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ORIGINAL RESEARCH article

Front. Chem., 22 July 2022
Sec. Theoretical and Computational Chemistry
This article is part of the Research Topic Recent Advances, New Perspectives and Applications of Conceptual Density Functional Theory View all 7 articles

Molecular interactions from the density functional theory for chemical reactivity: Interaction chemical potential, hardness, and reactivity principles

Ramn Alain Miranda-Quintana
Ramón Alain Miranda-Quintana1*Farnaz Heidar-ZadehFarnaz Heidar-Zadeh2Stijn FiasStijn Fias3Allison E. A. ChapmanAllison E. A. Chapman3Shubin LiuShubin Liu4Christophe MorellChristophe Morell5Tatiana Gmez
Tatiana Gómez6*Carlos Crdenas,
Carlos Cárdenas7,8*Paul W. Ayers
Paul W. Ayers3*
  • 1Department of Chemistry and Quantum Theory Project, University of Florida, Gainesville, FL, United States
  • 2Department of Chemistry, Queen’s University, Kingston, ON, Canada
  • 3Department of Chemistry and Chemical Biology, McMaster University, Hamilton, ON, Canada
  • 4Research Computing Center, University of North Carolina, Chapel Hill, NC, United states
  • 5Université de Lyon, Universit́e Claude Bernard Lyon 1, Institut des Sciences Analytiques, UMR CNRS 5280, Villeurbanne Cedex, France
  • 6Theoretical and Computational Chemistry Center, Institute of Applied Chemical Sciences, Faculty of Engineering, Universidad Autonoma de Chile, Santiago, Chile
  • 7Departamento de Fisica, Facultad de Ciencias, Universidad de Chile, Santiago, Chile
  • 8Centro para el desarrollo de la Nanociencias y Nanotecnologia, CEDENNA, Santiago, Chile

In the first paper of this series, the authors derived an expression for the interaction energy between two reagents in terms of the chemical reactivity indicators that can be derived from density functional perturbation theory. While negative interaction energies can explain reactivity, reactivity is often more simply explained using the “|dμ| big is good” rule or the maximum hardness principle. Expressions for the change in chemical potential (μ) and hardness when two reagents interact are derived. A partial justification for the maximum hardness principle is that the terms that appear in the interaction energy expression often reappear in the expression for the interaction hardness, but with opposite sign.

1 Introduction

Nearly 35 years ago, Parr recognized that density-functional theory (DFT) could be used not only as a formal alternative to wavefunction-based quantum chemistry and as a computational tool, but also as an interpretative tool through which chemical reactivity could be elucidated (Parr et al., 1978; Parr and Yang, 1989). The special power of the electron-density perspective arises because the mathematical structure of DFT naturally accommodates fractional numbers of electrons and therefore partial electron transfer (Janak, 1978; Parr and Bartolotti, 1982; Perdew et al., 1982; Yang et al., 2000; Ayers, 2008; Fuentealba and Cardenas, 2013; Miranda-Quintana and Bochicchio, 2014; Miranda-Quintana and Ayers, 2016a; Ayers and Mel, 2018), while in the wavefunction theory, the number of electrons is linked to the dimensionality of the wavefunction, and is inherently an integer. Ironically, the great utility of DFT for conceptual purposes arises from the same feature that is most problematic for computational applications (Merkle et al., 1992; Mori-Sanchez et al., 2006; Ruzsinszky et al., 2007; Cohen et al., 2008a; b;Mori-Sanchez et al., 2008; Yang et al., 2016). However, the difficulty of treating fractional electrons computationally in DFT is not important in the context of this paper, where one can assume that an exact (or otherwise accurate ab initio) density functional is used (Levy, 1979; Lieb, 1983; Bartlett et al., 2005; Ayers, 2006).

The use of density functional theory for chemical reactivity (DFT-CR), often called conceptual DFT or chemical DFT (Parr and Yang, 1989; Chermette, 1999; Geerlings et al., 2003; Ayers et al., 2005; Chattaraj et al., 2006; Gazsquez, 2008; Liu, 2009a; Johnson et al., 2012; De Proft et al., 2014; Fuentealba and Cardenas, 2015; Miranda-Quintana, 2018), is now well established, both mathematically, conceptually, and computationally. The greatest successes of DFT-CR are probably linked to the definition of a quantitative scale for the chemical hardness, (Parr and Pearson, 1983), which has led to increased understanding of the hard/soft acid/base (HSAB) theory (Nalewajski, 1984; Nalewajski et al., 1988; Chattaraj et al., 1991; Gazquez and Mendez, 1994; Mendez and Gazquez, 1994; Chattaraj, 2001; Melin et al., 2004; Ayers, 2005; Ayers et al., 2006; Anderson et al., 2007a; Ayers, 2007; Cardenas and Ayers, 2013; Miranda-Quintana R. A., 2017) and an entirely new principle for chemical reactivity and molecular stability, the maximum hardness principle (MHP) (Pearson, 1987; Zhou and Parr, 1989; Parr and Chattaraj, 1991; Pearson and Palke, 1992; Pearson, 1994; Chattaraj, 1996; Pearson, 1999; Ayers and Parr, 2000; Torrent-Sucarrat et al., 2001, 2002; Miranda-Quintana R. A., 2017). One of the subtlest results in DFT-CR is that the HSAB principle is, in fact, an inexorable consequence of the maximum hardness principle (Chattaraj et al., 1991; Parr and Chattaraj, 1991; Chattaraj and Ayers, 2005; Chattaraj et al., 2007), with a similar result also connecting the HSAB and minimum electrophilicity principles (Chattaraj and Nath, 1994; Chattaraj et al., 2001; Chattaraj, 2007; Noorizadeh, 2007; Liu, 2009b; Morell et al., 2009; Pan et al., 2013; Miranda-Quintana, 2017a; Miranda-Quintana and Ayers, 2018b; 2019). Here it is important to clarify that what is usually understood as a “principle” in chemistry is different from the definition in physics. For example, Heisenberg’s uncertainty principle does not admit violations. The principles of chemical reactivity, on the other hand, may have exceptions. Therefore, instead of speaking of principles, one should speak of “rules of thumb” for chemical reactivity. However, for consistency with the literature, here we will use both terms as synonyms.

Almost all applications and theory in DFT-CR have been based on a one-reagent approach: the response functions of a reactant molecule are computed, and then used to predict its reactivity. Despite the usefulness of this approach, it sometimes fails. That is, sometimes understanding the inherent reactivity of a molecule is insufficient; one must discern how well-matched two reagents are. Early attempts at quantifying well-matched-ness were made by Berkowitz, Geerlings, and then, much later, by one of the present authors (Berkowitz, 1987; Langenaeker et al., 1995; Ayers et al., 2006; Anderson et al., 2007a; Ayers, 2007; Anderson et al., 2007b; Ayers and Cardenas, 2013). In the first paper of this series (Miranda-Quintana et al., 2022b), we derived a general expression for the interaction energy between two reagents using DFT-CR and drew the links to other, more computational, DFT theories like density-functional embedding (Cortona, 1991; Vaidehi et al., 1992; Wesolowski and Warshel, 1993; Govind et al., 1999; Wesolowski, 2004; Wesolowski and Leszczynski, 2006), electronegativity equalization molecular mechanics (Yang and Mortier, 1986; Mortier et al., 1985; Mortier et al., 1986; Yang and Mortier, s1986; Mortier, 1987; Rappe and Goddard, 1991; Bultinck et al., 2002a; Bultinck et al., 2002b; Verstraelen et al., 2013), and density-based energy decomposition analysis (Wu et al., 2009).

While the interaction energy provides the most fundamental perspective on chemical reactivity, sometimes it is simpler to understand chemical reactivity using alternative reactivity rules. For example, the “|dμ| big is good” (DMB) rule of Parr and Yang (Parr, 1994; Miranda-Quintana R. A. and Ayers P. W., 2018; Miranda-Quintana et al., 2018) states that favorable chemical interactions are usually associated with a large change in the chemical potential (Miranda-Quintana et al., 2018; Miranda-Quintana and Ayers, 2019; Miranda-Quintana et al., 2021). Likewise, Sanderson’s electronegativity equalization principle (Mortier et al., 1986) and Pearson’s HSAB (Pearson, 1968a; Pearson, 1968b; Nalewajski, 1984; Nalewajski et al., 1988; Chattaraj et al., 1991; Miranda-Quintana R. A., 2017; Miranda-Quintana R. A. et al., 2017), and the more recent minimum electrophilicity principles (Fuentealba et al., 2000b; Morell et al., 2009; Miranda-Quintana R. A., 2017) put these reactivity descriptors in center stage by telling us how to use them to understand and predict chemical reactivity (Sanderson, 1951; Parr et al., 1978). It is not farfetched to say that the biggest triumph of DFT-CR is not only to provide mathematically precise definitions for the reactivity descriptors, but to also give us a robust framework to derive new ones (Ayers et al., 2018; Geerlings et al., 2020). However, more often than not, these derivations have been solely based on the venerable parabolic model of Parr and Pearson (Parr and Bartolotti, 1982; Parr and Pearson, 1983; Chattaraj et al., 1995; Ayers and Parr, 2008; Alain Miranda-Quintana and Ayers, 2016; Heidar-Zadeh et al., 2016b; Miranda-Quintana and Ayers, 2016b; Cárdenas et al., 2016; Franco-Pérez et al., 2018). While powerful and hugely influential, this model can be seen as the simplest representation of electron transfer during a chemical reaction. This simplicity has often been (rightfully) argued as one of its key advantages, but this also means that elementary proofs of the HSAB (Pearson, 1968a; Chattaraj et al., 1991), minimum electrophilicity, maximum hardness, and DMB principles usually ignore electrostatic and polarization effects, and do not include charge transfer effects beyond second order (Chattaraj et al., 1991; Parr and Chattaraj, 1991; Ayers and Parr, 2000; Ayers, 2005; Chattaraj and Ayers, 2005; Ayers and Cardenas, 2013; Miranda-Quintana et al., 2018; Ayers et al., 2022). Recently, some of these approximations have been relaxed (Alain Miranda-Quintana et al., 2016; Miranda-Quintana and Ayers, 2016a; Miranda-Quintana R. A., 2017; Miranda-Quintana et al., 2021; Miranda-Quintana et al., 2022a), which strengthens the support for these principles, but the more realistic two-reagent picture remains largely unexplored (Ayers et al., 2006; Ayers, 2007; Chattaraj et al., 2007; Miranda-Quintana R. A. et al., 2017).

The chemical potential (Parr et al., 1978),

μ=(EN)v(r)(1)

measures the intrinsic Lewis acid/base strength of a molecule and can be considered to be minus one times the electronegativity. The essence of the rule, then, is that favorable molecular interactions between acids and bases “quench” the acidity/basicity of the reagents as much as possible (forming, in the extreme case, nearly inert salts). Alternatively, favorable chemical changes are associated with large changes in molecular electronegativity. Even though the “|dμ| big is good” rule was first formulated more than 30 years ago, its theoretical provenance has only recently begun to be elucidated (Miranda-Quintana et al., 2018; Miranda-Quintana and Ayers, 2019; Miranda-Quintana et al., 2021).

The maximum hardness principle (MHP) is a particularly interesting case, with roots that seem less transparent than many of the aforementioned reactivity rules. (Pearson, 1987; Pearson and Palke, 1992; Pearson, 1993; Chattaraj, 1996; Pearson, 1999). The maximum hardness principle indicates that more stable conformations are associated with a large hardness,

η=(2EN2)v(r)(2)

A corollary of this principle is that the harder a molecule is, the more stable it is. The problem is that the mathematical assumptions under which the maximum hardness principle has been proved (fixed molecular geometry and either constant electron number or constant chemical potential) do not match the conditions under which the principle is usually applied, because the MHP is most commonly used to study molecular rearrangements (Torrent-Sucarrat et al., 2001; 2002). For example, there has been substantial recent interest in using the initial hardness response (the change in hardness associated with the initial approach of two reagents) to study pericyclic reactions (De Proft et al., 2006; Ayers et al., 2007; De Proft et al., 2008; Geerlings et al., 2012).

In the remainder of this paper, we differentiate the energy expression in Eq. 33 of the first paper in this series with respect to the number of electrons. This gives the change in chemical potential (related to the first derivative) hardness (second derivative) due to the interactions between two reagents. These expressions are then used to mathematically justify the “|dμ| big is good” and maximum hardness principles.

The key expression [Eq. 33 from (Miranda-Quintana et al., 2022b)] is

In this equation, ΔUAB is the change in the total energy (U = E + Vnn, where Vnn is the nuclear-nuclear repulsion energy) when the reagents A and B come together. ΔN is the change in the number of electrons in the reagent. Superscript “0” indicates that the term is evaluated for the isolated reagent; subscripts index the reagents. As explained in (Miranda-Quintana et al., 2022b), Eq. 3 can be iterated. The only difference between Eq. 3 and the equation Eq. 33 in (Miranda-Quintana et al., 2022b), is that last it was assumed that ΔNA=ΔNB. That is, every electron that leaves one reagent goes to the other reagent, and the number of electrons in the combined system does not change. In this paper, we will extend this analysis to consider changes in the total number of electrons,

ΔN=ΔNA+ΔNB(4)

Most of the reactivity indicators that enter into Eq. 3 are well-known in DFT-CR: the Fukui function f(r) (Parr and Yang, 1984; Yang et al., 1984; Ayers P. W. and Levy M., 2000; Heidar-Zadeh et al., 2016a; Fuentealba et al., 2016), the dual descriptor f(2)(r) (Fuentealba and Parr, 1991; Morell et al., 2005, 2006; Ayers et al., 2007; Cardenas et al., 2009b; Geerlings et al., 2012), and the electron density ρ(r). The change in energy and density upon polarization of one reagent by another are defined through,

ΔEpol12Δv(r)Δv(r)χ(r,r)drdr12Δρ(pol)[Δv;r]Δv(r)dr.(5)

where χ(r,r’) is the linear-response (or polarizability) kernel χ(r,r’). For convenience, Eq. 3 is written in terms of the nuclear charge density instead of the external potential, (Ayers et al., 2009)

v(r)=z(r)|rr|dr(6)

The molecular electrostatic potential (Politzer, 1980; Politzer and Truhlar, 1981; Sjoberg and Politzer, 1990; Gadre et al., 1992; Shirsat et al., 1992; Murray et al., 1996; Suresh and Gadre, 1998; Politzer and Murray, 2002),

Φ(r)=z(r)ρ(r)|rr|dr(7)

is not traditionally considered a reactivity indicator in DFT-CR, but it can be placed in a DFT context by differentiating the total energy (including Vnn) with respect to the external potential (Ayers and Parr, 2001; Anderson et al., 2007a). The non-additive kinetic and exchange-correlation energies in the first line of Eq. 3 capture electron-pairing and steric effects, (Gordon and Kim, 1972; Wesolowski and Warshel, 1993; Wesolowski and Warshel, 1994; Liu, 2007; Wu et al., 2009)

Tsnon-add[ρA,ρB]Ts[ρAB]Ts[ρA]Ts[ρB](8)
Excnon-add[ρA,ρB]Exc[ρAB]Exc[ρA]Exc[ρB].(9)

1.1 The interaction hardness and chemical potential

To elucidate the MHP, we need to compute the change in hardness due to the interactions between A and B. To do this, we define the interaction hardness

ΔηAB=ηABηA0ηB0(10)

and notice that this quantity can be computed by differentiating the interaction energy,

ΔηAB=(2ΔUABN2)vAB(r)=(2UABN2)vAB(r)(2UA0N2)vA0(r)(2UB0N2)vB0(r)(11)

The treatment of the chemical potential is a bit more nuanced, because since it is an intensive property the correct expression is (Miranda-Quintana et al., 2018):

ΔμAB=μAB12(μA+μB)(12)

with

ΔμAB=(ΔUABN)vAB(r)=(UABN)vAB(r)12{(UA0N)vA0(r)+(UB0N)vB0(r)}(13)

The total energy can be used in the previous equations because the nuclear-nuclear repulsion energy does not depend on the number of electrons.

As complicated as it is, Eq. 3 already contains assumptions, most notably assumptions about the “effective external potential” that electrons in one reagent feel due to the electrons and nuclei in the second reagent (Ayers and Parr, 2001; Ayers et al., 2005; Cohen and Wasserman, 2007; Cohen et al., 2009; Liu et al., 2009; Elliott et al., 2010; Osorio et al., 2011). We also must assume that the higher-order terms in the Taylor series (which are implicitly neglected or averaged over in a Taylor-series-with-remainder strategy) are negligible. Extension to include higher-order terms can be made, with commensurate increased complexity in Eq. 3 (Senet, 1996; Geerlings and De Proft, 2008; Cardenas et al., 2009a; Heidar-Zadeh et al., 2016b). Finally, we must assume that the derivatives exist, which implicitly requires that the system is not quasi-degenerate for perturbations of the strength relevant for the analysis. Quasi-degeneracy (even exact degeneracy) can be treated, however, if the derivatives are reinterpreted as differentials (Cardenas et al., 2011; Bultinck et al., 2013a; Bultinck et al., 2013b; Pino-Rios et al., 2017; Cerón et al., 2020; Bultinck and Cárdenas, 2022; Cárdenas et al., 2022). Other effects (e.g., temperature-dependence, spin-specificity) can likewise be treated without essential difficulty, merely by an extension of definition and notation (Galvan et al., 1988; Ghanty and Ghosh, 1994; Ayers and Yang, 2006; Garza et al., 2006; Perez et al., 2008; Franco-Perez et al., 2015a; Franco-Perez et al., 2015b; Alain Miranda-Quintana and Ayers, 2016; Miranda-Quintana R. A. and Ayers P. W., 2016; Franco-Perez et al., 2017a; Franco-Perez et al., 2017b; Franco-Pérez et al., 2017; Robles et al., 2018; Gázquez et al., 2019). The following analysis can also be treated at an atom (or functional-group) condensed level: the integrations over space are merely replaced by sums over atom labels (Yang and Mortier, 1986; Fuentealba et al., 2000a; Ayers et al., 2002; Tiznado et al., 2005; Bultinck et al., 2007; Fuentealba et al., 2016; Echegaray et al., 2017). That provides a more computationally practical form for these results and draws the link to electronegativity equalization methods more strongly.

To obtain expressions for µAB and ηAB that are simple enough to be useful, some further assumptions are needed. Before proceeding, let us rewrite Eq. 3 using a shorter, more convenient notation that will help us with the upcoming manipulations:

ΔUAB[ΔNA,ΔNB]=μAΔNA+μBΔNB+12ηA(ΔNA)2+12ηB(ΔNB)2+hABΔNAΔNB+θA(ΔNA)2ΔNB+θBΔNA(ΔNB)2+14cAB(ΔNAΔNB)2(14)

Here hAB is the Coulomb interaction between the fragments’ Fukui functions, cAB is the Coulomb interaction between the fragments’ dual descriptors, and θA is the Coulomb interaction between the dual descriptor of fragment A and the Fukui function of fragment B. To obtain this expression, we just grouped terms according to the powers of ΔNA,ΔNB, and neglected the terms that are N-independent (since we are dealing with derivatives with respect to N these terms won’t be relevant). For instance, μA, denotes the chemical potential of fragment A in the presence of B and includes not only a contribution from the chemical potential of the isolated fragment A, but also contributions from the interaction of A and B (e.g., the term fA(r)ΦB0(r)dr). From Eq. 14 it is easy to see that the energy will be minimized if the coefficient of the ΔNAΔNB term is as big as possible (since ΔNAΔNB<0); while the coefficient of the (ΔNAΔNB)2 term is as small as possible (ideally, a negative number). That is:

hAB=fA(r)fB(r)|rr|drdr>0(15)
cAB=fA(2)(r)fB(2)(r)|rr|drdr<0(16)

In order to evaluate Eq. 11 we need to know the interaction energy as a function of the number of electrons. The number of electrons enters the expression for the interaction energy (Eq. 3) through the reactivity indicators and through the extent of electron transfer. We assume that the reactivity indicators in Eq. 3 do not depend on the number of electrons; this is reliable if the expression in Eq. 3 has already been iterated to convergence or, failing that, that the result of the first iteration (where all the reactivity indicators are computed for the isolated reagents) suffices. Typically, we would take ΔNA+ΔNB=0, and then solve for the amount of charge transfer that minimizes the interaction energy. However, this leads to expressions that are far too complicated to analyze. Minimizing Eq. 3 requires solving a cubic equation, and even when there is a clear indication of which root should be taken, the resulting expressions are of little help. Hence, we need to introduce a second approximation, assuming that the dependence of ΔNA and ΔNB on the number of electrons is linear,

ΔNA=ΔNA0+dAδΔNB=ΔNB0+dBδ(17)

Here, ΔNA0,ΔNB0 are just some convenient reference values used as starting points to expand the correct changes in particle numbers in A and B, respectively.

To calculate the chemical potential and hardness of the product, it is convenient to consider an excess charge on the products, namely (Miranda-Quintana R. A., 2017):

ΔNA+ΔNB=δ(18)

So this implies that

ΔNA=ΔNA0+dAδΔNB=ΔNB0+(1dA)δ(19)

Now we can substitute Eq. 19 in Eq. 14, consider an infinitesimal δ, and truncate at second order:

ΔUAB[ΔNA,ΔNB]=δ{12cAB(2dA1)(ΔNA0)3+ΔNA0[dA(ηA+ηB)2hABdA+hABηB]+μAdAμBdA+μB+(ΔNA0)2[3θAdA+θA+θB(3dA2)]}+14δ2{cAB[6(dA1)dA+1](ΔNA0)2+2ηA(dA)24hABdA(dA1)+2ηB(dA1)2+4ΔNA0[3(dA)2(θBθA)+2θAdA4θBdA+θB]}(20)

Notice that we have omitted the terms that do not depend on δ. The linear and quadratic coefficients are the equations for the chemical potential and (twice the) hardness of the reaction product, respectively. Namely,

μAB={12cAB(2dA1)(ΔNA0)3+ΔNA0[dA(ηA+ηB)2hABdA+hABηB]+μAdAμBdA+μB+(ΔNA0)2[3θAdA+θA+θB(3dA2)]}(21)
ηAB=12{cAB[6(dA1)dA+1](ΔNA0)2+2ηA(dA)24hABdA(dA1)+2ηB(dA1)2+4ΔNA0[3(dA)2(θBθA)+2θAdA4θBdA+θB]}(22)

These are the fundamental expressions of this manuscript, as they serve as the basis for our forthcoming analyses.

1.2 The maximum hardness principle

A key point in Eq. 17 is how to estimate ΔNA0 and dA. Perhaps the simplest route is to just use the standard parabolic model result, thus:

ΔNA0=μB0μA0ηA0+ηB0dA=ηB0ηA0+ηB0(23)

In the case of the hardness, this leads to a relatively simple expression:

ηAB(1ηA0+ηB0)2(((ηB0)2fA(2)(r)ΦB0(r)+(ηA0)2fB(2)(r)ΦA0(r))dr+((ηB0)2ΔρB(pol)[ΦA0,r]fA(2)(r)+(ηA0)2ΔρA(pol)[ΦB0,r]fB(2)(r))|rr|drdr+2ηAηBfA(r)fB(r)|rr|drdr+2(μA0μB0)ηA0+ηB0(ηA0(2ηB0ηA0)fA(r)fB(2)(r)ηB0(2ηA0ηB0)fB(r)fA(2)(r))|rr|drdr+((μA0μB0)2((ηA0)24ηA0ηB0+(ηB0)2)(ηA0+ηB0)2)×(fA(2)(r)fB(2)(r)|rr|drdr)+ηA0ηB0(ηA0+ηB0))(24)

where, for the sake of completeness, we have reverted back to the original notation, showing all the contributions to the interaction hardness in terms of both reagents.

Since ηAηB>0, and the reaction energy tends to decrease when hAB increases (Eq. 15), it is straightforward to corroborate that the “Fukui function” pairing that minimizes the energy (cf. Eq. 15) also guaranties a maximum hardness value.

Analyzing the term corresponding to the electrostatic interaction of the dual descriptors, namely (ηA0)24ηA0ηB0+(ηB0)2, is a bit more involved. Since the energy tends to decrease when cAB decreases (Eq. 16), the interaction between the dual descriptors that minimizes the energy will maximize the hardness if

(ηA0)24ηA0ηB0+(ηB0)2<0(25)

However, this will be true only when:

23<ηB0ηA0<2+3(26)

This might seem like an odd result, since at it (falsely) seems like it introduces an asymmetry between the reactants. However, because (23)(2+3)=1, one can rearrange this equation so that the symmetry of the expression with respect to permutation of the reactant labels is clear: 23<ηB0ηA0<2+323<ηA0ηB0<2+3. Notice that Eq. 26 means that the MHP will hold when the hardnesses of the reactants are not very different. This implies that there could be cases where minimizing the energy actually implies that the hardness will tend to decrease. Equation 26 is consistent with other results from the literature which indicates that in double-exchange reactions of acids and bases, the HSAB and DMB rules are be fulfilled only when the differences in hardness of the reactants are not too large (Cardenas and Ayers, 2013; Miranda-Quintana et al., 2018). This is not especially concerning as the restriction on the hardness values is rarely implicated. For example, excluding the (very hard) noble gas atoms, Eq. 26 is violated by very few atom pairs within the periodic table (Cárdenas et al., 2016), and the pairs that do violate the constraint (e.g. Cesium and Fluorine) are so extreme that there is little need for additional tools to elucidate their reactivity.

These results provide some support for the MHP, but they rely on the parabolic model (Eq. 23). We can obtain a more realistic estimate of ΔNA0 and dA if we work instead with a simplified version of Eq. 14 where we neglect all cross-terms (i.e., ΔNAΔNB,(ΔNA)2ΔNB,ΔNA(ΔNB)2,(ΔNAΔNB)2). Thus, we are still working with a parabolic model, but now the descriptors have some information regarding the perturbation induced by the other reagent. Therefore now we will have

ΔNA0=μBμAηA+ηBdA=ηBηA+ηB(27)

Now the expression for the energy reads:

ΔUAB[ΔNA,ΔNB]δ{cAB(ηBηA)(μAμB)32(ηA+ηB)4+hAB(ηBηA)(μAμB)(ηA+ηB)2+μAηB+μBηA(ηA+ηB)+(μAμB)2[ηA(θA2θB)+ηB(θB2θA)](ηA+ηB)3}+δ24(ηA+ηB)4{cAB(ηA24ηAηB+ηB2)(μAμB)2+2ηAηB(ηA+ηB)3+4hABηAηB(ηA+ηB)+4(ηA+ηB)(μAμB)[ηBθA(ηB2ηA)ηAθB(ηA2ηB)]}(28)

Hence

ηAB=12(ηA+ηB)4{cAB(ηA24ηAηB+ηB2)(μAμB)2+2ηAηB(ηA+ηB)3+4hABηAηB(ηA+ηB)+4(ηA+ηB)(μAμB)[ηBθA(ηB2ηA)ηAθB(ηA2ηB)]}(29)

In this case we can obtain support for the MHP in the same way as we did before: the coefficient of the hAB term is positive, therefore the “Fukui function pairing” that minimizes the energy also guaranties a maximum hardness value.

As for the dual descriptor interactions, now the MHP will be true if:

23<ηBηA<2+3(30)

As we saw previously, this result also indicates that just maximizing the hardness might not always lead to more favorable interactions between various reagents. Only when the harder reactant is no more than ∼3.7 times higher than the hardness of the softer reagent might the formation of the hardest product will be favored.

1.3 The “|dμ| big is good” principle

Given the generally more complicated nature of the expressions involved in the treatment of the DMB principle, we will only consider the reference ΔNA0 and νA presented in Eq. 27. Substituting these expressions into Eq. 20:

ΔμAB=(ηBηA)(μAμB)2(ηA+ηB){1cAB(μAμB)2(ηA+ηB)3+2hAB(ηA+ηB)}+(μAμB)2[ηA(θA2θB)+ηB(θB2θA)](ηA+ηB)3(31)

As it was the case in the last part of the discussion on the MHP, we will discard the cross-terms corresponding to the (ΔNA)2ΔNB,ΔNA(ΔNB)2 factors, which means that we can write:

ΔμAB=(ηBηA)(μAμB)2(ηA+ηB){1cAB(μAμB)2(ηA+ηB)3+2hAB(ηA+ηB)}(32)

With this approximation, the energy model reduces to:

ΔUAB[δ](μAμB)22(ηA+ηB){2(ηA+ηB)2(ηA+ηB+2hAB)cAB(μAμB)22(ηA+ηB)3}δ{cAB(ηBηA)(μAμB)32(ηA+ηB)4+hAB(ηBηA)(μAμB)(ηA+ηB)2+μAηB+μBηA(ηA+ηB)}+δ24(ηA+ηB)4{cAB(ηA24ηAηB+ηB2)(μAμB)2+2ηAηB(ηA+ηB)3+4hABηAηB(ηA+ηB)}(33)

Here we have elected to explicitly include the constant (the δ-independent term):

ΔUAB=ΔUAB[0](μAμB)22(ηA+ηB){2(ηA+ηB)2(ηA+ηB+2hAB)cAB(μAμB)22(ηA+ηB)3}.(34)

Equations 32 and 34 can be rewritten as:

ΔμAB=ΔμAB(PP){1cAB(μAμB)2(ηA+ηB)3+2hAB(ηA+ηB)}(35)
ΔUAB=ΔUAB(PP){2(ηA+ηB)2(ηA+ηB+2hAB)cAB(μAμB)22(ηA+ηB)3},(36)

where the (PP) index indicates that these are the expressions obtained using the Parr-Pearson parabolic model. (Parr and Pearson, 1983)

Without losing any generality, we can assume that A is the acid, namely: μA<μB. Then, since, the terms in brackets in Eqs 35, 36 are always positive (cf. Eq. 16, proving the DMB is equivalent to showing that:

(ΔUΔμ)>0whenηA<ηB(37)
(ΔUΔμ)<0whenηA>ηB(38)

which can be rephrased as:

sgn(ΔUΔμ)=sgn(ηBηA)(39)

Closely following the strategy employed in previous approaches to DMB (Miranda-Quintana et al., 2018), we can take (notice that equivalent expressions for the change in reactant B can be obtained by simply exchanging the indices in the following equations)

ΔUΔμ=ΔUμAμAΔμ+ΔUηAηAΔμ(40)

resulting in:

ΔUΔμ=T+TA,(41)

where:

T=(μAμB){2[(ηA+ηB)2(ηA+ηB+2hAB)cAB(μAμB)2](ηAηB)[(ηA+ηB)2(ηA+ηB+2hAB)3cAB(μAμB)2]}(42)
TA=(μAμB)[(ηA+ηB)2(ηA+ηB+2hAB)2cAB(μAμB)2]{ηA[3cAB(μAμB)2+2ηB2(5hAB+3ηB)]+ηB[5cAB(μAμB)22ηB2(3hAB+ηB)]+2ηA2[ηA(hABηB)ηB(hAB+3ηB)]}(43)

It is easy to check that the sign of T only depends on the sign of ηBηA, so we only need to show that sgn(TA)=sgn(ηBηA). For this we only need to analyze the sign of the expression:

T˜A=ηA[3cAB(μAμB)2+2ηB2(5hAB+3ηB)]ηB[5cAB(μAμB)22ηB2(3hAB+ηB)]2ηA2[ηA(hABηB)ηB(hAB+3ηB)](44)

which we can rewrite in a form that makes its dependence on the sign of ηBηA more apparent,

T˜A=2(ηBηA)[ηA2(hABηB)+ηB2(5hAB+7ηB)+4ηAηB232cAB(μAμB)2][2cABηB(μAμB)2+16ηB3(ηB+hAB)](45)

Ignoring, for the moment, the term on the second line, DMB follows if:

ηA2(hABηB)+ηB2(5hAB+7ηB)+4ηAηB23cAB(μAμB)2>0(46)

This inequality is very likely to hold in most cases (particularly, in the weakly-interacting regime). The last three terms are always positive. The first term is likewise positive if that hAB<ηB. This can always be ensured by taking the initial separation of reagents to be sufficiently large (Yañez et al., 2021). For instance, for Fukui functions localized on two atomic sites separated by 5 Å, hAB is less than 3 eV. Hence, it is safe to assume that inequality 46) holds.

It remains to analyze the term [2cABηB(μAμB)2+16ηB3(ηB+hAB)] , which cannot be factored in terms of ηBηA.

These results largely point out to the validity of the DMB principle. The only potential incongruity comes in the form of the hard-to-factor terms in the expressions of T˜A and T˜B. However, these terms appear because we decided to perform a more rigorous mathematical treatment of the foundations of this principle. Should we have chosen to go with more qualitative arguments (as it was the case for the MHP, and some other discussions of reactivity principles), the evidence in favor of DMB would have been even stronger. Just note that, in accordance with Eqs 15, 16, the hAB>0 and cAB<0 conditions that guarantee a minimum interaction energy, are also the ones that, following Eq. 31, will tend to maximize |Δμ|. However, due to the central role of DMB in chemical reactivity, it is illustrative to perform a more detailed analysis of the conditions supporting its validity. Our analysis here indicates that the DMB principle is valid where certain terms become negligible. The confounding terms become negligible when reagents are sufficiently far apart, suggesting that failures of the DMB principle to predict reactivity are most likely to occur in cases where the activated complex in a chemical reaction is tightly bound or does not exist (e.g., barrierless reactions). This is consistent with the (already well-established) reduction of the efficacy of conceptual density functional theory in such cases, due mainly to the importance of higher-order terms in the perturbative expansion.

2 Summary

In this work we have shown that starting from an expression for the interaction energy between reactants deduced in the first part of this series of papers, it is possible to find a theoretical support for the maximum hardness principle. The main difference between this work and other related papers is that here the perturbation of one reactant on another is explicitly accounted for. In summary, the MHP is fulfilled if 1) the electrostatic interaction of the Fukui functions of the reactants is positive, (Berkowitz, 1987; Ayers P. W. and Levy M., 2000; Osorio et al., 2011), 2) the electrostatic interaction of the dual descriptors of the reactants is negative and, (Ayers et al., 2007; Cardenas et al., 2009b), 3) the relation between the hardnesses of the reactants is bounded by the inequality (30).

Similarly, we provided more arguments favoring the DMB principle, which unsurprisingly also seems to hold when we take in to account the full two-reagent picture. Contrary to the MHP case, establishing the validity of the DMB principle requires more caveats, and additional mathematical scrutiny is warranted. However, even simple qualitative discussion of the form of the expression for the change in reagents’ chemical potential support its validity, and also support the favorability of large Coulomb interactions between fragments’ Fukui functions and small Coulomb interactions between fragments’ dual descriptors.

Overall, the two-reagent model discussed in this and the previous contribution provides a more complete picture of chemical reactivity, encompassing several previous approaches, while also strengthening the arguments supporting several reactivity principles. Further applications of this framework are underway and will be presented elsewhere.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding authors.

Author contributions

PA, SL, CC, and RM contributed to conception and design of the study. PA and RM wrote the first draft of the manuscript. CC, RM, TG wrote sections of the manuscript. All authors contributed to manuscript revision, read, and approved the submitted version.

Conflict of interest

The reviewer (WT) declared a past co-authorship with the author (CC) to the handling editor.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Acknowledgments

PWA thanks the Canada Research Chairs, NSERC, and Compute Canada for funding. FH-Z was supported by NSERC and Compute Canada. SL was supported as part of the UNC EFRC: Solar Fuels, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award No. DE-SC0001011. CC acknowledges financial support from Fondecyt through Grant Nos.1220366 and also by Centers of Excellence With Basal-Conicyt Financing, Grant FB0807. RM-Q thanks support from the University of Florida in the form of a start-up grant.

References

Anderson, J. S. M., Melin, J., and Ayers, P. W. (2007a). Conceptual density-functional theory for general chemical reactions, including those that are neither charge- nor frontier-orbital-controlled. 1. Theory and derivation of a general-purpose reactivity indicator. J. Chem. Theory Comput. 3, 358–374. doi:10.1021/ct600164j

PubMed Abstract | CrossRef Full Text | Google Scholar

Anderson, J. S. M., Melin, J., and Ayers, P. W. (2007b). Conceptual density-functional theory for general chemical reactions, including those that are neither charge- nor frontier-orbital-controlled. 2. Application to molecules where Frontier molecular orbital theory fails. J. Chem. Theory Comput. 3, 375–389. doi:10.1021/ct6001658

PubMed Abstract | CrossRef Full Text | Google Scholar

Ayers, P. W. (2005). An elementary derivation of the hard/soft-acid/base principle. J. Chem. Phys. 122, 141102. doi:10.1063/1.1897374

PubMed Abstract | CrossRef Full Text | Google Scholar

Ayers, P. W., Anderson, J. S. M., and Bartolotti, L. J. (2005). Perturbative perspectives on the chemical reaction prediction problem. Int. J. Quantum Chem. 101, 520–534. doi:10.1002/qua.20307

CrossRef Full Text | Google Scholar

Ayers, P. W. (2006). Axiomatic formulations of the Hohenberg-Kohn functional. Phys. Rev. A 73, 012513. doi:10.1103/physreva.73.012513

CrossRef Full Text | Google Scholar

Ayers, P. W., and Cárdenas, C. (2013). Communication: A case where the hard/soft acid/base principle holds regardless of acid/base strength. J. Chem. Phys. 138, 181106. doi:10.1063/1.4805083

PubMed Abstract | CrossRef Full Text | Google Scholar

Ayers, P. W., Fias, S., and Heidar-Zadeh, F. (2018). The axiomatic approach to chemical concepts. Comput. Theor. Chem. 1142, 83–87. doi:10.1016/j.comptc.2018.09.006

CrossRef Full Text | Google Scholar

Ayers, P. W., and Levy, M. (2000a). Perspective on "Density functional approach to the frontier-electron theory of chemical reactivity" - Parr RG, Yang W (1984. J. Am. Chem. Soc. 106, 4049–4050.

Google Scholar

Ayers, P. W., and Levy, M. (2000b). Perspective on "Density functional approach to the frontier-electron theory of chemical reactivity". Theor. Chem. Accounts 103, 353–360. doi:10.1007/978-3-662-10421-7_59

CrossRef Full Text | Google Scholar

Ayers, P. W., Liu, S., and Li, T. (2009). Chargephilicity and chargephobicity: Two new reactivity indicators for external potential changes from density functional reactivity theory. Chem. Phys. Lett. 480, 318–321. doi:10.1016/j.cplett.2009.08.067

CrossRef Full Text | Google Scholar

Ayers, P. W., and Mel, L. (2018). Levy constrained search in fock space: An alternative approach to noninteger electron number. 物理化学学报 34, 625–630.

Google Scholar

Ayers, P. W., Mohamed, M., and Heidar‐Zadeh, F. (2022). The hard/soft acid/base rule: A perspective from conceptual density‐functional theory. Concept. Density Funct. Theory Towards a New Chem. React. Theory 1, 263–279. doi:10.1002/9783527829941.ch14

CrossRef Full Text | Google Scholar

Ayers, P. W., Morell, C., De Proft, F., and Geerlings, P. (2007). Understanding the Woodward-Hoffmann rules by using changes in electron density. Chem. Eur. J. 13, 8240–8247. doi:10.1002/chem.200700365

PubMed Abstract | CrossRef Full Text | Google Scholar

Ayers, P. W., Morrison, R. C., and Roy, R. K. (2002). Variational principles for describing chemical reactions: Condensed reactivity indices. J. Chem. Phys. 116, 8731–8744. doi:10.1063/1.1467338

CrossRef Full Text | Google Scholar

Ayers, P. W., and Parr, R. G. (2008). Local hardness equalization: Exploiting the ambiguity. J. Chem. Phys. 128, 184108. doi:10.1063/1.2918731

PubMed Abstract | CrossRef Full Text | Google Scholar

Ayers, P. W., Parr, R. G., and Pearson, R. G. (2006). Elucidating the hard/soft acid/base principle: A perspective based on half-reactions. J. Chem. Phys. 124, 194107. doi:10.1063/1.2196882

PubMed Abstract | CrossRef Full Text | Google Scholar

Ayers, P. W., and Parr, R. G. (2000). Variational principles for describing chemical reactions: The fukui function and chemical hardness revisited. J. Am. Chem. Soc. 122, 2010–2018. doi:10.1021/ja9924039

CrossRef Full Text | Google Scholar

Ayers, P. W., and Parr, R. G. (2001). Variational principles for describing chemical reactions. Reactivity indices based on the external potential. J. Am. Chem. Soc. 123, 2007–2017. doi:10.1021/ja002966g

PubMed Abstract | CrossRef Full Text | Google Scholar

Ayers, P. W. (2008). The dependence on and continuity of the energy and other molecular properties with respect to the number of electrons. J. Math. Chem. 43, 285–303. doi:10.1007/s10910-006-9195-5

CrossRef Full Text | Google Scholar

Ayers, P. W. (2007). The physical basis of the hard/soft acid/base principle. Faraday Discuss. 135, 161–190. doi:10.1039/b606877d

PubMed Abstract | CrossRef Full Text | Google Scholar

Ayers, P. W., and Yang, W. (2006). Legendre-transform functionals for spin-density-functional theory. J. Chem. Phys. 124, 224108. doi:10.1063/1.2200884

PubMed Abstract | CrossRef Full Text | Google Scholar

Bartlett, R. J., Lotrich, V. F., and Schweigert, I. V. (2005). Ab initiodensity functional theory: The best of both worlds? J. Chem. Phys. 123, 062205. doi:10.1063/1.1904585

PubMed Abstract | CrossRef Full Text | Google Scholar

Berkowitz, M. (1987). Density functional approach to Frontier controlled reactions. J. Am. Chem. Soc. 109, 4823–4825. doi:10.1021/ja00250a012

CrossRef Full Text | Google Scholar

Bultinck, P., and Cárdenas, C. (2022). Chemical response functions in (Quasi‐)Degenerate states. Concept. Density Funct. Theory Towards a New Chem. React. Theory 1, 93–109. doi:10.1002/9783527829941.ch6

CrossRef Full Text | Google Scholar

Bultinck, P., Cardenas, C., Fuentealba, P., Johnson, P. A., and Ayers, P. W. (2013a). Atomic charges and the electrostatic potential are ill-defined in degenerate ground states. J. Chem. Theory Comput. 9, 4779–4788. doi:10.1021/ct4005454

PubMed Abstract | CrossRef Full Text | Google Scholar

Bultinck, P., Cardenas, C., Fuentealba, P., Johnson, P. A., and Ayers, P. W. (2013b). How to compute the fukui matrix and function for systems with (Quasi-)Degenerate states. J. Chem. Theory Comput. 10, 202–210. doi:10.1021/ct400874d

PubMed Abstract | CrossRef Full Text | Google Scholar

Bultinck, P., Fias, S., Van Alsenoy, C., Ayers, P. W., and Carbó-Dorca, R. (2007). Critical thoughts on computing atom condensed Fukui functions. J. Chem. Phys. 127, 034102. doi:10.1063/1.2749518

PubMed Abstract | CrossRef Full Text | Google Scholar

Bultinck, P., Langenaeker, W., Lahorte, P., De Proft, F., Geerlings, P., Van Alsenoy, C., et al. (2002a). The electronegativity equalization method II: Applicability of different atomic charge schemes. J. Phys. Chem. A 106, 7895–7901. doi:10.1021/jp020547v

CrossRef Full Text | Google Scholar

Bultinck, P., Langenaeker, W., Lahorte, P., De Proft, F., Geerlings, P., Waroquier, M., et al. (2002b). The electronegativity equalization method I: Parametrization and validation for atomic charge calculations. J. Phys. Chem. A 106, 7887–7894. doi:10.1021/jp0205463

CrossRef Full Text | Google Scholar

Cárdenas, C., Ayers, P. W., and Cedillo, A. (2011). Reactivity indicators for degenerate states in the density-functional theoretic chemical reactivity theory. J. Chem. Phys. 134, 174103–174113. doi:10.1063/1.3585610

PubMed Abstract | CrossRef Full Text | Google Scholar

Cárdenas, C., and Ayers, P. W. (2013). How reliable is the hard-soft acid-base principle? An assessment from numerical simulations of electron transfer energies. Phys. Chem. Chem. Phys. 15, 13959–13968. doi:10.1039/c3cp51134k

PubMed Abstract | CrossRef Full Text | Google Scholar

Cárdenas, C., Echegaray, E., Chakraborty, D., Anderson, J. S. M., and Ayers, P. W. (2009a). Relationships between the third-order reactivity indicators in chemical density-functional theory. J. Chem. Phys. 130, 244105. doi:10.1063/1.3151599

CrossRef Full Text | Google Scholar

Cárdenas, C., Echeverry, A., Novoa, T., Robles‐Navarro, A., Gomez, T., and Fuentealba, P. (2022). The fukui function in extended systems: Theory and applications. Concept. Density Funct. Theory Towards a New Chem. React. Theory 2, 555–571. doi:10.1002/9783527829941.ch27

CrossRef Full Text | Google Scholar

Cárdenas, C., Heidar-Zadeh, F., and Ayers, P. W. (2016). Benchmark values of chemical potential and chemical hardness for atoms and atomic ions (including unstable anions) from the energies of isoelectronic series. Phys. Chem. Chem. Phys. 18, 25721–25734. doi:10.1039/c6cp04533b

PubMed Abstract | CrossRef Full Text | Google Scholar

Cárdenas, C., Rabi, N., Ayers, P. W., Morell, C., Jaramillo, P., and Fuentealba, P. (2009b). Chemical reactivity descriptors for ambiphilic reagents: Dual descriptor, local hypersoftness, and electrostatic potential. J. Phys. Chem. A 113, 8660–8667. doi:10.1021/jp902792n

CrossRef Full Text | Google Scholar

Cerón, M. L., Gomez, T., Calatayud, M., and Cárdenas, C. (2020). Computing the fukui function in solid-state chemistry: Application to alkaline earth oxides bulk and surfaces. J. Phys. Chem. A 124, 2826–2833. doi:10.1021/acs.jpca.0c00950

PubMed Abstract | CrossRef Full Text | Google Scholar

Chattaraj, P. K. (2007). A minimum electrophilicity perspective of the HSAB principle. Indian J. Phys. Proc. Indian Assoc. Cultiv. Sci. 81, 871–879.

Google Scholar

Chattaraj, P. K., Ayers, P. W., and Melin, J. (2007). Further links between the maximum hardness principle and the hard/soft acid/base principle: Insights from hard/soft exchange reactions. Phys. Chem. Chem. Phys. 9, 3853–3856. doi:10.1039/b705742c

PubMed Abstract | CrossRef Full Text | Google Scholar

Chattaraj, P. K., and Ayers, P. W. (2005). The maximum hardness principle implies the hard/soft acid/base rule. J. Chem. Phys. 123, 086101. doi:10.1063/1.2011395

PubMed Abstract | CrossRef Full Text | Google Scholar

Chattaraj, P. K. (2001). Chemical reactivity and selectivity: Local HSAB principle versus Frontier orbital theory. J. Phys. Chem. A 105, 511–513. doi:10.1021/jp003786w

CrossRef Full Text | Google Scholar

Chattaraj, P. K., Lee, H., and Parr, R. G. (1991). HSAB principle. J. Am. Chem. Soc. 113, 1855–1856. doi:10.1021/ja00005a073

CrossRef Full Text | Google Scholar

Chattaraj, P. K., Liu, G. H., and Parr, R. G. (1995). The maximum hardness principle in the Gyftopoulos-Hatsopoulos three-level model for an atomic or molecular species and its positive and negative ions. Chem. Phys. Lett. 237, 171–176. doi:10.1016/0009-2614(95)00280-h

CrossRef Full Text | Google Scholar

Chattaraj, P. K., and Nath, S. (1994). Maximum hardness and HSAB principles: An ab initio SCF study. Indian J.Chem., Sect.A Inorg. Bio-inorg., Phys. theor.anal.chem. 33A, 842–843.

Google Scholar

Chattaraj, P. K., Pérez, P., Zevallos, J., and Toro-Labbé, A. (2001). Ab initio SCF and DFT studies on solvent effects on intramolecular rearrangement reactions. J. Phys. Chem. A 105, 4272–4283. doi:10.1021/jp0021345

CrossRef Full Text | Google Scholar

Chattaraj, P. K., Sarkar, U., and Roy, D. R. (2006). Electrophilicity index. Chem. Rev. 106, 2065–2091. doi:10.1021/cr040109f

PubMed Abstract | CrossRef Full Text | Google Scholar

Chattaraj, P. K. (1996). The maximum hardness principle: An overview. Proc. Indian Natl. Sci. Acad. Part A Phys. Sci. 62, 513–531.

Google Scholar

Chermette, H. (1999). Chemical reactivity indexes in density functional theory. J. Comput. Chem. 20, 129–154. doi:10.1002/(sici)1096-987x(19990115)20:1<129::aid-jcc13>3.0.co;2-a

CrossRef Full Text | Google Scholar

Cohen, A. J., Mori-Sánchez, P., and Yang, W. (2008a). Fractional charge perspective on the band gap in density-functional theory. Phys. Rev. B 77, 115123. doi:10.1103/physrevb.77.115123

CrossRef Full Text | Google Scholar

Cohen, A. J., Mori-Sánchez, P., and Yang, W. (2008b). Insights into current limitations of density functional theory. Science 321, 792–794. doi:10.1126/science.1158722

PubMed Abstract | CrossRef Full Text | Google Scholar

Cohen, M. H., Wasserman, A., Car, R., and Burke, K. (2009). Charge transfer in partition theory. J. Phys. Chem. A 113, 2183–2192. doi:10.1021/jp807967e

PubMed Abstract | CrossRef Full Text | Google Scholar

Cohen, M. H., and Wasserman, A. (2007). On the foundations of chemical reactivity theory. J. Phys. Chem. A 111, 2229–2242. doi:10.1021/jp066449h

PubMed Abstract | CrossRef Full Text | Google Scholar

Cortona, P. (1991). Self-consistently determined properties of solids without band-structure calculations. Phys. Rev. B 44, 8454–8458. doi:10.1103/physrevb.44.8454

PubMed Abstract | CrossRef Full Text | Google Scholar

De Proft, F., Ayers, P. W., Fias, S., and Geerlings, P. (2006). Woodward-Hoffmann rules in density functional theory: Initial hardness response. J. Chem. Phys. 125, 214101. doi:10.1063/1.2387953

PubMed Abstract | CrossRef Full Text | Google Scholar

De Proft, F., Chattaraj, P. K., Ayers, P. W., Torrent-Sucarrat, M., Elango, M., Subramanian, V., et al. (2008). Initial hardness response and hardness profiles in the study of Woodward-Hoffmann rules for electrocyclizations. J. Chem. Theory Comput. 4, 595–602. doi:10.1021/ct700289p

PubMed Abstract | CrossRef Full Text | Google Scholar

Echegaray, E., Toro-Labbe, A., Dikmenli, K., Heidar-Zadeh, F., Rabi, N., Rabi, S., et al. (2017). “Negative condensed-to-atom fukui functions: A signature of oxidation-induced reduction of functional groups,” in Correlations in condensed matter under extreme conditions (Springer), 269–278. doi:10.1007/978-3-319-53664-4_19

CrossRef Full Text | Google Scholar

Elliott, P., Burke, K., Cohen, M. H., and Wasserman, A. (2010). Partition density-functional theory. Phys. Rev. A 82, 024501. doi:10.1103/physreva.82.024501

CrossRef Full Text | Google Scholar

Franco-Pérez, M., Ayers, P. W., Gázquez, J. L., and Vela, A. (2015a). Local and linear chemical reactivity response functions at finite temperature in density functional theory. J. Chem. Phys. 143, 244117. doi:10.1063/1.4938422

CrossRef Full Text | Google Scholar

Franco-Pérez, M., Ayers, P. W., Gázquez, J. L., and Vela, A. (2017a). Thermodynamic responses of electronic systems. J. Chem. Phys. 147, 094105. doi:10.1063/1.4999761

CrossRef Full Text | Google Scholar

Franco-Pérez, M., Gázquez, J. L., Ayers, P. W., and Vela, A. (2015b). Revisiting the definition of the electronic chemical potential, chemical hardness, and softness at finite temperatures. J. Chem. Phys. 143, 154103. doi:10.1063/1.4932539

CrossRef Full Text | Google Scholar

Franco-Pérez, M., Gázquez, J. L., Ayers, P. W., and Vela, A. (2018). Thermodynamic justification for the parabolic model for reactivity indicators with respect to electron number and a rigorous definition for the electrophilicity: The essential role played by the electronic entropy. J. Chem. Theory Comput. 14, 597–606. doi:10.1021/acs.jctc.7b00940

PubMed Abstract | CrossRef Full Text | Google Scholar

Franco-Pérez, M., Heidar-Zadeh, F., Ayers, P. W., Gázquez, J. L., and Vela, A. (2017b). Going beyond the three-state ensemble model: The electronic chemical potential and fukui function for the general case. Phys. Chem. Chem. Phys. 19, 11588–11602. doi:10.1039/c7cp00224f

CrossRef Full Text | Google Scholar

Franco-Pérez, M., Polanco-Ramírez, C.-A., Ayers, P. W., Gázquez, J. L., and Vela, A. (2017). New Fukui, dual and hyper-dual kernels as bond reactivity descriptors. Phys. Chem. Chem. Phys. 19, 16095–16104. doi:10.1039/c7cp02613g

PubMed Abstract | CrossRef Full Text | Google Scholar

Fuentealba, P., and Cardenas, C. (2015). “Density functional theory of chemical reactivity,” in Chemical modelling: Volume 11. Editor J.-O. J. Michael Springborg (London: The Royal Society of Chemistry), 151–174.

Google Scholar

Fuentealba, P., and Cárdenas, C. (2013). On the exponential model for energy with respect to number of electrons. J. Mol. Model. 19, 2849–2853. doi:10.1007/s00894-012-1708-5

PubMed Abstract | CrossRef Full Text | Google Scholar

Fuentealba, P., Cardenas, C., Pino-Rios, R., and Tiznado, W. (2016). “Topological analysis of the fukui function,” in Applications of topological methods in molecular chemistry. Editors C. L. Remi Chauvin, S. Bernard, and E. Alikhani (Springer International Publishing), 227–241. doi:10.1007/978-3-319-29022-5_8

CrossRef Full Text | Google Scholar

Fuentealba, P., and Parr, R. G. (1991). Higher‐order derivatives in density‐functional theory, especially the hardness derivative ∂η/∂N. J. Chem. Phys. 94, 5559–5564. doi:10.1063/1.460491

CrossRef Full Text | Google Scholar

Fuentealba, P., Pérez, P., and Contreras, R. (2000a). On the condensed Fukui function. J. Chem. Phys. 113, 2544–2551. doi:10.1063/1.1305879

CrossRef Full Text | Google Scholar

Fuentealba, P., Simón-Manso, Y., and Chattaraj, P. K. (2000b). Molecular electronic excitations and the minimum polarizability principle. J. Phys. Chem. A 104, 3185–3187. doi:10.1021/jp992973v

CrossRef Full Text | Google Scholar

Gadre, S. R., Kulkarni, S. A., and Shrivastava, I. H. (1992). Molecular electrostatic potentials: A topographical study. J. Chem. Phys. 96, 5253–5260. doi:10.1063/1.462710

CrossRef Full Text | Google Scholar

Galvan, M., Vela, A., and Gazquez, J. L. (1988). Chemical reactivity in spin-polarized density functional theory. J. Phys. Chem. 92, 6470–6474. doi:10.1021/j100333a056

CrossRef Full Text | Google Scholar

Garza, J., Vargas, R., Cedillo, A., Galván, M., and Chattaraj, P. K. (2006). Comparison between the frozen core and finite differences approximations for the generalized spin-dependent global and local reactivity descriptors in small molecules. Theor. Chem. Acc. 115, 257–265. doi:10.1007/s00214-005-0002-3

CrossRef Full Text | Google Scholar

Gázquez, J. L., Franco‐Pérez, M., Ayers, P. W., and Vela, A. (2019). Temperature‐dependent approach to chemical reactivity concepts in density functional theory. Int. J. Quantum Chem. 119, e25797. doi:10.1002/qua.25797

CrossRef Full Text | Google Scholar

Gazquez, J. L., and Mendez, F. (1994). The hard and soft acids and bases principle: An atoms in molecules viewpoint. J. Phys. Chem. 98, 4591–4593. doi:10.1021/j100068a018

CrossRef Full Text | Google Scholar

Gazquez, J. L. (2008). Perspectives on the density functional theory of chemical reactivity. J. Mexican Chem. Soc. 52, 3–10.

Google Scholar

Geerlings, P., Ayers, P. W., Toro-Labbé, A., Chattaraj, P. K., and De Proft, F. (2012). The woodward-hoffmann rules reinterpreted by conceptual density functional theory. Acc. Chem. Res. 45, 683–695. doi:10.1021/ar200192t

PubMed Abstract | CrossRef Full Text | Google Scholar

Geerlings, P., Chamorro, E., Chattaraj, P. K., De Proft, F., Gázquez, J. L., Liu, S., et al. (2020). Conceptual density functional theory: Status, prospects, issues. Theor. Chem. Accounts 139, 1–18. doi:10.1007/s00214-020-2546-7

CrossRef Full Text | Google Scholar

Geerlings, P., and De Proft, F. (2008). Conceptual DFT: The chemical relevance of higher response functions. Phys. Chem. Chem. Phys. 10, 3028–3042. doi:10.1039/b717671f

PubMed Abstract | CrossRef Full Text | Google Scholar

Geerlings, P., De Proft, F., and Langenaeker, W. (2003). Conceptual density functional theory. Chem. Rev. 103, 1793–1874. doi:10.1021/cr990029p

PubMed Abstract | CrossRef Full Text | Google Scholar

Ghanty, T. K., and Ghosh, S. K. (1994). Spin-Polarized generalization of the concepts of electronegativity and hardness and the description of chemical binding. J. Am. Chem. Soc. 116, 3943–3948. doi:10.1021/ja00088a033

CrossRef Full Text | Google Scholar

Gordon, R. G., and Kim, Y. S. (1972). Theory for the forces between closed‐shell atoms and molecules. J. Chem. Phys. 56, 3122–3133. doi:10.1063/1.1677649

CrossRef Full Text | Google Scholar

Govind, N., Wang, Y. A., and Carter, E. A. (1999). Electronic-structure calculations by first-principles density-based embedding of explicitly correlated systems. J. Chem. Phys. 110, 7677–7688. doi:10.1063/1.478679

CrossRef Full Text | Google Scholar

Heidar-Zadeh, F., Miranda-Quintana, R. A., Verstraelen, T., Bultinck, P., and Ayers, P. W. (2016a). When is the fukui function not normalized? The danger of inconsistent energy interpolation models in density functional theory. J. Chem. Theory Comput. 12, 5777–5787. doi:10.1021/acs.jctc.6b00494

PubMed Abstract | CrossRef Full Text | Google Scholar

Heidar-Zadeh, F., Richer, M., Fias, S., Miranda-Quintana, R. A., Chan, M., Franco-Pérez, M., et al. (2016b). An explicit approach to conceptual density functional theory descriptors of arbitrary order. Chem. Phys. Lett 660, 307–312. doi:10.1016/j.cplett.2016.07.039

CrossRef Full Text | Google Scholar

Janak, J. F. (1978). Proof that∂E∂ni=εin density-functional theory. Phys. Rev. B 18, 7165–7168. doi:10.1103/physrevb.18.7165

CrossRef Full Text | Google Scholar

Johnson, P. A., Bartolotti, L. J., Ayers, P. W., Fievez, T., and Geerlings, P. (2012). “Charge density and chemical reactivity: A unified view from conceptual DFT,” in Modern charge density analysis. Editors C. Gatti, and P. Macchi (New York: Springer), 715–764.

Google Scholar

Langenaeker, W., De Proft, F., and Geerlings, P. (1995). Development of local hardness-related reactivity indices: Their application in a study of the SE at monosubstituted benzenes within the HSAB context. J. Phys. Chem. 99, 6424–6431. doi:10.1021/j100017a022

CrossRef Full Text | Google Scholar

Levy, M. (1979). Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v -representability problem. Proc. Natl. Acad. Sci. U.S.A. 76, 6062–6065. doi:10.1073/pnas.76.12.6062

PubMed Abstract | CrossRef Full Text | Google Scholar

Lieb, E. H. (1983). Density functionals for Coulomb systems. Int. J. Quantum Chem. 24, 243–277. doi:10.1002/qua.560240302

CrossRef Full Text | Google Scholar

Liu, S. B. (2009a). Conceptual density functional theory and some recent developments. Acta Physico-Chimica Sin. 25, 590–600.

Google Scholar

Liu, S. B. (2009b). “Electrophilicity,” in Chemical reactivity theory: A density functional view. Editor P. K. Chattaraj (Boca Raton: Taylor & Francis), 179. doi:10.1201/9781420065442.ch13

CrossRef Full Text | Google Scholar

Liu, S., Li, T., and Ayers, P. W. (2009). Potentialphilicity and potentialphobicity: Reactivity indicators for external potential changes from density functional reactivity theory. J. Chem. Phys. 131, 114106. doi:10.1063/1.3231687

PubMed Abstract | CrossRef Full Text | Google Scholar

Liu, S. (2007). Steric effect: A quantitative description from density functional theory. J. Chem. Phys. 126, 244103. doi:10.1063/1.2747247

PubMed Abstract | CrossRef Full Text | Google Scholar

Melin, J., Aparicio, F., Subramanian, V., Galván, M., and Chattaraj, P. K. (2004). Is the fukui function a right descriptor of Hard−Hard interactions? J. Phys. Chem. A 108, 2487–2491. doi:10.1021/jp037674r

CrossRef Full Text | Google Scholar

Mendez, F., and Gazquez, J. L. (1994). Chemical reactivity of enolate ions: The local hard and soft acids and bases principle viewpoint. J. Am. Chem. Soc. 116, 9298–9301. doi:10.1021/ja00099a055

CrossRef Full Text | Google Scholar

Merkle, R., Savin, A., and Preuss, H. (1992). Singly ionized first‐row dimers and hydrides calculated with the fully‐numerical density‐functional program numol. J. Chem. Phys. 97, 9216–9221. doi:10.1063/1.463297

CrossRef Full Text | Google Scholar

Miranda-Quintana, R. A., Chattaraj, P. K., and Ayers, P. W. (2017a). Finite temperature grand canonical ensemble study of the minimum electrophilicity principle. J. Chem. Phys. 147, 124103. doi:10.1063/1.4996443

PubMed Abstract | CrossRef Full Text | Google Scholar

Miranda-Quintana, R. A., and Ayers, P. W. (2018a). Dipolar cycloadditions and the “| Δμ| big is good” rule: A computational study. Theor. Chem. Accounts 137, 1–7. doi:10.1007/s00214-018-2391-0

CrossRef Full Text | Google Scholar

Miranda-Quintana, R. A., and Ayers, P. W. (2016a). Fractional electron number, temperature, and perturbations in chemical reactions. Phys. Chem. Chem. Phys. 18, 15070–15080. doi:10.1039/c6cp00939e

PubMed Abstract | CrossRef Full Text | Google Scholar

Miranda-Quintana, R. A., and Ayers, P. W. (2016). Fractional electron number, temperature, and perturbations in chemical reactions. Phys. Chem. Chem. Phys. 18, 15070–15080. doi:10.1039/c6cp00939e

PubMed Abstract | CrossRef Full Text | Google Scholar

Miranda-Quintana, R. A., and Ayers, P. W. (2016b). Interpolation of property-values between electron numbers is inconsistent with ensemble averaging. J. Chem. Phys. 144, 244112. doi:10.1063/1.4953557

PubMed Abstract | CrossRef Full Text | Google Scholar

Miranda-Quintana, R. A., and Ayers, P. W. (2018b). Note: Maximum hardness and minimum electrophilicity principles. J. Chem. Phys. 148, 196101. doi:10.1063/1.5033964

PubMed Abstract | CrossRef Full Text | Google Scholar

Miranda-Quintana, R. A., and Ayers, P. W. (2016c). Systematic treatment of spin-reactivity indicators in conceptual density functional theory. Theor. Chem. Accounts 135, 1–18. doi:10.1007/s00214-016-1995-5

CrossRef Full Text | Google Scholar

Miranda-Quintana, R. A., and Ayers, P. W. (2019). The “|Δμ| big is good” rule, the maximum hardness, and minimum electrophilicity principles. Theor. Chem. Accounts 138. doi:10.1007/s00214-019-2435-0

CrossRef Full Text | Google Scholar

Miranda-Quintana, R. A., and Bochicchio, R. C. (2014). Energy dependence with the number of particles: Density and reduced density matrices functionals. Chem. Phys. Lett. 593, 35–39. doi:10.1016/j.cplett.2013.12.071

CrossRef Full Text | Google Scholar

Miranda-Quintana, R. A. (2018). Density functional theory for chemical reactivity. Toronto: Apple Academic Press.

Google Scholar

Miranda-Quintana, R. A., Deswal, N., and Roy, R. K. (2022a). Hammett constants from density functional calculations: Charge transfer and perturbations. Theor. Chem. Accounts 141, 1–10. doi:10.1007/s00214-021-02863-5

CrossRef Full Text | Google Scholar

Miranda-Quintana, R. A., Heidar-Zadeh, F., and Ayers, P. W. (2018). Elementary derivation of the "|Δμ| big is good" rule. J. Phys. Chem. Lett. 9, 4344–4348. doi:10.1021/acs.jpclett.8b01312

PubMed Abstract | CrossRef Full Text | Google Scholar

Miranda-Quintana, R. A., Heidar-Zadeh, F., Fias, S., Chapman, A. E. A., Liu, S., Morell, C., et al. (2022b). Molecular interactions from the density functional theory for chemical reactivity: The interaction energy between two-reagents. Front. Chem. 10. doi:10.3389/fchem.2022.906674

CrossRef Full Text | Google Scholar

Miranda-Quintana, R. A., Kim, T. D., Cárdenas, C., and Ayers, P. W. (2017b). The HSAB principle from a finite-temperature grand-canonical perspective. Theor. Chem. Acc. 136, 135. doi:10.1007/s00214-017-2167-y

CrossRef Full Text | Google Scholar

Miranda-Quintana, R. A., Martínez González, M., and Ayers, P. W. (2016). Electronegativity and redox reactions. Phys. Chem. Chem. Phys. 18, 22235–22243. doi:10.1039/c6cp03213c

PubMed Abstract | CrossRef Full Text | Google Scholar

Miranda-Quintana, R. A. (2017a). Note: The minimum electrophilicity and the hard/soft acid/base principles. J. Chem. Phys. 146, 046101. doi:10.1063/1.4974987

PubMed Abstract | CrossRef Full Text | Google Scholar

Miranda-Quintana, R. A. (2017b). Perturbed reactivity descriptors: The chemical hardness. Theor. Chem. Accounts 136, 1–8. doi:10.1007/s00214-017-2109-8

CrossRef Full Text | Google Scholar

Miranda‐Quintana, R. A., Ayers, P. W., and Heidar‐Zadeh, F. (2021). Reactivity and charge transfer beyond the parabolic model: The “| Δμ| big is good” principle. ChemistrySelect 6, 96–100.

Google Scholar

Morell, C., Grand, A., and Toro-Labbé, A. (2005). New dual descriptor for chemical reactivity. J. Phys. Chem. A 109, 205–212. doi:10.1021/jp046577a

PubMed Abstract | CrossRef Full Text | Google Scholar

Morell, C., Grand, A., and Toro-Labbé, A. (2006). Theoretical support for using the Δf(r) descriptor. Chem. Phys. Lett. 425, 342–346. doi:10.1016/j.cplett.2006.05.003

CrossRef Full Text | Google Scholar

Morell, C., Labet, V., Grand, A., and Chermette, H. (2009). Minimum electrophilicity principle: An analysis based upon the variation of both chemical potential and absolute hardness. Phys. Chem. Chem. Phys. 11, 3417–3423. doi:10.1039/b818534d

PubMed Abstract | CrossRef Full Text | Google Scholar

Mori-Sánchez, P., Cohen, A. J., and Yang, W. (2008). Localization and delocalization errors in density functional theory and implications for band-gap prediction. Phys. Rev. Lett. 100, 146401. doi:10.1103/physrevlett.100.146401

PubMed Abstract | CrossRef Full Text | Google Scholar

Mori-Sánchez, P., Cohen, A. J., and Yang, W. (2006). Many-electron self-interaction error in approximate density functionals. J. Chem. Phys. 125, 201102. doi:10.1063/1.2403848

PubMed Abstract | CrossRef Full Text | Google Scholar

Mortier, W. J. (1987). Electronegativity equalization and its applications. Struct. Bond. 66, 125–143.

Google Scholar

Mortier, W. J., Ghosh, S. K., and Shankar, S. (1986). Electronegativity-equalization method for the calculation of atomic charges in molecules. J. Am. Chem. Soc. 108, 4315–4320. doi:10.1021/ja00275a013

CrossRef Full Text | Google Scholar

Mortier, W. J., Van Genechten, K., and Gasteiger, J. (1985). Electronegativity equalization: Application and parametrization. J. Am. Chem. Soc. 107, 829–835. doi:10.1021/ja00290a017

CrossRef Full Text | Google Scholar

Murray, J. S., Brinck, T., and Politzer, P. (1996). Relationships of molecular surface electrostatic potentials to some macroscopic properties. Chem. Phys. 204, 289–299. doi:10.1016/0301-0104(95)00297-9

CrossRef Full Text | Google Scholar

Nalewajski, R. F. (1984). Electrostatic effects in interactions between hard (soft) acids and bases. J. Am. Chem. Soc. 106, 944–945. doi:10.1021/ja00316a020

CrossRef Full Text | Google Scholar

Nalewajski, R. F., Korchowiec, J., and Zhou, Z. (1988). Molecular hardness and softness parameters and their use in chemistry. Int. J. Quantum Chem. 34, 349–366. doi:10.1002/qua.560340840

CrossRef Full Text | Google Scholar

Noorizadeh, S. (2007). Is there a minimum electrophilicity principle in chemical reactions? Chin. J. Chem. 25, 1439–1444. doi:10.1002/cjoc.200790266

CrossRef Full Text | Google Scholar

Osorio, E., Ferraro, M. B., Oña, O. B., Cardenas, C., Fuentealba, P., and Tiznado, W. (2011). Assembling small silicon clusters using criteria of maximum matching of the Fukui functions. J. Chem. Theory Comput. 7, 3995–4001. doi:10.1021/ct200643z

PubMed Abstract | CrossRef Full Text | Google Scholar

Pan, S., Solà, M., and Chattaraj, P. K. (2013). On the validity of the maximum hardness principle and the minimum electrophilicity principle during chemical reactions. J. Phys. Chem. A 117, 1843–1852. doi:10.1021/jp312750n

PubMed Abstract | CrossRef Full Text | Google Scholar

Parr, R. G., and Bartolotti, L. J. (1982). On the geometric mean principle for electronegativity equalization. J. Am. Chem. Soc. 104, 3801–3803. doi:10.1021/ja00378a004

CrossRef Full Text | Google Scholar

Parr, R. G., and Chattaraj, P. K. (1991). Principle of maximum hardness. J. Am. Chem. Soc. 113, 1854–1855. doi:10.1021/ja00005a072

CrossRef Full Text | Google Scholar

Parr, R. G. (1994). Companions in the search. Int. J. Quantum Chem. 49, 739–770. doi:10.1002/qua.560490515

CrossRef Full Text | Google Scholar

Parr, R. G., Donnelly, R. A., Levy, M., and Palke, W. E. (1978). Electronegativity: The density functional viewpoint. J. Chem. Phys. 68, 3801–3807. doi:10.1063/1.436185

CrossRef Full Text | Google Scholar

Parr, R. G., and Pearson, R. G. (1983). Absolute hardness: Companion parameter to absolute electronegativity. J. Am. Chem. Soc. 105, 7512–7516. doi:10.1021/ja00364a005

CrossRef Full Text | Google Scholar

Parr, R. G., and Yang, W. (1984). Density functional approach to the frontier-electron theory of chemical reactivity. J. Am. Chem. Soc. 106, 4049–4050. doi:10.1021/ja00326a036

CrossRef Full Text | Google Scholar

Parr, R. G., and Yang, W. (1989). Density-functional theory of atoms and molecules. New York: Oxford UP.

Google Scholar

Pearson, R. G. (1968a). Hard and soft acids and bases, HSAB, part 1: Fundamental principles. J. Chem. Educ. 45, 581–587. doi:10.1021/ed045p581

CrossRef Full Text | Google Scholar

Pearson, R. G. (1968b). Hard and soft acids and bases, HSAB, part II: Underlying theories. J. Chem. Educ. 45, 643–648. doi:10.1021/ed045p643

CrossRef Full Text | Google Scholar

Pearson, R. G. (1999). Maximum chemical and physical hardness. J. Chem. Educ. 76, 267–275. doi:10.1021/ed076p267

CrossRef Full Text | Google Scholar

Pearson, R. G., and Palke, W. E. (1992). Support for a principle of maximum hardness. J. Phys. Chem. 96, 3283–3285. doi:10.1021/j100187a020

CrossRef Full Text | Google Scholar

Pearson, R. G. (1994). Principle of maximum physical hardness. J. Phys. Chem. 98, 1989–1992. doi:10.1021/j100058a044

CrossRef Full Text | Google Scholar

Pearson, R. G. (1987). Recent advances in the concept of hard and soft acids and bases. J. Chem. Educ. 64, 561–567. doi:10.1021/ed064p561

CrossRef Full Text | Google Scholar

Pearson, R. G. (1993). The principle of maximum hardness. Acc. Chem. Res. 26, 250–255. doi:10.1021/ar00029a004

CrossRef Full Text | Google Scholar

Perdew, J. P., Parr, R. G., Levy, M., and Balduz, J. L. (1982). Density-functional theory for fractional particle number: Derivative discontinuities of the energy. Phys. Rev. Lett. 49, 1691–1694. doi:10.1103/physrevlett.49.1691

CrossRef Full Text | Google Scholar

Pérez, P., Chamorro, E., and Ayers, P. W. (2008). Universal mathematical identities in density functional theory: Results from three different spin-resolved representations. J. Chem. Phys. 128, 204108. doi:10.1063/1.2916714

PubMed Abstract | CrossRef Full Text | Google Scholar

Pino‐Rios, R., Yañez, O., Inostroza, D., Ruiz, L., Cardenas, C., Fuentealba, P., et al. (2017). Proposal of a simple and effective local reactivity descriptor through a topological analysis of an orbital‐weighted fukui function. J. Comput. Chem. 38, 481–488.

PubMed Abstract | Google Scholar

Politzer, P. (1980). Electrostatic potential-electronic density relationships in atoms. II. J. Chem. Phys. 73, 3264–3267. doi:10.1063/1.440521

CrossRef Full Text | Google Scholar

Politzer, P., and Murray, J. S. (2002). The fundamental nature and role of the electrostatic potential in atoms and molecules. Theor. Chem. Accounts Theory, Comput. Model. Theor. Chimica Acta) 108, 134–142. doi:10.1007/s00214-002-0363-9

CrossRef Full Text | Google Scholar

Politzer, P., and Truhlar, D. (1981). Chemical applications of atomic and molecular electrostatic potentials. New York: Plenum.

Google Scholar

Proft, F. D., Ayers, P. W., and Geerlings, P. (2014). “The conceptual density functional theory perspective of bonding,” in The chemical bond: Fundamental aspects of chemical bonding. Editors S. Shaik, and G. Frenking (Darmstadt: Wiley), 233–270. doi:10.1002/9783527664696.ch7

CrossRef Full Text | Google Scholar

Rappe, A. K., and Goddard, W. A. (1991). Charge equilibration for molecular dynamics simulations. J. Phys. Chem. 95, 3358–3363. doi:10.1021/j100161a070

CrossRef Full Text | Google Scholar

Robles, A., Franco-Pérez, M., Gázquez, J. L., Cárdenas, C., and Fuentealba, P. (2018). Local electrophilicity. J. Mol. Model. 24, 245. doi:10.1007/s00894-018-3785-6

PubMed Abstract | CrossRef Full Text | Google Scholar

Ruzsinszky, A., Perdew, J. P., Csonka, G. I., Vydrov, O. A., and Scuseria, G. E. (2007). Density Functionals that are one- and two- are not always many-electron self-interaction-free, as shown for H2+, He2+, LiH+, and Ne2+. J. Chem. Phys. 126, 104102. doi:10.1063/1.2566637

PubMed Abstract | CrossRef Full Text | Google Scholar

Sanderson, R. T. (1951). An interpretation of bond lengths and a classification of bonds. Science 114, 670–672. doi:10.1126/science.114.2973.670

PubMed Abstract | CrossRef Full Text | Google Scholar

Senet, P. (1996). Nonlinear electronic responses, Fukui functions and hardnesses as functionals of the ground‐state electronic density. J. Chem. Phys. 105, 6471–6489. doi:10.1063/1.472498

CrossRef Full Text | Google Scholar

Shirsat, R. N., Bapat, S. V., and Gadre, S. R. (1992). Molecular electrostatics. A comprehensive topographical approach. Chem. Phys. Lett. 200, 373–378. doi:10.1016/0009-2614(92)87006-b

CrossRef Full Text | Google Scholar

Sjoberg, P., and Politzer, P. (1990). Use of the electrostatic potential at the molecular surface to interpret and predict nucleophilic processes. J. Phys. Chem. 94, 3959–3961. doi:10.1021/j100373a017

CrossRef Full Text | Google Scholar

Suresh, C. H., and Gadre, S. R. (1998). A novel electrostatic approach to substituent constants: Doubly substituted benzenes. J. Am. Chem. Soc. 120, 7049–7055. doi:10.1021/ja973105j

CrossRef Full Text | Google Scholar

Tiznado, W., Chamorro, E., Contreras, R., and Fuentealba, P. (2005). Comparison among four different ways to condense the fukui function. J. Phys. Chem. A 109, 3220–3224. doi:10.1021/jp0450787

PubMed Abstract | CrossRef Full Text | Google Scholar

Torrent-Sucarrat, M., Luis, J. M., Duran, M., and Solà, M. (2002). Are the maximum hardness and minimum polarizability principles always obeyed in nontotally symmetric vibrations? J. Chem. Phys. 117, 10561–10570. doi:10.1063/1.1517990

CrossRef Full Text | Google Scholar

Torrent-Sucarrat, M., Luis, J. M., Duran, M., and Solà, M. (2001). On the validity of the maximum hardness and minimum polarizability principles for nontotally symmetric vibrations. J. Am. Chem. Soc. 123, 7951–7952. doi:10.1021/ja015737i

PubMed Abstract | CrossRef Full Text | Google Scholar

Vaidehi, N., Wesolowski, T. A., and Warshel, A. (1992). Quantum‐mechanical calculations of solvation free energies. A combinedabinitiopseudopotential free‐energy perturbation approach. J. Chem. Phys. 97, 4264–4271. doi:10.1063/1.463928

CrossRef Full Text | Google Scholar

Verstraelen, T., Ayers, P. W., Van Speybroeck, V., and Waroquier, M. (2013). ACKS2: Atom-condensed Kohn-Sham DFT approximated to second order. J. Chem. Phys. 138, 074108. doi:10.1063/1.4791569

PubMed Abstract | CrossRef Full Text | Google Scholar

Wesolowski, T. A., and Leszczynski, J. (2006). “One-electron equations for embedded electron density: Challenge for theory and practical payoffs in multi-scale modelling of complex polyatomic molecules,” in Computational chemistry: Reviews of current trends (Singapore: World Scientific), 1–82.

Google Scholar

Wesolowski, T. A. (2004). Quantum chemistry 'without orbitals' - an old idea and recent developments. Chimia 58, 311–315. doi:10.2533/000942904777677885

CrossRef Full Text | Google Scholar

Wesolowski, T. A., and Warshel, A. (1993). Frozen density functional approach for ab initio calculations of solvated molecules. J. Phys. Chem. 97, 8050–8053. doi:10.1021/j100132a040

CrossRef Full Text | Google Scholar

Wesolowski, T., and Warshel, A. (1994). Ab initio free energy perturbation calculations of solvation free energy using the frozen density functional approach. J. Phys. Chem. 98, 5183–5187. doi:10.1021/j100071a003

CrossRef Full Text | Google Scholar

Wu, Q., Ayers, P. W., and Zhang, Y. (2009). Density-based energy decomposition analysis for intermolecular interactions with variationally determined intermediate state energies. J. Chem. Phys. 131, 164112. doi:10.1063/1.3253797

PubMed Abstract | CrossRef Full Text | Google Scholar

Yañez, O., Báez-Grez, R., Inostroza, D., Pino-Rios, R., Rabanal-León, W. A., Contreras-García, J., et al. (2021). Kick-Fukui: A fukui function-guided method for molecular structure prediction. J. Chem. Inf. Model. 61, 3955–3963. doi:10.1021/acs.jcim.1c00605

PubMed Abstract | CrossRef Full Text | Google Scholar

Yang, W., and Mortier, W. J. (1986). The use of global and local molecular parameters for the analysis of the gas-phase basicity of amines. J. Am. Chem. Soc. 108, 5708–5711. doi:10.1021/ja00279a008

PubMed Abstract | CrossRef Full Text | Google Scholar

Yang, W., Parr, R. G., and Pucci, R. (1984). Electron density, Kohn-Sham Frontier orbitals, and Fukui functions. J. Chem. Phys. 81, 2862–2863. doi:10.1063/1.447964

CrossRef Full Text | Google Scholar

Yang, W., Zhang, Y., and Ayers, P. W. (2000). Degenerate ground states and a fractional number of electrons in density and reduced density matrix functional theory. Phys. Rev. Lett. 84, 5172–5175. doi:10.1103/physrevlett.84.5172

PubMed Abstract | CrossRef Full Text | Google Scholar

Yang, X. D., Patel, A. H. G., Miranda-Quintana, R. A., Heidar-Zadeh, F., González-Espinoza, C. E., and Ayers, P. W. (2016). Communication: Two types of flat-planes conditions in density functional theory. J. Chem. Phys. 145, 031102. doi:10.1063/1.4958636

PubMed Abstract | CrossRef Full Text | Google Scholar

Zhou, Z., and Parr, R. G. (1989). New measures of aromaticity: Absolute hardness and relative hardness. J. Am. Chem. Soc. 111, 7371–7379. doi:10.1021/ja00201a014

CrossRef Full Text | Google Scholar

Keywords: DFT‐density functional theory, chemical reactivity, HSAB (hard-soft-acid-base) concept, chemical potential, variational principle

Citation: Miranda-Quintana RA, Heidar-Zadeh F, Fias S, Chapman AEA, Liu S, Morell C, Gómez T, Cárdenas C and Ayers PW (2022) Molecular interactions from the density functional theory for chemical reactivity: Interaction chemical potential, hardness, and reactivity principles. Front. Chem. 10:929464. doi: 10.3389/fchem.2022.929464

Received: 26 April 2022; Accepted: 27 June 2022;
Published: 22 July 2022.

Edited by:

Juan Frau, University of the Balearic Islands, Spain

Reviewed by:

William Tiznado, Andres Bello University, Chile
Daniel Glossman-Mitnik, Centro de Investigación de Materiales Avanzados (CIMAV), Mexico

Copyright © 2022 Miranda-Quintana, Heidar-Zadeh, Fias, Chapman, Liu, Morell, Gómez, Cárdenas and Ayers. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Ramón Alain Miranda-Quintana, quintana@chem.ufl.edu; Tatiana Gómez, tatiana.mgc@gmail.com   Carlos Cárdenas, cardena@uchile.cl; Paul W. Ayers, payers@mcmaster.ca

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