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Indian Journal of Chemistry Vol. 53A, Aug-Sept 2014, pp. 1052-1057 Polarization, reactivity and quantum molecular capacitance: From electrostatics to density functional theory Patrice Delarue & Patrick Senet* Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS-Université de Bourgogne, Dijon, France Email: [email protected] Received 31 March 2014; revised and accepted 6 May 2014 The charge distribution induced by an inhomogeneous electric potential applied to a molecule is in fact the sum of two terms: polarization (localized) and chemical (delocalized) charge distributions. The chemical induced charge distribution is proportional to the inhomogeneous response of the molecule to an electron transfer (Fukui function). Analogy with the electrostatic Thomson theorem for the perfect conductors permits to define the quantum molecular capacitance. Keywords: Theoretical chemistry, Density functional calculations, Polarization, Chemical reactivity, Electrostatics, Quantum molecular capacitance, Fukui function In the Born-Oppenheimer approximation, a chemical reaction between two molecules can be described as the variation of the electronic density of the ensemble of the two interacting molecules due to the motions of their nuclei. This electronic density variation is an internal charge reorganization (polarization) of the whole system of reagents which can be formally and arbitrarily decomposed in responses (reactivity indexes) of the individual molecules. Indeed, consider for example two isolated molecules A and B with ground-state densities ρA(r) and ρB(r), respectively. When the molecules are at finite distance, the density of the whole system A+B reaches the value ρAB(r). The difference between the density ρAB(r) and the density of each of the isolated reagents, i.e., ∆ρ(r) = ρAB(r)-ρA(r)-ρB(r), contains information on the respective reactivity of the individual moieties. For example, the charge transfer on A due to B can be calculated as ∆ ≡ Ω ρ , where Ω is a (arbitrary) partition function1,2. For nonzero charge transfer, one may define a linear reactivity index ≡ Ω ∆ρ(r)/∆ . In the density-functional theory (DFT) of the chemical reactivity3-9, one aims to evaluate the variation of the electronic density of a molecule induced by its reaction with another molecule from the ground-state properties of the individual reagents. The perturbation induced by one molecule on another can be either described by an external potential (physical perturbation or polarization) or by a charge transfer (chemical perturbation)3. In the example described above, we aim to evaluate an approximation of which does not depend on actual position of B and on the choice of Ω . The so-called Fukui function is this quantity3,10. It represents the linear variation of the electronic density when A is “charged” with ∆ in absence of B. However, there is no unique definition of . The Fukui function can be defined as a derivative of the density relative to the number of electrons (Eq. (1)). = …(1) For an actual charge transfer to an isolated molecule, the variation of the electron number, ∆ , is a positive or a negative integer. Therefore, the Fukui function defined by Eq. (1) must be computed by the finite difference between the molecular electronic densities of the neutral and positively or negatively charged molecules or approximated by the HOMO or the LUMO orbitals4,5,7. Two Fukui functions3,10 are thus defined: which describes the molecular electrophilic reactivity (uptake of electrons, ∆ > 0) and which describes the molecular nucleophilic reactivity (transfer of electrons, ∆ < 0). There is a vast interest in the DELARUE & SENET: POLARIZATION, REACTIVITY & QUANTUM MOLECULAR CAPACITANCE 1053 applications of these electrophilic and nucleophilic Fukui functions to understand reactivity,4,5,11 ion solvation12, biochemical properties13-15 and to probe reaction paths16. If ∆ results from an internal charge transfer as described above, there is no reason to approximate the Fukui function by the response to an actual charge transfer of an integer number of electrons. In this case, is related to response of the chemical potential of the molecule to a potential, we named the polarization Fukui function, defined in density functional theory of reactivity3,17,18 by Eq. (2). = …(2) The polarization Fukui function (Eq. (2)) is related to the linear electronic response by Dyson equations8,19,20. For an energy functional ! " which is a continuous function of , the charge transfer Fukui function (Eq. (1)) and the polarization Fukui function (Eq. (2)) are identical because they are related to each other by a Maxwell relation (Eq. (3)). = # = $ # % …(3) The relation (3) is broken once approximations are made to compute the derivative for a non-differentiable model of ! ". We demonstrate here that a Fukui function for the polarization can be also defined in classical physics by using the Thomson theorem for perfect conductors. The analogy with the electrostatic Fukui function permits to define a quantum molecular capacitance and to demonstrate that the variation of the molecular electronic density induced by an external potential can be decomposed in a localized and delocalized (Fukui) density variations associated to a fictitious internal charge transfer (perturbation at constant N). This clarifies the physical meaning of in Eq. (2) and its relation with the molecular capacitance and the so-called chemical hardness34. The Thomson Theorem and the Electrostatic Fukui Function and Hardness In order to apply electrostatics, one represents an isolated molecule by a perfect conductor in vacuum. What is the electric charge distribution at the equilibrium, −' "( , when the molecule is charged exactly with N electrons ( = "( )? The solution can be found by applying the variational Thomson theorem: the energy of the actual electrostatic field of charged conductors at equilibrium is a minimum relative to the energies of the fields which could be produced by any other charge distribution on or in the conductors21. According to this theorem, the charge distribution at equilibrium of an isolated conductor charged with N electrons is the one which minimizes the Coulomb energy (EC) (Eq. (4)), !) = *+ . |0| , Ω ′ …(4) under the constraint that = " and where Ω is the volume of the conductor. The solution is obtained by introducing a Lagrange multiplier λ, 2 !) − 3 −' = 0 3 = −' Ω 4 5. 6 ′ |. | …(5) …(6) where Eq. (6) is readily obtained by the functional differentiation of the Eq. (4) relative to the charge distribution −'" . The Eq. (6) states that the electrostatic potential generated by the charge added (i.e., −' "( ) is constant at every point r on and inside the conductor. In other words, there is no electric field inside the conductor, as it must be for a perfect conductor (which would create otherwise the instability of the charge density added). The Eq. (6) can be written as follows (Eq. (7)), 7* = ℎ* , 0 * 0 ′ …(7) where we have defined a normalized function * and an energy 7* , * = 4 …(8) :* 7* = − …(9) and where the kernel ℎ* , 0 is *+ ℎ* , 0 = |0| …(10) By definition, * is the electron density at equilibrium corresponding to a charge of exactly one electron on the conductor. The quantity 7* is twice the INDIAN J CHEM, SEC A, AUG-SEPT 2014 energy needed to charge the cluster with one electron. Indeed, using Eq. (6) in Eq. (4), one finds ; !) "( = , 7* , …(11) According to the definition of electrostatic capacitance C of an isolated conductor, one deduces21,22 7* = *+ ) …(12) It is worth noting that the Eqs (11) and (12) were first derived by Max Born in its seminal work on the volume and hydration of ions23. We name * and 7* the electrostatic Fukui function and hardness, respectively. The analogy between the chemical hardness in DFT of reactivity and the electrostatic capacitance (Eq. (12)) was pointed by Pearson24. For a spherical perfect conductor, the capacitance < is the radius of the sphere21. Based on this argument, Eq. (12) leads to a practical formula to compute the hardness of atoms as shown numerically by Islam and Ghosh25. As shown in electrostatics, for a perfect conductor, * is nonzero only at the surface of the conductor21. Therefore, using Eq. (7) and assuming that an atom is represented by a conducting sphere with a radius equal to its (covalent) radius (=> ), one obtains the following equation (Eq. (13)). 7* = * + ? . ′ |@ . |AB |0| C …(13) By replacing * 0 by the atomic frontiers orbitals in the Eq. (13), one recovers an earlier result derived by Chattaraj et al.26 Surprisingly, the electrostatic potential generated by the Fukui function27 (Eqs (1) and (2)) evaluated at the nucleus of an atom is also a measure of its chemical hardness28. It is interesting to point out that the electrostatic Fukui function is also the so-called shape function29-31 associated to the electronic density "( added to the molecule and solution of Eq. (6), and is also the first term in the density gradient expansion for the Fukui function26. The Thomson theorem states that the electrostatic Fukui function (the density induced by charging a conductor) is solution of a variational principle (Eq. (7)) which is closely related to the variational principle for the Fukui function32 derived in DFT. 1054 The electrostatic Fukui function * is related to the polarization. Indeed, the equilibrium density of electrons 2"( induced within a conductor (representing the molecule) by an external applied electric potential Φ*EF (constant N) can be deduced from the Thomson theorem as follows. One considers the molecule in contact with an infinite reservoir of electrons, i.e., one considers a grounded conductor. According to the Thomson theorem, the charge distribution −'2"( minimizes the total electrostatic energy !, ! = *+ , -Ω . 0 |. | − ' Ω 2" Φ*EF …(14) that is, Ω *4 5. 6 0 |. | = Φ*EF [HIJK'] …(15) In contrast, the functional differentiation of the Eq. (14) relative to −'2" for an isolated conductor under the constraints that the conductor remains neutral (Eq. (16)), 2 M! + O' Ω 2" P = 0 …(16) gives Eq. (17), −' Ω 4 5. 6 0 |. | + Φ*EF = O …(17) where O is a Lagrange multiplier associated to the charge conservation. The particular solution of Eq. (17) for O = 0, we name 2"QA( 0 , corresponds to the polarization density if the conductor is grounded, i.e. if the molecule is connected to an infinite reservoir of electrons (Eq. (15)). From Eq. (17), one realizes that the electron density 2"( induced by the external potential can be formulated in function of 2"QA( 0 and of the electrostatic Fukui function (Eq. (18)), 2"( = 2"QA( − * Δ …(18) where Δ ≝ Ω 2"QA( . For a grounded conductor, Δ is the amount of charge which is transferred from the ground (O = 0) to the conductor due to the polarization of the molecule by the external potential. The validity of Eq. (18) is established DELARUE & SENET: POLARIZATION, REACTIVITY & QUANTUM MOLECULAR CAPACITANCE 1055 by inserting (18) in (17), which gives two equations, Eqs (19) and (20), −' Ω TU4 5. 6 |. | 0 + Φ*EF = 0 Δ' Ω ? 0 0 |. | V Δ * W* ) = O …(19) …(20) By definition, 2"QA( is solution of the Eq. (19) (equivalent to Eq. (15)). Using Eqs (7) and (12), one finds the value of the Lagrange multiplier. = = O …(21) The Eq. (17) can be rewritten in an illuminating form: Ω *4 5. 6 0 |. | = Φ*EF − W* ) (22) Equations (18) and (22) provide a clear physical meaning of the electrostatic Fukui function and of its role in the electric polarization. Indeed, the density induced by the external potential if the molecule is connected to an infinite reservoir of electrons (conductor grounded) is 2"QA( . The spatial variation of this density depends on the external potential. For example, if the external potential is the electrostatic potential generated by a point charge, the density induced (2"QA( will be localized in the vicinity of the point charge and it will decrease with the distance from this point perturbation. The integration of this density, Δ, represents the number of electrons transferred from the reservoir to the molecule to build Δ = Ω 2"QA( . Because of the conservation of the total number of electrons, the electron number of the reservoir changes by −Δ. For an isolated molecule, the reservoir is the molecule itself and the Δ electrons arises from all the regions of the molecule according to −* Δ (Eq. (18)). The number of electrons −Δ can be interpreted as a virtual charge transfer of −Δ electrons from the molecule to itself in order to build the polarization charge, 2"QA( . Because there is no actual charge transfer, Δ is a continuous variable and can be fractional. Because the molecule is (virtually) charged with – Δ electrons, the external electric W* potential is shifted by − ) (Eq. (22)). Consequently, Eq. (18) tells us that an external electric potential has two effects: it induced a polarization density 2"QA( and a chemical density −* Δ if Δ ≠ 0. For the particular case of the electrostatic potential generated by a point charge, 2"QA( will be localized in the vicinity of the perturbation point charge whereas −* Δ is delocalized and does not depend on the position of the point charge (except through the value Δ). It means that a point charge will induce longrange modifications of the molecular electronic density depending on the values of the electrostatic Fukui function * . As shown in electrostatics, for a perfect conductor, * is nonzero only at the surface of the conductor (at the surface of the molecule in the present conducting simple model). Of course, a perfect conductor is a priori a very bad model of the dielectric properties of a molecule. However, as we show vide infra, the physics of the Eqs (18) and (22) is preserved in more accurate DFT models as discussed next. The Quantum Molecular Capacitance and the Fukui Function One applies to a molecule (in its ground-state) an electric external potential ΔZ*EF = −'Φ*EF . According to the first-order DFT perturbation theory, the electronic density induced by ΔZ*EF is the solution of Eq. (23)3, ℎ , 0 2"( 0 0 = eΦ*EF + 2\ …(23) where 2\ is the shift of the electron chemical potential of the molecule due to the perturbation. The hardness kernel ℎ , 0 is defined by the second derivative of the Hohenberg-Kohn functional (F) relative to the electronic density at constant external potential33, i.e., +] ℎ , 0 = 0 …(24) Using the Eq. (2) for the polarization Fukui function, one has Eq. (25). 2\ = −' 0 Φ*EF 0 ′ …(25) By definition, the electron density which would be induced by the external potential if the molecule would be connected to an infinite reservoir of electrons is solution of Eq. (23) with 2\ = 0, i.e., ℎ , 0 2" 0 0 A( = eΦ*EF …(26) INDIAN J CHEM, SEC A, AUG-SEPT 2014 1056 The Fukui function obeys the following equation (Eq. (27)), can be written as the response to an effective self-consistent “electric” potential Φ_`a @, ℎ , 0 0 0 = 7 Ω …(27) where η is the so-called chemical hardness of the molecule34 which plays a fundamental role in the DFT theory of reactivity22,35-41 and in the equalization principle of electronegativity42,43. Therefore, using Eqs (25) to (27), one gets Eq. (28), 2\ = − 2" 0 A( ′ = − 7Δ …(28) in which Δ represents the number of electrons induced on the molecule by the electric potential to build 2" A( . Because the molecule is its own reservoir of electrons (isolated molecule), the Δ electrons arises from all the regions of the molecule according to − Δ. The number of electrons −Δ can be interpreted as a virtual charge transfer of −Δ electrons from the molecule to itself in order to build the polarization charge 2" A( as shown by Eq. (28). Because there is no actual charge transfer, Δ is a continuous variable and can be fractional. Using Eq. (28), the first-order DFT equation reads as ℎ , 0 2"( 0 0 = eΦ*EF − 7Δ …(29) or ; * ℎ , 0 2"( 0 0 = Φ*EF − *W ) …(30) where we define the quantum molecular capacitance < = ' , /7 by analogy with Eq. (12). It is readily shown that Eq. (31), 2"( = 2" A( − Δ *4 5. 6 0 |. | W* ) …(32) where Ω is the entire space and 1 ℎ , 0 E> 2"( 0 0 Φb>? = Φ*EF − d ' ; − * ℎ , 0 e 2"( 0 0 …(33) by defining ℎ , 0 E> and ℎ , 0 e as the second functional derivatives of the exchange-correlation and kinetic energy functionals, respectively. Equation (32) shows that the molecule can be viewed as a perfect conductor with a capacitance < responding to an effective “electric” potential Φb>? . Eq. (32) permits to formulate the condition for a metal (molecule) to behave as a perfect classical conductor. Indeed, comparing Eqs (32) and (33) with Eq. (22), one deduces that a system will behave as a perfect conductor if [ℎ , 0 E> + ℎ , 0 e ]2"( 0 0 = 0 …(34) Because of the charge conservation, Eq. (34) is equivalent to Eq. (35), ℎ , 0 E> + ℎ , 0 e = f …(35) where K is an arbitrary constant. The Eq. (35) is a nontrivial condition. For example, for a metal described by the Thomas-Fermi-Dirac (TFD) model8, the condition (35) is not fulfilled. In TFD, one has (Eq. (36)), ℎ g]h , 0 E + ℎ g]h , 0 e ,j.l(m =i …(31) is the solution of Eq. (30). Equation (31) clearly shows the physical meaning of the polarization Fukui function (defined by Eq. (2)); it controls the internal electron flows needed to build the polarization charge density 2" A( induced by an external potential. The quantity − Δ is a delocalized chemical charge distribution induced by an external potential. Then Eq. (30) resembles the electrostatic Eq. (22) and = Φb>? − n o . − j.p,p + q 2 − ′ o . …(36) where the units of the kernel are eV Å2. The condition (34) reads as ,j.l(m i n o . − j.p,p + q 2"( = 0 o . …(37) The term in brackets should be zero for a perfect conductor which is not the case in the TFD model. 1057 DELARUE & SENET: POLARIZATION, REACTIVITY & QUANTUM MOLECULAR CAPACITANCE This means that the electric potential inside a conductor described by the TFD model is not constant. Conclusions We demonstrate that the electron density induced in a molecule by an external potential can be decomposed in the sum of two contributions (Eq. (31)); 2" A( and − Δ and involved a (generally nonzero) virtual charge transfer, – Δ. By analogy with electrostatics, this charge transfer can be interpreted as the amount of charge which would be transferred from an infinite reservoir of electrons in order to build the polarization charge, 2" A( . Because an isolated molecule is its own reservoir, this charge transfer arises from the molecule itself from all regions of the molecule proportionally to the value of the (polarization, Eq. (2)) Fukui function, . A high value of within a molecule means that this region is a local reservoir of electrons for any perturbation. Consequently, a localized perturbation may have an effect far away from its vicinity, in the regions of the local reservoirs of electrons, if – Δ ≠ 0. The local reservoirs of electrons should play a direct role in the chemical reactivity of the molecules. The analogy with electrostatics allows us to define a quantum molecular capacitance C. As in electrostatics (Eq. (7)), the potential associated to the charge distribution −' Δ is a constant equal to W* − ) . For any external potential, the linear response of the molecule can be formulated as the response to an effective electric potential (Φb>? ) as shown by Eq. (32). Therefore, one may hope that appropriate models inspired from electrostatics can be used to model the reactivity of (large) molecules. Implementation of such models and of the Eqs (31) and (32) are currently being undertaken. 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