Electrochemical Electron Transfer and Proton-Coupled Electron

Article pubs.acs.org/JCTC

Electrochemical Electron Transfer and Proton-Coupled Electron Transfer: Effects of Double Layer and Ionic Environment on Solvent Reorganization Energies Soumya Ghosh, Alexander V. Soudackov, and Sharon Hammes-Schiffer* Department of Chemistry, 600 South Mathews Avenue, University of Illinois at Urbana−Champaign, Urbana, Illinois 61801, United States S Supporting Information *

ABSTRACT: Electron transfer and proton coupled electron transfer (PCET) reactions at electrochemical interfaces play an essential role in a broad range of energy conversion processes. The reorganization energy, which is a measure of the free-energy change associated with solute and solvent rearrangements, is a key quantity for calculating rate constants for these reactions. We present a computational method for including the effects of the double layer and ionic environment of the diffuse layer in calculations of electrochemical solvent reorganization energies. This approach incorporates an accurate electronic charge distribution of the solute within a molecular-shaped cavity in conjunction with a dielectric continuum treatment of the solvent, ions, and electrode using the integral equations formalism polarizable continuum model. The molecule-solvent boundary is treated explicitly, but the effects of the electrode-double layer and double layer-diffuse layer boundaries, as well as the effects of the ionic strength of the solvent, are included through an external Green’s function. The calculated total reorganization energies agree well with experimentally measured values for a series of electrochemical systems, and the effects of including both the double layer and ionic environment are found to be very small. This general approach was also extended to electrochemical PCET and produced total reorganization energies in close agreement with experimental values for two experimentally studied PCET systems.

1. INTRODUCTION Electron transfer (ET) and proton-coupled electron transfer (PCET) reactions at electrochemical interfaces play an important role in a variety of catalytic processes.1−8 The reorganization energy is a key parameter in the electrochemical ET and PCET rate constant expressions that are derived within the framework of Marcus theory and its extensions.9−17 This parameter is a measure of the free-energy penalty associated with the solute and environmental rearrangements that accompany charge transfer between a solute and an electrode in an electrochemical system. Under experimental conditions, the solvent and electrolyte ions form a partially ordered structure, denoted a double layer, near the electrode surface.18 The inner Helmholtz plane (IHP) is associated with the layer of solvent molecules closest to the electrode surface. In this layer, the dipoles of most of the solvent molecules are oriented in a particular direction, and hence the dielectric constant is considerably lower than that of bulk solvent. The outer Helmholtz plane (OHP) is associated with the next layer, which is comprised of less-structured solvent molecules and partially solvated electrolyte ions and has a dielectric constant larger than the first layer but still significantly lower than the bulk solvent. Beyond the OHP lies the diffuse layer, in which the solvent molecules and electrolyte ions move around more freely. The objectives of this paper are to capture the physical effects of the double layer and ionic environment in calculations of the © XXXX American Chemical Society

solvent reorganization energy for ET, as well as to extend these computational methods to electrochemical PCET. A variety of models have been devised for calculating solvent reorganization energies of homogeneous and electrochemical ET reactions analytically.11,19−25 Typically, these models represent the solute as a point charge immersed in a dielectric continuum solvent and also include an electrode surface for electrochemical ET. Such models enable the analytical solution of the associated classical electrostatic equations. Moreover, molecular-shaped cavities have been introduced for the calculation of solvent reorganization energies for homogeneous ET within the framework of the polarizable continuum model (PCM).23,26,27 Recently, we introduced computational methods to calculate electrochemical solvent reorganization energies within the framework of the PCM, including a molecular-shaped cavity, as well as the effects of the electrode.28 These methods utilize the integral equations formalism PCM (IEF-PCM),29−33 in which the molecule-solvent boundary is treated explicitly, while the effects of other boundaries are included through an external Green’s function. This approach has also been extended to treat a self-assembled monolayer (SAM)-modified electrode surface.34 The main advance in the present paper is the further extension of this approach to include the effects of the double layer and ionic environment. We apply this approach to a series Received: March 1, 2016

A

DOI: 10.1021/acs.jctc.6b00233 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

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Journal of Chemical Theory and Computation

the relatively slow rotational and translational solvent degrees of freedom. Thus, the solvent response is divided into the faster, predominantly electronic component and the slower, inertial component. The inertial polarization potential is defined to be the difference between the total polarization potential, which is dependent on the static dielectric constant ε0, and the electronic polarization potential, which is dependent on the optical dielectric constant ε∞. The free energy of the solvated molecule in either the oxidized or reduced state is a functional of this inertial polarization potential, leading to the parabolic curves depicted in Figure 1. The solvent reorganization energy is defined as the difference between the nonequilibrium and equilibrium free energies for either the reduced or the oxidized state of the solute, as illustrated in Figure 1.

of small organic molecules and compare the results to values obtained from simpler models and to experimentally measured values. We also extend this general methodology to electrochemical PCET and investigate two PCET reactions for which the electrochemical reorganization energy has been measured experimentally. An outline of the paper is as follows. Section 2 presents the theoretical and computational methods. Section 2A describes the methods for calculating the inner-sphere reorganization energy and provides a brief description of our recently developed method to calculate the electrochemical solvent reorganization energy within the IEF-PCM framework. Section 2B introduces the effects of a double layer and an ionic environment. Section 2C explains how electrochemical solvent reorganization energies can be calculated for PCET systems with this methodology. The computational details are provided in Section 2D. Section 3 presents the results from applying these methods to experimentally studied systems. The concluding remarks are presented in Section 4.

2. THEORY AND COMPUTATIONAL METHODS A. Inner-Sphere and Solvent Reorganization Energies. Electrochemical electron transfer is associated with structural changes in both the solute and the surrounding solvent environment. The energy penalty associated with the solute is denoted the inner-sphere reorganization energy (λi), whereas the free-energy penalty associated with the solvent is denoted outer-sphere or solvent reorganization energy (λs). The solvent reorganization energy also includes the effects of the ions and electrode. Usually, the inner- and outer-sphere reorganization energies are assumed to be uncoupled and, thus, additive.11 Consequently, the total reorganization energy is obtained by summing λi and λs, and the solvent reorganization energy is calculated at a fixed solute geometry. Various methods can be employed to calculate the innersphere reorganization energy. For instance, Marcus’s original formulation was based on the harmonic approximation, in which the force constants for the normal modes of the oxidized and reduced species and the shifts in the equilibrium values of the corresponding normal mode coordinates were utilized to calculate the inner-sphere reorganization energy.11,35,36 Alternatively, this quantity can also be obtained from standard quantum chemical calculations of the oxidized and reduced species in the gas phase:37 1 λ i = [λ i(ox) + λ i(red)] 2 1 red ox ox red = [Eox (R eq ) − Eox (R eq ) + Ered(R eq ) − Ered(R eq )] 2

Figure 1. Free-energy curves associated with the oxidized (blue) and reduced (red) states of the solute molecule depicted along the collective solvent coordinate used in Marcus theory for electron is transfer. The electrochemical solvent reorganization energy λ(ox) s defined as the difference between the nonequilibrium and equilibrium solution-phase free energies of the solute in the oxidized state (blue). The analogous definition applies to the reduced state (red). In practice, these two quantities may differ slightly because of differences in the optimized geometries for the oxidized and reduced species, and the solvent reorganization energy is calculated as the average of these two quantities. (Figure modified with permission from ref 28. Copyright 2014. American Chemical Society, Washington, DC.)

Recently, computational methods have been developed to calculate electrochemical solvent reorganization energies within the IEF-PCM framework.28,34 Within this formalism, the surface of the molecular-shaped cavity is divided into small surface elements called tesserae, and the solvent polarization potential is represented by apparent surface charges at the centers of these tesserae. These charges are obtained by solving the relevant electrostatic equations with the appropriate boundary conditions. The molecule−solvent boundary is treated explicitly, while the effect of the electrode is incorporated through an external Green’s function. The procedure for obtaining solvent reorganization energies with this formulation requires the initial calculation of the equilibrium free energies for the oxidized and reduced states and the subsequent calculation of the nonequilibrium free energies, as illustrated in Figure 1. In the remainder of this subsection, we outline the essential elements of this procedure. A more detailed description, as well as a complete flowchart and all of the essential working equations, is provided in ref 28. To calculate the inertial solvent polarization potential, the apparent surface charges must be computed twice. First, they are calculated for state i (i.e., the oxidized or reduced state)

(1)

λi(ox)

λi(red)

In this expression, and are the inner-sphere reorganization energies associated with the oxidized and red reduced species, respectively, Rox eq and Req are the optimized equilibrium geometries of the oxidized and reduced species, respectively, and Eox and Ered are the energies of the oxidized and reduced states, respectively, evaluated at the designated geometry. However, the calculation of the solvent reorganization energy is more challenging. Within Marcus theory, the free-energy curves along a collective solvent coordinate for the oxidized and reduced states of the solute are parabolic, as depicted in Figure 1. Here, the collective solvent coordinate corresponds to the inertial solvent polarization potential, which is associated with B


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Journal of Chemical Theory and Computation with the static dielectric constant of the solvent, where both the charge density of the solute and the total solvent polarization potential are equilibrated to each other to obtain the discretized electrostatic potential associated with the solute charge density, {Vik}, and the apparent surface charges associated with the total polarization potential, {qik}. Second, they are calculated with the optical dielectric constant, while keeping the solute charge density fixed to that obtained in the first step, to obtain the apparent surface charges associated with the electronic polarization potential, {q∞,i k }. The differences in the apparent surface charges obtained by the above procedure represent the apparent surface charges associated with the inertial polari ∞,i ization potential, {qin,i k = qk − qk }. This set of calculations is performed for both the oxidized and reduced states to obtain the apparent surface charges for both states, denoted {q∞,ox }, k ∞,red },{qin,red }, and the discretized electrostatic poten{qin,ox k },{qk k tials corresponding to the solute charge densities, denoted red {Vox k },{Vk }. The nonequilibrium free energy for the oxidized or reduced state at the inertial polarization potential equilibrated for the other state is obtained by appropriately combining these quantities, as described in ref 28. The solvent reorganization energy is calculated as the difference between this nonequilibrium free energy and the equilibrium free energy for that state. When the ionic environment is included, the calculation of the surface charges with the optical dielectric constant does not include the effects of the ions.11 B. Double Layer and Ionic Environment. In this subsection, the methodology described above is extended to calculate electrochemical solvent reorganization energies within the IEF-PCM framework, while incorporating the effects of both the double layer and the ionic environment of the diffuse layer. The double layer can be represented as a dielectric medium with an effective dielectric constant εd obtained from a model based on the charging of parallel-plate capacitors. The IHP and OHP are assumed to be located at distances lIII and (lII + lIII), respectively, from the electrode surface, and the dielectric constant of the second layer is assumed to change from εIII to εII at a distance (lIII + l′) from the electrode surface. This setup can be modeled by two parallel-plate capacitors in series with widths (lIII + l′) and (lII − l′) and dielectric constants εIII and εII, respectively, as shown in Figure 2. The effective dielectric constant εd of this system is given by38 εd =

The double layer does not contain any mobile ions; hence, the electrostatic potential is given by a solution of the Laplace equation. In contrast, the diffuse layer that constitutes the solvent beyond the OHP contains mobile ions. Assuming sufficiently low concentration of ions, the linearized Poisson−Boltzmann equation (LPBE) must be solved to obtain the electrostatic potential in the diffuse layer. Figure 3 depicts the model that

Figure 3. Definition of parameters for a simplified system with a unit point charge placed at the origin in the diffuse layer. Note that the zaxis corresponds to the normal to the electrode surface passing through the origin, with the positive z-direction associated with the vector pointing from the origin toward the surface.

includes the double layer and a unit point charge at the origin in the diffuse layer. For this model, the solution of the LPBE (i.e., the potential ϕI) in cylindrical coordinates (ρ,z) is given by GI(r; r′ = 0) ≡ ϕI(ρ , z) =

∫0

∞

dk

k J (ρk)[e−P |z| + A(k)e−P(2d − z)] P0

(3)

with A (k ) = −

e−2k(lII + lIII) + δ δ e−2k(lII + lIII) + 1

where

(lII + lIII)ε IIε III (lII − l′)ε III + (lIII + l′)ε II

1 εI

δ= (2)

ε d k − ε IP ε d k + ε IP

and ρ2 = x 2 + y 2 P 2 = k2 + κ 2

As indicated in eq 3, this potential is also defined as the Green’s function GI(r;r′) for this model. The details of this solution are given in Appendix A. A slightly modified version of this Green’s function is employed to calculate the relevant matrix elements in the IEF-PCM formalism, as described below. The IEF-PCM approach requires the matrix elements of the integral operators29 in the tessera basis inside the cavity (internal, SIij and DIij) and outside the cavity (external, SEij and DEij ). The matrix elements of the internal operators remain the same, irrespective of the environment, and have been discussed previously.29,30 Evaluation of the matrix elements outside the cavity requires the Green’s function corresponding to the solution of the LPBE at the tessera center sj with Cartesian

Figure 2. Depiction of layers with different dielectric constants and widths associated with the double layer near the electrode surface modeled by two parallel-plate capacitors. C


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coordinates (xj,yj,zj) with a unit point charge at the center of another tessera center si with Cartesian coordinates (xi,yi,zi), along with the boundary conditions that account for discontinuous jumps in the value of the dielectric constant. Similar to the solution given by eq 3, this Green’s function is given by G E(si ; sj) =

1 εI

∫0

∞

⊕ AH ··· B XooooY  A• ··· HB    +e [red]

⎡ ⎤ k dk J0 (ρij k)⎢e−P |zi − zj| + A(k)e−P(2d − (zi + zj))⎥ ⎣ ⎦ P (4)

ρij 2 = (xi − xj)2 + (yi − yj )2

To evaluate the matrix elements of the external integral operators, we employ the following relationships between the operators and the Green’s function: aiaj 4π

G E(si , sj)

DijE = ε I nj ·∇j SijE

(5)

The relevant matrix elements are given by aiai

SiiE = SiiE ,ionic + SijE = SijE ,ionic + SijE ,ionic =

aiaje

4πε aiaj

I

∫0

4πε

I

∫0

∞

dk

k A(k)e−P(2d − 2zi) P

dk

k A(k)J0 (ρij k)e−P(2d − (zi + zj)) P

∞

−κ |si − sj|

4πε I|si − sj|

DiiE = DiiE ,ionic +

aiai ( 2π

∫0

∞

dk kA(k)e−P(2d − 2zi))z ̂·ni

∞ aiaj ⎧ ⎪ k2 ⎨ dk A(k)J1(ρij k)e−P(2d − (zi + zj)) = + 4π ⎪ P ⎩ 0 ⎤ ⎡⎛ x − x ⎞ ⎛y − y⎞ i j i j ⎥ ⎢ ⎟ ⎜ ⎟ ⎜ × ⎜ ⎟ x ̂ · n j + ⎜ ρ ⎟ y ̂ · n j⎥ ⎢ ρ ⎠ ⎝ ⎠ ⎝ ij ij ⎦ ⎣ ⎫ ∞ ⎪ −P(2d − (zi + zj)) dkkA(k)J0 (ρij k)e )z ̂·nj⎬ +( ⎪ 0 ⎭

DijE

DijE ,ionic

(7)

In general, the charges on A and B could be different, and the radical could be localized on a different species. Assuming separation of the inner- and outer-sphere reorganization energies, the reduced and oxidized states are defined in such a way that the structures of A and B are fixed except for the change in position of the transferring proton. Upon oxidation, the proton position is shifted from the donor to the acceptor and is optimized while fixing all other nuclei to their positions in the optimized geometry for the reduced state. The small shift in proton position results in a slight change in the volume and shape of the molecular cavity. For the systems studied herein, this change in the cavity did not change the number of tesserae on the molecular surface and, hence, the numerical procedure employed for calculating solvent reorganization energies for ET reactions was still applicable. The analogous procedure was used to model the reduction process for the optimized geometry in the oxidized state. The solvent reorganization is defined as the average of the solvent reorganization energies calculated for the reduced and oxidized geometries. D. Computational Details. All of the calculations were performed with a development version of the GAMESS quantum chemistry software package.39,40 The electronic structure calculations were performed with density functional theory (DFT) using a modified version of the B3LYP functional,41,42 implemented in GAMESS as B3LYPV3,43,44 and the 6-31G** basis set.45,46 The molecular cavity was generated using van der Waals radii of the atoms for the ET systems and the Simplified United Atom Radii (SUAHF)47 for the PCET systems. The tesserae were generated with the FIXPVA tessellation scheme48 with 60 and 240 tesserae per sphere for the ET and PCET systems, respectively. The dielectric constants of the double layer (εd0 and εd∞) were III estimated with eq 2, where the dielectric constants εIII 0 and ε∞ were both assumed to be ε∞ of the corresponding solvent. The static and optical dielectric constants εII0 and εII∞ were assumed to be ε0/2 and ε∞, respectively, of the corresponding solvent. The widths of the different layers given in eq 2 were estimated as follows: lII was equal to the radius of the solvated cation of the electrolyte, and lIII and l′ were each equal to the radius of a solvent molecule. The various solute, solvent, and electrolyte parameters are provided in the Supporting Information. The radii of the solvent molecules and the solvated electrolyte ions, Bu4N+ or Et4N+,49 used to obtain these widths are given in Tables S1 and S2 in the Supporting Information. The static and optical dielectric constants of the solvents for the ET and PCET systems are given in the Supporting Information (Tables S1 and S4, respectively). The static dielectric constants for the double layer are provided in Table S3 in the Supporting Information. The radii of the spheres representing the solute molecules in eq S1 in the Supporting Information are calculated from the cavity volumes obtained from PCM calculations and are provided in Table S5 in the Supporting Information. The width of the double layer and the distance of the center of mass of the molecule from the outer Helmholtz plane are given in Table S6 in the Supporting Information. To illustrate the individual effects of different factors influencing the electrochemical reorganization energy for ET, we applied the methodology described above to the set of

where

SijE =

[ox]

∫

∫

⎡ a a e−κ |si − sj|(1 + κ |s − s |) ⎤ i j i j ⎥(si − sj) ·nj DijE ,ionic = ⎢ ⎢⎣ ⎥⎦ 4π |si − sj|3 (6)

where ai and aj are the areas of the tesserae with centers si and sj, respectively, x̂, ŷ, and ẑ are the unit vectors in the x, y, and z directions, respectively, and ni is the unit vector normal to the ith tessera at the center si. The expressions for the matrix elements with subscripts ij in eq 6 are valid only for i ≠ j. The diagonal elements SE,ionic and, consequently, DE,ionic do not have ii ii closed form analytical expressions but can be evaluated numerically, as described in Appendix B. C. PCET Reactions. We also extended this approach to calculate electrochemical solvent reorganization energies for PCET systems. In this case, the reduced and oxidized states can be defined as follows: D


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respectively. In other words, R̃ ox eq is the optimized equilibrium geometry of the oxidized species, except that the transferring proton is optimized for the reduced state. A schematic depiction of these four geometries for PCET system 1 is provided in Figure 6.

molecules shown in Figure 4. The center of mass of each molecule was placed at the origin with the z-axis normal to the

Figure 4. Molecules for which the electrochemical ET reorganization energies were calculated. (Figure reproduced with permission from ref 28. Copyright 2014. American Chemical Society, Washington, DC.)

electrode surface, and the parameter ZMAXTS was set to the largest z-coordinate of the surface tesserae. Each molecule was placed in the diffuse layer, such that the distance between the origin and the OHP was −5

d = ZMAXTS + 10

(in Bohr)

Figure 6. Schematic representation of the gas phase structures employed in the calculation of the inner-sphere reorganization energies for PCET system 1 using eq 9. 1red and 1ox represent fully optimized structures in the gas phase with overall charges of 0 and +1, respectively. The tilde symbol (∼) on the 1̃red (1̃ox) term indicates that the position of the transferring hydrogen was optimized for the oxidized (reduced) state while keeping all other nuclei fixed to their positions in the optimized geometry for the reduced (oxidized) state.

(8)

−5

where the addition of 10 ensures that the molecular surface does not touch the OHP. The planar molecules are placed parallel to the electrode surface. We also studied the PCET reactions depicted in Figure 5 with the simpler model without the double layer and the ionic

3. RESULTS AND DISCUSSION Previously, we calculated the electrochemical solvent reorganization energies with34 and without28 a SAM attached to the electrode, using the method outlined briefly in section 2A, and compared our results to experimental values available in the literature. Herein, we provide a more comprehensive analysis of the combined data. To illustrate this analysis, Figure 7 depicts the calculated versus experimental total reorganization energies. These data indicate that this method is able to produce accurate results with a root-mean-square deviation of 0.09 eV for the systems with a SAM and 0.1 eV for the systems without a SAM. The numerical values of the calculated and experimental total electrochemical reorganization energies with and without the SAM, as depicted in Figure 7, are provided in the Supporting Information (Tables S7 and S8, respectively). As discussed in our previous work,28 inclusion of the electrode in the IEF-PCM calculations is essential for obtaining even qualitatively accurate solvent reorganization energies. In this paper, we extend this methodology to include the effects of the double layer and the ionic environment. The models studied are as follows: SOLV, which includes only bulk solvent without ions; DL, which includes a double layer at the electrode surface; IONS, which includes ions in the solvent; and DL+IONS, which includes both the double layer and the ionic environment. Table 1 compares the electrochemical solvent reorganization energies with and without a double layer between the molecule and the electrode surface, denoted the DL and SOLV models, respectively. While the effects are quite small, our calculations show that the double layer reduces the solvent reorganization energy. Corresponding results employing the analytical expressions derived by Marcus (eq S1 in the Supporting Information) and Liu and Newton (eq S3 in the

Figure 5. Electrochemical PCET reactions for which the electrochemical PCET reorganization energies were calculated and compared to experimental values.

solution. In these calculations, the center of mass of each molecule was placed at the OHP. The inner-sphere reorganization energies were calculated using a slightly modified version of eq 1 to account for proton transfer:50 1 ox red λ i = [Eox (R̃ eqred) − Eox (R eq ) + Ered(R̃ eqox ) − Ered(R eq )] 2 (9)

where Eox and Ered are the energies of the oxidized and reduced states, respectively, evaluated at the designated geometry. Here, red Rox eq and Req are the optimized equilibrium geometries of the ̃ red oxidized and reduced species, respectively, and R̃ ox eq and Req are the same optimized geometries, except that the transferring hydrogen is optimized for the reduced and oxidized state, E


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Supporting Information) agree qualitatively with these numerical results and are given in the Supporting Information (Tables S9 and S10, respectively). The inclusion of ions in the solution without a double layer, denoted the IONS model, leads to a small increase of the solvent reorganization energy, as shown in Table 1. Moreover, the contribution of the ionic environment increases with decreasing dielectric constant of the solvent. Corresponding results employing the analytical expression in eq S2 in the Supporting Information follow the same qualitative trend and are given in Table S11 in the Supporting Information. When both the double layer and the ionic environment are included, denoted the DL+IONS model, the effects are decreased in the majority of cases studied here, as shown in Table 1. Because the effects of the double layer and ionic environment are of opposite sign, they have a tendency to cancel each other, leading to a negligible overall effect on the solvent reorganization energy. As the molecule is moved away from the electrode surface, however, the double layer effects diminish and the contribution of the ionic strength becomes dominant, as illustrated in Figure 8.

Figure 7. Calculated versus experimental total electrochemical reorganization energies (a) without and (b) with a SAM attached to the electrode surface. The systems studied in the absence of a SAM are (1) tBuNO2 in DMF (N,N-dimethylformamide); (2) PhCOPh in DMSO (dimethyl sulfoxide); (3) PhCOPh in DMA (dimethylacetamide); (4) OPhO in DMF; (5) OPhO in ACN (acetonitrile); (6) PhCN in DMF; (7) NCPhCN in DMF; (8) NCPhCN in ACN; (9) NCPhCN in ATO (acetone); and (10) NCPhCN in DCM (dichloromethane). The systems studied in the presence of a SAM in water are (1) TEMPO+; (2) [Ru(NH3)6]3+; (3) [Ru(NH3)5py]3+; (4) [Ru(bpy)3]3+; (5) [FeCp2]+; (6) [Fe(bpy)3]3+; (7) [Fe(bpy)2(CN)2]+; (8) [Fe(dMbpy)2(CN)2]+; (9). [Fe(bpy) (CN)4]−; (10) [Fe(dMbpy) (CN)4]−; (11) [Fe(CN)6]3−; and (12) [Mo(CN)8]3−. The numerical values of the reorganization energies are given in Tables S7 and S8 of the Supporting Information.

Figure 8. Distance dependence of the electrochemical solvent reorganization energy of tBuNO2 in DMF. In this figure, a value of d = 0 indicates that the center of mass of the molecule is placed at a distance (ZMAXTS+10−5) Bohr from the OHP.

We also extended our method to the PCET systems depicted in Figure 5. PCET system 1 involves intramolecular proton transfer during electrochemical oxidation/reduction. In this

Table 1. Comparison of Electrochemical Solvent Reorganization Energiesa Calculated with SOLV, DL, IONS, and (DL+IONS) Models solvent

λSOLV s

λDL s

ΔDLb

c λIONS s

ΔIONSd

c λ(DL+IONS) s

Δ(DL+IONS)e

t

BuNO2

DMF

1.02

0.99

−0.03

1.03

0.01

1.02

0.00

PhCOPh

DMSO DMA

0.72 0.75

0.70 0.72

−0.02 −0.03

0.72 0.76

0.00 0.01

0.72 0.75

0.00 0.00

OPhO

DMF ACN

0.91 1.01

0.88 0.99

−0.03 −0.02

0.92 1.02

0.01 0.01

0.91 1.02

0.00 0.01

PhCN

DMF

0.89

0.87

−0.02

0.90

0.01

0.90

0.01

NCPhCN

DMF ACN ATO DCM

0.82 0.92 0.87 0.67

0.80 0.90 0.84 0.65

−0.02 −0.02 −0.03 −0.02

0.83 0.93 0.89 0.74

0.01 0.01 0.02 0.08

0.83 0.92 0.89 0.76

0.01 0.00 0.02 0.09

solute

SOLV c Reorganization energies are given in units of eV. bΔDL = λDL . The ionic strengths of all solutions are 0.1 M. dΔIONS = λIONS − λSOLV . s − λs s s (DL+IONS) SOLV (DL+IONS) Δ = λs − λs .

a e

F


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insights. Nevertheless, the dielectric continuum treatment provides semiquantitative reorganization energies that can be used as input for the calculation of ET and PCET rate constants.

case, the reduced state corresponds to the proton on oxygen, while the oxidized state corresponds to the proton on the nitrogen. In PCET system 2, the oxidized state corresponds to a water molecule hydrogen bonded to the singly reduced benzophenone, while the reduced state corresponds to a proton transferred from the water molecule to the doubly reduced benzophenone. As shown in Table 2, the calculated electro-

■

APPENDIX A. ELECTROSTATIC POTENTIAL OF A POINT CHARGE IN THE DIFFUSE LAYER In this appendix, we derive the solution given by eq 3. The equations to be solved are

Table 2. Comparison of Calculated and Experimental Electrochemical Reorganization Energiesa for PCET Systems in Figure 4

(∇2 − κ 2)ϕ(r) = −

Calculated Value PCET system

λs

λi

λtotal

λexptb

1 2 2-ETc

0.67 0.48 0.46

0.47 0.62 0.38

1.14 1.10 0.84

1.06 1.0

∇2 ϕ(r) = 0

4πδ(r) εI

r ∈ diffuse layer

r ∈ double layer and electrode

(A1)

The general solution in the three regions, diffuse (I), double layer (II), and electrode (elec), in cylindrical coordinates (ρ,z) are given by38,51

a Reorganization energies are given in units of eV. bExperimental values for 1 and 2 were obtained from refs 52 and 53, respectively. c2ET corresponds to the reorganization energies calculated for PCET system 2 with only ET.

ϕI =

1 εI

∫0

∞

dk

k J (ρk)[e−P |z| + A1(k)e Pz] P0

P 2 = k2 + κ 2 ∞ 1 dk J0 (ρk)[A 2 (k)e−k |z| + A3(k)ekz] ϕII = d 0 ε

chemical total reorganization energies are in excellent agreement with the experimentally measured values. To estimate the impact of proton transfer on the reorganization energy, we calculated the reorganization energy of PCET system 2 for ET without proton transfer. Our calculations indicate that the proton transfer has a negligible effect on the solvent reorganization energy but a more significant effect on the inner-sphere reorganization energy.

∫

ϕelec = 0

(A2)

The boundary conditions are given by

ϕI = ϕII|z = d

4. CONCLUSIONS In this paper, we presented computational methods to calculate electrochemical solvent reorganization energies for models that include the effects of the double layer and the ionic strength of the solvent. This approach combines accurate electronic distributions of the solute in a molecular-shaped cavity with a dielectric continuum treatment of the solvent, ions, and electrode. For the systems studied here, the effects of the double layer and ionic environment on the solvent reorganization energy are relatively small and are of opposite sign, thereby virtually canceling each other in most cases. As shown previously, the calculated total reorganization energies agree well with experimentally measured values both with and without a SAM attached to the electrode. Inclusion of the electrode is essential for these calculations of electrochemical solvent reorganization energies. This approach was also extended to electrochemical PCET and was applied to two experimentally studied PCET systems. The calculated total reorganization energies agree well with the experimentally measured values for the PCET systems, and the proton transfer was found to significantly impact only the inner-sphere reorganization energy. These approaches enable the calculation of reorganization energies for a wide range of ET and PCET systems under various experimental conditions. The dielectric continuum treatment neglects the effects of explicit hydrogen bonding of the solvent to the solute, as well as the detailed structure of the solvent and ions at the electrode/solvent interface. Moreover, it also neglects the dynamical effects of the system and the electronic structure of the electrode. Molecular dynamics studies with explicit solvent and ions, combined with an atomic-level treatment of the electrode, would provide further

εI

∂ϕI ∂ϕII = εd ∂z ∂z

z=d

II

ϕ |z = d + lII + lIII = 0

(A3)

Substituting eq A2 into eq A3 and applying Hankel’s Theorem, which is expressed as f (k ) =

∫0

∞

dρ ρJ0 (ρk)

∫0

∞

dx xJ0 (ρx)f (x)

we obtain the following linear system of equations for determination of the unknown functions A1(k) , A2(k) , and A3(k): −kd kd e−Pd + A1e Pd ε I ⎡ A e + A3e ⎤ ⎥ = d⎢ 2 ⎥⎦ P k ε ⎢⎣

−e−Pd + A1e Pd = − A 2 e−kd + A3ekd A 2e−k(d + lII + lIII) + A3ek(d + lII + lIII) = 0

(A4)

Solving the above set of equations leads to eq 3 with A1(k) = A(k)e−2Pd.

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APPENDIX B. NUMERICAL EVALUATION OF DIAGONAL MATRIX ELEMENTS OF EXTERNAL AND DE,ionic INTEGRAL OPERATORS SE,ionic ii II In this appendix, we outline the numerical calculation of the diagonal elements of the integral operators corresponding to the ionic environment. These matrix elements are given by the following surface integrals over tesserae Ti: G


Article

Journal of Chemical Theory and Computation SiiE ,ionic =

∫T da ∫T da′ GE ,ionic(s′; s) i

DiiE ,ionic = ε I

■

i

∫T da ∫T da′∂GE ,ionic(s′; s) i

i

(B1)

e−κ |si − s| 4πε I|si − s|

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

(B2)

ACKNOWLEDGMENTS This research was supported as part of the Center for Molecular Electrocatalysis, an Energy Frontier Research Center, funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences.

Using the approximation of a piecewise constant charge density located at the center of the ith tessera, the double integrals above can be converted to the following single integrals: SiiE ,ionic = ai

∫T da GE ,ionic(si; s)

■

i

DiiE ,ionic = aiε I

∫T

da ∂G E ,ionic(si ; s)

i

REFERENCES

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(B3)

where si is the center of the ith tessera and s represents any point on the surface of the ith tessera with area ai. DE,ionic in eq ii B3 is calculated by evaluating the normal derivative of the Green’s function given in eq B2 (n̂i·∇iGE,ionic(si;s)). In this implementation, the tesserae are approximated as equilateral spherical triangles. The angles at the vertices of each of these triangles are obtained from the area of the tessera. A local polar coordinate system (r,φ) is set up at the center si, where φ ranges from 0 to 2π while the range of r is obtained from standard spherical trigonometry, as illustrated in Figure B1. Using this local polar coordinate system, the integrals in eq B3 are evaluated numerically with standard Gaussian quadrature rules.

Figure B1. An equilateral spherical triangle is used to approximate a tessera. The local polar coordinates (r,φ), as well as the center of the triangle, are shown. The point si is the center of the tessera.

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AUTHOR INFORMATION

Corresponding Author

where the Green’s function (GE,ionic) is given by G E ,ionic(si ; s) =

models, and coordinates of optimized molecules studied (PDF)

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.6b00233. Solvent and electrolyte ion properties and parameters, solute radii, double layer widths and molecular distances, calculated and experimental electrochemical total reorganization energies with and without SAM, comparison of electrochemical solvent reorganization energies calculated analytically and numerically for different H


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Electrochemical Electron Transfer and Proton-Coupled Electron

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