Classical Density Functional Theory via Levy's Constrained Search Method
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r Variational Principle of Classical Density Functional Theory via Levy’s ConstrainedSearch Method
Wipsar Sunu Brams Dwandaru
H. H. Physics Laboratory, University of Bristol, Royal Fort, Tyndall Avenue, Bristol, BS8 1TL, UK andJurusan Fisika, Universitas Negeri Yogyakarta, Bulaksumur, Yogyakarta, Indonesia
Matthias Schmidt
H. H. Physics Laboratory, University of Bristol, Royal Fort, Tyndall Avenue, Bristol, BS8 1TL, UK andTheoretische Physik II, Physikalisches Institut, Universit¨at Bayreuth, Universit¨atsstraße, D-95440 Bayreuth, Germany (Dated: 15 April 2011, to appear in Phys. Rev. E)We show that classical density functional theory can be based on the constrained search method[M. Levy,
Proc. Natl. Acad. Sci. , 6062 (1979)]. From the Gibbs inequality one first derives avariational principle for the grand potential as a functional of a trial many-body distribution. Thisfunctional is minimized in two stages. The first step consists of a constrained search of all many-body distributions that generate a given one-body density. The result can be split into internaland external contributions to the total grand potential. In contrast to the original approach byMermin and Evans, here the intrinsic Helmholtz free energy functional is defined by an explicitexpression that does not refer to an external potential in order to generate the given one-bodydensity. The second step consists of minimizing with respect to the one-body density. We show thatthis framework can be applied in a straightforward way to the canonical ensemble. Keywords:
I. INTRODUCTION
The variational principle of density functional theory(DFT) was originally formulated for ground-state prop-erties of quantum systems by Hohenberg and Kohn in1964 [1]. The extension to non-zero temperatures wasperformed by Mermin in the following year [2], here stillformulated for quantum systems. The application to thestatistical mechanics of classical systems, i.e. the devel-opment of classical DFT, was initiated about a decadelater through the work of Ebner, Saam, and Stroud [3–5]. The generality of the framework was fully realized byEvans [6]. His 1979 article continues to be the standardreference on the subject; there are more recent review[6–9] and textbook [10] presentations.The Hohenberg-Kohn theorem applies to one-particledensity distributions that correspond to a particular ex-ternal (one-body) potential v in the Hamiltonian, inwhich the kinetic and internal interaction are those of thetrue system [1, 2]. One refers to v -representability of theone-particle density, i.e. the condition that a one-particledensity is generated by some external potential v . How-ever, it was realized, already in the original Hohenberg-Kohn paper, that v -representability is not guaranteed foran arbitrarily chosen density ρ [1, 11, 12]. One arguesthat this does not pose any problems in the practical ap-plications of DFT to quantum systems [13]. In the devel-opment of the theory, it turned out that there indeed ex-ist non- v -representable densities, i.e. one-body densitiesthat are not associated to any ground-state wave func-tion [14]. The original Hohenberg-Kohn theorem doesnot apply to these.In 1979 Levy introduced an alternative foundation ofDFT for quantum systems, based on a constrained, two- stage search [11]. Here a weaker condition, known as N -representability, is used, where the density distributionmay be directly obtained from some anti-symmetric N -body wave function, although an external potential thatgenerates this wave function need not exist [11, 14, 15].One defines an exchange-correlation functional that de-mands searching all wave functions that return the fixed(trial) one-body density. The latter need not be v -representable. Subsequently, a method similar to Levy’swas proposed by Lieb [16], called the generalized Legen-dre transform [12]. Instead of searching all wave func-tions, the functional searches all possible external poten-tials that correspond to a fixed density. Kohn adoptedthe constrained search for his Nobel lecture [13], and itis viewed as an important theoretical contribution to thefoundation of DFT for electronic structure. Practical ap-plications of constrained search functionals are of ongo-ing research interest, see e.g. [12, 17]. Levy gives a briefhistoric account of the development of his ideas in Ref.[18].Given the significance of Levy’s and Lieb’s methodsfor electronic structure, it is somewhat surprising thatthere are very few studies that point to the use of these inclassical systems. One example is the work by Weeks [19],where the v -representability of the one-body density insome finite region of space is investigated through theGibbs inequality. Although Weeks cites Levy’s originalpaper [11], and makes a remark that his formulation isin spirit similar to that of Levy, it seems that his methodis related more to Lieb’s generalized Legendre transformmethod. Earlier work has been carried out in order toinvestigate the existence of an external potential that isassociated to a given equilibrium one-body density [20].It is concluded that there is such an external potentialthat produces any given density for any (classical) systemwithout hard core interaction. Although one might guessfrom general arguments that the constrained search canbe applied to classical DFT, to the best of our knowledge,this procedure has not been spelled out explicitly in theliterature.In the present article we show how to formulate thevariational principle of classical DFT based on Levy’sconstrained search method. This alternative can providefurther insights into the foundation of classical DFT. Inparticular the intrinsic free energy functional is definedhere without implicit reference to an external potential v . A more relaxed condition for ρ , similar to that of N -representability, is imposed. Here, the one-body densityis only required to be obtained from an arbitrarily chosenmany-body probability distribution f . We refer to thiscondition as f -representability of a given ρ . While dis-tinguishing between the different type of representabilityin practical DFT calculations seems unnecessary, we findthe discrimination very useful for conceptual purposesand hence point out throughout the manuscript whichof the conventions is followed in the reasoning. We alsoshow that Levy’s method can be applied in a straight-forward way to the canonical ensemble. There is consid-erable current interest in the theoretical description ofthe behaviour of small systems, where the grand and thecanonical ensembles are inequivalent in general, and thelater might model certain (experimental) realizations ofstrongly confined systems more closely. Several relativelyrecent contributions address the problem of formulatingDFT in the canonical ensemble [21–26]. The authors ofthese papers consider the important problem of how toobtain DFT approximations that make computations inthe canonical ensemble feasible. Our present article hasa much lower goal: We are only concerned with formu-lating the variational principle in an alternative way.This article is organized as follows. We start by defin-ing the grand potential as a functional of the many-bodyprobability distribution in Sec. II. This is a necessarystep and our presentation follows [6–8, 10]. In Sec. IIIwe give a brief overview of the standard proof of DFT, ex-pressing the free energy as a functional of the one-bodydensity based on a one-to-one correspondence betweenthe one-body density and the external potential. Thefull derivation is widely known and can be found in nu-merous references [6–8, 10, 27]. We proceed, in Sec. IV,by formulating the intrinsic free energy functional via theconstrained search method; our presentation is similar toLevy’s original work [11]. Our central result is the def-inition (22) of the intrinsic free energy functional, with-out reference to an external potential. We summarizethe essence of Levy’s argument as a double minimization[18] in Sec. V. In Sec. VI we apply this to the canonicalensemble and we conclude in Sec. VII. II. GRAND POTENTIAL FUNCTIONAL OFTHE MANY-BODY DISTRIBUTION
In the grand canonical ensemble of a system of classicalparticles, the equilibrium probability distribution for N particles at temperature T is assumed to exist and to begiven by f = Ξ − exp ( − β ( H N − µN )) , (1)where H N is the Hamiltonian of N particles, µ is thechemical potential, and β = 1 / ( k B T ), with k B being theBoltzmann constant. The normalization constant is thegrand canonical partition sumΞ = Tr cl exp ( − β ( H N − µN )) , (2)where Tr cl represents the classical trace, i.e. the sum overtotal particle number and integral over all degrees of free-domTr cl = ∞ X N =0 h N N ! Z d r . . . d r N Z d p . . . d p N , (3)where h is the Planck constant, r , . . . , r N are the po-sition coordinates and p , . . . , p N are the momenta ofparticles 1 , . . . , N .One introduces the grand potential as a functional ofthe many-body probability distribution,Ω[ f ] = Tr cl f (cid:0) H N − µN + β − ln f (cid:1) , (4)where f is a variable trial probability distribution thatsatisfies the normalization conditionTr cl f = 1 . (5)Note that f as an argument of functional (4) can be quitegeneral and need not be linked to an external potentialat this stage. Inserting the equilibrium probability dis-tribution (1) into (4) one obtainsΩ[ f ] = Tr cl f (cid:0) H N − µN + β − ln f (cid:1) = Tr cl f (cid:0) H N − µN + β − [ − ln Ξ − β ( H N − µN )] (cid:1) = − β − ln Ξ ≡ Ω , (6)where Ω is the equilibrium grand potential. An im-portant property of functional (4) is that it satisfies thevariational principleΩ[ f ] > Ω[ f ] , f = f , (7)which may be proven using the Gibbs-Bogoliubov rela-tion as follows. First, from (1) and (6), Ω[ f ] of Eq. (4)can be written asΩ[ f ] = Ω[ f ] + β − Tr cl f ln (cid:18) ff (cid:19) . (8)According to the Gibbs inequality [6, 10], f ln (cid:18) f f (cid:19) < f (cid:18) f f − (cid:19) , (9)and hence Tr cl f ln (cid:18) ff (cid:19) > Tr cl ( f − f ) . (10)Since f and f are normalized, i.e. satisfy (5), the RHSof the inequality above vanishes, and β − Tr cl f ln (cid:18) ff (cid:19) > . (11)Thus the second term on the RHS of (8) is positive, andthe inequality (7) follows.For classical particles the Hamiltonian may be re-stricted to the form H N = N X i =1 p i m + U ( r , . . . , r N ) + N X i =1 v ( r i ) , (12)where the first term is the total kinetic energy, with thesquared momentum p i = p i · p i of the i -th particle, U isthe interatomic potential between the particles, v is an(arbitrary) external one-body potential and m is the par-ticle mass. The equilibrium one-body density at position r is given as a configurational average ρ ( r ) = Tr cl f ˆ ρ ( r ) , (13)where the density operator for N particles is defined asˆ ρ ( r ) = N X i =1 δ ( r − r i ) . (14)The functional form (4) was originally introduced byMermin [2] for (finite temperature) quantum systems,where the grand potential is a functional of a (trial) den-sity matrix. The variational principle (7) will be used inSecs. III and IV below, where we present two alternativederivations of the intrinsic free energy as a functional ofthe one-body density. III. MERMIN-EVANS DERIVATION OF THEFREE ENERGY FUNCTIONAL
Evans gave a formal proof that the intrinsic free en-ergy of a system of classical particles is a functional ofthe one-body density [7]. Here we briefly lay out his chainof arguments. The many-body distribution f as given in(1) is a functional of the external potential v through Eq.(12), and therefore ρ is a functional of v via (13). This,in principle, requires solution of the many-body problemand the dependence is in accordance with physical intu-ition, i.e. it is the action of v that generates the shape ofthe density profiles ρ .However, the more useful result that can be de-duced [7], is that once the interatomic interaction po-tential U is given, f is a functional of ρ . The proofof this statement rests on reductio ad absurdum [7, 10], where for a given interaction potential U , v is uniquelydetermined by ρ . The resultant v then determines f via (1) and (12). Hence, f is a functional of ρ .An important consequence in this reasoning is that forgiven interaction potential U , F [ ρ ] = Tr cl f N X i =1 p i m + U + β − ln f ! (15)is a unique functional of the (trial) one-body density ρ .Here, the dependence of f on the external potential, v ,is now only implicit through the one-body density, ρ . Wewill comment on this sequence of dependences in the con-clusions, after having laid out Levy’s alternative methodto define a free energy functional in Sec. IV. Further-more, using a Legendre transform, the grand potentialfunctional is obtained for a given external potential asΩ v [ ρ ] = F [ ρ ] + Z d r ( v ( r ) − µ ) ρ ( r ) . (16)The functional Ω v [ ρ ] returns its minimum value if ρ = ρ ,i.e. if the trial density is the true equilibrium one-bodydensity of the system under the influence of v . The valueis the grand potential Ω . The existence of the min-imum value of Ω v [ ρ ] may be proven by considering an-other equilibrium density ρ ′ associated with a probabilitydistribution f ′ of unit trace, such thatΩ[ f ′ ] = Tr cl f ′ (cid:0) H N − µN + β − ln f ′ (cid:1) = F [ ρ ′ ] + Z d r ( v ( r ) − µ ) ρ ′ ( r ) (17)= Ω v [ ρ ′ ] , where F [ ρ ′ ] = Tr cl f ′ N X i =1 p i m + U + β − ln f ′ ! . (18)However, it is known from Eq. (7) that Ω[ f ′ ] > Ω[ f ], for f ′ = f , thus it follows thatΩ v [ ρ ′ ] > Ω v [ ρ ] . (19)In other words, the correct equilibrium one-body density, ρ , minimizes Ω v [ ρ ] over all density functions that can beassociated with a potential v .This important result may be stated as a functionalderivative δ Ω v [ ρ ] δρ ( r ) (cid:12)(cid:12)(cid:12)(cid:12) ρ = 0 , (20)and Ω v [ ρ ] = Ω . (21)To conclude, the formal argument for the definition(15) of the intrinsic free energy functional, F [ ρ ], is basedon v -representability of the one-body density. A v -representable ρ is one which is associated with a proba-bility distribution, f , of the given Hamiltonian H N withexternal potential v [11, 13]. This condition was origi-nally introduced for quantum systems, and is implicit inthe current approach. It is used to prove the chain of de-pendency outlined above (15), and confirmed for a largeclass of (classical) systems in [20]. IV. FREE ENERGY FUNCTIONAL VIA LEVY’SCONSTRAINED SEARCH METHOD
Here we show how one may alternatively define a freeenergy functional via Levy’s method. This is basedon the weaker condition of f -representability, wheretrial density fields ρ need not necessarily be associatedwith some external potential. We define the intrinsicHelmholtz free energy functional as F L [ ρ ] = min f → ρ " Tr cl f N X i =1 p i m + U + β − ln f ! , (22)where the minimization searches all probability distribu-tions f , that are normalized according to (5), and thatyield the fixed trial one-body density ρ via ρ ( r ) = Tr cl f ˆ ρ ( r ) . (23)The notation f → ρ in (22) indicates the relationship(23). Note that i) in general there will be many differentforms of f that yield the same ρ , and ii) no further con-ditions on f are imposed, apart from its normalization.In particular, the form of f need not be of Boltzmann-type containing the interaction potential U (as was thecase in Sec. III). Hence ρ need only be f -representable,but not necessarily v -representable. F L [ ρ ] returns a min-imum value by choosing the probability distribution thatminimizes the term in brackets in (22). Note that thefunctional form of this term is formally equivalent to (15)and that it is a sum of contributions due to kinetic en-ergy, internal interaction energy U , and (negative) en-tropy k B f ln f multiplied by T .The grand potential functional for a given external po-tential is thenΩ L [ ρ ] = F L [ ρ ] + Z d r ( v ( r ) − µ ) ρ ( r ) . (24)This functional possesses two important properties. i) Atthe equilibrium density it yields the equilibrium grandpotential Ω L [ ρ ] = Ω , (25)where ρ is given by (13) and Ω by (6). This value alsoconstitutes the minimum such thatΩ L [ ρ ] ≥ Ω . (26) In order to prove (25) and (26), we introduce additionalnotation. Let f ρ min be the probability distribution thatsatisfies the RHS of Eq. (22). Then it follows that F L [ ρ ] = Tr cl f ρ min N X i =1 p i m + U + β − ln f ρ min ! , (27)and for the case of the equilibrium density F L [ ρ ] = Tr cl f ρ min N X i =1 p i m + U + β − ln f ρ min ! . (28)First we proof the inequality (26). By its very defini-tion (24), the LHS of (26) may be rearranged into Z d r ( v ( r ) − µ ) ρ ( r ) + F L [ ρ ]= Z d r ( v ( r ) − µ ) ρ ( r )+Tr cl f ρ min N X i =1 p i m + U + β − ln f ρ min ! (29)= Tr cl f ρ min (cid:0) H N − µN + β − ln f ρ min (cid:1) . But according to the inequality (7),Tr cl f ρ min (cid:0) H N − µN + β − ln f ρ min (cid:1) ≥ Ω . (30)Thus, combining (29) and (30), the inequality (26) isrecovered. In order to prove (25), it is obvious from (7)that Tr cl f ρ min (cid:0) H N − µN + β − ln f ρ min (cid:1) ≥ Ω , (31)or, recalling (6),Tr cl f ρ min (cid:0) H N − µN + β − ln f ρ min (cid:1) ≥ Tr cl f (cid:0) H N − µN + β − ln f (cid:1) . (32)But f ρ min and f generate the same one-body density ρ ,hence from Z d r ( v ( r ) − µ ) ρ ( r )+Tr cl f ρ min N X i =1 p i m + U + β − ln f ρ min ! ≥ Z d r ( v ( r ) − µ ) ρ ( r )+Tr cl f N X i =1 p i m + U + β − ln f ! , (33)we obtainTr cl f ρ min N X i =1 p i m + U + β − ln f ρ min ! ≥ Tr cl f N X i =1 p i m + U + β − ln f ! . (34)However, by the very definition of f ρ min , the followinginequality should also hold:Tr cl f ρ min N X i =1 p i m + U + β − ln f ρ min ! ≤ Tr cl f N X i =1 p i m + U + β − ln f ! . (35)The above two inequalities hold simultaneously, if andonly if equality is attained,Tr cl f ρ min N X i =1 p i m + U + β − ln f ρ min ! = Tr cl f N X i =1 p i m + U + β − ln f ! . (36)Inserting (28) into (36) yields F L [ ρ ] = Tr cl f N X i =1 p i m + U + β − ln f ! . (37)Furthermore, asΩ = Tr cl f (cid:0) H N − µN + β − ln f (cid:1) = Z d r ( v ( r ) − µ ) ρ ( r )+Tr cl f N X i =1 p i m + U + β − ln f ! , (38)inserting (37) into (38) returns (25), which completes theproof. Equation (37) implies that if ρ is v -representable,then F L [ ρ ] = F [ ρ ]. Moreover, f = f ρ min means that f may be obtained directly from ρ even if v is un-known: find the probability distribution which yields ρ and which minimizes (22).Finally, the inequality (26) implies that the functionalderivative of the grand potential functional vanishes atequilibrium, δ Ω L [ ρ ] δρ ( r ) (cid:12)(cid:12)(cid:12)(cid:12) ρ = 0 . (39)For completeness we mention that it is convenient tosplit F L [ ρ ] into two terms, viz. the ideal and excess freeenergy functionals, F id [ ρ ] and F exc [ ρ ], respectively, suchthat F exc [ ρ ] ≡ F L [ ρ ] − F id [ ρ ] , (40)where the free energy of the ideal gas (with no interactionpotential present, U = 0) is given by F id [ ρ ] = β − Z d r ρ ( r ) (cid:0) ln (cid:0) λ ρ ( r ) (cid:1) − (cid:1) , (41) where λ = (cid:0) h β/ (2 mπ ) (cid:1) / . Thermodynamics enters byrealizing that F L [ ρ ] is the ‘intrinsic’ Helmholtz free en-ergy of the system, such that the total free energy is thesum of internal and external contributions, F L [ ρ ] + Z d r ρ ( r ) v ( r ) . (42) V. TWO-STAGE MINIMIZATION
The essence of the derivation presented in Sec. IV isa double minimization of the grand potential functional(4) of the many-body distribution. In the following, wespell this out more explicitly. From Sec. II we know thatΩ = min f Tr cl f (cid:0) H N − µN + β − ln f (cid:1) . (43)We decompose the RHS into a double minimizationΩ = min ρ min f → ρ Tr cl f (cid:0) H N − µN + β − ln f (cid:1) , (44)where the inner minimization is a search under the con-traint the f generates ρ (via relationship (23)). ForHamiltonians of the form (12) the above can be writtenas Ω = min ρ min f → ρ Tr cl f N X i =1 p i m + U + N X i =1 v ( r i ) − µN + β − ln f ! . (45)In the expression aboveTr cl f N X i =1 v ( r i ) − µN ! = Z d r ( v ( r ) − µ ) ρ ( r ) , (46)because f → ρ . So we may re-write (45) asΩ = min ρ (cid:26)Z d r ( v ( r ) − µ ) ρ ( r ) (47)+ min f → ρ Tr cl f N X i =1 p i m + U + β − ln f !) or Ω = min ρ (cid:26)Z d r ( v ( r ) − µ ) ρ ( r ) + F L [ ρ ] (cid:27) , (48)where F L [ ρ ] is given by (22). Clearly (48) is equivalentto (25) and (26). VI. DFT IN THE CANONICAL ENSEMBLE
One benefit of Levy’s method is that it allows straight-forward generalization to the canonical ensemble, as wedemonstrate in the following. In the canonical ensem-ble (i.e. for fixed number of particles, N ) the equilibriummany-body distribution functions is f N, = Z − exp( − βH N ) , (49)where the canonical partition sum is Z = Tr N exp( − βH N ) , (50)with the canonical traceTr N = 1 h N N ! Z d r . . . d r N Z d p . . . d p N . (51)In analogy to (4) we define the functional F [ f N ] = Tr N f N (cid:0) H N + β − ln f N (cid:1) , (52)where f N is an arbitrary N -body distribution that sat-isfies Tr N f N = 1. It is easy to show that the (total)Helmholtz free energy F = − β − ln Z is obtained byinserting the equilibrium distribution (49) into the func-tional (52), hence F = F [ f N, ] . (53)Reasoning based on the Gibbs-Bogoliubov inequality,completely analogous to the arguments presented inSec. II, yields F = min f N F [ f N ] . (54)We decompose this into a double minimization F = min ρ N min f N → ρ N F [ f N ] , (55)where the canonical one-body density distribution thatis generated by f N is ρ N ( r ) = Tr N f N ˆ ρ ( r ) , (56)with the density operator ˆ ρ ( r ) defined by (14). Clearlythe density defined in this way satisfies R d r ρ N ( r ) = N ,and there are no fluctuations in the total number of parti-cles. For Hamiltonians of the form (12), Eq. (55) becomes F = min ρ N min f N → ρ N Tr N f N N X i =1 p i m + U + N X i =1 v ( r i ) + β − ln f N ! . (57)In the above expressionTr N f N N X i =1 v ( r i ) = Z d r v ( r ) ρ N ( r ) , (58) because f N → ρ N via (56). Hence F = min ρ N (cid:26)Z d r v ( r ) ρ N ( r ) (59)+ min f N → ρ N Tr N f N N X i =1 p i m + U + β − ln f N !) , which we write as F = min ρ N (cid:26)Z d r v ( r ) ρ N ( r ) + F N [ ρ N ] (cid:27) , (60)where the intrinsic Helmholtz free energy functional inthe canonical ensemble is defined as F N [ ρ N ] = min f N → ρ N Tr N f N N X i =1 p i m + U + β − ln f N ! , (61)which is formally equivalent to the definition (22) of F L inthe grand ensemble upon identifying the different tracesand different types of many-body distributions. It isclear that the density distribution ρ N, that minimizesthe RHS of (60) is the true equilibrium distribution inthe canonical ensemble ρ N, ( r ) = Tr N f N, ˆ ρ ( r ) . (62)and that F = F N [ ρ N, ]. The variational principle (60)implies that δF N [ ρ N ] δρ N ( r ) (cid:12)(cid:12)(cid:12)(cid:12) ρ N, + v ( r ) = 0 , (63)where the derivative is taken under the constraint R d r ρ N ( r ) = N . VII. DISCUSSION AND CONCLUSION
The formulation of DFT rests on the existence anduniqueness of the intrinsic free energy as a functional ofthe one-body density for a given classical system. Wehave described two methods for defining this quantity,via Eq. (15) based on the Mermin-Evans argument [2,7], and via (22) based on Levy’s constrained search [11].Following the derivations presented in Secs. III and IV,it is clear that these methods are different in procedureand underlying principles.In the Mermin-Evans sequence of arguments it is for-mally proved that the equilibrium many-body probabil-ity distribution, f is a functional of the equilibrium one-body density, ρ . The existence of this functional rests ona sequence of functional dependencies. For given inter-atomic potential U and given one-body density ρ , thereis a unique external potential v , that generates this ρ .When input into the form of the many-body distributionin the grand ensemble (1), this uniquely determines f as used on the RHS of the definition (15) of the intrinsicfree energy functional F [ ρ ]. This chain of dependencyis implicit in order to properly define the free energyfunctional via (15). Note that the naive view that theequilibrium probability distribution, f , is a function ofthe external potential, v , such that (15) should also de-pend on v , gives the impression that functional (15) is notindependent of the external potential energy. Certainlythis is not the case–as one may recall the argument above(15).On the other hand, Levy’s method does not rely onthe above rather subtle argument. An appealing featureof the constrained search method is the definition (22) of F L [ ρ ]. Here the intrinsic free energy functional is explic-itly independent of the external potential, which is notas easily observed from F [ ρ ] of Eq. (15). Kohn [13] andLevy [18] describe the constrained search method as atwo step minimization procedure, and we have laid outanalogous reasoning in Sec. V.The underlying principle of the Mermin-Evans methodof defining the intrinsic free energy functional is v -representability of the trial density, whereas Levy’sfunctional is based on the weaker condition of f -representability. However, one may restrict the con-strained search to the class of one-body densities thatis v -representable. In this case Levy’s functional (22)becomes equal to the Mermin-Evans functional (15).Hence, the constrained search method reduces to find-ing the equilibrium one-body density which correspondto an (equilibrium) external potential, v , that minimizes functional (24) over all one-body densities, ρ , each associ-ated to a specific v . Furthermore, applying the Legendretransform upon functional (24) and minimizing a set ofexternal potentials which yields a fixed one-body density,gives Week’s free energy functional [19].In practice, minimizing Levy’s version of the free en-ergy functional (22) will certainly not be easier thansolving the many-body problem itself. Hence, whetherthe definition (22) helps to construct approximations forgrand-canonical free energy functionals remains an openquestion. For the case of the canonical ensemble we pointthe reader to the very significant body of work that hasbeen carried out to formulate a computational schemethat permits to capture the effects that arise due to theconstraint of fixed number of particles [21–26]. While thegeneralization to equilibrium mixtures is straightforward,we expect the application of Levy’s method to DFT forquenched-annealed mixtures [28–31] to constitute an in-teresting topic for future work. Acknowledgments
We thank R. Evans, A.J. Archer and P. Maass for use-ful discussions. WSBD acknowledges funding through anOverseas Research Student Scholarship of the Universityof Bristol. This work was supported by the EPSRC underGrant EP/E065619/1 and by the DFG via SFB840/A3. [1] P. Hohenberg and W. Kohn, Phys. Rev. , B 864(1964).[2] N. D. Mermin, Phys. Rev. , A1441 (1965).[3] C. Ebner, W. F. Saam, and D. Stroud, Phys. Rev. A ,2264 (1976).[4] W. F. Saam and C. Ebner, Phys. Rev. A , 2566 (1977).[5] C. Ebner and W. F. Saam, Phys. Rev. Lett. , 1486(1977).[6] R. Evans, Adv. Phys. , 143 (1979).[7] R. Evans, in Fundamentals of Inhomogeneous Fluids (ed . D . Henderson) (Dodrecht: Kluwer, New York, 1997),pp. 85–175.[8] R. Evans, in
Les Houches Session XLVIII: Liquids AtInterfaces (North-Holland, Amsterdam, 1988), pp. 30–66.[9] R. Roth, J. Phys.: Condens. Matter , 063102 (2010).[10] J.-P. Hansen and I. R. MacDonald, Theory of Simple Liq-uids, 3rd Edition (Academic Press (Elsevier), London,2006).[11] M. Levy, Proc. Natl. Acad. Sci. , 6062 (1979).[12] P. W. Ayers and M. Levy, J. Chem. Sci. , 507 (2005).[13] W. Kohn, Rev. Mod. Phys. , 1253 (1999).[14] M. Levy, Phys. Rev. A , 1200 (1982).[15] A. J. Coleman, Rev. Mod. Phys. , 668 (1963).[16] E. H. Lieb, Int. J. Quantum Chem. , 243 (1983).[17] S. M. Valone and M. Levy, Phys. Rev. A , 042501(2009). [18] M. Levy, Int. J. Quant. Chem. , 3140 (2010).[19] J. D. Weeks, J. Stat. Phys. , 1209 (2003).[20] J. T. Chayes, L. Chayes, and E. H. Lieb, Commun. Math.Phys. , 57 (1884).[21] J. A. White, A. Gonz´alez, F. L. Rom´an, and S. Velasco,Phys. Rev. Lett. , 1220 (2000).[22] J. A. White and S. Velasco, Phys. Rev. E , 4427 (2000).[23] J. A. Hernando, J. Phys.: Condensed Matter , 303(2002).[24] A. Gonz´alez, J. A. White, F. L. Rom´an, and S. Velasco,J. Chem. Phys. , 10634 (2004).[25] J. A. White and A. Gonz´alez, J. Phys.: Condensed Mat-ter , 11907 (2002).[26] J. A. Hernando and L. Blum, J. Phys.: Condensed Matter , L577 (2001).[27] J. S. Rowlinson and B. Widom, International Series ofMonographs on Chemistry: Molecular Theory of Capil-larity (Oxford University Press, Oxford, 1982).[28] M. Schmidt, Phys. Rev. E , 041108 (2002).[29] H. Reich and M. Schmidt, J. Stat. Phys. , 1683(2004).[30] M. Schmidt, E. Sch¨oll-Paschinger, J. K¨ofinger, and G.Kahl, J. Phys.: Condensed Matter , 12099 (2002).[31] L. Lafuente and J. A. Cuesta, Phys. Rev. E74