Conceptual Surface Electrochemistry and New Redox Descriptors

Article pubs.acs.org/JPCC

Conceptual Surface Electrochemistry and New Redox Descriptors J.-S. Filhol*,†,‡ and M.-L. Doublet†,‡ †

Institut Charles Gerhardt, CNRS and Université Montpellier II, Place Eugène Bataillon, 34 095 Montpellier Cédex 5, France Réseau sur le Stockage Electrochimique de l’Energie (RS2E), CNRS FR3459, 80039 Amiens Cedex, France

‡

S Supporting Information *

ABSTRACT: This paper aims at enlightening the link between surface properties, conceptual density functional theory (DFT), and electrochemistry. The energy of a metallic charged surface is expanded into classical terms such as electrochemical potential or chemical hardness and into new terms such as electromechanical coupling vector, electrochemical Hessian matrix change, etc. These quantities are shown to be crucial parameters to understand interface electrochemistry and in particular the surface zero-charge potential, the capacitance and its derivatives, and the surface structural polarization at the atomic scale. Furthermore, the Fukui function initially introduced to rationalize molecular reactivity is here shown to directly probe the electrochemical activity of the surface. First, it gives the spatial repartition on the surface of the charge added during an electrochemical step, and second it quantifies the electrochemical parameters of the system: the electrochemical potential and the electromechanical coupling vector mainly arise from the interaction of the Fukui function with the local surface dipole and the nucleus electric field, while the capacitance mostly comes from the Fukui function self-electrostatic interaction. The electrochemical Hessian change matrix controls part of the bond modifications induced by the electrochemical process and directly depends on the Fukui function derivatives with respect to atomic displacements. This description not only allows linking the electrochemical bond modification with conceptual DFT and molecular orbital theories but can also be used quantitatively to extract electrochemical properties from response theory.

I. INTRODUCTION Electrochemistry is highly important in many industrial fields from metal production by electrolysis, corrosion protection, metal coating, or energy storage. For more than a decade, density functional theory (DFT) has been successfully applied to model electrochemical reactions at interfaces, therefore reproducing vibration frequency change, phase diagram, or reactivity as a function of the applied potential or electric field1−10 even if problems still exist.11 However, if the modeling correctly reproduces macroscopic experimental data,12 such as dissociation energies, working voltage, etc., the fundamental understanding of the microscopic phenomena associated with an electrochemical reaction is still unclear. Many conceptual approaches exist for rationalizing chemical reactivity of molecular or extended systems such as density analysis,13 molecular orbital approaches,14 conceptual DFT15,16 (Fukui functions...), and dband center.17 No such developments are as much advanced in surface electrochemistry. As an example, the shift in the Pt(111) surface reactivity toward CO adsorption upon electrochemical charging is captured neither by the d-band center approach nor by the band structure.18 Thus, the reactivity descriptors commonly used in chemistry to rationalize the reaction mechanisms need to be adapted to include the electrochemical dimension. A step toward this direction is nicely exemplified in the Schmickler and Santos book,19 in which most of the electrochemical effects are linked with general quantum parameters. We also have shown that band theory, molecular © 2014 American Chemical Society

orbital, and conceptual DFT models can be qualitatively mixed to understand different aspects of electrochemical reactivity,20 but formal relations need to be clarified. We recently started this development in the simple case of a CO molecule adsorbed on a metal surface.21 In this study, we showed that the modification of the C−O bond strength upon charging mainly arises from electrochemical effects associated with the change in the molecular orbital occupation and from electromechanical effects due to the interplay between the structural deformation induced by the electric field and the system anharmonicity. In the present paper, we want to generalize this decomposition to any surface/ interface. We use a DFT conceptual-like approach inspired by Parr and Yang’s work22 to explicitly account for the electronic structure of the charged interface and to decompose its energetics within the large capacitance limit of the double layer. First, we focus on the energy expansion of a charged metallic surface as a function of the charge (N) and the atomic positions (R⃗ ). Then, we extract the system deformation with charge and the associated energy change using conceptual DFT indexes and introducing new descriptors which are relevant to electrochemistry. Finally, we establish direct links between the electrochemical parameters and the Fukui function or its derivatives. This approach provides a meaningful and powerful Received: March 6, 2014 Revised: July 24, 2014 Published: July 28, 2014 19023

dx.doi.org/10.1021/jp502296p | J. Phys. Chem. C 2014, 118, 19023−19031

The Journal of Physical Chemistry C

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0). It is worth noting here that in classical electrochemistry the parameters generally used are the surface energy γa and the average surface charge σa, leading to

tool to discriminate and rationalize the electrochemical response of a charged surface in terms of energy, potential, deformation, and bonding. We believe this approach and its associated descriptors extend the classical one for electrified interfaces19 and open new routes for apprehending surface/interface electrochemistry in many different systems such as fuel cells, Li-ion batteries, or any other electrochemical device in which interface electrochemistry may affect the system performances.

σa + ηSσa 2 − ηS(2)σa 3 e ⎧J ⎛→ ⎛σ ⎞ ⎛σ ⎞ ⎞ 1 + ⎜βS σa 2 − ⎜ a ⎟→ α ⎟ΔR⃗ + t(ΔR⃗)⎨ − ⎜ a ⎟Ξ ⎝e⎠ ⎝e⎠ ⎠ ⎩S ⎝ 2

γa(σa , R⃗) = γa (σa = 0, R⃗ 0) − μ0 0

II. GENERAL EQUATIONS FOR A CHARGED SURFACE The energy variation of a surface is directly linked to the modification of its electrochemical potential μ due to the added charge N and to the effect of the variation of the external r ) (caused by the change in the nuclei positions) on potential v(→ r )15,23−25 the charge density ρ(→ dE = μdN +

∭ ρ(→r )dv(→r )d→r

+

→ 2 3 → (2) 2 3 with ηS = ηS/e2, η(2) S = (η S /e ), and β S = ((β )S /e ). B). Geometry Reorganization Imposed by the Charging. At equilibrium, N and R⃗ are coupled since for each charge N the system reaches a new equilibrium R⃗ (N) where the forces acting on atoms vanish. Then, for every atomic component j, the energy checks the implicit expression

(1)

This can be written for a total surface area S as

⎛ ∂E(N , R⃗(N )) ⎞ ⎜⎜ ⎟⎟ = 0 ∂R j ⎝ ⎠

μdσa 1 + ρ(→ r )dv(→ r )d→ r e S 1 ρ(→ = ϕdσa + r )dv(→ r )d→ r S

∭

dγa = −

∭

This gives the implicit matrix condition

with γa = E/S being the average surface energy, σa = N/S the average surface charge, and ϕ the surface Galvani potential (vide inf ra). A). Expression of the Surface Energy. Using a DFT conceptual theory derived from the Evans approach,26 the fundamental electronic energy variation is controlled by two terms: the number of excess charge from neutrality and the nucleus position R⃗ which sets the external potential. For a metallic N-charged surface characterized by the geometry R⃗ , the expansion of the energy as a function of these two parameters is written as follows

⎛ ⎞ 1 Nα⃗ + β ⃗N 2 + ⎜J + N Ξ + Λ(ΔR⃗)⎟(ΔR⃗) = 0⃗ ⎝ ⎠ 2

{

}

⎛⎛ ⎞ ⎞ 1 ΔR⃗ = −NJ −1α⃗ + N 2J −1⎜⎜Ξ + Λ(J −1α⃗)⎟(J −1α⃗) − β ⃗⎟ ⎝ ⎠ ⎝ ⎠ 2 (4)

This can be simplified in first-order to give the elastic limit (characterized by the matrix J) as the charging creates a force Nα⃗ leading to the new equilibrium structure ΔR⃗ = −NJ −1α⃗

(5)

The second-order term involves both the modification of the Hessian matrix due to the electrochemical effect (Ξ), the anharmonicity (Λ), and the nonlinear electromechanical response (β⃗). If one restricts the energy expansion to the thirdorder, then only the linear term impacts the surface energy change with charging. The energy of the system at the new equilibrium position given in eq 5 is a function of the total added charge N and is given by

(2)

In this expression η = 1/2 (∂ E/∂N ) is the chemical hardness; η(2) = 1/6(∂3E/∂N3) is the chemical hyper-hardness, i.e., the higher-order term corresponding to the hardness; J = (J)ij = [(∂2E)/(∂Ri∂Rj)] is the force constant matrix, i.e., the Hessian matrix commonly used in vibration modes calculations; Λ = (Λ)ijk = [(∂3E)/(∂Ri∂Rj∂Rk)] is the third-order anharmonic mechanical tensor including first-order anharmonic effects; α⃗ = (α)i = (∂3E/∂Ri∂N) = grα⃗ d(μ) = −(∂F⃗/∂N) is the electromechanical coupling vector which quantifies the mechanical response of the system to the charge; β⃗ = (β)i = 1/2(∂3E/ ∂Ri∂N2) = −1/2(∂2F⃗/∂N2) is the second-order electromechanical coupling vector; and Ξ = (Ξ)ij = [(∂3E)/(∂Ri∂Rj∂N)] is the electrochemical Hessian change matrix which quantifies the bond strength change upon charging. This expression is the general expression of the surface energy for any charge N and any geometry R⃗ , these two variables being independent. It is referenced to the neutral (N = 0) surface in its equilibrium structure (R⃗ 0) for which the forces acting on atoms are nil (F⃗ = 2

(3)

which can be linearized in the small charging limit into

E (N , R ⃗ ) = E 0 (N = 0, R⃗ 0) + μ0 N + ηN2 + η(2)N3 → 1 + (N→ α + β N 2)ΔR⃗ + t(ΔR⃗) 2 1 J + N Ξ + Λ(ΔR⃗) (ΔR⃗) 3

⎫ 1 Λ(ΔR⃗)⎬(ΔR⃗) ⎭ 3S

2

⎛ ⎞ 1 E(N ) = E(0) + μ0 N + ⎜η − αJ⃗ −1α⃗⎟N 2 ⎝ ⎠ 2 1⎛ + ⎜η(2) + t(αJ⃗ −1)Ξ(αJ⃗ −1) − 2β ⃗J −1α⃗ 2⎝ ⎞ 1 − t(αJ⃗ −1)(Λ(αJ⃗ −1))(αJ⃗ −1)⎟N3 + Ο(N 4) ⎠ 3

(6)

from which we can extract the key parameters commonly used in electrochemistry. The first one is the electrochemical potential which is given by 19024


The Journal of Physical Chemistry C μ=

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⎛ ∂E ⎞ ⎜ ⎟ = μ + (2η − αJ ⃗ − 1 α ⃗ )N 0 ⎝ ∂N ⎠

Interestingly, C is proportional to the invert of the positive chemical hardness η reduced by the term α⃗ J−1α⃗ which here corresponds to a structural polarization, i.e., the mechanical work associated with the charge-induced response of the system. It is the scalar product of the charge-induced force Nα⃗ by the chargeinduced displacement NJ−1α⃗ . As we will see now, the physical meaning of these parameters lies in their relation with the electron density. C). Expression of Electrochemical Parameters. a). Expansion of the Density. In the absence of a double layer, the electron density variation of a charged metallic system for a frozen geometry is given by

3 ⎛ (2) 1 ⎜η + (αJ⃗ −1)Ξ(αJ⃗ −1) − 2β ⃗J −1α⃗ − t(αJ⃗ −1) 2⎝ 3 ⎞ (Λ((αJ⃗ −1)))(αJ⃗ −1)⎟N 2 + Ο(N3) ⎠ (7) +

This expression allows recovering the usual linear behavior between the surface charge and the electrochemical potential at low charging which turns to a parabolic dependence at higher charging. The electrochemical potential (per electron) can be linked to the surface Galvani potential ϕ, if the potential reference is set to the vacuum potential reference, by the expression

1 ρ (N , → r ) = ρ(0, → r ) + f (→ r )N + h(→ r )N 2 + Ο(N3) 2 r ) = (∂ρ(→ r )/∂N) and h(→ r ) = (∂f(→ r )/∂N) = (∂2ρ(→ r )/ with f(→ → 2 r) ∂N ). In this expression, f( r ) is the Fukui function22 and h(→

μ = −eϕ

Using this gauge, the electrochemical potential is equal to the opposite of the surface work function.27 The free electrochemical energy F(N) = E(N) − μN is the relevant energy at constant potential and is proportional to the free electrochemical surface energy of the system as (F/S) = γfree a = γa + ((μσa)/e) = γa − ϕσa. F(N) can then be expanded as

its second-order. These functions can be complex to compute in some molecular cases,28 but in the present case of a nonmagnetic metallic infinite surface, the Fukui function is unique because of the continuous electronic structure at the Fermi level: therefore the classical Fukui functions are equal (f+ = f− = f 0). The Fukui function is mostly positive and normalized to 1: +∞

⎛ ⎞ 1 F(N ) = E(0) − ⎜η − αJ⃗ −1α⃗⎟N 2 ⎝ ⎠ 2 ⎛ − ⎜η(2) + t(αJ⃗ −1)Ξ(αJ⃗ −1) − 2β ⃗J −1α⃗ ⎝ −

⎞ 1 t −1 (αJ⃗ )(Λ(αJ⃗ −1))(αJ⃗ −1)⎟N3 + Ο(N 4) ⎠ 3

∫−∞

(8)

2 1 (μ − μ0 ) 2 (2η − αJ⃗ −1α⃗)

⎛ − ⎜η(2) + t(αJ⃗ −1)Ξ(αJ⃗ −1) − 2β ⃗J −1α⃗ ⎝ −

3 ⎞ (μ − μ0 ) 1 t −1 (αJ⃗ )(Λ(αJ⃗ −1))(αJ⃗ −1)⎟ ⎠ (2η − αJ⃗ −1α⃗) 3

+ Ο(N 4)

(9)

F(μ) has an inverted parabola shape centered at the zero-charge electrochemical potential μ0 as imposed by the thermodynamics. It is worth noting here that for large potential away from μ0 a nonparabolic behavior may appear due to non-negligible thirdorder terms. The second important parameter that may be extracted from these relations is the differential capacitance of the system C. It is defined as S S ⎛ ∂σ ⎞ ⎛ ∂N ⎞ 1 C = 2⎜ a⎟ = ⎜ ⎟ = 2 e e ⎝ ∂ϕ ⎠ ⎝ ∂μ ⎠ 2η − αJ⃗ −1α⃗ t

−

t

3η(2) + 3 (αJ⃗ −1)Ξ(αJ⃗ −1) − 6β ⃗J −1α⃗− (αJ⃗ −1)(Λ(αJ⃗ −1))(αJ⃗ −1) (2η − αJ⃗ −1α⃗)3

(μ − μ0 )

with the zero-charge differential capacitance C0 =

e2 1 = −1 S S(2η − αJ⃗ α⃗) 2ηS − 2 αJ⃗ −1α⃗ e

+∞

∫−∞

∂N ρ(→ r )d→ r = =1 ∂N

Negative contributions, although smaller than positive contributions, are associated with the electronic polarization of the zero-charge electron density in response to the added charge. In that sense, the Fukui function is clearly a many-body response function including not only the frontier orbitals (HOMO/ LUMO) response to the charge but also the polarization effects directly induced by the added charge on the lower- and higherlying orbitals. The Fukui function is unambiguous (in opposite to charge and orbitals) as it is an observable quantity associated with the difference of two observable charge densities. In the particular case of electrochemistry, the Fukui function is meaningful as it captures the spatial distribution of the added charge. To exemplify this, we have performed periodic calculations on various metallic surfaces using the DFT formalism within the PBE29 generalized gradient approximation and projected augmented wave (PAW) pseudopotentials30,31 as implemented in the VASP code6,31 following an electrochemical approach previously developed.20 The Fukui function can be computed using any of the electrochemical scheme allowing a modification of the surface charge. Pt(111) and Pd(111) surfaces were built with a vacuum of about 15 Å. Molecules were adsorbed as most symmetrically as possible on both sides of the surface to avoid long-range electrostatic interactions between periodic images arising from periodic boundary conditions. The energy cutoff was set to 450 eV. All structures were fully relaxed up to residual forces on atoms lower than 0.01 eV Å−1. The first- and second-order Fukui functions were computed from finite surface metallic charging differences of 0.1 |e|/unit cell (over a total number of electrons of the order of 300) at constant surface geometry

Using eq 6, the μ-dependence of the free energy is written F(μ) = E(μ0 ) −

∂ f (→ r )d→ r = ∂N

(10) 19025

f ( r ⃗) =

ρ( r ⃗)(N + 0.1) − ρ( r ⃗)(N ) 0.1

h( r ⃗) =

ρ( r ⃗)(N + 0.1) + ρ( r ⃗)(N − 0.1) − 2ρ( r ⃗)(N ) (0.1)2 dx.doi.org/10.1021/jp502296p | J. Phys. Chem. C 2014, 118, 19023−19031


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As there is no bulk charging/no electric field inside a conductor, → the electric field created by the nuclei charges Zi at R i and the p r ) can be approximated to a surface dipole → electron density ρ(→ → expansion (in which r is the total charge barycenter) interacting

As illustrated in the different cases of Figure 1, the added charge mainly localizes in the vicinity of the surface and exponentially

0

with the added charge represented by the Fukui function. Hence, the electrochemical potential depends not only on the surface electrostatic dipole, as it is generally assumed in the literature,18,32 but also on the variation of the kinetic energy density and on the exchange-correlation potential. These additional termsand more specifically the dominant kinetic termmay significantly affect the electrochemical potential under certain conditions. They are nevertheless moderate at a vacuum−metal interface. Indeed, the kinetic contribution which is the product of the Fukui function by the electron density r )(∂T[ρ]/∂ρ) ∼ ∫ f(→ r )ρ3/2(→ r )d→ r ) within the Thomas(∫ f(→ Fermi modelalmost vanishes in vacuum conditions since we r ) is maximal (respectively minimal) in the have seen that f(→ r ) is minimal (respectively maximal). This spatial area where ρ(→ has been recently quantified for the first time in the case of the CO/Pt(111) surface,10 where a maximal deviation from the pure dipole of 3−4% was found. This recent quantification justifies the general assumption that the electrochemical potential, in many cases, is governed by the interaction between the surface electrostatic dipole and the added charge. In the case of solvent− metal interfaces such as the one presented in Figure 1, the Fukui function has an important contribution on the atoms of the solvent leading to possible non-negligible terms of the kinetic part that would have to be evaluated. This expression allows also obtaining the Galvani potential for the surface that is

Figure 1. Fukui function f(r)⃗ and second-order Fukui function h(r)⃗ for (a) the bare Pt(111) surface; (b) the Pt(111)/CO surface; (c) the water monolayer/Pd(111) surface; and (d) the water multilayer/Pd(111) surface. Blue atoms are metallic; red are oxygen; gray are carbon; and white are hydrogen. The blue and yellow isosurfaces illustrate the positive and negative contributions to the distribution, respectively.

fades into the vacuum (or solvent) due to the potential barrier or into the metal bulk due to the high screening of the conducting electrons. Therefore, the Fukui function is a particular descriptor of the interfacial electrochemistry. The second-order Fukui function h(r)⃗ is also shown for different surfaces in Figure 1. Its spatial average value being nil (∫ +∞ ‑∞ h(r)⃗ = 0): there are as many positive and negative contributions. h(r)⃗ becomes important for large surface charging that cannot be only described by the linear response given by the Fukui function. It gives higher-charging spatial distribution, and its sign is qualitatively the opposite of the Fukui function moderating the first-order charging. These two functions are key parameters that allow expressing and computing all the other electrochemical properties that are crucial to understand interface electrochemistry at the atomic scale and its impact on the system reactivity and/or performance. This will be exemplified below. b). Electrochemical Potential. As shown in eq 6, the firstorder energy variation of a charged surface is governed by the zero-charge electrochemical potential μ0. Using the Fukui descriptor, the electrochemical potential in the DFT formalism is written μ0 =

μ0 =

⎛

∫

⎞

for an atom i with S being the surface area. However, in this expression, the Fukui dependency is lost. This can be recovered using the charge derivation of the Hellman−Feynman scheme36 to obtain the expression of the electromechanical forces

→ ⎞⎞ ρ ( r ′ ) → ⎟⎟ → d r ′ → ⎟⎟d r |→ r − r′ | ⎠⎠ ⎛

∫ f (→r )⎜⎜⎝ ∂T∂[ρρ] + Vxc[ρ] −

→ p (→ r −→ r )⎞ f (→ r ) ⎛ ∂T[ρ] ⎜⎜ r + Vxc[ρ] − → →0 ⎟⎟d→ e ⎝ ∂ρ | r − r0 | ⎠

⎛ ∂ 2E ⎞ ⎛ ∂ 2E ⎞ ε0S εS ≃ 0 (α ) X i (Z B)Xi , εz = ⎜ ⎟ ≃ −⎜ ⎟ e ⎝ ∂Xi ∂N ⎠ e ⎝ ∂Xi ∂εz ⎠

⎛ ⎛ Z ∂T[ρ] + Vxc[ρ] + ⎜⎜∑ → i f (→ r )⎜⎜ → ∂ ρ ⎝ i |R i − r | ⎝

∫

∫

c). Electromechanical Forces. The electromechanical vector → ρ can be seen as a surface-specific case of the Born effective charge tensor33 ZB used to study electric polarization effects in condensed matter and plays a central role in particular for modeling LO/TO splitting.34,35 It is proportional to the z component of ZB following the relation

∫ f (→r )⎜⎝ ∂∂Eρ ⎟⎠d→r

+

μ0 ≃

ϕ≃−

⎛ ⎛ ∂E ⎞ x − Xi → r) (F )xi = −⎜ ⎟ = −Zi⎜ ρ(→ → dr ⎜ → ⎝ ∂Xi ⎠ | r − R i|3 ⎝

∫

→ p (→ r −→ r0 ) ⎞ → ⎟⎟d r r −→ r0 | ⎠ |→

Xj − Xi ⎞⎟ − ∑ Zi → → | R j − R i|3 ⎟⎠ j≠i

(11) 19026



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x−X ∂ ∫ ρ(→ r ) → →i 3 d→ r ⎛ ∂FXi ⎞ | r − R i| (α)xi = −⎜ ⎟ = Zi ∂N ⎝ ∂N ⎠ x − Xi → r) = Zi f (→ → dr → | r − R i|3

contributions to the hardness are negligible under vacuum conditions. The chemical hardness of the surface is then controlled by two integrals (see eq 13) which correspond to the added charge electrostatic self-interaction and the interaction of the second-order charge modification with the surface dipole. In most cases, the self-interaction term dominates over the second term. The electrochemical capacitance then arises mainly from a charge distribution self-interaction similarly to the classical electrostatics of a capacitor, with the main difference that the Fukui function is setting the charge distribution. e). Higher-Order Terms. The hyperhardness expression, as deduced from eq 13 for the sake of simplification is then written

∫

(12)

The electromechanical coupling term arises from the electric field induced on the nucleus by the added charge distribution which, in turn, is controlled by the Fukui function. A representation of the computed electromechanical vector is shown in Figure 2 for a water monolayer on a Pd(111) surface.

6η(2) =

⎛ ∂ ⎜ ∂ 3E ≈ ∂N ⎜⎝ ∂N3

→ f (→ r )f ( r′ ) → → → d r′ d r r − r′ | |→

∬

⎞ → p (→ r −→ r) ⎟ h(→ r ) → → 30 d→ r⎟≈2 | r − r0 | ⎠

−

∫

−

∫ f (3) (→r )

→ f (→ r )h( r′ ) → → → d r′ d r |→ r − r′ |

∬

→ p (→ r −→ r0 ) → dr → → | r − r |3

(14)

0

The first term corresponds to the interaction between the firstr ) and the order added charge given by the Fukui function f(→ r ); the last term corresponds to the second-order given by h(→ r ) in interaction with the surface third-order Fukui function f(3)(→ dipole. The second-order electromechanical response is

Figure 2. (a) Fukui function and electromechanical vector for a H-up water monolayer on Pd(111). (b) Labels for the different atoms.

x−X ∂ 2 ∭ ρ(→ r ) → →i 3 d→ r ⎛ ∂ 2F ⎞ | r − R i| xi (β)xi = − ⎜⎜ 2 ⎟⎟ = Zi ∂N2 ⎝ ∂N ⎠ ⎛ ∂ 2ρ(→ r ) ⎞ x − Xi → = Zi dr ⎜ 2 ⎟ → ⎝ ∂N ⎠ | r − → R i|3 x − Xi → = Zi h(→ r) → dr → | r − R |3

For the two oxygen atoms O2 and O3, the Fukui function is positive upward (+z) and negative downward (−z). The electromechanical vector is directed from the positive part of the Fukui function toward the negative part. The electromechanical response of the system can then be predicted through the computation of the Fukui function for the uncharged system. d). Chemical Hardness. The chemical hardness is written ∂μ 2η = = ∂N =

∫

∫

∭

⎛ ∂ 2E ⎞ ⎛ ∂E ⎞ r ))2 ⎜ 2 ⎟ + h(→ r )⎜ ⎟d→ r (f (→ ⎝ ∂ρ ⎠ ⎝ ∂ρ ⎠

⎛ ⎛ ∂ 2T[ρ] ∂Vxc[ρ] ⎞ + f (→ r )⎜⎜f (→ r )⎜ ⎟+ 2 ∂ρ ⎠ ⎝ ∂ρ ⎝ ⎛ ∂T[ρ] → + Vxc(→ h( r )⎜ r)+ ⎜ ∂ρ ⎝ → ⎞ ρ0 ( r′ ) →⎟ → ′ r d dr → ⎟ |→ r − r′ | ⎠

r + d→

+

∫

∭

∫

∑ i

∫

i

It corresponds to the force induced on the ions by the charge distribution associated with the second-order Fukui function. In other words, it represents the modification of the atomic forces due to the nonlinear term of the added charge. The last electrochemical element is Ξ and corresponds to the electrochemical change of the Hessian matrix. This matrix can be seen as controlling the change induced by the charge N on the Hessian matrix J, giving an effective Hessian matrix Jeff(N) = J + NΞ. This element was shown to be important to predict the chemical bonding under electrochemical conditions.21 It only depends on the Fukui function and on its derivative with respect to atomic displacements

→ ⎞ f ( r ′ ) →⎟ ′ r d → ⎟ |→ r − r′ | ⎠

Zi → → |R i − r |

→ → p (→ r −→ r0 ) → f (→ r )f ( r′ ) → → → r r h r d d ( ) dr 2η ≃ ′ − → → → → | r − r0 |3 | r − r′ | ⎛ ∂T[ρ] ⎞ ∂ f (→ r )⎜ + Vxc(→ r )⎟d→ r + ∂ρ ⎝ ∂ρ ⎠ → → p (→ r −→ r) f (→ r )f ( r′ ) → → ≃ r ) → → 30 d→ r d r′ d r − h(→ → → | r − r | 0 | r − r′ |

∬

∫

⎛ ⎛ ∂(J )X , X ⎞ i j ⎟ = Zi⎜ (Ξ)Xi , Xj = ⎜⎜ ⎟ ⎜ ⎝ ∂N ⎠ ⎝

∫

∬

(15)

∭

r ) x − Xi ∂f (→ → ∂Xj |→ r − R i|3

→ ⎞ ⎛ 3(x − Xi)2 − |→ r − R i|2 ⎞⎟ →⎟ r) d r + δ(Xi − Xj)⎜⎜f (→ ⎟ ⎟ → r − R i|5 |→ ⎝ ⎠ ⎠

∫

(16)

The bond change between the components Xi of atom i and the component Xj of atom j is given by the off-diagonal term Zi → r ))/(∂X ))((x − X )/(|→ r − R |3))d→ r which directly ∫ ∫ ∫ ((∂f(→

(13)

For the same reasons as those previously discussed for the electrochemical potential, the kinetic and exchange-correlation

j

19027

i

i



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Figure 3. Graphical representation of the Hessian matrix J element entries and of the electrochemical change matrix (Ξ) for a water monolayer/Pd(111) surface. The atom labels are shown in Figure 4.

destabilizing added charge self-interaction characterized by the Fukui function. In other words, the more localized the Fukui function, the larger the self-interaction and therefore the smaller the capacitance. This is illustrated by the comparison between the bare metal surface and the surface on which we have adsorbed a CO molecule, a water monolayer, or a water multilayer. While the surface charge is mainly localized on the metal atoms located at the surface, it becomes more and more delocalized in the other cases. As shown in Figure 1, the surface hardness (capacitance) thus decreases (increases) from bare to adsorbed surfaces and from low to high surface coverage. The second effect is the α J−1→ α . As J is a Hessian matrix close to structural polarization → the surface equilibrium position, all its eigenvalues are positive; α J−1→ α is negative. It leads to a reduction of the therefore −→ hardness and an energy stabilization induced by the structural polarization. Hence, the stronger the polarization, the larger the capacitance. Generally speaking, polarizable surfaces which are susceptible to accommodate more charge than nonpolarizable surfaces should be efficient for electrochemical reactivity/ applications. As an example, the water multilayer surface allows a strong structural polarization which stabilizes the added charge, thus increasing by 3.3 the capacitance compared to the water bilayer surface. The increase of structural polarization increases strongly the surface capacitance allowing important charging and electrochemical effects. Note that the present approach does not explicitly account for the electrochemical layer structure which may be highly important to understand the electrochemical effects: we then discuss the effect of the double layer in the next paragraph. B). Extension to Realistic Electrochemical Surfaces. In a real electrochemical system, the total capacitance arises not only from the local surface capacitance CS (mostly associated with the inner Helmholtz plane) but also from the one of the outer and diffuse layer Cout leading to a total capacitance that can be approximated at first order to CT = CSCout/(CS + Cout). As this total capacitance is dominated by the smallest capacitance, high Cout and CS are needed to obtain a substantial surface charging and therefore electrochemical activity. This is usually achieved by electrochemists in their typical setup through the use of

depends on the derivative of the Fukui function with respect to the atomic positions.

III. DISCUSSION An important point in electrochemistry is to determine how the chemical bond is modified with the surface charging. As the modification of the 0-charge force constant J with charge is given α )), a modification in the bonding caused by by N(Ξ − 1/3Λ(J−1→ the charging can have three origins. The first origin lies in the ability of the surface to get charged and will be discussed in the next section; the second lies in the 1/3Λ(J−1→ α ) term which corresponds to the conjunction of the structural modifications induced by the electric field generated by the added charge distribution and the bond anharmonicity. As previously shown, the added charge induces a structural modification NJ−1→ α which, in turn, induces a weakening or a strengthening of the different bonds due to bond anharmonicity.21,37 The third origin lies in the Ξ term which corresponds to the chemical bond modification induced by the charging: the added charge flows into orbitals with a bonding or antibonding character which triggers bond strengthening or lengthening. However, usually the second and third terms (Ξ − 1/3Λ(J−1→ α )) are small compared to the force constant matrix J, and therefore large charging is required to modify the surface bonding. This charging ability is controlled by the surface capacitance. A). Surface Capacitance and Fukui Function. The ability to charge a surface is given by the surface capacitance C0 (see eq 10) which is primarily controlled by the surface hardness: the harder (large η) the system, the smaller the capacitance. As electrochemistry only occurs with large enough charging, a first criterion for an electrochemical reaction to occur is that the system is soft. This hardness is linked to the Fukui function through a charge−charge repulsion-like term 1/2 r )f(→ r ′))/(|r − r′|)]d→ r ′d→ r possibly modulated by a ∫ ∫ [( f(→ r )[(p⃗(→ r − weaker charge surface−dipole interaction ∫ h(→ → → → 3 r→0))/(| r − r 0| )]d r . Therefore, the factor decreasing the capacitance of a metallic surface can be roughly evaluated from a 19028



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electrolytes with high concentration and polar solvents to maximize Cout and gets CT ≈ CS. The electrochemical surface properties in these conditions become intrinsic to the surface itself and weakly dependent on the double layer. Therefore, CS can be mainly associated with the inner Helmholtz layer and mostly depends on the local surface/interface electronic properties. Another way to achieve high Cout is to approach a scanning tunneling microscope (STM) tip close to a surface. In this case, the surface charging can be important enough to create potentialinduced structural modifications. Therefore, the observed STM tip induced reactions38 can suggest that electrochemistry can even be done under ultrahigh vacuum conditions. More generally, electrochemical effects should appear when high capacitance structures are present along with different polar surfaces. These effects should occur on nanoparticles18 at the interface of polar surfaces20 but also at edges,39 steps,40 tips, or interfaces.41,42 A high enough surface charge is then a necessary condition to achieve electrochemical reactivity but not sufficient as it should also modify the bonding. C). Electrochemical Bond Modification. Explicitly defining bonding interactions within multiatomic systems can be challenging.43 One iteratively improvable way to quantitatively describe bonding between atoms is to use the Hessian matrix. A bond between two atom components i and k should be associated with a force constant term Jik. Then, the Hessian matrix can be used to extract the connectivity in the system. In Figure 3, we give a graphical representation of the Hessian matrix for the water monolayer/Pd(111) surface. The Hessian matrices and their charge derivatives were obtained by discrete calculation using 0.01 Å displacement steps for different charging of 0.1|e|. A high value of the off-diagonal matrix elements suggests a bond between atoms as seen between O2 and H4, H5 and O3, with H6 and H7. The electrochemical change matrix (Ξ)XiXj shown in Figure 3 gives the modification of the force constants between atoms upon oxidation (electron removal). (Ξ)XiXj can be seen as a good approximation to bond modification upon charging. In the present case, it shows that the O3−Pd9 bond strengthens as the force constant between these two atoms globally increases with the electron depletion, in agreement with previous studies.2,20 This indirectly informs on the bonding vs antibonding character of the orbitals involved in the electrochemical process (here antibonding). Some more details are given in the Supporting Information. This force constant change is caused by the offdiagonal terms of the Ξ matrix that can be directly expressed from the Fukui function derivatives with the displacement (see Figure 4). A positive surface charging (N > 0) and a positive displacement of the oxygen atoms in the z direction lead to a r )/∂Xj) that is mostly negative close to charge distribution N(∂f(→ Pd9, inducing a supplementary force upward the Pd atom leading to an increase of the force constant between the atoms O3 and Pd9. A negative charging has the opposite effect by weakening the force constant. This Fukui derivative can be used to predict the change into the chemical bonding with the electrochemical effect and gives a result similar to molecular orbital analysis.21 Then, the Fukui function plays a central role in the modification of the chemical bonding. This function remains close to the frontier orbitals (HOMO/LUMO) used in molecular chemistry but includes many-body effects and can be linked equivalently with

Figure 4. Isosurface of the function N(∂f(→ r )/∂Xj) for displacement of O3 in the z-direction. The labels for the different atoms are given. H7 is hidden by H6.

the charging of the bonding/antibonding orbital as previously discussed.2,20,21 In this paper, we have shown that the Fukui function and its derivative are powerful and easy-handling tools which provide both a quantitative rationalization of surface electrochemistry and a qualitative understanding of the factors affecting the redox reactivity. Given the importance of surface/interface electrochemistry in worldwide energy storage applications and the new challenges that have emerged with the nanostructuration of the electrochemically active materials where interface electrochemistry plays a central role, these tools could be very helpful in rationalizing and designing new performance devices.

IV. CONCLUSION In the present paper, we have shown that electrochemical effects can be rationalized using tools derived from a conceptual DFTlike approach. More particularly, the Fukui function plays a central role in the electrochemical process: it is a many-body function mostly corresponding to the spatial distribution of the electrochemically added charge but also including the polarization response of all the other electrons to that added charge. It controls (i) the surface electrochemical potential μ through its interaction with the surface dipole, (ii) the surface structural α under charging through its interaction with polarization NJ−1→ the nucleus electric field, and (iii) the surface hardness η mainly through its electrostatic self-interaction. We have shown that it also controls part of the bond modification through the electrochemical tensor Ξ as it is associated with the bonding/ nonbonding/antibonding character of molecular orbitals. Higher surface charging is given by higher capacitance, arising mainly from the chemical hardness reduced by the structural relaxation under charge. The capacitance increases with a larger spreading of the added charge (given by the Fukui function) and with an important structural relaxation. Bond modification with potential is linked to the complex balance between electrochemical and electromechanical effects. The surface charging leads to a change in the bond strength, but this charging also creates an added electric field that polarizes the surface structure leading to a bond strain. Because this strain can be large, the effects of anharmonicity can become strong and then also modify the bond strength. The global bond change is then modulated at first-order by the change in force constant given by α ). It can be used in particular to the matrix N(Ξ − 1/3Λ)(J−1→ 19029



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understand the Stark effect,44,45 beyond the simple case previously studied,21 in complex adsorbed molecules. The matrix can even be directly computed from the Fukui function. More generally, Fukui function can be systematically used to qualitatively rationalize electrochemical effects at the surface. It can also be used quantitatively to extrapolate the electrochemical behavior, not only using finite difference as done in this paper but also using response theory that would allow fast estimates of the relevant parameters. This could be done by using schemes close to the one used to extract the closely related parameters such as Born effective charges.34 In respect to recent advances and new challenges related to interface electrochemistry46 in many societal applications such as catalysis, corrosion, and energy storage, the chemical, electrochemical, and mechanical descriptors proposed in this paper should open new routes to apprehend, rationalize, or predict the performances of electrochemical cells and hopefully help in designing new devices.

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ASSOCIATED CONTENT

S Supporting Information *

Enlarged representations of the Hessian matrix and electrochemical tensor for the H-up water phase on Pd(111) are given. This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: fi[email protected]. Tel.: 33-4-67-14-46-19. Fax: 334-67-14-48-39. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge support from the French computational resource centers IDRIS and CINES under Contract No. 0911750.

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Conceptual Surface Electrochemistry and New Redox Descriptors

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