Treatments of the exchange energy in density-functional theory

Transcription

1 Int. J. Mod. Phys. B (008) 5 arx: Treatments of the echange energy n densty-functonal theory Tamás Gál Secton of Theoretcal Physcs, Insttute of uclear Research of the Hungaran Academy of Scences, 400 ebrecen, Hungary Abstract: Followng a recent work [Gál, Phys. Re. A 64, (00)], a smple deraton of the densty-functonal correcton of the Hartree-Fock equatons, the Hartree- Fock-Kohn-Sham equatons, s presented, completng an ntegrated ew of quantum mechancal theores, n whch the Kohn-Sham equatons, the Hartree-Fock-Kohn-Sham equatons and the ground-state Schrödnger equaton formally stem from a common ground: densty-functonal theory, through ts Euler equaton for the ground-state densty. Along smlar lnes, the Kohn-Sham formulaton of the Hartree-Fock approach s also consdered. Further, t s ponted out that the echange energy of densty-functonal theory bult from the Kohn-Sham orbtals can be gen by degree-two homogeneous -partcle densty functonals (=,,...), formng a sequence of degree-two homogeneous echange-energy densty functonals, the frst element of whch s mnus the classcal Coulomb-repulson energy functonal. Emal address: galt@phys.undeb.hu

2 I. Introducton ensty-functonal theory (FT) [,] forms the eact theoretcal background for partcle-densty based approaches to the quantum mechancal many-body problem, such as theores of the Thomas-Ferm knd [3-7] and of the Slater knd [8-0]. The man dea behnd FT s to replace the quantum mechancal wae-functon wth the partcle densty as basc arable from whch all propertes of a quantum system can be ganed, whch means the use of densty functonals nstead of wae-functon functonals. Ths dea s justfed for ground states formally by the Hohenberg-Kohn theorems [], but n practcal applcatons FT appears as an appromate theory because of the lack of knowledge of an eact epresson for the ground-state energy densty functonal E [ ρ] = F [ ρ] + ρ( r ) ( r ) dr, () the mnmum poston of whch ges the ground-state densty for a gen eternal potental ( r ) and partcle number = ρ dr, () yeldng the Euler equaton or δe[ ρ] = 0, δ ρ( r ) (3a) δf[ ρ] + r ( ) = µ (3b) δρ( r ) [wth µ beng the Lagrange multpler correspondng to the -conserng constrant on ρ( r ), Eq.()], for the determnaton of the ground-state densty. In Eq.(), F[ ρ ], characterstc to the nteracton between the (dentcal) Fermons of the consdered system, s a unersal functonal of the densty ρ( r ) [not dependng on the eternal potental, ( r ) ], whch represents the unknown n E [ ρ ]. To treat a major part of F[ ρ ] eactly, the Kohn-Sham (KS) method [] ntroduces sngle-partcle orbtals [ u ] nto FT, separatng n F the knetc energy

3 Ts = u ( ) u dr (4) = of a ground-state system of nonnteractng fermons wth the densty ρ( r ) of the gen nteractng fermon system, gettng a set of sngle-partcle equatons, KS u + u = ε u =,, (5) wth KS for the determnaton of the (ground-state) densty KS δ ( F[ ρ] Ts [ ρ]) = + r ( ), (6) δρ( r ) ρ = u u, (7) = nstead of usng Eq.(3) drectly. Wth the subtracton of the classcal Coulomb repulson from F[ ρ] [ ρ], T s J[ ρ] = ρ ρ( r ) drdr r r (8) F[ ρ] = T [ ρ] + J[ ρ] + E [ ρ], (9) s then only the relately small echange-correlaton (c) part of the energy remans to be appromated. The echange part of E c s usually defned by u uj uj( r ) u( r ) E = drdr. (0) r r, j= A further reducton of need for appromaton can be acheed by utlzng the defnton Eq.(0) of the echange energy by sngle-partcle orbtals, and settng up the sngle-partcle equatons uj uj( r ) u( r ) u dr + ( r ) u ( r ) = ε u ( r ) =,,, () r r j= wth the local (multplcate) potental c δ( F[ ρ] Ts [ ρ] E [ ρ]) δj[ ρ] δec [ ρ] = + r ( ) = + + r ( ) δρ( r ) δρ( r ) δρ( r ), () whch ges a densty-functonal correcton of the Hartree-Fock (HF) equatons, yeldng the Hartree-Fock-Kohn-Sham method [,3]. The solutons { u } = of the 3

4 dfferent sets of sngle-partcle equatons, Eqs.(5) and (), correspondng to the same ρ( r ), are dfferent. ote that though wth the approach a hgher degree of eactness of practcal calculatons can be reached, t means another step of backng away from a pure densty-functonal theory; n addton to the ntroducton of orbtals, now a nonlocal potental appearng. (The net step would be to ntroduce manypartcle wae-functons to treat correlaton eactly as well, reganng the Schrödnger equaton tself.) Followng the sprt of a recent work by the author [4], n ths artcle a smple deraton of the equatons wll be presented, placng them n a unfed scheme of quantum mechancal theores n whch the Kohn-Sham equatons, the Hartree-Fock- Kohn-Sham equatons and the ground-state Schrödnger equaton orgnate from one root: the Hohenberg-Kohn theorems, through the Euler equaton Eq.(3). Besde the scheme, another mture of FT and the Hartree-Fock theory, the Kohn-Sham formulaton of the Hartree-Fock scheme, wll also be consdered. In the second part of the paper, the concept of -partcle echange-energy densty functonals wll be ntroduced, followng also [4] (see also [5]), and t wll be shown that they emerge naturally as second-degree homogeneous functonals, for eery partcle number. For smplcty, spn s not consdered throughout the paper. II. The Hartree-Fock-Kohn-Sham scheme, and the Kohn-Sham formulaton of the Hartree-Fock theory Buldng on the dea of the constraned search defnton of densty functonals [6-8], the eact ncorporaton of correlaton nto the Hartree-Fock method n the form of a multplcate, densty-dependent potental (whch s a densty-functonal derate), that s, the so-called Hartree-Fock-Kohn-Sham method, can be dered drectly from densty-functonal theory n a smple way. The startng pont s the defnton of the functonal [7] F [ ρ] = mn{ T[ ψ ] + V[ ψ ]}, (3) ψ ρ 4

5 whch, for a ρ( r ), ges the mnmum of the sum T[ ψ ] + V[ ψ ] of wae-functon functonals and = T [ ψ] = ψ ψ (4) V [ ψ] = ψ ψ (5) < j r rj oer the set of normalzed Slater determnants (ψ ) that yeld ρ( r ), that s, ψ $( ρ r) ψ = ρ whle F[ ρ ], defned by F[ ρ] = mn{ T[ ψ ] + V[ ψ ]} ψ ρ, (6) searches oer the space of normalzed (antsymmetrc) wae-functons not restrcted to be determnants. Smlar to the case of the nonnteractng knetc-energy densty functonal Eq.(3), by the mappng T[ ρ] = mn T[ ψ ], (7) s ψ ρ ρ ψ : ρ[ ψ ] = ρ, T[ ψ ] + V[ ψ ] = F[ ρ], (8) assocates eery ρ( r ) wth a fcte nonnteractng system of densty ρ( r ), composed of sngle-partcle orbtals u ( r ) of number = ρ dr. Wrtng Eq.(3) eplctly wth the orbtals, F [ ρ] = mn T [ u, u,..., u, u ] + E [ u, u,..., u, u ] + J[ ρ], (9) { s } { u } ρ wth Ts[ u, u,..., u, u] beng defned by Eq.(4) and E[ u, u,..., u, u] by Eq.(0). The orbtals u ( r ) correspondng to a densty ρ( r ) can be determned through an Euler-Lagrange mnmzaton procedure [6,9,0] (see also p. 5 and p. 87 of []), wth the constrants of fed densty, and orthonormalzaton, u uj dr = δ j, (0) yeldng the Euler-Lagrange equatons u ( r ) j= u ( r ) u ( r j j ) u r r ( r ) dr + λ ρ ( r ) u ( r ) = ε u ( r ) =,, () 5

6 (n canoncal form), wth the Lagrange multplers ε securng the normalzaton of u and λ ρ ( r ) securng the fulflment of Eq.(7) (wth ρ( r ) fed). Ths approach of the mnmzaton problem posed by Eq.(9), howeer, tells nothng more about λ ρ ( r ), about ts connecton to ρ( r ), and hence to ( r ) ; ths s because, as has been ponted out n [4], t does not utlze the defnton Eq.(9), the mplct defnton of the u s correspondng to a gen ρ( r ), fully. Wth a dfferent approach, namely, mnmzng the dfference F = s + [ u, u,..., u, u ]: T [ u, u,..., u, u ] E [ u, u,..., u, u ] ( F J)[ ρ[ u, u,..., u, u ]] () under the orthonormalzaton constrant only (nstead of the aboe mnmzaton of Ts[ u, u,..., u, u] + E[ u, u,..., u, u] ), the ρ-dependence of the multpler that forces the orbtals u to yeld a gen ρ( r ) can be dentfed: the Euler-Lagrange equatons resultng from the mnmzaton of [ u, u,..., u, u ] F, under normalzaton constrant on u ( r ) s, are u j= uj uj( r ) δ( F[ ρ] J[ ρ]) u ( r ) dr u = ε u =,,. (3) r r δρ The bass for ths mnmzaton s that, followng from the defnton of F [ ρ ], [,,...,, ] F 0 u u u u (4) s, the equalty holdng for the orbtals { } for any normalzed u u ( r ) assocated = to ρ( r ) by Eq.(8); thus only the normalzaton of u s has to be reckoned wth as eternal constrant. Wth Eq.(3) then the equatons Eq.() of the method arse mmedatelly through the Hohenberg-Kohn Euler equaton, Eq.(3), that s, δ ( F[ ρ] J[ ρ]) + = µ δρ( r ) wth δ{ F[ ρ] ( F[ ρ] J[ ρ])} = + r ( ) δρ( r ) [Eq.()], whch brngs the eternal potental ( r ) that correspondng to ( r ) ). (5) (6) nto the equatons, fng ρ( r ) (as 6

7 It s mportant to underlne that the correlaton energy functonal of the scheme s not the same as that of the Kohn-Sham theory, snce t s defned by whle n the Kohn-Sham scheme E [ ρ] = F[ ρ] F [ ρ], (7) c E [ ρ] = F[ ρ] T [ ρ] J[ ρ] E [ ρ], (8) c s but F [ ρ] T [ ρ] + J[ ρ] + E [ ρ] (9) s [ E [ ρ ] s defned through Eq.(0) wth u ( r ) s beng the (ρ-dependent) orbtals of the Kohn-Sham scheme]. In partcular, snce F [ ρ ] mnmzes T [ u, u,..., u, u ] + E [ u, u,..., u, u ] n total [under Eq.(7)], s from whch F [ ρ] T [ ρ] + J[ ρ] + E [ ρ], (30) s E c [ ρ] E [ ρ], (3) whch means that, besdes the need for appromaton of E [ ρ ], n the Kohn-Sham method, there s also a greater correlaton part n the energy densty functonal to be appromated. (ote that both knds of correlaton energes are always negate or zero [3].) Besde the densty-functonal amplfcaton of Hartree-Fock theory, dscussed so far, there s another mture of FT and Hartree-Fock theory: the Kohn-Sham formulaton of the Hartree-Fock scheme [9,7,,3] (the Kohn-Sham-Hartree-Fock scheme), where the nonlocal echange operator of the Hartree-Fock equatons s replaced by a local potental whch s a densty-functonal derate. Ths theory ges the sound justfcaton of early attempts n that drecton. As the Hartree-Fock method s an appromate theory by defnton, neglectng correlaton completely by narrowng the space of wae-functons to Slater determnants, ts Kohn-Sham formulaton s not eact ether. Here, the densty functonal F[ ρ ] of eact FT s replaced by F [ ρ ], and the energy functonal HF E [ ρ ] = F[ ρ] + ρ( r ) ( r ) dr (3) c 7

8 s mnmzed, under the -conseraton constrant, gng the Hartree-Fock groundstate energy, and densty. Ths yelds the Euler equaton δf [ ρ] δρ( r ) HF + ( r ) = µ (33) for the determnaton of the Hartree-Fock ground-state densty. ote that, wth E HF [ρ], E [ρ ] arses as HF E [ ρ] = E [ ρ] + E [ ρ]. (34) Separatng T s [ ρ ] n E HF [ ρ ], analogously to eact FT, a major part of the unknown F [ ρ ] can be treated eactly, obtanng the sngle-partcle Schrödnger equatons HF KSHF u + u = ε u =,,, (35) wth Eq.(7), for the determnaton of the densty, where HF KS KS c δ ( F[ ρ] Ts[ ρ]) ( r ) = + ( r ). (36) δρ( r ) These equatons can be obtaned n a smple manner, utlzng the dea descrbed aboe n connecton wth the method; namely, mnmzng the dfference T [ u s, u,..., u, u ]: = T s [ u, u,..., u, u ] T s [ ρ [ u, u,..., u, u ]] (37) under orthonormalzaton constrant on u s (just as n the case of the deraton of the Kohn-Sham equatons gen n [4]), then usng the pure densty-functonal Euler δt [ ] equaton Eq.(33) to substtute for s ρ n the obtaned Euler-Lagrange equatons, δρ( r ) hereby fng ρ( r ) by the eternal potental. The tradtonal quantum mechancal equatons of the Hartree-Fock approach, the Hartree-Fock equatons themseles, can be recoered as well from Eq.(33), followng the aboe procedure, but now F [ u, u,..., u, u] = Ts [ u, u,..., u, u ] + E[ u, u,..., u, u ] ( F J)[ ρ [ u, u,..., u, u]] (38) has to be mnmzed (followed by the nserton of Eq.(33)); just as can the (groundstate) Schrödnger equaton of eact quantum mechancs be dered from the Hohenberg-Kohn Euler equaton [Eq.(3)] [4]. ote that the dfference functonals Eq.() and Eq.(38) are the same; the dfference between the sngle-partcle equatons of the and the HF theores arses from the dfferent [eact, or appromate 8

9 (HF)] physcs behnd the equatons, that s, dfferent FT backgrounds, characterzed by the densty-functonal Euler equatons Eq.(3) and Eq.(33), respectely. Regardng the deratons of Schrödnger equatons from FT, some words need to be sad about the densty-functonal derates (local potentals) appearng n these equatons snce the densty functonals defned through wae-functons are defned only for denstes of nteger norm. Ths problem can be elmnated wth the help of the concept of -partcle densty functonals and -conserng functonal dfferentaton [] (for a dscusson of ts mathematcs, see [3]). In [4], t has been ponted out that the Euler-Lagrange mnmzaton procedure proposed there, and used aboe as well, to determne the correspondng orbtals for a densty ρ( r ), can be carred out by usng an -partcle densty functonal A [ ρ ] nstead of the gen densty functonal A[ ρ ], defnng the dfference functonal wth A [ ρ ], A [ u, u,..., u, u ]: = Au [, u,..., u, u ] A [ ρ [ u, u,..., u, u ]], snce the normalzaton of the orbtals and the faton of ther number consere the norm of ρ( r ). An A [ ρ ] defned for eery ρ( r ) can then be chosen; the tral constant shftng [] (or other homogeneous etenson) of A[ ρ ], e.g., s one proper choce, beng (unconstraned) dfferentable f A[ ρ ] s dfferentable oer the space of ρ( r ) s of the gen norm. In the orbtal equatons resultng from the mnmzaton of [ u, u,..., u, u ], addng δa [ ρ] ρ( r ) dr u r ( ) to both sdes of the equatons δρ( r ) A [], the unconstraned derate of A [ ρ ] s replaced by the -conserng functonal derate, δ A ρ [ ], for whch [] δ ρ( r ) δa[ ρ] δa[ ρ ] =. (39) δ ρ( r ) δ ρ( r ) After usng Eq.(39), fnally, the replacement of the unconstraned derate by the correspondng -conserng derate (of A[ ρ ]) s acheed n the orbtal equatons (for ρ (r ) 's), thereby restrctng the dfferentaton to the allowed doman of ρ( r ) s. ote that smlar consderatons hold also f A[ ρ ] comes from a wae-functon 9

10 functonal A[ ψ ] that s not an orbtal functonal. Wth -conserng densty-functonal derates, the deratons of Schrödnger equatons from FT thus look lke: from (e.g.), usng u u j= j= uj uj( r ) δ( F[ ρ] J[ ρ]) u ( r ) dr u = ε u (40) r r δ ρ δ F[ ρ] + r ( ) = µ, (4) δ ρ( r ) u u ( r j j ) δ J[ ρ ] δ Ec [ ρ ] u ( r ) dr ( r) u = ε u. (4) r r δ ρ( r) δ ρ( r) Another way of treatng the problem of A[ ρ ] (= T s [ ρ ], F [ρ ], or F[ ρ ]) beng defned only for nteger 's s to utlze that a chan rule, namely, embracng δa[ ρ[ g]] δa[ ρ] δρ( ) = d, δ g( ) δρ( ) δ g( K ) δa[ ρ[ g]] δa[ ρ] δρ( ) = d, δ g( ) δ ρ( ) δ g( K ) stll holds (as proed n the Append of [3]) for cases A[ ρ [ g]] where the functonal ρ [g] s such that ρ ( )[ g] satsfes the gen K-conseraton constrant here ρ ( r ) dr = for any (allowed) g ( ). In the present cases, g u, u,..., u, u ) [or = ( g = ( ψ, ψ ) ], and the normalzaton of u (r) 's ensures the fulflment of ρ ( r ) dr =. Thus, δ F[ ρ [ u j, u j ]] δ u u ( r ) u ( r ) dr = u ( r ) u ( r ) dr = δ F[ ρ ] δ ρ( r ) = dr, (43a) δ ρ( r ) δ u or δ F[ ρ [ u j, u j ]] δ u δ F[ ρ ] δ ρ( r ) = dr δ ρ( r ) δ u u ( r ) u ( r ) dr = u ( r ) u ( r ) dr =. (43b) ote that ths chan rule s essental also for the aratonal deraton of the Kohn- Sham equatons themseles. 0

11 It s mportant to note here fnally that, hang the -conserng derates (as potentals) n the dered wae-functon equatons, the fractonal-partcle-number generalzaton [4] of densty functonals can be substtuted for the orgnal densty functonals, obtanng the possblty of relang the constrant of -conseraton on the functonal dfferentaton and usng the unconstraned (left or rght) derates of densty functonals. III. Second-degree homogeneous -partcle echange-energy densty functonals To construct appromatons for densty functonals, knowledge about the structure of the functonals s essental. In ths secton, a natural formal constructon of the echange-energy densty functonal E [ ρ ] of Kohn-Sham FT, a second-degree homogeneous -partcle echange-energy densty functonals, wll be proposed. It s known that E [ ρ ], defned through Eq.(0) wth u ( r ) as the (ρdependent) Kohn-Sham orbtals, scales homogeneously of degree one wth coordnate scalng [5], that s, from whch 3 E [ λρλ ] = λe [ ρ], (44) E δe [ ρ] [ ρ] = ρ r dr δρ( r ) emerges [6]. Besde ths property, the queston of homogenety of E [ ρ ] n ρ tself may also arse, consderng that the classcal Coulomb repulson functonal J[ ρ ], the negate of whch s an eact epresson for E for one-partcle denstes, s (45) homogeneous of degree two,.e. J[ λρ] = λ J[ ρ]. (46) The queston of homogenety wth respect to the densty, and related, or affntate, propertes of the arous components of the energy, has been addressed n seeral works recently [7-39], homogenety beng a ery attracte (strong) analytcal property for functonals. For the nonnteractng knetc-energy densty functonal

12 T s [ ρ ], the frst-degree homogenety n ρ [33] has been shown to be an ncorrect proposal [37,38], though the Wezsäcker (one-partcle knetc-energy) functonal, r T [ ] ρ( ) W ρ = dr 8 ρ( r ) (47) (whch s an eact epresson for T s [ ρ ] for one-partcle denstes), s homogeneous of degree one (n ρ). In the case of E [ ρ ], the naldty of homogenety (of second, or any other, degree) n ρ can be proen n a smple way, as for T s [ ρ ] [37]; snce f homogenety of a degree k held then k ρ ρ E[ ρ] = E J = J[ ρ] =, (48) that s, E [ ρ ] would be k J[ ρ]. ote that from the second-degree homogenety of E [ ρ ] (n ρ), Eq.(5) of [39], dered from the result of [3] for the full electronelectron-nteracton energy densty functonal, would also follow. (Chan and Handy [37] hang shown that the homogenety results of [3-34] are ncorrect n the case of the knetc energy, Joubert [39] has also shown that for the correlaton energy functonal after, n [38], t was ponted out that an essental queston was not treated properly n [3-34], therefore ther results cannot be consdered to be ald.) Wth the ntroducton of the concept of -partcle densty functonals, that s, functonals that are ald epressons for a gen densty functonal for -partcle denstes, howeer, t can be shown that the echange energy can be gen by densty functonals homogeneous of degree two, as n the case of T s, whch can be gen by frst-degree homogeneous -partcle densty functonals, for all partcle numbers [4]. By utlzng the result n [4] that the KS orbtals arse most naturally as degree- homogeneous functonals of the densty, that s, ( ),..., ( ) [ ] = ( ),..., ( ) [ ρ], (49) ( u r u r ) kρ k( u r u r ) E arses as a second-degree homogeneous functonal of the densty, through Eq.(0). The resultant densty functonal s an -partcle densty functonal, E [ ρ ], snce ts form depends on the number of orbtals used n ts constructon. Ths E [ ρ ] thus satsfes

13 that s, E [ λρ] = λ E [ ρ], (50) E δe [ ρ] [ ρ] = ρ dr. (5) δρ( r ) ote that an -partcle echange-energy densty functonal (as any other -partcle densty functonal) can be consdered as a two-arable functonal, E[, ρ ], and the general echange-energy densty functonal emerges as E [ ρ] = E [ ρ, ρ], (5) for whch thus the homogenety Eq.(50) yelds the property E[ ρ] = E[ ρ, λρ] λ. (53) Besde Eq.(5), also the analogue of Eq.(45) can be wrtten for E [ ρ ], for - partcle denstes [ ρ ( r ) ], E δe [ ρ ] [ ρ] = ρ r dr, (54) δρ( r ) snce the dfference between the derates δ E ρ [ ] and δ E ρ [ ] s r -ndependent δρ( r ) δρ( r ) [,3]. Eq.(54) s n accordance wth the fact that coordnate scalng conseres the normalzaton of the densty, wth the use of whch E [ λρ 3 ( λr)] E[ λρ 3 = ( λr)] = λe[ ρ] = λe [ ρ ] (55) [gng also Eq.(54)]; see the Append for the proof for a general ρ( r ). It s worth notng that requrng -partcle densty functonals to be homogeneous of some degree elmnates the ambguty present n ther defnton [5], meanng that the only possble -partcle nonnteractng knetc-energy and echangeenergy densty functonals homogeneous of degree one and two, respectely, are those constructed n [4] and here (see also the Append). The smplest echange energy densty functonal arsng from the condtons of homogenety of degree two n densty scalng [Eq.(50)] and homogenety of degree one n coordnate scalng [Eq.(55), see also Eq.(A6)] s mnus the Coulomb functonal J[ ρ ] [whch s just (the degree-two homogeneous) E [ ρ ], [ ρ] = J[ ρ] ]; just as the E 3

14 Wezsäcker functonal s the smplest epresson for a knetc energy densty functonal satsfyng the two homogenety scalng-condtons for T s [ ρ ] [4] (see also [40]). That J[ρ] s the smplest echange energy functonal satsfyng the homogenety condtons s of course stated from a physcal pont of ew: a "real" echange-energy densty functonal, on one hand, has to be a two-partcle functonal (regardng ts structure), and on the other hand, a term, characterzng the r r nteracton, has to appear eplctly besde the densty-functonal part, snce the echange energy s obtaned a Eq.(0), the orbtals u formng the ρ-dependent part n the functonal; thus, t has to hae a form (or a combnaton of forms lke) f [ ρ] f [ ρ( r )] dr dr. (56) r r ote that wth the separaton of a factor ρ( ) ρ( r r ) n the ntegrand, r r ρ ρ( r ) g[ ρ] g[ ρ( r )] dr dr, (57) r r the remanng part g[ ρ( r )] g[ ρ( r )] (or, smply g [ ρ( r )]) has to be homogeneous of degree zero n ρ( r ) for Eq.(57) to satsfy the scalng condtons. Fnally, the queston of separablty [4] for -partcle densty functonals may be worth takng a look at, referrng also to a recent work [4], whch nestgates the separablty problem of (eplctly) -dependent densty functonals. It can be seen easly that -partcle densty functonals A [ ρ ] correspondng to a separable densty functonal A[ ρ ] are separable for -partcle denstes (only for whch they hae physcal releance): A [ ρ + ρ ] = A[ ρ + ρ ] = A[ ρ ] + A[ ρ ] = A [ ρ ] + A [ ρ ], (58) where + =. IV. Summary Varous treatments of echange n the framework of densty-functonal theory hae been nestgated. Frst, the so-called Hartree-Fock-Kohn-Sham method, whch s 4

15 the densty-functonal amplfcaton of the Hartree-Fock approach, has been consdered. In Sec.II, a smple deraton of the Hartree-Fock-Kohn-Sham equatons has been presented, followng the sprt of [4], showng how these equatons, together wth the Kohn-Sham equatons and the ground-state Schrödnger equaton [4], formally orgnate from one common root: densty-functonal theory, through ts Hohenberg-Kohn Euler equaton. Besde the Hartree-Fock-Kohn-Sham scheme, along the same lnes, n Sec.II another mture of densty-functonal theory and Hartree-Fock theory, the Kohn-Sham formulaton of the Hartree-Fock approach (wth the Kohn- Sham-Hartree-Fock equatons), has also been consdered. In Sec.III, t has been ponted out that the echange energy of ground-state systems of dentcal fermons naturally emerges as a second-degree homogeneous -partcle densty functonal E [ ρ ], for any partcle number, yeldng a = ρ constructon E [ ρ] E [ ρ] of the echange-energy densty functonal, for whch Eq.(53) holds. The frst element of the arsng sequence (that s, =,,...) of degreetwo homogeneous -partcle echange-energy densty functonals E [ ρ ] s just mnus the classcal Coulomb-repulson energy functonal. It has further been shown that E [ ρ ] (homogeneous n ρ) retans the coordnate scalng property of the general echange-energy densty functonal, E [ ρ ], and generally, any -partcle densty functonal homogeneous n the densty has the same coordnate scalng behaour as the general densty functonal to whch t corresponds. Thus, the scalng propertes of E [ ρ ] proposed n Sec.III are just the reerse of those of T s [ ρ ] of [4], whch s frst-degree homogeneous n ρ and second-degree homogeneous wth respect to coordnate scalng. Append: Unqueness of homogeneous -partcle densty functonals, and ther coordnate scalng Gen a densty functonal A[ ρ ], -partcle densty functonals A [ ρ ] can be defned, for whch A [ ρ ] = A[ ρ ] (A) 5

16 (wth ρ dr = ), gng the alue of A for -partcle denstes ρ. The defnton Eq.(A) s of course ambguous. E.g., the free choce of generalzaton of normalzaton represents the ambguty n the case of -partcle nonnteractng knetcenergy densty functonals constructed as descrbed n [4]. Requrng an A [ ρ ] to be homogeneous of some degree k, whch means A k [ λρ] = λ A [ ρ], (A) howeer, leads to a unque defnton for ths A [ ρ ]. For, utlzng Eqs.(A) and (A), k ρ ρ ρ A[ ρ] = A A ρ =, (A3) ρ makng use of that the norm of ρ s [5]. From Eq.(A3) then ρ A [ ρ] = ρ k ρ A, (A4) ρ whch shows that the faton of (on the rght) leads to the -partcle nature. Eq.(A4) s the unque epresson for a degree-k homogeneous -partcle densty functonal correspondng to an A[ ρ ]. Ths unqueness, n the case of the densty functonals T s [ ρ ] and E [ ρ ], means that the homogeneous -partcle densty functonals T s [ ρ ] and E [ ρ ] constructed n [4] and Sec.III of the present artcle are the only frst-degree homogeneous -partcle nonnteractng knetc-energy and second-degree homogeneous -partcle echange-energy densty functonals, respectely. Thus, e.g., T W [ ρ ] and J[ρ] are the only possble frst-degree homogeneous and second-degree homogeneous densty functonals that ge the eact nonnteractng knetc energy and the eact echange energy, respectely, for oneelectron systems. Wth Eq.(A4), the aldty of Eq.(54) [and Eq.(55)] for arbtrary ρ( r ) (not just for ρ( r ) s of norm ),.e. E [ ρ] [ ρ] δe = ρ r δρ( r ) dr, (A5) 6

17 3 can be justfed easly. Utlzng λρλ ( r ) dr = ρ( r ) dr and the homogenety of degree one of E [ ρ ] n coordnate scalng, E ρ [ λρλ ] E λ ( r ) E E [ ] ρ ρλ ρ 3 3 λ ρ ρ λ ρ = = =, (A6) whch means that E [ ρ ] tself, too, s homogeneous of degree one n coordnate scalng, yeldng Eq.(A5). A smlar proof apples for T s [ ρ ] of [4] as well, meanng that T s [ ρ ] has the same coordnate scalng property as T s [ ρ ], that s, homogenety of degree two. Acknowledgements: Grants from the Hungaran Hgher Educaton And Research Foundaton and from OTKA (048675) are gratefully acknowledged. References [] Parr, R. G., and Yang, W., ensty Functonal Theory of Atoms and Molecules (Oford Un. Press, ew York, 989). [] agy, Á., Phys. Rep. 98, (998). [3] Thomas, L. H., Proc. Cambrdge Phl. Soc. 3, 54 (97). [4] Ferm, E., Z. Phys. 48, 73 (98). [5] rac, P. A. M., Proc. Cambrdge. Phl. Soc. 6, 376 (930). [6] on Wezsäcker, C. F., Z. Phys. 96, 43 (935). [7] March,. H., Ad. Phys. 6, (957). [8] Slater, J. C., Phys. Re. 8, 385 (95). [9] Gáspár, R., Acta Phys. Hung. 3, 63 (954) [for Englsh translaton, see J. Mol. Struct. Theochem 50, (000)]. [0] Slater, J. C., The Self-Consstent Feld for Molecules and Solds, Quantum Theory of Molecules and Solds, Vol. 4 (McGraw-Hll, ew York, 974). [] Hohenberg, P., and Kohn, W., Phys. Re. 36, B864 (964). [] Kohn, W., and Sham, L. J., Phys. Re. 40, A33 (965). [3] Holas, A., and March,. H., n ensty Functonal Theory, Topcs n Current 7

18 Chemstry, Vol. 80, ed. R. F. alewajsk (Sprnger, Hedelberg, 996), p. 57. [4] Gál, T., Phys. Re. A 64, (00). [5] Gál, T., Int. J. Quantum Chem. 07, 586 (007). [6] Percus, J. K., Int. J. Quantum Chem. 3, 89 (978). [7] Ley, M., Proc. atl. Acad. Sc. USA 76, 606 (979). [8] Leb, E. H., Int. J. Quantum Chem. 4, 43 (983). [9] Payne, P. W., J. Chem. Phys. 7, 490 (979). [0] Freed, K. F., and Ley, M., J. Chem. Phys. 77, 396 (98). [] Holas, A., March,. H., Takahash, Y., Zhang, C., Phys. Re. A 48, 708 (993). [] Gál, T., Phys. Re. A 63, 0506 (00). [3] Gál, T., J. Math. Chem. 4, 66 (007) [eprnt arx:math-ph/060307]. [4] Perdew, J.P., Parr, R.G., Ley, M., Balduz, J.L., Phys. Re. Lett. 49, 69 (98). [5] Ley, M., and Perdew, J. P., Phys. Re. A 3, 00 (985). [6] Append n Ghosh, S. K., and Parr, R. G., J. Chem. Phys. 8, 3307 (985). [7] Zhao, Q., Morrson, R. C., and Parr, R. G., Phys. Re. A 50, 38 (994). [8] Parr, R. G., and Ghosh, S. K., Phys. Re. A 5, 3564 (995). [9] Lu, S., and Parr, R. G., Phys. Re. A 53, (996). [30] Morrson, R. C., and Parr, R. G., Phys. Re. A 53, R98 (996). [3] Lu, S., and Parr, R. G., Phys. Re. A 55, 79 (997). [3] Parr, R. G., and Lu, S., Chem. Phys. Lett. 76, 64 (997). [33] Lu, S., and Parr, R. G., Chem. Phys. Lett. 78, 34 (997). [34] Parr, R. G., and Lu, S., Chem. Phys. Lett. 80, 59 (997). [35] Joubert,. P., Chem. Phys. Lett. 88, 338 (998). [36] Tozer,. J., Phys. Re. A 58, 354 (998). [37] Chan, G. K.-L., and Handy,. C., Phys. Re. A 59, 670 (999). [38] Gál, T., Phys. Re. A 6, (000). [39] Joubert,., Phys. Re. A 64, (00). [40] Gál, T., and agy, Á., J. Mol. Struct. Theochem 50, 67 (000). [4] Perdew, J. P., Ad. Quantum Chem., 3 (990). [4] Pérez-Jménez, A. J., Moscardó, F., Sancho-García, J. C., Aba, L. P., San-Fabán, E., and Pérez-Jordá, J. M., J. Chem. Phys. 4, 0 (00). 8

19 Append B. Summary of the connectons between the densty-functonal Euler equaton and the arous wae-functon equatons, ndcatng the functonal the constranedsearch defnton of whch s utlzed to get to the gen wae-functon equaton(s) Eact quantum mechancs: Kohn-Sham equatons T s [ ρ ] δf[ ρ] + r ( ) = µ F δρ( r ) [ ρ ] Hartree-Fock-Kohn-Sham eqs. F[ ρ ] Schrödnger equaton Hartree-Fock appromaton: Kohn-Sham-Hartree-Fock eqs. T s [ ρ ] δf[ ρ] + ( r ) = µ δρ( r ) HF F [ ρ ] Hartree-Fock equatons 9