Why the traditional concept of local hardness does not work

Transcription

1 Theor Chem Acc (2012) 131:1223 DOI /s x REGULAR ARTICLE Why the traditional concept of local hardness does not work Tamás Gál Receied: 19 October 2011 / Accepted: 14 April 2012 Ó Springer-Verlag 2012 Abstract Finding a proper local measure of chemical hardness has been a long-standing aim of density functional theory. The traditional approach to defining a local hardness index, by the deriatie of the chemical potential l with respect to the electron density nðr * subject to the constraint of a fixed external potential ðr *, has raised seeral questions, and its chemical applicability has proed to be limited. Here, we point out that the only actual possibility to obtain a local hardness measure in the traditional approach emerges if the external potential constraint is dropped; consequently, utilizing the ambiguity of a restricted chemical potential deriatie is not an option to gain alternatie definitions of local hardness. At the same time, howeer, the arising local hardness concept turns out to be fatally undermined by its inherent connection with the asymptotic alue of the second deriatie of the uniersal density functional. The only other local hardness concept one may deduce from the traditional approach, ½nŠ=dnðr * * ð r, is the one that gies a constant alue, the global hardness itself, throughout an electron system in its ground state. Consequently, the traditional approach is in principle incapable of deliering a local hardness indicator. The parallel case of defining a local ersion of the chemical potential itself is also outlined, arriing at similar conclusions. Keywords Reactiity indices Restricted deriaties Local hardness Local chemical potential T. Gál (&) Quantum Theory Project, Uniersity of Florida, Gainesille, FL 32611, USA galt@phys.unideb.hu 1 Introduction Chemical reactiity indices [1 9], defined within the framework of density functional theory (DFT) [1], hae found successful application in the study of chemical phenomena. The three most well-known reactiity descriptors, the electronegatiity [10 13], or in the language of DFT, minus the chemical potential [13], the chemical hardness, and its inerse, the softness [14 17], are basic constituents of essential principles goerning chemical reactions the electronegatiity equalization principle [13, 18], the hard/ soft acid/base principle [14 17, 19 24], and the maximum hardness principle [25 30]. An important aim of chemical reactiity theory [2 9] is to establish local ersions of the global indices, on the basis of which predictions can be made regarding the molecular sites a gien reaction happens at. Defining a local softness can be done in a natural way [31], by replacing the electron number with the electron density nðr * in the definition of softness as the deriatie of with respect to the chemical potential l. Howeer, defining a local counterpart [32, 33] of hardness, the multiplicatie inerse of softness, has met essential difficulties [34 40], undermining the applicability of the local hardness concept. This may not seem to be a substantial problem, as the concepts of hardness and softness are simple complementers; howeer, ery recently, een the definition of local softness sðr * has been found to fail to properly signify the soft sites in the case of hard systems [41, 42], and een before, the interpretation of small sðr * alues as indicators of locally hard sites, preferred in hard hard interactions [43], had been put into question [44 46]. Therefore, the question of a possible existence of a proper local hardness indicator has a renewed significance.

2 Page 2 of 14 Theor Chem Acc (2012) 131:1223 In this study, we will re-examine the idea of defining a local hardness concept ia differentiation of the chemical potential with respect to the density subject to the constraint of a fixed external potential ðr *, in the iew of the questions as to (1) why this traditional way of defining a local hardness concept could not yield a (generally) correct local indicator of chemical hardness, and (2) whether there is any possibility at all to gain such a local index from this approach. We will find that the only possibility to obtain a local hardness measure in the traditional approach emerges if the external potential constraint on the differentiation is dropped. The arising local hardness concept, howeer, will be shown to be fatally undermined by the necessary inolement of the asymptotic fixation of the external potential. At the same time, we will show that the constant local hardness of Ghosh [33] emerges as l s unique constrained deriatie with respect to nðr * corresponding to the fixed-ðr * constraint but this local hardness concept cannot be a local reactiity measure because of its constancy. Our conclusion will be that the traditional approach to defining a local hardness index is, in fact, not capable of deliering a local hardness measure; therefore, an essentially new approach to this problem needs to be applied (like that proposed in [40], which originates a local hardness index ia a local chemical potential a chemical potential density). We will also consider the analogous case of defining a local counterpart of the chemical potential itself, haing releance (1) regarding the definition of a local electronegatiity index and (2) sering as a potential basis for an alternatie local hardness definition. The results will be similar to the local hardness case in particular, the idea of defining a local chemical potential through the deriatie of the ground-state energy with respect to the density subject to the constraint of a fixed external potential yields the constant local chemical potential concept of Parr et al. [13] as the only feasible option. 2 The traditional concept of local hardness The chemical concept of hardness has been quantified by Parr and Pearson [15] as g ol : ð1 ð * r In contrast with its inerse chemical quantity, the softness, S ol ; ð2 ð * r defining a local counterpart for hardness has met essential difficulties, due to the fact that there is no such obious way to do this as in the case of Eq. (2). For Eq. (2), a corresponding local quantity can be readily introduced [31]: sðr * onðr* ; ð3 ol ð * r which has been termed local softness. This has a direct connection with the Fukui function [47] f ðr * onðr* ; ð4 ð * r a well-established chemical reactiity index: Applying the chain rule of differentiation, one obtains sðr * onðr* ð r * ol f ðr * S: ð * r ð5 sðr * integrates to S (just as the Fukui function integrates to 1), and it is natural to interpret it as a pointwise, that is, local, softness [31]. A local hardness concept has been introduced by Berkowitz et al. [32], who defined the local hardness as gðr * : ð6 dnðr * ð r * Equation (6) has since been the basis for practically all inestigations concerning the local counterpart of hardness; therefore, we will term it the traditional concept of, or (since it actually embraces a class of concrete local hardness concepts, with different concrete quantitatie formulae) traditional approach to, local hardness. This local index is not a local quantity in the sense the local softness is, since it does not integrate to the hardness; consequently, its integral oer a region in the molecule will not gie a regional hardness. In fact, gðr * times the Fukui function is what gies g by integration oer the whole space, gðr * f ðr * dr * g; ð7 which emerges ia an application of the chain rule, as can be seen from the definitions (4) and (6). The biggest difficulty with the local hardness defined by Eq. (6) has been that it is not clear how to understand the fixed external potential [ðr * ] condition on the differentiation in Eq. (6). If we consider that the hardness is defined by Eq. (1) as the partial deriatie of the chemical potential l½; Š (a function(al) of the electron number and the external potential) with respect to, Eq. (6) suggests that ðr * as one of the ariables in l½; Š should be fixed when differentiating with respect to the electron density nðr *. Howeer, this yields

3 Theor Chem Acc (2012) 131:1223 Page 3 of 14 ol½; Š d gðr * ð * r dnðr * g; ð8 that is, the local hardness equals the global hardness at eery point in space. If one utilizes the DFT Euler Lagrange equation df½nš dnðr * þ l; ð9 ðr* emerging from the minimization principle for the groundstate energy density functional E ½nŠ F½nŠþ nðr * ðr * dr * ð10 for the determination of the ground-state density corresponding to a gien ðr *, l½; Š ð oe½; Š= can be obtained as l½; Š df dnðr * ½n½; ŠŠ þ : ð11 ðr* Differentiating this expression with respect to yields ol½; Š d 2 F onðr *0 g dnðr * dnðr * 0 dr * 0 d 2 F dnðr * dnðr * 0 f ðr *0 *0 dr : ð12 On the basis of this, then, it is natural to identify the local hardness yielding Eq. (8) with d 2 F gðr * dnðr * dnðr * 0 f ðr *0 *0 dr : ð13 This local hardness definition was proposed by Ghosh [33] and was discoered to be a constant giing the global hardness eerywhere by Harbola et al. [34]. Equation (13) thus cannot be a local counterpart of hardness on the basis of which one could differentiate between molecular sites. Howeer, it still is a useful conceptual and practical tool since a hardness equalization principle can be based on it [48 51], which says that gðr * of Eq. (13) should be constant for the whole system for the ground-state density but only for that density. This principle is closely related with the long-known chemical potential (or electronegatiity) equalization principle [13, 52]. To gain other definition for the local hardness than the one giing the global hardness in eery point of space, one may consider the fixed-ðr * constraint in Eq. (6) as a constraint on the differentiation with respect to the density, ½½nŠ; ½nŠŠ gðr * ; ð14 dnðr * * ð r instead of a simple fixation of the ariable ðr * of l½; Š. That is, the density domain oer which the differentiation is carried out is restricted to the domain of densities that yield the gien ðr *, through the first Hohenberg Kohn theorem [1], which constitutes a unique nðr * ðr * mapping, that is, a ðr * ½nŠ functional. The result will be an ambiguous restricted deriatie (see Sec.II of [53]), similarly to the case of deriaties restricted to the domain of densities of a gien normalization, which deriaties are determined only up to an arbitrary additie constant [1, 54]. Harbola et al. [34], to characterize the ambiguity of the local hardness concept of Eq. (6), first pointed out by Berkowitz and Parr [55], hae gien the explicit form d 2 F gðr * dnðr * dnðr * 0 uðr *0 *0 dr ð15 for the possible local hardness candidates, where uðr * is an arbitrary function that integrates to 1. The second deriatie of F½nŠ, appearing in Eq. (15), is called the hardness kernel [55], which also seres as a basis for a minimization theorem determining the Fukui function [56]. The choice uðr * fðr * gies back Eq. (13), while another natural choice is uðr * nðr * =, which yields the original local hardness formula of Berkowitz et al. [32], gðr * 1 d 2 F dnðr * dnðr * 0 nðr *0 *0 dr ; ð16 who deduced it as an alternatie form of Eq. (6). Besides the aboe two definitions for gðr *, another one, termed the unconstrained local hardness, has been proposed by Ayers and Parr [28, 38]: ½½nŠ; ½nŠŠ gðr * ; ð17 dnðr * where the fixed-ðr * constraint on the differentiation with respect to nðr * is simply dropped. A substantial difficulty with this definition as regards practical use [38] is the explicit appearance of the deriatie of ðr * 0 with respect to nðr *, as can be seen by gðr * ol ½; Š þ dðr *0 dðr * 0 dnðr * dr*0 g þ f ðr * 0 dðr *0 dnðr * dr*0 ; ð18 where the well-known fact de½; Š nðr * ð19 dðr * and Eqs. (1) and (4) hae been utilized. ote that Eq. (17), too, is embraced by Eq. (14), since for a restricted deriatie, a triial choice is the unrestricted deriatie itself

4 Page 4 of 14 Theor Chem Acc (2012) 131:1223 (if exists), being alid oer the whole functional domain, hence oer the restricted domain too. (That gðr * of Eq. (8) [that is, of Eq. (13)] and of Eq. (16), or generally, of Eq. (15), are also embraced by Eq. (14) will be shown at the end of Sect. 3.) 3 Excluding the ambiguity of the local hardness concept of Eq. (14) A proper local hardness is expected to yield proper regional hardness alues, on the basis of which one can predict the molecular region (or site) a reaction with another species happens at. The only plausible way of obtaining regional hardnesses from an gðr * defined by Eq. (14) is g X X gðr * onðr* dr * X gðr * f ðr * dr * ; ð20 that is, the integral in Eq. (7) is carried out oer a gien region X of space instead of the whole space. Equation (20) has been applied in practical calculations to characterize the hardness of atomic regions or functional groups in molecules (for recent examples, see for instance [57 62]), and as a special case [in the form of Eq. (7)], to ealuate the global (that is, total) hardness itself [63 70]. Equation (20) represents an extensie hardness concept: The total hardness of a molecule can be obtained as a sum of its regional hardness corresponding to a gien (arbitrary) diision of the molecule into regions. That is, roughly saying, a molecule that contains regions haing high alues of hardness in a majority will hae a high global hardness, while a molecule that contains mainly soft regions, with low gðx, will hae a low global hardness. Of course, in a strict sense, the hardness will not be an extensie property, since for the determination of the hardness of a gien region on the basis of Eq. (20), the whole of the electronic system needs to be inoled (a change in the electron number induces a change in the electron density distribution as a whole) howeer, we cannot expect more in quantum mechanics, since there is no sense in asking how much a gien property of a segment of a system changes due to the addition of a fraction d of electrons to, and only to, that segment. A problem with this local hardness/regional hardness scheme is that if g is extensie, with regional hardnesses gien by Eq. (20), the quantity gðr * f ðr * should be considered to be the local hardness instead of gðr * [of Eq. (14)]. A local quantity qðr * corresponding to a gien extensie global quantity H emerges as qðr * * DH lim DV r DV, implying HðX R X qðr* dr *. At the same time, howeer, it seems plausible to take gðr * of Eq. (14) as the local hardness since it characterizes the change of the chemical potential induced by a small (infinitesimal) change of the electron density nðr * at a gien point of space in a gien external potential this seems to be a proper local counterpart of the hardness, gien by Eq. (1). Although this iew is intuitiely appealing, one should be careful with such an approach, because then we may argue that a change (een if infinitesimal) of the density at a single * r will yield a discontinuous density, so why should one bother himself with chemical potential changes corresponding to unphysical density changes? This point is just to show the dangerous side of intuitie arguing regarding a functional deriatie but there is a physical/real argument against the aboe interpretation of Eq. (14) as the local counterpart of hardness. If we add a small fraction d of number of electrons to a molecule, it will be distributed oer the whole molecule, no matter where we added that d of electrons. Consequently, only a change of l that is induced by a density change that is caused by a d makes sense directly. dnð r * is only an intermediate quantity that deliers the infinitesimal change in l due to an infinitesimal change of or some other quantity determining the gien electron system and hence nðr *, through ol dnðr * onðr * dr * ; ð21 or dnðr *0 dðr * dnðr * 0 dr * 0 ; ð22 dðr * for example. (Proided it exists, an unrestricted deriatie of l with respect to nðr *,, may be used in both of the dnð * r aboe equations in the place of the restricted deriaties). Thus, instead of =dnðr * j, and =dnðr * j, alone, the whole of the integrands in the aboe equations should be considered the local quantities corresponding to the quantities on the left-hand sides. It may then be more appropriate to term Eq. (14), for example, as local hardness factor, instead of local hardness, which indicates its role in deliering the actual local hardness gðr * f ðr * and regional hardnesses. Of course, this is just a matter of terminology (and why should we change a name nearly 30 years old?); howeer, the releant point here is that one should not expect gðr * of Eq. (14) itself to be a measure of local hardness. The question of considering gðr * f ðr * a local hardness measure instead of gðr * was first raised by Langenaeker et al. [71] (to get a proper complementer quantity of local softness sðr * ), who called gðr * f ðr * hardness density. The latter term, of course, is an

5 Theor Chem Acc (2012) 131:1223 Page 5 of 14 appropriate name for gðr * f ðr * as this integrates to the hardness and een more appropriate if g is indeed extensie. Howeer, if gðr * f ðr * proed to be a proper hardness density distribution indeed (with larger alues in harder regions), it should be termed also local hardness, since it would then be a local measure of hardness. But if (some choice of) gðr * of Eq. (14) itself turned out to be a proper local hardness measure, it would be gðr * what should be termed local hardness (but in this case, een terming gðr * f ðr * hardness density, just because it integrates to the hardness, would become strongly questionable). gðr * and gðr * f ðr * simultaneously cannot be a correct measure of local hardness. We note that a local hardness index does not hae to be a property density [72] but if Eq. (20) isto delier regional hardnesses, then it does hae to be, and the local hardness cannot be gðr * of Eq. (14) itself. ow, the question is as to which of the choices of Eq. (14), that is, which way of fixing the external potential while differentiating with respect to the density, is (are) the proper one(s) to obtain a local quantity gðr * f ðr * that may correctly delier regional hardnesses. As we will see, the only possible concrete choice of Eq. (14) is the unconstrained local hardness (factor) of Ayers and Parr. Consider Eqs. (21) and (22) with the integrals taken only oer a gien region of space. We are interested (directly) only in the case of Eq. (21), but by the example of Eq. (22), some more insight may be gained; therefore, it is worth considering it, too, in parallel with Eq. (21). Thus, we hae, on one hand, Eq. (20), and on the other hand, dnðr *0 f X ðr * dnðr * 0 dr * 0 ; ð23 dðr * X which is a regional Fukui function, as the left-hand side of Eq. (22) is just the Fukui function, f ðr * ; ð24 dðr * due to Eq. (19). What do these regional integrals tell us? They can be iewed as entities that gie the contributions, to the infinitesimal change of l, that come from the change of the density oer the gien region X due to an increment of and ðr *, respectiely. To ease understanding, compare this with the finite-dimensional example of a function gðxðt; yðt (with a deriatie _g og dx ox dt þ og dy oy dt, with respect to t), for which a regional integral, or partial sum, means _g x og dx ox dt that is, the part of _g that is due to the x part of the full change of gðxðt; yðt with respect to t. Thus, an infinitesimal change of,orðr *, induces a density change dnðr *, and then the regional integral Eq. (20), or Eq. (23), tells us how much the part of dnðr * that falls on the gien domain X contributes to the whole change ol of l due to dnðr *, ia ðol X R X dnð * r dnðr* dr * (multiply Eq. (20) and Eq. (23) by and R dr * dðr *, respectiely). This shows that Eq. (20) is indeed a plausible way to obtain a regional hardness measure but only if the unrestricted deriatie of l is applied, as will be pointed out below. ote that Eq. (23) gies a natural decomposition of the Fukui function f ðr *, P i f X i ðr * fðr *. It gies how much contribution to f ðr *, at any gien * r, can be attributed to a gien region X of the molecule (which does not necessarily include * r ). To understand why the unrestricted deriatie is the dnð * r only possible choice in Eqs. (20) and (23) to obtain proper regional measures, it is important to see where the ambiguity of restricted deriaties emerges from. The deriatie of a functional A½qŠ, as used in physics, is defined by da½qš dqðx 0 Dqðx0 dx 0 DðA½q; DqŠ; ð25a which has to hold for any Dqðx, and where DðA½q; DqŠ denotes the Fréchet, or Gâteaux, differential of A½qŠ for Dqðx; see for example [53] for details. Equation (25a) may be written less rigorously as da½qš dqðx 0 dqðx0 dx 0 A½q þ dqš A½qŠ; ð25b where dqðx denotes a first-order, that is, infinitesimal, increment of qðx. ow, if we restrict the functional domain by the requirement that the qðx s of the domain hae to satisfy some constraint C½qŠ C (i.e., we are not expecting the functional deriatie to be alid oer the whole domain of qðx s), more than one function da½qš dqðx will be capable of deliering DðA½q; DqŠ for any Dqðx [that is in accordance with the constraint, D C qðx]. amely, if da½qš dqðx fulfills Eq. (25), any other da½qš dqðx þ k dc½qš dqðx will fulfill it, too, oer the gien restricted domain, since dc½qš dqðx 0 d Cqðx 0 dx 0 0; ð26 emerging from C½q þ d C qš C½qŠ 0. Denoting a restricted deriatie by da½qš dqðx, while resering the notation da½qš C dqðx for the unrestricted deriatie (alid oer the unrestricted domain), this ambiguity can be expressed as da½qš dqðx da½qš C dqðx þ k dc½qš ð27 dqðx (with k being an arbitrary constant), proided, of course, that the unrestricted deriatie exists. As has been proed in the Appendix of [53], in the chain rule of differentiation

6 Page 6 of 14 Theor Chem Acc (2012) 131:1223 of a composite functional A½q½qŠŠ, the full deriatie da½qš dqðx may be replaced by any choice of the restricted deriatie da½qš dqðx, C da½q½qšš da½qš dqðx 0 ½qŠ dqðx dqðx 0 dx 0 ; ð28 C dqðx in the case qðx½qš is such that it satisfies the gien constraint C½qŠ C for all qðx s which is the case for Eqs. (21) and (22). It is crucial for both of the aboe cancellations of the ambiguity of restricted deriaties (yielding a unique A½q þ d C qš A½qŠ and a unique da½q½qšš dqðx ) that the integrals [in Eqs. (25a) and (28)]aretakenoerthewholespace.Inthecaseof applications of a deriatie da½qš dqðx where the ambiguity of the corresponding restricted deriatie under a gien constraint does not cancel [like in the case of Eqs. (20) and(23)], the unrestricted deriatie cannot be replaced by another choice da½qš dqðx. We should keep in mind that only the unrestricted C deriatie is capable of deliering the correct change of A½qŠ due to a change of its ariable at a gien point x 0 induced by a change of a function qðx that qðx 0 depends on either qðx 0 ½qŠ obeys some constraint or not. An additional term þk dc½qš dqðx just unnecessarily, and incorrectly, modifies the result gien by da½qš dqðx. To gain more insight into this, one may consider again the example of a composite function gðxðt; yðt, with ðxðt; yðt now obeying the constraint x 2 ðtþy 2 ðt c, for example. Under this constraint on g s ariables, the ambiguous ogðx;y restricted deriatie ox ; ogðx;y oy þ kð2x; 2y; with any choice of k, will correctly delier the full first-order change of g due to a change in ðx; y that is in accordance with the constraint but not a partial first-order change, such as og ox dxðt dt dt: We hae pointed out aboe that (1) it is gðr * f ðr * what may delier a correct local hardness measure instead of gðr * of Eq. (14) alone, and (2) gðr * should be an unrestricted deriatie in order to correctly obtain regional hardnesses by integration of gðr * f ðr * oer molecular regions. Thus, we conclude that a correct local hardness measure may be deliered only by ½½nŠ; ½nŠŠ gðr * f ðr * : ð29 dnðr * Howeer, there is an inherent problem with, as will be dnð * r pointed out in the following section. We should add here that the aboe local quantity may not quite be a local counterpart of hardness, since ðr * in l½; Š is explicitly fixed when obtaining g½; Š: Howeer, the unrestricted deriatie of l with respect to nðr * that keeps ðr * explicitly fixed is the deriatie ½½nŠ;Š, that is, the dnð * r deriatie in Eq. (8). Hence, it is not capable of giing a local measure of hardness. It would only yield a local quantity that is proportional to the Fukui function itself, gðr * gfðr *, which would therefore measure regional softnesses by integration oer molecular regions for soft molecules. (We note that this is precisely the reason for the numerical obserations of Torrent-Sucarrat et al. [41, 42], who found that the regional integrals calculated with Eq. (8) used in Eq. (20) predict high regional hardness for actually soft regions in the case of globally soft systems. This is then not surprising, since this is just what is expected from the Fukui function. The interesting fact, which gies the findings of Torrent-Sucarrat et al. high significance, is that this local hardness expression works well for hard systems [41, 42], which implies that the Fukui function actually indicates local hardness instead of softness in the case of globally hard systems. Therefore, the interpretation of the Fukui function as a general local softness measure has to be reconsidered. But it is clear that gðr * gfðr * also cannot be a local hardness measure.) To close this section, it is worth exhibiting the ambiguity of the regional integrals Eqs. (20) and (23) that would be caused by the ambiguity of dnð * r and dnð * r, respectiely, if the use of those restricted deriaties, instead of the unrestricted deriatie, was actually allowed in the dnð * r case of integrals not coering the whole space. In the case of Eq. (23), the ambiguity of the restricted deriatie appears in the form of a simple additie constant; that is, in the place of a gien dnð * r, any other dnð * r þk can be taken as a choice for the chemical potential deriatie oer the -restricted domain of nðr * s. We may exhibit this ambiguity as dnðr * dnðr * þ k: ð30 This ambiguity then leads to an ambiguity of þk d X dð * r in f X ðr *. The ambiguity Eq. (30) may be expressed with other particular choices of dnð * r replacing in Eq. dnð * r (30). Such a choice is ½; ½nŠŠ ½; Š dðr *0 dnðr * dðr * 0 dnðr * dr*0 f ðr * 0 dðr *0 dnðr * dr*0 ; ð31 which is the analogue of Eq. (8). With this, then, we may also write

7 Theor Chem Acc (2012) 131:1223 Page 7 of 14 dnðr * f ðr * 0 dðr *0 dnðr * dr*0 þ k ð32 (emphasizing that k denotes an arbitrary constant throughout, not to be taken to be identical when appearing in different equations). By inserting Eq. (32) in Eq. (23), we obtain f X ðr * f ðr * 00 dðr * 00 dnðr *0 X dnðr * 0 dr * 0 *00 dr dðr * d X þ k : ð33 dðr * (It can be seen that if X is chosen to be the whole space, Eq. (33) gies back the Fukui function). As regards, dnð r * it is determined only up to a term þ R kðr * 0 dð *0 r 0 dnð * r dr* (with kðr * arbitrary), emerging from the fixed-ðr * constraint, ðr * 0 ½nðr *Š ðr * 0 which can be considered as an infinite number of constraints ( numbered by * 0 r ) on the nðr *domain. This ambiguity may be exhibited as dnðr * dnðr * þ kðr * 0 dðr *0 dnðr * dr*0 ; or with the particular choice Eq. (8) instead of g þ dnðr * kðr * 0 dðr *0 dnðr * dr*0 : dnð r *,as ð34 ð35 With Eq. (35), for example, the ambiguity of Eq. (20) may then be gien as g X g f ðr * 0 *0 dr þ kðr * 00 dðr * 00 dnðr * 0 f ðr *0 *0 *00 dr dr : ð36 X Equation (35) gies back Eq. (17) with the choice kðr * fðr *, as can be seen from Eq. (18). From Eq. (34), one can get back Eq. (15) if kðr * is chosen to be a function uðr * that integrates to 1, utilizing dnð * r R *0 uðr 0 dnð * r dr* and Eq. (9). This then shows that the possible choices of Eq. (14) are een more numerous than has been belieed on the basis of Eq. (15). 4 Ill-definedness of the chemical potential s deriatie with respect to the density For any possible application of Eq. (29), a proper method to ealuate the deriatie of ðr * with respect to the density X is necessary, as reealed by Eq. (18). ðr * is gien as a functional of nðr * by Eq. (9) itself; namely, ðr * 0 df½nš ½nŠ l½nš dnðr * 0 : ð37 That is, in order to obtain the deriatie of Eq. (37) with respect to nðr * to determine dnð r * through Eq. (18), we already need to hae We cannot determine dnð * r dnð * r without further information on l½nš, since from Eqs. (18) and (37), ½nŠ dnðr * g þ g þ ½nŠ dnðr * f ðr * 0 d l½nš df½nš dnðr * dnðr * 0 dr * 0 f ðr * 0 d 2 F½nŠ dnðr * dnðr * 0 dr *0 ; ð38 which is an identity, inoling Eq. (12). l is determined as a functional of the density by a boundary condition in Eq. (37). In the case of potentials bounded at infinity, this will be according to the asymptotic condition ð1 0 on the external potentials, yielding l½nš df½nš dnð1 : ð39 (ote that nðr * 1is taken along one gien direction, just as ð1 needs to be fixed only along one direction which then allows the extension to an een wider domain of external potentials.) We emphasize that there is no other way to determine l as a functional of nðr * than the aboe, since l [either as the chemical potential, that is, the deriatie of E½; Š with respect to, or as the Lagrange multiplier in Eq. (9)] emerges directly as l½; Š, which leaes l½½nš; ½nŠŠ undetermined, as seen aboe. With Eq. (39), then, we obtain ½nŠ dnðr * d2 F½nŠ dnðr * dnð1 : ð40 It is worth obsering that Eq. (40) corresponds to the choice uðr * 0 *0 dðr 1in Eq. (15). Equation (40) seems to offer an easy way to ealuate : Just take the hardness kernel and consider its limit as dnð * r (any) one of its ariables approaches infinity. Howeer, a problem immediately arises. With using approximations for F½nŠ that construct F½nŠ simply in a form F½nŠ R * gðnðr ; rnðr * ; r 2 nðr * ;...dr * (which is common in practical calculations), delta functions dðr * 1appear as multipliers of constant components on the right of Eq. (40), which cannot yield a useful local index. One may argue that this is only an issue of the quality of approximation for

8 Page 8 of 14 Theor Chem Acc (2012) 131:1223 F½nŠ. For example, as has been pointed out by Tozer et al. [73 75], a proper density functional F½nŠ (if continuously differentiable) should yield an exchange correlation potential that has a nonanishing asymptotic alue, in contrast with the commonly used E xc [n] s, with the aboe construction. The problem, howeer, is more fundamental. Consider the (exact) one-electron ersion of the DFT Euler Lagrange equation Eq. (9), dt W ½nŠ dnðr * þ I; ð41 ðr* where T W ½nŠ is the Weizsäcker functional T W ½nŠ 1 8 R jrnð r * j 2 nð r * dr *, exactly alid as F[n] for one-particle densities, while I denotes the ionization potential, which is just minus the ground-state energy for one-particle systems. It is important that T W ½nŠ is not only an exact functional for one-particle densities, in which case its deriatie would possibly differ from the generally alid df½nš dnð * r by a (nðr * -dependent) constant, but in the zerotemperature grand canonical ensemble extension of the energy for fractional electron numbers [76] (see [77] for the spin-polarized generalization), it is the exact F functional for densities with 1[78], implying dt W ½n 1 Š dnðr * df½n 1Š dnðr * (with no difference by a constant), and Ið 1 l ð 1; ð42 ð43 where the minus sign in the subscripts denotes that a leftside deriatie is taken (in the zero-temperature ensemble scheme, the two one-sided deriaties are different in general, implying the existence of deriatie discontinuities [76, 77]). We then hae for ground-state-representable n 1 ðr * s (and nðr * s with 1) l ½nŠ dt W½nŠ dnð1 : ð44 (For one-particle densities that correspond to excited states of the external potential deliered by Eq. (41), on the lefthand side of Eq. (44), only -I[n] can be written.) The deriatie of Eq. (44) with respect to nðr *, d 2 T W ½nŠ dnðr * 0 1 ðrnð1 2 dnð1 4 ðnð1 3 r2 nð1 ðnð1 2 dð1 * 0 r þ 1 rnð1 4 ðnð1 2 rdð1 r* nð1 r2 dð1 * 0 r ð45 (where the corresponding asymptotic limits are to be taken), howeer, is ill-defined for electronic densities. The exponential asymptotic decay e ffiffiffi 2 p 2I r [79, 80] of such densities leads to infinite alues of the factors of the delta functions aboe. (ote though that een without this, the delta functions would not make Eq. (45) a useful local descriptor.) This is not only a formal problem that can be aoided by writing Eq. (45) with the arguments 1 and * 0 r interchanged. The deriatie of l ½nŠ does not exist for electronic densities This can be seen by considering the infinitesimal increment R ½nŠ dnð * r dnðr* dr * of l in a case where the ionization potential corresponding to an electronic density nðr * decreases, that is, the decay of ~nðr * nðr * þdnðr * is slower than nðr * s. In such case, as can be checked readily, Eq. (45) leads to an infinite, whereas it should be I ~I, and this outcome remains the same een if we consider the full Taylor expansion of l, that is, a full change Dl. (The increasing I case may also be considered, with all terms containing ~I anishing in the Taylor expansion.) The Weizsäcker-functional deriatie is not only a one-particle example, but dt W ½nŠ df½nš, a component of in the dnð * r dnð * r general case, itself gies I (which equals l [76]) in the case of finite electron systems, which can be seen if one inserts the density decay e ffiffiffi 2 p 2I r [79, 80] in dt W ½nŠ dt W ½nŠ dnðr * rnðr * 1 r 2 nðr * nðr * 4 nðr * r1 dnð r *, I: ð46 Equation (45) then implies that the component of l ½nŠ that is the most essential for electronic densities yields an illdefined contribution to ½n dnð * r for such densities. It is important to point out that the aboe finding is not only some peculiar feature of the ensemble extension [76] of the energy for fractional s. In the case of other (possibly continuously differentiable) extensions, the deriaties of T W ½nŠ and F½nŠ may differ only by a (densitydependent) constant [53] at a one-particle density n 1 ðr * (since the two functionals are equal for any n 1 ðr * ). This implies that their second deriaties may differ only by some cðr * þcðr * 0, as can be seen by applying (1) the aboe constant-difference rule of deriaties to the deriatie of df½n 1 Š dt W ½n 1 Š þ C½n 1Š itself and (2) the symmetry property dnð * r dnð * r of second deriaties in * r and * 0 r. Then, to obtain ½n 1 Š dnð * r corresponding to a gien fractional generalization of F½nŠ, cðr * 0 þcð1 needs to be added to Eq. (45), where the

9 Theor Chem Acc (2012) 131:1223 Page 9 of 14 function c depends on the generalization. Thus, the problematic Eq. (45) will remain as a component of =dnðr *. A ery recent finding by Hellgren and Gross (HG) [81] gies further support of our conclusion regarding the illdefinedness of =dnðr *. These authors hae showed that the right-side second deriatie of the exchange correlation (xc) component of F½nŠ of the ensemble generalization for fractional s [76] dierges (exponentially) as r 1, by which they hae also placed earlier findings regarding the asymptotic diergence of the xc kernel [82] onto sound theoretical grounds. This diergent behaior has been pointed out to emerge from the integer discontinuity of the xc kernel [81]. Since the left- and the right-side deriatie at a gien nðr * may differ only by a constant (see Appendix of [83] for a proof), the difference between the left- and the right-side second deriatie may only be some cðr * þcðr * 0, on similar grounds as aboe (note that the left-side deriatie and the right-side deriatie of a functional at a gien nðr * may be considered as the deriaties of two different, continuously differentiable functionals that intersect on a subset of nðr * s of a gien ). HG has found that gðr * of gðr * þgðr * 0 : d 2 E xc ½nŠ dnð * r dnð *0 r d2 E xc ½nŠ þ dnð * r dnð *0 r, which is the so-called discontinuity of the xc kernel at integer electron number, dierges exponentially as r 1. F½nŠ is decomposed as F½nŠ T s ½nŠþE xch ½nŠ, with T s ½nŠ being the noninteracting kinetic-energy density functional and E xch ½nŠ the sum of E xc ½nŠ and the classical Coulomb repulsion, or Hartree, functional. Since the latter is continuously differentiable, E xch ½nŠ s discontinuity properties are the same as d E xc ½nŠ s. The diergent behaior of 2 E xch ½nŠ dnð * r dnð *0 r is closely þ related with long-range correlation effects [81, 82], therefore d it is unlikely to be canceled by 2 T s ½nŠ dnð * r dnð *0 r ; consequently, þ d 2 F½nŠ dnð * r dnð *0 r dierges asymptotically, too. This then immediately gies that ½nŠ dnð * r d2 F½nŠ dnð * r dnð1 is ill-defined, being þ þ þ infinite at eery * r Thus, the unrestricted deriatie of l with respect to the density is ill-defined at least, as long as we insist that the zero of energy should be fixed according to ð1 0 for Coulombic potentials. If we chose some other, een though nonphysical, fixation such as R gðr * ðr * dr * 0, for example, (where gðr * is some fixed function that integrates to one and tends fast to zero with * r 1), we would obtain l½nš R gðr * df½nš dnð * r dr* for any potentials, which, then, would yield a proper deriatie but not of the real chemical potential. We refer to [84] for further insight into this issue and for a discussion of the related issue of the ground-state energy as a functional solely of the density. Since the appearance of a preliminary ersion of the present work as an arxi preprint (arxi: ), a related study has been published by Cueas-Saaedra et al. [85]. These authors deal with the problem of how to calculate the unconstrained local hardness Eq. (17) and conclude from similar contradictions as those pointed out in [84] that this local hardness concept is infinitely ill-conditioned and argue further that it dierges exponentially fast asymptotically. Our conclusions thus go further; Eq. (17) is completely ill-defined for electronic systems. 5 Local hardness as a constrained deriatie with respect to the density It has thus been found that one cannot obtain a local hardness measure by gðr * ½½nŠ;½nŠŠ dnð * r f ðr *, since one of the two mathematically allowed forms, Eq. (29), is illdefined, while the other one, gðr * gfðr *, is simply a measure of local softness in the case of soft systems. Howeer, one may raise the question: Could we consider Eq. (14) directly as some local hardness measure, irrespectie of it being able to delier a proper regional hardness concept or not? That is, one would not be interested in getting hardness alues corresponding to regions of molecules, but only in obtaining a pointwise measure, which, besides, should delier the global hardness [ia Eq. (7)] but not regional ones. Although this is a questionable concept [see the argument aboe Eq. (21)], it may still seem to be plausible to consider Eq. (14) some kind of local counterpart of hardness due to its intuitie interpretation as a measure of how the chemical potential changes if the number of electrons is increased locally (by an infinitesimal amount) in a gien external potential setting. Therefore, we will explicitly examine this option, too. So, we are interested in finding a fixation of the ambiguity of Eq. (14) that would properly characterize the chemical potential change due to a density change at * r when the density domain is restricted to densities corresponding to a gien ðr * 0. This requires a proper modification of the unconstrained gradient, which leads us dnð * r to the concept of constrained deriaties [86 88]. (ote the difference of the names restricted deriatie and constrained deriatie [53]. This is not a canonized terminology yet, but the names should be different for these two conceptually, and also manifestly, different entities.) To see how this concept works, consider the case of the simple

10 Page 10 of 14 Theor Chem Acc (2012) 131:1223 R * -conseration constraint, nðr dr * ; that is, the domain of nðr * s is restricted to those integrating to a gien. The functional deriatie da½nš is obtained from the firstorder differential Eq. (25a) by inserting Dnðr * 0 dnð * r dðr * 0 *. r That is, we obtain the functional deriatie (i.e., gradient) by weighting all (independent) directions in the functional domain equally. In a case the functional domain is restricted by some constraint C½nŠ C, the allowed directions are restricted by Eq. (26); consequently, dðr * 0 * r cannot be inserted in Eq. (25a). We need to find a modification of dðr * 0 *, r or in general, of Dnðr * 0, that is in accordance with the constraint. For the -conseration constraint, this is achieed by D nðr * 0 R *0 *00 dðr r uðr * 0 *00 *00 Dnðr dr [86 88], where uðr * is a function that integrates to one. Inserting this D nðr * 0 in Eq. (25a) and taking Dnðr * 00 *00 dðr * r yields the proper modification of a deriatie da½nš : da½nš da½nš R uðr * 0 da½nš 0. The dnð * r d nð * r dnð * r dnð *0 r dr* key for obtaining the constrained deriatie for a gien constraint C½nŠ C, thus, is to find the D C nðr * 0 s that obey the constraint, that is, C½n þ D C nš C½nŠ 0. ow, consider the domain determined by the fixed-ðr * constraint. This domain of nðr * s will be a ery thin domain literally; it will be a single chain of densities nðr * ½; Š, with only changing (nondegeneracy is assumed, of course, which is a basic requirement when dealing with nðr * ½Š). Consequently, there is not much choice in writing a proper D nðr * 0. The only possible form is D nðr * 0 onðr *0 ½; Š D þ higher-order terms: ð47 Inserting this in Eq. (25a), oa½n½; ŠŠ DðA½n; D nš D þ higher-order terms ð48 arises ia an application of the chain rule of differentiation. By utilizing D R Dnðr * 0 *0 *0 *0 dr and taking Dnðr dðr * r (while neglecting the terms higher-order in Dnðr *, which appear due to the nonlinearity of the constraint [88]), from Eq. (48), we then obtain da½nš d nðr * oa½n½; ŠŠ ð49 as the constrained deriatie corresponding to the ðr * - conseration constraint. Interestingly, though not surprisingly (considering the ery restrictie nature of the fixed-ðr * constraint), there is no ambiguity at all in this expression contrary to the -consering deriatie, for example, where uðr * represents an ambiguity. Thus, we obtain that the only mathematically allowed deriatie of l with respect to the density under the fixedðr * constraint (that may bear releance in itself) is d nðr * ol½; Š ð50 (that is, the ðr * -constrained, or ðr * -consering, deriatie of the chemical potential with respect to the density is simply its partial deriatie with respect to ). ote that l½½n½; ŠŠ; ½n½; ŠŠŠ l½; Š. It turns out, thus, that the seere ambiguity of Eq. (14), embodied in Eq. (35), can be narrowed down to the single choice of kðr * 0 which is the constant local hardness of Eq. (8). That is, while the definition of a functional deriatie leads to an ambiguity, Eq. (35), under a fixed-ðr * constraint, this ambiguity disappears if one wishes to use this deriatie in itself as a physical quantity. Howeer, the constant local hardness will not gie a local counterpart of hardness. We can sum up our findings so far as: Here, we hae shown that Eq. (8) is the only mathematically allowed choice if one wishes to obtain a local hardness measure directly by Eq. (14), while preiously we hae shown that if we want to hae a local hardness measure by dnð * r f ðr *, in order to hae regional hardnesses as well, then the only allowed choices are gf ðr * and Eq. (29) but the former cannot be a (general) local hardness measure because of its proportionality to the Fukui function. 6 The parallel problem of defining a local chemical potential Defining a local hardness ia Eq. (14) naturally raises the idea of defining a local counterpart of the chemical potential l itself in a similar fashion. By a oe½;š ð r * local counterpart of l, we mean a local index that indicates the local distribution of l within a gien ground-state system, that is, not some * r -dependent chemical potential concept, like that of [13], that yields l as its special, ground-state, case. We may introduce the following local quantity: de½½nš; ½nŠŠ ~lðr * ; ð51 dnðr * * ð r which parallels Eq. (14). Of course, we then hae the same kind of ambiguity problem as in the case of Eq. (14).

11 Theor Chem Acc (2012) 131:1223 Page 11 of 14 Fixing ðr * as one of the ariables of E½; Š will not yield a ~lðr * that is a local measure of the chemical potential, similarly to Eq. (8), since this ~lðr * will be constant in space the chemical potential itself: oe½; Š d ~lðr * ð * r dnðr * l: ð52 Equation (52) may be obtained in another way as well, since the ground-state energy as a functional of the groundstate density can be obtained ia two routes: E½nŠ E½½nŠ; ½nŠŠ E ½nŠ ½nŠ: ð53 The first route is through E½; Š, while the second is through the energy density functional Eq. (10) of DFT in both cases, the functional dependence of ðr * on nðr * is inserted into the corresponding places. Then, specifically, de½nš de dnð * r may be ½nŠ, which equals l on the basis of dnð * r Eq. (9), giing the ~lðr * of Eq. (52). We note here that the idea of a local chemical potential concept has been raised preiously by Chan and Handy [89], as a limiting case of their more general concept of shape chemical potentials; howeer, they automatically took the energy deriatie with respect to the density as the constant d E ½nŠ dnð * r ignoring other possibilities. The constant local chemical potential concept of Eq. (52) is of course not without use; it may be considered as an equalized * r -dependent chemical potential, defined by lðr * _ df½nš þ [13], which gies a dnð * r ðr* formal background for the electronegatiity equalization principle [13]. The latter lðr *, howeer, is not a local chemical potential in the sense that it would be the local counterpart of a global property (l), but it is rather a kind of intensie quantity, which becomes constant when reaching equilibrium (here, ground state). Similar can be said of the * r -dependent, generalized hardness concept defined by Eq. (13) for general densities. A general property of a ~lðr * defined through Eq. (51) is de½nš ~lðr * f ðr * dr * dnðr * onðr * ½; Š dr * l;, ð54 that is, it gies the chemical potential after integration when multiplied by the Fukui function analogously to Eq. (7). We emphasize again that in spite of the great extent of ambiguity in Eq. (51), all choices will indeed gie l in Eq. (54), due to the fact that the density in onð r* is aried with the external potential fixed, and in cases like this, the ambiguity of the inner deriatie of the composite functional cancels out [53]. An appealing choice of the restricted deriatie in Eq. (51) may be the unrestricted deriatie, de½½nš; ½nŠŠ ~lðr * : ð55 dnðr * This quantity gies to what extent the ground-state energy changes when the density is changed by an infinitesimal amount at a gien point in space. There will be places * r in a gien molecule where an infinitesimal change of nðr * (at the gien * r ) would imply a greater change of the energy, while at other places, it would imply a smaller change in E, going together with a higher and a lower local alue of j~lðr * j, respectiely. The most sensitie site of a molecule toward receiing an additional amount of electron (density) will be the site with the lowest alue of ~lðr *, implying the biggest decrease in the energy due to an increase in the density at * r by an infinitesimal amount but only if the external potential changes accordingly. [ote that since the fixed external potential setting is an inherent part of the concept of the chemical potential, Eq. (55) cannot be a local counterpart of the chemical potential, but when multiplied by the Fukui function, below, a fixation of ðr * appears in the emerging lðr *.] Equation (55) can be ealuated as oe½; Š de½; Š ~lðr * þ dðr *0 dðr * 0 dnðr * dr*0 l þ nðr * 0 dðr *0 dnðr * dr*0 ; ð56a or alternatiely, ~lðr * de ½nŠ dnðr * þ de ½nŠ dðr *0 dðr * 0 dnðr * dr*0 l þ nðr * 0 dðr *0 dnðr * dr*0 ; ð56b where Eqs. (9) and (10) hae been utilized. ote that the second term of Eqs. (56a, 56b) integrates to zero when multiplied by f ðr *,asðr * is independent of. Equation (55) is not only an appealing choice in Eq. (51), but also, on the basis of the argument gien in the case of the local hardness in Sect. 3, it is one of the two mathematically allowed choices to obtain a local chemical potential concept. The emerging local chemical potential is lðr * de dnðr * f ; ðr* which gies regional chemical potentials ia l X lðr * dr * : X ð57 ð58

12 Page 12 of 14 Theor Chem Acc (2012) 131:1223 (Just as in the case of Eq. (29), applying other choices of ~lðr * of Eq. (51) in Eq. (57) would lead to an improper modification of the regional chemical potential alues.) de Unfortunately, howeer, the ealuation of meets the dnð r * same principal problem as the ealuation of. Inserting dnð * r Eq. (37) in Eq. (56) gies de½nš dnðr * l þ ½nŠ dnðr * nðr * 0 d 2 F½nŠ dnðr * dnðr * 0 dr *0 ; ð59 which shows that l½nš s deriatie appears as a component in E½nŠ s deriatie. It is interesting to obsere that the last term of Eq. (59) is just the original local hardness expression of Berkowitz et al. [32], Eq. (16), times. Equation (59) indicates that large (positie) alues of Eq. (16) imply that the global alue l is more decreased by them at the gien points in space. This throws more light upon the recent finding [90] that Eq. (16) is a local indicator of sensitiity toward perturbations, which goes against the essence of the concept of local hardness. (The latter is not surprising in the iew of Sects. 3 and 5 actually nothing supports Eq. (16) asa formula for local hardness.) The other possible way to obtain a local chemical potential measure is de½; Š lðr * f ðr * lfðr * ; ð60 dnðr * similar to the case of local hardness. In that case, gðr * gf ðr * could not gie a correct local hardness measure since the Fukui function f ðr * is actually not an indicator of hard sites, while here the question is as to whether f ðr * can be considered an indicator of local electronegatiity, lðr * ; or not (note that l is negatie, and minus the chemical potential is the electronegatiity). A positie answer would imply, for example, that two soft systems interact through their highest-local-electronegatiity sites. Howeer, to judge the appropriateness of such a possible role of f ðr *,it should first be clarified what to expect from a local electronegatiity concept a matter well-worth of future studies. We note that Eq. (60) gies a possible local hardness index ia gðr * olð r* Also just as in the case of the local hardness, one may examine the question as to what choices of Eq. (51) are allowed if one wishes to use Eq. (51) itself as a local chemical potential measure, without obtaining regional chemical potentials ia l X R X ~lðr* f ðr * dr *. Similarly, as in Sect. 5, it can be shown that actually the only possible choice to fix Eq. (51) s ambiguity is gien by the unique constrained deriatie of the energy (with respect to the : density) corresponding to the fixed-ðr * constraint, which turns out to be de d nðr * oe½; Š ; ð61 that is, the constant ~lðr * of Eq. (52). That is, Eq. (51) cannot be taken as the direct definition of a local chemical potential, as it will only gie back the chemical potential itself. Of course, as noted earlier, a constant local chemical potential can still be a special, equalized, case of a generalized, * r -dependent, chemical potential concept [13] but it will not gie a local reactiity index, characterizing molecular sites within indiidual species. We emphasize that Eq. (61) is not a triial result obtained by the explicit fixation of ðr * de½½nš;š of E½½nŠ; Š, that is, by, but it dnð * r is the deriatie of E½½nŠ; ½nŠŠ with respect to nðr * under the constraint of fixed ðr *. Finally, in parallel with Sect. 3, we may consider the external potential deriatie of the energy, de de½; ½nŠŠ dnðr *0 dðr * dnðr * 0 dr * 0 ; ð62 dðr * which gies the density, Eq. (19). External potential-based reactiity indices hae proed to be useful and hae been much inestigated [91 101]. The regional contributions to Eq. (62) are de½; ½nŠŠ dnðr *0 n X ðr * dnðr * 0 dr * 0 : ð63 dðr * X Equation (63) gies a density component that can be iewed as the contribution of the gien region X to nðr *. Here, an interesting application of Eq. (63) may be worth mentioning (disregarding the fact that ðde=dnðr * is illdefined). A natural decomposition of the density is the one in terms of the occupied Kohn Sham orbitals, nðr * X i1 ju i ðr * j 2 : ð64 One may then look for regions X i (i = 1,,) of the gien molecule that contribute n Xi ðr * ju i ðr * j 2 to nðr * on the basis of Eq. (63). Of course, this may imply a highly ambiguous result; howeer, the number of possible diisions of the molecule into X i s can be significantly reduced by searching for X i s around the intuitiely expectable regions where the single n i ðr * s are dominant. In this way, one would find a spatial diision of a molecule into subshells. To go een further, one might assume that by applying the regions X i found in this way in Eq. (23), the