Thermodynamic Justification for the Parabolic Model for Reactivity

Article pubs.acs.org/JCTC

Cite This: J. Chem. Theory Comput. 2018, 14, 597−606

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 Marco Franco-Pérez,*,†,‡ José L. Gázquez,† Paul W. Ayers,‡ and Alberto Vela§ †

Departamento de Química, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, Ciudad de México, 09340, México ‡ Department of Chemistry, McMaster University, Hamilton, Ontario L8S 4M1, Canada § Departamento de Química, Centro de Investigación y de Estudios Avanzados, Av. Instituto Politécnico Nacional 2508, Ciudad de México, 07360, México ABSTRACT: The temperature-dependence of the Helmholtz free energy with respect to the number of electrons is analyzed within the framework of the Grand Canonical Ensemble. At the zero-temperature limit, the Helmholtz free energy behaves as a Heaviside function of the number of electrons; however, as the temperature increases, the profile smoothens and exhibits a minimum value at noninteger positive values of the fractional electronic charge. We show that the exact average electronic energy as a function of the number of electrons does not display this feature at any temperature, since this behavior is solely due to the electronic entropy. Our mathematical analysis thus indicates that the widely used parabolic interpolation model should not be viewed as an approximation for the average electronic energy, but for the dependence of the Helmholtz free energy upon the number of electrons, and this analysis is corroborated by numerical results. Finally, an electrophilicity index is defined for the Helmholtz free energy showing that, for a given chemical species, there exists a temperature value for which this quantity is equivalent to the electrophilicity index defined within the parabolic interpolation of the electronic energy as a function of the number of electrons. Our formulation suggests that the convexity property of the energy versus the number of electrons together with the entropic contribution does not allow for an analogous nucleophilicity index to be defined.

1. INTRODUCTION

μ=

Chemical reactivity theory based on density functional theory (CR-DFT) has proven to be a very useful tool to analyze several aspects associated with the reactivity of chemical species.1−8 From the CR-DFT perspective, some information regarding the chemical reactivity of the electronic systems can be captured by the chemical reactivity response indexes that quantify how a molecule or an atom is initially perturbed by an approaching reagent. Each of these response indexes has been connected with any of the partial derivatives of the electronic energy with respect to its natural variables, the number of electrons N and the external potential υ(r). The first order indexes are thus defined from the following exact differential:1,9−11 dE =

⎛ ∂E ⎞ ⎜ ⎟ dN + ⎝ ∂N ⎠υ(r)

⎛

⎞

∫ ⎜⎝ δυδ(Er) ⎟⎠

(2)

is the electronic chemical potential and ⎛ δE ⎞ ρ(r) = ⎜ ⎟ ⎝ δυ(r) ⎠ N

(3)

is the ground state electronic density. In eq 1, E is the exact ground state electronic energy functional, and thus, the partial or functional derivatives obtained from eq 1 at any order can be considered as true responses of a chemical species in their corresponding fundamental electronic states. It is expected that as more coefficients are evaluated for a given chemical species, a better understanding of its corresponding chemical reactivity is obtained. Nevertheless, the rigorous foundation of CR-DFT has been overshadowed by the problems coming from the exact dependence of the energy density functional with respect to the

δυ(r) dr N

⎛ ∂E ⎞ ⎜ ⎟ ⎝ ∂N ⎠υ(r)

(1) Received: September 6, 2017 Published: December 21, 2017

where © 2017 American Chemical Society

597

DOI: 10.1021/acs.jctc.7b00940 J. Chem. Theory Comput. 2018, 14, 597−606

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Journal of Chemical Theory and Computation

reasonable to define the electrophilicity using eq 7, but multipled by minus one. Numerical results have proven that the electrophilicity index defined in eq 7 is closely related to experimental properties for a plethora of chemical phenomena.31−35 Furthermore, theoretical investigations have brought some insight about possible fundamental principles related to this quantity. The minimum electrophilicity principle asserts that a chemical transformation must evolve toward the state with the lowest electrophilicity value,36−38 whereas the electrophilicity equalization principle states that at equilibrium, all the constituents of a chemical species must display the same electrophilicity value as the species as a whole.39,40 Recently, an equivalency between the minimum electrophilicity principle and the Hard/Soft Acid/ Base (HSAB) principle41 has been suggested, after analyzing the equilibrium conditions of the same double-exchange reaction scheme previously used by Ayers in his fundamental derivation of the HSAB principle.42−44 The electrophilicity index has been conceived as a measure of the electro-accepting “potency” of the chemical species, and new concepts like the electrodonating and the electroaccepting powers have been developed by distinguishing between the potency of a reagent in donating (accepting) charge-transfer processes.45,46 Recently, Miranda-Quintana has highlighted several issues regarding the definition of the electrophilicity index given in eq 7.47 He discussed, for example, that for systems with very low electron affinities the “parabolic” electrophilicity index would be mainly determined by the ionization potential of the species, an inconsistent result since this quantity is related to an electron release and not to an electron attachment process. Given the parab mathematical similarities between ΔEparab min (eq 7) and ΔNmax (eq 8), this last statement is also true for the maximum charge accepted by an electronic species. He also pointed out that eq 7 is only valid for systems close to electron saturation and cannot be applied for highly charged cations. This analysis encouraged him to define the electrophilicity index within the framework of the Helmholtz free energy potential, here denoted as “A,” mainly focusing on the zero temperature limit case. He obtained a general zero temperature limit working equation in which the electrophilicity is expressed as the sum of all the possible electron affinities consistent with the number of electrons in the chemical species under consideration. This definition, called the thermodynamic electrophilicity, avoids issues that appear in the parabolic electrophilicity index, since it does not depend on the ionization potential and can be applied for cations with any charge value. Nevertheless, it is not clear how this new definition can be used to obtain new information regarding the chemical reactivity of the species or how this exact result can be connected to the widely (and successfully) used parabolic electrophilicity index. In this work, we present a systematic and detailed analysis of the behavior of A as a function of the number of electrons at any temperature. It will be shown that A(N) resembles a quadratic function in the number of electrons even at relatively low temperature values. Through the minimization of the Helmholtz free energy with respect to the number of electrons, we obtain an electrophilicity index which strongly depends on (but is not equal to) the electron affinity of the electronic species. Through this approach, we were able to find the temperature conditions under which the parabolic and our electrophilicity index are equivalent. As a final remark, we also show that within our formalism, a nucleophilicity index defined

number of electrons, which is known to be a set of straight lines interconnecting contiguous N-integer pure states.12−14 This dependence has brought difficulties in the evaluation of higher order response indexes with respect to N, since it is well documented that the corresponding second order response function, called chemical hardness, behaves as a Dirac delta function in the electron number.1,14 As an alternative, it is common to consider interpolation models between the integer values of N, with the prerequisite of being N-differentiable. There exist several flavors of these models in the literature,15−20 each of them carrying its own issues,21 but the most widely used is the parabolic interpolation. The parabolic interpolation of the E vs N profile was initially proposed by Parr and Pearson16 and has the following expression: EN = E N0 −

1 1 (IP + EA)[N − N0] + (IP − EA) 2 2

[N − N0]2

(4)

where IP = EN0−1 − EN0 and EA = EN0 − EN0+1 are the vertical first ionization potential and the vertical first electron affinity relative to the EN0 reference state. (We use EN to denote the energy of the N-electron ground state at the geometry of the reference state.) From eq 4 one obtains the following results at first and second order: μ=

⎛ ∂E ⎞ 1 ⎜ ⎟ = −χ = − (IP + EA) ⎝ ∂N ⎠υ(r) 2

(5)

and η=

⎛ ∂μ ⎞ ⎜ ⎟ = ηPP = IP − EA ⎝ ∂N ⎠υ(r)

(6)

where μ = −χ is the negative of Mulliken’s electronegativity and η = ηPP is the definition of chemical hardness proposed by Parr and Pearson.16,23−25 A few years later, based on a report by Maynard et al.,26 Parr proposed a new reactivity indicator called the electrophilicity index,27−30 which corresponds to the change in the energy associated with the maximum amount of electronic charge that a chemical species can accept; i.e., the electrophilicity is the difference in energy between the system in its reference state and the system when it is saturated with electrons. The electrophilicity index therefore corresponds to the minimum energy value in the E vs N parabolic interpolation function: 9,22

parab ΔEmin =−

μ2 2η

(7)

where μ and η are given in eqs 5 and 6, respectively, and we have used the superscript “parab” to indicate that this quantity has been defined within the E vs N parabolic model. The maximum charge accepted by a particular chemical species dictates the position of the minimum in the E vs N quadratic function, and it can be obtained from μ parab ΔNmax =− η (8) In both of these equations, it is assumed that the chemical potential is negative. If it is positive, the reference system is supersaturated with electrons and, consequently, it is 598


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Journal of Chemical Theory and Computation analogously to the electrophilicity index is not justifiable even at very high temperatures.

A[ρ ̅ (r)] = E[ρ ̅ (r)] − TST [ρ ̅ (r)] = μBath N[ρ ̅ (r)] − β −1 ln Ξ

2. THEORETICAL DEVELOPMENT 2.1. Helmholtz Free Energy in the Grand Canonical Formulation. It is more convenient to study the dependence of the Helmholtz free energy on the number of electrons within the framework of the Grand Canonical Potential (GCP), Ω, because then the language of fractional electron number is naturally introduced.48,49 In the GCP formalism, molecules are treated as open reactive systems, able to exchange energy and electrons with external reservoirs with temperature T and chemical potential μBath. In several recent reports, we have exploited the mathematical features of the GCP in the study of chemical reactivity of electronic systems.50−57 Within this framework, we have found that the N-differentiability of the average energy density functional with respect to the average number of electrons is guaranteed, ensuring the existence of arbitrarily high-order response coefficients for both the average electronic energy51 and the average electron density.50 Likewise, we have demonstrated that a plausible definition of “problematic” quantities, like the local hardness and the local chemical potential, can be formulated within this framework.54,58 Using the GCP formalism, it has also been possible to develop a new reactivity index associated with the energetic interactions between electronic species at the very earlier stages of a chemical reaction, namely the global heat capacity of the electronic systems,53 providing a deeper understanding of how a chemical reaction proceeds. At a given temperature T and modified potential u(r) = υ(r) − μBath, the grand potential is a unique functional of the equilibrium electron density ρ̅(r):1,59,60 Ω[ρ ̅ (r)] = E[ρ ̅ (r)] − TST [ρ ̅ (r)] − μBath N[ρ ̅ (r)]

as the Helmholtz Free Energy. 2.1.1. Helmholtz Free Energy As a Function of the Chemical Potential of the Reservoir. To obtain a practical expression for A as a function of μBath, it is necessary to define the size of our ensemble. In a recent study,55 we have shown that the three-state ensemble model, constituted by the ground states of the N0, N0 − 1, and N0 + 1 electron systems, provides a satisfactory description of the reactivity of the species at any temperature value of chemical interest. For this ensemble model, eq 11 can be expressed as Ω[ρ ̅ (r)] = E N0 − μBath N0 − β −1 ln{1 + exp[β(EA + μBath )] + exp[−β(IP + μBath )]}

(14)

Inserting eq 14 into eq 13, one obtains ΔA[ρ ̅ (r)] = ΔE[ρ ̅ (r)] − T ΔST [ρ ̅ (r)] = μBath ΔN[ρ ̅ (r)] − β −1 ln{1 + exp[β(EA + μBath )] + exp[−β(IP + μBath )]}

(15)

where ΔX[ρ̅(r)] = X[ρ̅(r)] − XN0 measures the deviation of the average property X[ρ̅(r)] from the value of the corresponding reference state ψN0. It is known that for the three state ensemble model under consideration51 ΔN[ρ ̅ (r)] = ω =

(9)

exp[β(EA + μBath )] − exp[−β(IP + μBath )] 1 + exp[β(EA + μBath )] + exp[−β(IP + μBath )]

(16)

where E[ρ̅(r)], ST[ρ̅(r)], and N[ρ̅(r)] are the average electronic energy, the entropy, and the average-electron number density functionals, respectively. (See ref 50 for detailed definitions of these quantities.) One of the most important features of the GCP electron density is that it has been designed to integrate to fractional values of the electron number:

Inserting eq 16 into eq 15, one obtains ΔA[ρ ̅ (r)] = ⎧ exp[β(EA + μ )] − exp[−β(IP + μ )] ⎫ Bath Bath ⎬ − β −1 μBath ⎨ ⎩ 1 + exp[β(EA + μBath )] + exp[−β(IP + μBath )] ⎭ ⎪

⎪

⎪

⎪

ln{1 + exp[β(EA + μBath )] + exp[−β(IP + μBath )]}

(17)

∫ ρ̅ (r) dr = N0 + ω[ρ̅ (r)]

(10)

which explicitly depends on the chemical potential of the reservoir, μBath, and on experimental properties of the reference electronic state. In the zero temperature limit, the second term at the right-hand side of eq 17 vanishes, and the first term shows the same behavior of ω in this limit, that is50,61

where N0 is an integer, N0 ≡ ⌊N[ρ̅(r)]⌋, and −1 ≤ ω[ρ̅(r)] < 1 is the fractional part of the electrons number. Similarly, the GCP can be expressed in terms of the Grand Canonical partition function, Ξ:

⎧−1 for μ < −I Bath ⎪ ⎪ 1 ⎪− 2 for μBath = −I ⎪ ⎪ lim ω = ⎨ 0 for − I < μBath < −A T→0 ⎪ ⎪ 1 for μ = −A Bath ⎪2 ⎪ ⎪1 for μ > −A ⎩ Bath

Ω[ρ ̅ (r)] = −β −1T ln Ξ = −β −1 ln ∑ exp[β −1(EN , i − μBath N )] N ,i

(11)

where we have introduced the usual notation for the thermodynamic beta, β = 1/kBT (kB is Boltzmann’s constant) and the electronic energy of state i of the N-electron system, EN,i. Combining eqs 9 and 11, one gets E[ρ ̅ (r)] − TST [ρ ̅ (r)] = μBath N[ρ ̅ (r)] − β

(13)

−1

ln Ξ

(18)

for the three-state ensemble model under consideration. Therefore, at the zero temperature limit, the Helhomltz free energy behaves as a Heaviside function in the chemical

(12)

where one can identify 599


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(although A does not display precisely a quadratic dependence on μBath). 2.1.2. Helmholtz Free Energy As a Function of the Number of Electrons. The main benefit of defining the Helmholtz free energy as a functional of the GCP equilibrium electron density is evident from eq 10. Now, for a fixed value of temperature T and modified potential u(r), any equilibrium value of A can be characterized by a given value of the fractional electrons number ω. Nevertheless, in order to analytically write A as a function of the fractional charge, one must isolate μBath from eq 16, a task that is practical only for the three-state ensemble model mentioned above, since the order of the polynomial to be solved increases with the number of pure states considered to build the ensemble. The dependence of μBath against the fractional charge ω for the ensemble model under consideration splits in the following two roots:51

potential of the electrons reservoir. Nevertheless, when the temperature increases, the contributions from the second term on the right-hand side of eq 17 smoothen the A[ρ̅(r)] vs μBath profile. In Figure 1a, we display the A[ρ̅(r)] vs μBath profile of

⎡ (α − ω) ⎤ − μBath = −IP − β −1 ln⎢ ⎥ ⎣ 2(1 + ω) ⎦

(19)

⎡ (α + ω) ⎤ + μBath = −EA + β −1 ln⎢ ⎥ ⎣ 2(1 − ω) ⎦

(20)

where α = (ω 2 + 4(1 − ω 2) exp[β(IP − EA)])1/2

(21)

Note that eqs 19 and 20 provide results only for nonpositive and non-negative values of ω, respectively, and consequently, both equations are needed in order to describe the chemical potential of the bath over the full range of fractional charge and temperature values. Thus, according to the results in eqs 19 and 20, one may have two different expressions for A as a function of the fractional charge, one for negative values, using eq 19 and another for positive values, using eq 20. Equations 19 and 20 can be equivalently expressed as − exp[−β(I + μBath )] =

− exp[β(A + μBath )] =

α−ω 2(1 + ω) α+ω 2(1 − ω)

(22)

(23)

respectively. Equations 22 and 23 can be now inserted into eq 11 to obtain the following expression for the Grand Potential: ⎡1+α ⎤ Ω = E N0 − μBath N0 − β −1 ln⎢ ⎣ 1 − ω 2 ⎥⎦

(24)

Inserting eq 24 into eq 13, the Helmholtz free energy is thus expressed as follows:

Figure 1. (a) A[ρ̅(r)] vs μBath, (b) A[ρ̅(r)] vs ω, and (c) E[ρ̅(r)] vs ω profiles for the carbon atom (μe0 = −1/2(IP + AE) = 6.26 eV), at the indicated temperature ranges.

⎡1+α ⎤ ΔA = μBath ω − β −1 ln⎢ ⎣ 1 − ω 2 ⎥⎦

the carbon atom for a range of temperature values up to 30 000 K. It can be observed that, as the system reaches a given temperature value (≈ 5000 K), the A[ρ̅(r)] vs μBath profile smoothens and the curve displays a minimum. This minimum is located in the positive fractional charge interval (μBath > −(IP + EA)/2), and as will be discussed in the next subsection, it is a consequence of the electronic entropy. For higher temperatures, where the entropy contribution is not negligible, the A[ρ̅(r)] vs μBath profile exhibits clearly a parabolic shape

(25)

where ω is defined in eq 16. The following expressions for A for negative fractional charge ⎧ ⎡ (α − ω) ⎤ ⎡ 1 + α ⎤⎫ ⎬ ΔA− = −ωIP − β −1⎨ω ln⎢ ⎥ + ln⎢ ⎣ 1 − ω 2 ⎥⎦⎭ ⎣ 2(1 + ω) ⎦ ⎩ (26)

and for positive fractional charge 600


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Journal of Chemical Theory and Computation ⎧ ⎡ 1 − ω 2 ⎤⎫ ⎡ (α + ω) ⎤ ΔA+ = −ω EA + β −1⎨ω ln⎢ ⎥⎬ ⎥ + ln⎢ ⎣ 2(1 − ω) ⎦ ⎣ 1 + α ⎦⎭ ⎩

temperature limit). One may verify, for example, that the exact result obtained for the finite temperature electronic chemical potential given in eq 29 can be reproduced if one takes the partial derivative of eq 32 with respect to the fractional charge ω, at constant temperature and external potential (this result was previously obtained by defining E+ and E− through eqs 19 and (20) respectively; then, performing the corresponding partial differentiation, one recovers eq 29).51 In Figure 1c, we report the E vs ω profile for the carbon atom at temperature values up to 60 000 K, using eq 32. Unlike the A vs μBath and the A vs ω profiles, the E vs ω profile does not display a minimum, but the expected result ΔEmin = −EA, which is observed even at very high temperature values, implying the minimum observed for the A[ρ̅(r)] vs ω profile, cannot be attributed to the energy contribution. To analyze the entropic contribution to the Helmholtz free energy, we conveniently consider the analytic and simpler dependence of A with respect to positive values of the fractional charge, eq 27. To simplify our treatment, we also consider the low temperature expression of eq 21, where β−1 > (I − A) and therefore α ≈ ω. Under these conditions, eq 27 can be simplified to

(27)

are obtained simply by inserting eq 19 or 20, respectively, into eq 25. For temperature values for which β−1 is small compared to the band gap (IP − EA), the expression μBath ≅ μe + β −1

ω ⎡ (α + |ω|) ⎤ ln⎢ ⎥ α ⎣ 2(1 − |ω|) ⎦

(28)

provides the same numerical results as eqs 19 and 20, in their corresponding interval values of fractional charge. In eq 28, |ω| represents the absolute value of the fractional charge whereas μe, the electronic chemical potential, is the response function of the average electronic energy, E, with respect to changes in the average number of electrons at constant temperature and external potential, which for the three state ensemble model is known to have the following mathematical expression:51 μe =

⎛ ∂E ⎞ ω 1 ⎜ ⎟ = − (IP + EA) + (IP − EA) ⎝ ∂N ⎠T , υ(r) α 2

(29)

ΔA = −ω EA + β −1{ω ln ω + (1 − ω) ln(1 − ω)}

Thus, through eq 28, one can obtain a unique expression for A as a function of ω, valid for the mentioned temperature range

(33)

The first term at the right-hand side of eq 33 can be identified as ΔE, i.e., ΔE = −ω EA.51 The second term in eq 33 represents the entropic contribution to A. One can verify that this term displays the same mathematical expression as the entropy of a Fermi−Dirac distribution,48,49 and therefore

⎧ ω 2 ⎡ (α + |ω|) ⎤ ⎡ 1 − ω 2 ⎤⎫ ln⎢ ΔA ≅ −ωμe + β −1⎨ ⎥⎬ ⎥ + ln⎢ ⎣ 1 + α ⎦⎭ ⎩ α ⎣ 2(1 − |ω|) ⎦ (30)

Figure 1b shows the A vs ω profile for the carbon atom, for a range of temperature values up to 30 000 K. At low temperatures, the dependence of A[ρ̅(r)] as a function of ω resembles a straight-line profile. Nevertheless, as the temperature increases, this profile smoothens and A(ω) displays a minimum for temperatures beyond ≈5000 K. As in the case of the A vs μBath, this minimum is located in the interval of positive values of the fractional charge. Which contribution to A is responsible for the minimum value observed in A(ω)? As can be seen in eq 15, the Helmholtz free energy is constituted by the sum of two terms, the average energy, E, and the entropy times the temperature, TS. Within the three states ensemble model under consideration, the average electronic energy is given by51 ΔE =

ΔST + = kB{ω ln ω + (1 − ω) ln(1 − ω)}

Equation 34 has a minimum at ω = 1/2, and an extremum value for the entropic contribution is thus expected. The presence of this extremum value explains the curvature of the A vs ω profile for temperature values where the entropy contribution is non-negligible. When the temperature value is close to the zero temperature limit, the minimum in the A vs ω is located at ω = 1 or correspondingly, ΔAmin = −EA, which is the same result obtained by Miranda-Quintana.47 As the temperature increases, the entropic term becomes more important and the minimum is gradually displaced toward the value ω = 1/2. Recall that the entropic term, as a function of ω, has a linear dependence on the temperature only for relatively low temperature values (see eq 33), where the approximation α ≈ ω is still valid. For situations where this approximation cannot be used, the temperature dependent term present in the expression of α (see eq 21) must be also considered. Consequently, the position of the minimum in the A vs ω profile will depend on the temperature dependence of α at higher temperatures. To end this section, we want to show that under the lowtemperature conditions, the A vs ω profile can be reasonably approximated by a polynomial in ω. To do so, we expand in series the two logarithm functions in eq 33. The expression in eq 33 has divergent first derivatives at the edges of the interval 0 ≤ ω ≤ 1 caused by the terms ω ln ω (when ω → 0) and (1 − ω) ln(1 − ω) (when ω → 1). Thus, a series expansion is not reliable. We will instead use the approximation

IP exp[−β(IP + μBath )] − EA exp[β(EA + μBath )] 1 + exp[β(EA + μBath )] + exp[−β(IP + μBath )] (31)

To obtain an explicit and unique expression for the dependence of the average electronic energy as a function of the fractional charge, one inserts eqs 22 and 23 into eq 31, and after some algebraic manipulations, one finds that E = E N0 −

1 1 α + ω2 (IP + EA)ω + (IP − EA) 2 2 1+α

(34)

(32)

Notice that this would be exactly Parr and Pearson’s quadratic model for the energy except that α is a function of the fractional charge, ω. Equation 32 constitutes the exact dependence of the (average) energy density f unctional with respect to the (average) number of electrons, valid for the three states ensemble model under consideration at any temperature value (even in the zero

(1 − ω) ln(1 − ω) ≈ ω 2 ln ω 2 ∀ 0 < |ω| < 1

(35)

which gives a new (approximate) expression for the entropy term present in eq 34 601


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Journal of Chemical Theory and Computation ΔST + ≈ kB{ω ln ω + ω 2 ln ω 2} = kB{(ω + 2ω 2) ln ω} (36)

This expression still has a divergent first derivative as ω → 0, but one can expand this function about ω = 1, giving ΔA+ ≈ −ω EA ⎧ ⎫ 7 1 3(ω − 1) + (ω − 1)2 + (ω − 1)3 ⎪ ⎪ ⎪ ⎪ 2 2 ⎬ + β −1⎨ ⎪ ⎪− 1 (ω − 1)4 + 1 (ω − 1)5 + ... ⎪ ⎪ ⎭ ⎩ 12 60 (37)

As a second alternative, we have made a parabolic fit to the Helmholtz free energy by making use of the values obtained at the points ω = 0, ω = 1/2, and ω = 1, with eq 33. The resulting expression is ΔA+ ≈ −(EA + β −1 ln 16)ω + (β −1 ln 16)ω 2 ≈ −ω EA − β −1ω(1 − ω) ln 16

(38)

Thus, the quadratic interpolation predicts “a parabolic entropy” which is equal to the second term on the right-hand side of the second equality in eq 38. In Figure 3, we compare the A vs ω

approx Figure 3. Comparison between (a) ωexact and (b) ΔAexact max and ωmax min approx and ΔAmin , values for the carbon and the fluorine atoms and for the NO molecule, at temperatures up to 40 000 K (continuous lines represent exact profiles, whereas dotted lines correspond to the approximations).

2.2. The Electrophilicity Index at Finite Temperatures. 2.2.1. The Helmholtz Electrophilicity. In this section, we are going to establish a connection between the electrophilicity index and the extremum of A as a function of ω. This is motivated by the fact that A(ω) has a minimum for a fractional charge, 0 < ω < 1, at sufficiently high temperature, while E(ω) does not. The minimum in the A(ω) can be determined by finding the value of the fractional charge, ωmax, for which dA/dω = 0. In accord with standard statistical mechanics and, explicitly, from eq 25, one can verify that in equilibrium:48,49

Figure 2. Comparison between the exact A[ρ̅(r)] vs ω profile, the polynomial expansion in eq 37, evaluated at several orders of approximation, and the parabolic interpolation given in eq 38, for the carbon atom at T = 10 000 K.

profiles obtained with eq 37 using the second, third, fourth, and fifth order expansion approximations, and the parabolic one obtained from eq 38, with the exact result obtained through eq 27, for the carbon atom at T = 10 000 K. Since the quantity (ω − 1) is always fractional and because the expansion coefficients in eq 37 decrease, the contributions from high order terms quickly vanish, indicating that the A vs ω profile can be reliably approximated by a low-order polynomial at relatively low temperature values (see Figure 2). This last statement is supported by the parabolic interpolation given in eq 38, where it can be observed that the best quadratic function has a similar performance to that of the highest order polynomial (n = 5, Figure 2). Likewise, it can be observed that due to the divergent behavior of first derivatives of eq 33 mentioned above, eqs 37 and 38 are less accurate for integer values of the electronic charge, indicating that any polynomial interpolation will exhibit a small deviation from the exact thermodynamic profile, at low fractional charge values.

0=

∂A = μBath ∂ω

(39)

Equation 39 directly imposes the zero value for the thermodynamic chemical potential of the reservoir, consistent with the electron saturation condition. Thus, ωmax is the maximum charge that an electronic system is able to accept at a given temperature and can be obtained from eq 20, by finding the ω value for which 0 = α + ω(1 + 2 exp[β EA]) − 2 exp[β EA]

(40)

Due to the mathematical expression of α (eq 21), eq 40 must be solved numerically. Nevertheless, for temperature values 602


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Table 1. Comparison between Some Reactivity Indexes Obtained from the Parabolic Energy Profile, with Their Helmholtz Free Energy Counterpart Evaluated at T = Tparaba

a

species

IP

EA

−(IP + EA)/2

IP − EA

ΔEparab min

Tparab × 103

ωparab max

ΔAparab min

1/Sparab

C Cs F K Ni Si SO2 S2 CH3I NO

11.26 3.89 17.42 4.34 7.64 8.15 12.3 9.36 9.5 9.26

1.27 0.47 3.4 0.5 1.15 1.39 1.1 1.66 0.2 0.026

−6.26 −2.18 −10.41 −2.42 −4.39 −4.77 −6.70 −5.51 −4.85 −4.64

9.99 3.42 14.02 3.84 6.49 6.76 11.2 7.70 9.30 9.234

−1.96 −0.70 −3.87 −0.76 −1.49 −1.68 −2.00 −1.97 −1.27 −1.17

27.2 9.3 36.9 10.5 17.6 18.2 30.3 20.6 22.0 17.4

0.627 0.637 0.743 0.630 0.677 0.706 0.598 0.716 0.522 0.503

−2.52 −0.88 −4.49 −0.97 −1.83 −2.02 −2.61 −2.35 −1.52 −1.08

9.78 3.40 16.45 3.78 6.82 7.43 10.54 8.60 7.42 5.94

Temperature units are given in degrees Kelvin. Energy values are given in eV. Experimental data were taken from ref 70.

where α ≈ ω, the solution of eq 40 provides the following simple result: ωmax =

⎡β ⎤ 1 1 + tanh⎢ EA⎥ ⎣2 ⎦ 2 2

true temperature, but an effective temperature that models, in a simple but nonetheless useful way, the fact that the interactions between reacting systems force them away from the equilibrium state they occupy if they were isolated. There are several attempts to quantify this effect in the literature. The first was to define a local temperature due to the kinetic energy of electrons, which is very high.62−64 One can similarly define a local electronic temperature as the conjugate variable to a suitably defined local or information theoretic entropy.65−67 In these interpretations, approaching reagents’ electrons can impart significant kinetic energy to (or induce significant disorder in) the electrons of a substrate molecule, and so a high value for the “effective temperature” is justified. From an alternative perspective, the effect of an approaching reagent is to perturbatively mix in the wave functions of higher-energy states, just as temperature mixes in higher-energy wave function components.68,69 That is, the density matrix of a subsystem will not be a pure state even in the zero-temperature limit, and the occupation numbers for the subsystem density matrix can be (approximately) fit to a Fermi distribution function corresponding to a substantial electronic temperature.68 This is similar to our observation about the electronic heat capacity, which indicates that at the onset of a chemical reaction, there is an electronic flux between the reagents that substantially elevates the temperature of the reactants, regardless of the temperature value of the surroundings.53 In the absence of a simple, practical, and convincing strategy for computing the effective electronic temperature and of the way it varies during a particular chemical reaction, a pragmatic approach seems advisible. Therefore, we assume that ΔNparab max resembles the true ωmax at high temperature values. This gives one a rough approximation for the effective temperature, simply by solving ωmax(T) = ΔNparab max . We call this temperature the parabolic temperature Tparab, and the maximum fractional charge exchanged at this temperature is denoted as ωparab max . Then, the parab Helmholtz free energy evaluated at ωparab is denoted max and T parab parab parab as ΔAmin . In Table 1, we report T , ΔEmin , ωparab max , and ΔAparab values for six neutral atoms and for four simple min molecules. (When solving for Tparab, we used eq 40 and not the low-temperature approximation in eq 41.) To verify that the parab curvature of ΔAparab min resembles the curvature of ΔEmin , we are also comparing the band gap IP − EA with the inverse of the thermodynamic softness, S, evaluated at Tparab and ωparab max , which is known to be54,61

(41)

It is worth noting that according to eq 41, the maximum charge accepted by an electronic species only depends on its corresponding EA value, avoiding in this way the dependence on the IP value exhibited by ΔNparabol for chemical species with max EA values close to zero. Once ωmax is determined either by solving eq 40 numerically or directly from the (accurate) approximate result in eq 41, one can thus obtain ΔA min = ΔA|ωmax

(42)

that is, the Helmholtz free energy evaluated at ωmax. We call this quantity the Helmholtz Electrophilicity Index (HEI). In Figure 3a, we compare the ωmax values obtained by solving eq 40 approx (ωexact max ) with those directly determined from eq 41 (ωmax ) for the carbon and fluorine atoms, as well as for the diatomic molecule NO, at temperature values up to 40 000 K, whereas in exact Figure 3b we compare ΔAmin evaluated at ωexact max , ΔAmin , with approx approx ΔAmin evaluated at ωmax , ΔAmin , in the same temperature range for the three chemical species mentioned above. Figure 3a shows that the approximation given in eq 41 provides exactly the same results as the ones obtained by the exact numerical solution of eq 40 for temperature values up to ≈20 000 K for the carbon and fluorine atoms, and for the NO molecule both equations provide the same results for the whole temperature range under consideration. Figure 3b shows that the Helmholtz electrophilicity is even less sensitive to the approximation mentioned above, since the ΔAapprox vs T profile exactly min reproduces the ΔAexact vs T profile for the three chemical min species on the entire temperature interval studied. 2.2.2. The Temperature of the E vs N Parabolic Interpolation. As mentioned in our Introduction, both ΔNparab max and ΔEparab min indexes have been successfully used to describe many reactivity trends even though they are based on the ad hoc assumption of a parabolic energy model. The empirical utility of these indicators suggests that they may be implicitly, and certainly approximately, modeling some real chemical information about the system. On the other hand, one might consider that the temperature values where the A(ω) has a minimum at fractional charge are far higher than the temperatures at which most chemical phenomena are observed. However, it is now well-accepted that the “temperature” that one uses when exploring chemical reactivity concepts is not the 603


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Journal of Chemical Theory and Computation ∂ 2A ∂ω 2

= parab T parab , ωmax

=

display a minimum for a certain temperature value. However, since the ionization potential is bigger than the electron affinity, the temperature required to make the entropy contribution significant is expected to be much higher for the donation process than in the acceptation process. Nevertheless, when the temperature is high enough, the minimum exhibited by the exact entropy contribution (second term at the right-hand side of eq 26) is displaced toward ω = 0. The temperature required for the entropy contribution to overcome the energetic one (ΔE[ρ̅(r)] = −ωIP) is proportional to IP (with ω as the proportionality constant), whereas the minimum entropy value is displaced toward |ω| < 1/2 for temperature values proportional to (IP − EA)/2 (recall that the second term inside the parentheses on the right-hand side of eq 21 is primarily responsible for the minimum displacements, bigger than |ω| = 1/2). In conclusion, the convexity of the energy versus the number of electrons does not allow the entropy contribution to overcome the energetic contribution, so a nucleophilicity index cannot be defined in a way that is analogous to the electrophilicity index. As a final remark, we want to provide a short discussion regarding the electrophilicity index of polycations. At the first instance, the equilibrium condition in eq 39 shows that there exists one and only one electrophilicity index for every chemical species. On the other hand, our analysis indicates that the convexity of the energy ensures that the entropic contribution of any cation will never overcome the energetic one at any temperature value. Therefore, in electron-saturation conditions, the Helmholtz free energy of a multicharged cation will decay toward the k − 1 charge state, where k is the number of stable anions of the molecule. In the zero temperature limit, the value of ΔAmin will be thus determined by the energy cost from going from the multicharged cation state to the last stable anion state. For temperature values where the entropic contribution is not negligible, ΔAmin will be determined by the energy cost from going from the multicharged cation state to the k − 1 last stable state plus the electrophilicity index of the k − 1 last stable state. For the case of chemical species with a negative first electron affinity, the equilibrium condition of eq 39 will not be fulfilled by positive fractional charges but by the negative fractional charges consistent with N[ρ̅(r)] − Z = ω ∀ −1 ≤ ω ≤ 0, where Z is the nuclear charge. In these cases, the singly charged cation, not the neutral species, is the one that exhibits an electrophilicity index.

∂μBath ∂ω 1 Sparab

= β −1

parab T parab , ωmax


α(1 − α)(1 − ω 2) 1 − α2


(43)

for the three-state ensemble model under consideration. Table 1 shows that the values of Tparab range from 9.3 × 103 K to 36.9 × 103 K, with an average value of 21.0 × 103 K. It is worth it to mention that in our previous report55 we have also shown that even at these temperature conditions the three states ensemble model is a reliable approximation to describe the chemical reactivity of electronic systems. It can be observed that Tparab increases as the band gap IP − EA increases, indicating that a hard system requires a larger amount of energy to react. It is worth mentioning that the same dependence of temperature on band gap was observed from our study of the global heat capacity of electronic systems.53 It is interesting to note that, in parab almost all cases, the ΔAparab min values are very close to the ΔEmin values, and both of them follow exactly the same trend, R2 = 0.981. One can also see that the band gap IP − EA and 1/Sparab strongly resemble each other. These results thus confirm that at Tparab the widely used E vs N interpolation model is usually an accurate approximation of the exact A vs ω profile. The approximation is less accurate for the NO molecule, where the 1/Sparab value significantly differs from the corresponding band gap value, IP − EA. This discrepancy is a direct consequence of the very small EA value for this molecule, which causes the parabolic model to give a questionable description of both the electron attachment and the electrophilicity index, as was pointed out by Miranda-Quintana.47 2.3. The Inexistence of the Nucleophilicity Index. Unfortunately, it is impossible to define an analogous nucleophilicity index by extending the preceding arguments to negative fractional charge, − 1 < ω < 0. A(ω) never has a minimum for ω < 0 because the equilibrium condition given in eq 39 can only be fulfilled for positive values of the fractional charge, with ω → 0 only for high temperatures. The Helmholtz free energy can therefore only be minimized for the acceptation process of electronic charge and not for the donation process. While there is no minimum in the free energy for electrondonation, there is a minimum in the electronic entropy. To show this, we use the α ≈ ω low temperature approximation in eq 26, obtaining in this way the following expression for the Helmholtz free energy change for negative values of fractional charge ⎧ ⎛α − ω⎞ ⎟ + (1 + ω) ΔA = −ωIP − β −1⎨ω ln⎜ ⎝ 2 ⎠ ⎩ ⎫ ln(1 + ω)⎬ ⎭

3. CONCLUSIONS By exploring the dependence of the average electronic energy and the Helmholtz free energy dependencies on the average number of electrons, we demonstrate that the Helmholtz free energy, but not the average electronic energy, has an approximately parabolic dependence on the number of electrons. This provides a mathematical justification for the utility of the parabolic model for E(N) in conceptual quantum chemistry. To establish this result, we derived the exact dependence of the (average) energy density functional with respect to the (average) number of electrons for the reliable and widely used three states ensemble model. We showed that this function never has a minimum for any temperature or fractional charge value. In contrast the entropic contribution that is included in the Helmholtz free energy exhibits a minimum at ω = 1/2, and consequentially, when the temperature is high enough, the entropic contribution induces a minimum in A(ω) at fractional

(44)

This last equation can be equivalently expressed as follows: ΔA = |ω|IP + β −1{|ω| ln|ω| + (1 − |ω|) ln(1 − |ω|)} (45)

where |ω| represents the absolute value of the fractional charge. Equation 45 displays an entropic contribution numerically equivalent to eq 36, and thus, one might think that eq 26 would 604


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Journal of Chemical Theory and Computation charge. (At lower temperatures, the minimum is at ω = 1. No such minima are ever observed for negative values of the fractional charge. We used the extremum of A(ω) to generalize the electrophilicity index to finite temperatures; we call this quantity the Helmholtz electrophilicity index, HEI. The minimization of A(ω) is obtained when the chemical potential of the reservoir becomes zero, as was first stated by Parr et al.27 In the zero-temperature limit, ω = 1, and the HEI is equal to the electron affinity. However, this minimum is displaced toward ω = 1/2 for temperature values where the entropy contribution is not negligible. One can thus conclude that the existence of the (nontrivial) electrophilicity index is a consequence of the entropy term in the Helmholtz free energy. To explore the relationship between the E vs N parabolic interpolation model and the exact A vs ω function, we estimate the temperature where the maximum accepted charge obtained by minimizing A(ω), ωmax, is equivalent to the one obtained by minimizing E(N), ΔNparab max , for 10 different chemical species. This temperature value, Tparab, is system-dependent and ranges from 9300 to 37 000 K, with an average value of 21 000 K. At this temperature, the parabolic, ΔEparab min , and the Helmholtz, ΔAmin, electrophilicity indexes have very similar numerical values. The same is also true for the inverse value of the thermodynamic softness S and the band gap IP − EA. Whether this effective temperature value has broader implications for chemical reactivity theory is an interesting topic for further research, as it is the new possibility to develop “thermodynamic” interpolations, which might overcome the actual zerotemperature interpolation approximations, in the description of the chemical reactivity of electronic species.

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(6) Liu, S. B. Conceptual Density Functional Theory and Some Recent Developments. Acta Phys.-Chim. Sin. 2009, 25, 590−600. (7) Chattaraj, P. K. Chemical Reactivity Theory: A Density Functional View; CRC Press: Boca Raton, FL, 2009. (8) Johnson, P. A.; Bartolotti, L. J.; Ayers, P. W.; Fievez, T.; Geerlings, P. Charge density and chemical reactivity: A unified view from conceptual DFT. In Modern Charge Density Analysis; Gatti, C., Macchi, P., Eds.; Springer: New York, 2012; pp 715−764. (9) Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E. Electronegativity: the density functional viewpoint. J. Chem. Phys. 1978, 68, 3801−3807. (10) Nalewajski, R. F.; Parr, R. G. Legendre transforms and Maxwell relations in density functional theory. J. Chem. Phys. 1982, 77, 399− 407. (11) Parr, R. G.; Yang, W. T. Density-Functional Theory of the Electronic-Structure of Molecules. Annu. Rev. Phys. Chem. 1995, 46, 701−728. (12) Perdew, J. P.; Parr, R. G.; Levy, M.; Balduz, J. L., Jr. Densityfunctional theory for fractional particle number: derivative discontinuities of the energy. Phys. Rev. Lett. 1982, 49, 1691−1694. (13) Yang, W. T.; Zhang, Y. K.; Ayers, P. W. Degenerate ground states and fractional number of electrons in density and reduced density matrix functional theory. Phys. Rev. Lett. 2000, 84, 5172−5175. (14) Ayers, P. W. The continuity of the energy and other molecular properties with respect to the number of electrons. J. Math. Chem. 2008, 43, 285−303. (15) Parr, R. G.; Bartolotti, L. J. On the geometric mean principle for electronegativity equalization. J. Am. Chem. Soc. 1982, 104, 3801− 3803. (16) Parr, R. G.; Pearson, R. G. Absolute hardness: companion parameter to absolute electronegativity. J. Am. Chem. Soc. 1983, 105, 7512−7516. (17) Fuentealba, P.; Parr, R. G. Higher-order derivatives in densityfunctional theory, especially the hardness derivative. J. Chem. Phys. 1991, 94, 5559−5564. (18) Noorizadeh, S.; Shakerzadeh, E. A new scale of electronegativity based on electrophilicity index. J. Phys. Chem. A 2008, 112, 3486− 3491. (19) Noorizadeh, S.; Parsa, H. Evaluation of absolute hardness: a new approach. J. Phys. Chem. A 2013, 117, 939−946. (20) Fuentealba, P.; Cardenas, C. On the exponential model for energy with respect to number of electrons. J. Mol. Model. 2013, 19, 2849−2853. (21) Heidar-Zadeh, F.; Miranda-Quintana, R. A.; Verstraelen, T.; Bultinck, P.; Ayers, P. W. When is the Fukui Function Not Normalized? The Danger of Inconsistent Energy Interpolation Models in Density Functional Theory. J. Chem. Theory Comput. 2016, 12, 5777−5787. (22) Mulliken, R. S. A new electroaffinity scale: together with data on states and an ionization potential and electron affinities. J. Chem. Phys. 1934, 2, 782−793. (23) Ayers, P. W. The physical basis of the hard/soft acid/base principle. Faraday Discuss. 2007, 135, 161−190. (24) Pearson, R. G. The hardness of closed systems. In Chemical Reactivity Theory: A Density Functional View; Chattaraj, P. K., Ed.; CRC Press: Boca Raton, FL, 2009; pp 155−162. (25) Pearson, R. G. Chemical Hardness; Wiley-VCH: Weinheim, Germany, 1997. (26) Maynard, A.; Huang, M.; Rice, W.; Covell, D. Reactivity of the HIV-1 nucleocapsid protein p7 zinc finger domains from the perspective of density-functional theory. Proc. Natl. Acad. Sci. U. S. A. 1998, 95, 11578−11583. (27) Parr, R. G.; Szentpaly, L. v.; Liu, S. Electrophilicity index. J. Am. Chem. Soc. 1999, 121, 1922−1924. (28) Chattaraj, P. K.; Sarkar, U.; Roy, D. R. Electrophilicity index. Chem. Rev. 2006, 106, 2065−2091. (29) Chattaraj, P. K.; Roy, D. R. Update 1 of: Electrophilicity Index. Chem. Rev. 2007, 107, PR46−PR74.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

José L. Gázquez: 0000-0001-6685-7080 Paul W. Ayers: 0000-0003-2605-3883 Alberto Vela: 0000-0002-2794-8622 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS P.W.A. and M.F.-P. acknowledge support from the Canada Research Chairs, Compute Canada, and an NSERC Discovery Grant. M.F.-P. also thanks Conacyt for a postdoctoral fellowship. J.L.G. and A.V. thank Conacyt for grants 237045 and 128369, respectively. The authors are grateful to Dr. Ramon Alain Miranda-Quintana for a preprint of his manuscript, ref 47.

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Thermodynamic Justification for the Parabolic Model for Reactivity

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