Semiempirical modeling of electrochemical charge transfer · 2017-11-07 · Semiempirical modeling of electrochemical charge transfer Rebecca L. Gieseking, Mark A. Ratner and George

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Semiempirical modeling ofelectrochemical charge transfer

Rebecca L. Gieseking, Mark A. Ratner and George C. Schatz *

Received 20th November 2016, Accepted 22nd November 2016

DOI: 10.1039/c6fd00234j

Nanoelectrochemical experiments using detection based on tip enhanced Raman

spectroscopy (TERS) show a broad distribution of single-molecule formal potentials E�0

for large p-conjugated molecules; theoretical studies are needed to understand the

origins of this distribution. In this paper, we present a theoretical approach to determine

E�0 for electrochemical reactions involving a single molecule interacting with an

electrode represented as a metal nanocluster and apply this method to the Ag20–

pyridine system. The theory is based on the semiempirical INDO electronic structure

approach, together with the COSMO solvation model and an approach for tuning the

Fermi energy, in which the silver atomic orbital energies are varied until the ground

singlet state of Ag20–pyridine matches the lowest triplet energy, corresponding to

electron transfer from the metal cluster to pyridine. Based on this theory, we find that

the variation of E�0 with the structure of the Ag20–pyridine system is only weakly

correlated with changes in either the ground-state interaction energy or the charge-

transfer excited-state energies at zero applied potential, which shows the importance of

calculations that include an applied potential in determining the variation of formal

potential with geometry. Factors which determine E�0 include wavefunction overlap for

geometries when pyridine is close to the surface, and electrostatics when the

molecule-cluster separation is large.

1. Introduction

Nanoscale electrochemistry has attracted broad interest in the past several yearsas many experimental techniques have been developed to detect electrochemicalprocesses occurring at nanoscale electrodes1–4 or at individual nanoparticles,5–7

allowing electrochemical measurements to be performed at the single-moleculelimit.8–11 Although electrochemical measurements have traditionally involvedensemble measurements that average over many molecules and surface sites,these new techniques have revealed signicant variation in the potential at whichelectron transfer occurs, which has been proposed to be due to differences in theformal potential E�0 due to heterogeneity of surface sites on the electrode and the

Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, USA.

E-mail: [email protected]; Tel: +1-847-491-5657

This journal is © The Royal Society of Chemistry 2017 Faraday Discuss., 2017, 199, 547–563 | 547

http://orcid.org/0000-0001-5837-4740

http://dx.doi.org/10.1039/c6fd00234j

http://pubs.rsc.org/en/journals/journal/FD

http://pubs.rsc.org/en/journals/journal/FD?issueid=FD017199

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adsorption geometries of the molecule.12 However, these experiments leave manyquestions unanswered, and theoretical approaches are needed to providea deeper understanding of the surface site heterogeneity of E�0.

A variety of experimental approaches have been explored for single-moleculeor few-molecule electrochemical detection. Since typical electrochemical reac-tions involve the transfer of only one or a couple electrons per molecule, directelectrical detection of single molecules is currently only possible by amplifyingthe signal via redox cycling of the molecule,4,8,13,14 effectively producing anensemble of electron transfer events involving the same molecule. In contrast,optical techniques have the potential to detect single electron-transfer eventsbased on spectroscopic differences between the oxidized and reduced forms.9–11,15

One particularly promising approach toward single-molecule electrochemistryinvolves detection of the molecules using techniques based on surface-enhancedRaman spectroscopy (SERS).11,16,17 The interaction of light with a plasmonic metalnanostructure leads to the enhancement of the local electric elds by many ordersof magnitude,18 particularly in “hot spots” near sharp points of the nano-structures or few-nm gaps between two nanostructures.19–21 This electric eldenhancement as well as chemical effects due to direct metal–molecule interac-tions lead to an enhancement of the Raman signal,22–24 enabling single-moleculedetection.25,26 To obtain few-nm spatial sensitivity, a plasmonic scanning probemicroscopy tip can be used to probe molecules on a surface, referred to as tip-enhanced Raman spectroscopy (TERS).27–29

Recently, there has been preliminary development of electrochemical single-molecule SERS and TERS techniques for few-molecule and single-molecule elec-trochemical detection.11,12,30,31 In particular, in experiments performed by the VanDuyne group,30 Nile blue molecules are adsorbed onto a at indium tin oxide(ITO) electrode, and a plasmonic tip is brought close to the electrode surface.Reduction of Nile blue breaks the conjugation across the molecule and signi-cantly shis the absorption peaks. Under light resonant with the absorption peakof the oxidized form, the oxidized Nile blue molecules near the plasmonic tipproduce a very strong TERS signal that disappears upon reduction. As thepotential at the ITO electrode is changed, the intensity of the TERS signal changesin discrete steps corresponding to the reduction or oxidation of individualmolecules or of a few molecules in similar chemical environments. Notably, theredox events occur over a range of potentials (0.2–0.3 V), suggesting that there isa distribution of E�0 values for molecules adsorbed onto the electrode surface.However, the details of the adsorption geometries of individual molecules arechallenging to probe experimentally.

Theoretical studies can provide important insight into the molecular details ofthe electrochemical process that are challenging to probe experimentally.However, developing a theoretical approach to describe electrochemical systemsis quite challenging due to the complexity of the experimental environment.32–39

Even a nanoscale electrode has a near-continuum of states, whereas moleculeshave a much smaller number of discrete states. The experiments are typicallyperformed in electrolyte solutions, and the potential applied at the electroderesults in the formation of an electrical double layer near the electrode surface.Many simplications must be made to make the system feasible to study.

The primary interest in this paper is on the thermodynamics of adsorbedmolecule–electrode systems, with a focus on quantum mechanical approaches

548 | Faraday Discuss., 2017, 199, 547–563 This journal is © The Royal Society of Chemistry 2017


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combined with implicit solvation that can be used to describe electrochemicalreactions on nanoparticles in water. While there have been a few previousattempts to consider this direction of theory development, with most of the workconcerned with determining the redox reaction mechanisms for small moleculessuch as H2,40–42 OH

�,43–46 and CO2 (ref. 47–50) on at electrodes, we are motivatedin this work by the very recent SERS/TERS/electrochemistry studies mentionedabove, where it makes sense to consider the transfer of an electron betweena plasmonic silver (or gold) cluster, and a SERS-active molecule (which is usuallya heterocyclic aromatic). Given this, we have chosen Ag20 as the metal cluster, andpyridine as the molecule. The Ag20–pyridine model represents a signicantsimplication even on the systems that were recently studied, but it captures theessence of what is involved, and is simple enough that we can use it to test newtheoretical methods. The Ag20 cluster is small enough that it has a signicantband gap, however it has been shown that the optical properties of this and largerplasmonic Ag nanoclusters extrapolate smoothly to match electrodynamicsresults for much larger nanoparticles.51 Also, silver electrodes have been widelyused in electrochemical SERS experiments since the 1970s. Although the elec-trochemistry of pyridine is irreversible and quite complex,52,53 pyridine has beenwidely used for theoretical studies of SERS, and its redox properties on the silvercluster are similar to more complicated molecule/metal systems (such as Nileblue on ITO) relevant to the recent experimental studies.54–56

Since we are particularly interested in molecules adsorbed onto nanoparticlesurfaces, the computational method must predict reasonable charge-transferenergies in strongly interacting systems. Typical density functional theory (DFT)approaches are well-known to over-delocalize the wavefunction and providespuriously low charge-transfer energies,57,58 whichmakes them unsuitable for thisapplication. Although various approaches such as constrained DFT and fragment-based approaches such as frozen-density embedding provide localized chargesand more accurate charge-transfer energies,59–61 these approaches are limited tothe regime where the wavefunction overlap between the two moieties is small.62

Instead, we use the semiempirical INDO approach, which has been shown to yieldmore accurate charge-transfer energies than typical DFT approaches.63,64 We haverecently developed parameters that yield accurate INDO/SCI plasmonic excitedstates in Ag clusters65 and implemented the implicit COSMO solvent model66,67

with INDO/SCI,68 so key elements of theory needed for the electrochemicalapplication with SERS detection are now available.

Since this is an electrochemical system, we must also tune the potential of theAg20 electrode. In principle, varying the potential requires dynamically modu-lating the electrode charge;69–71 however, in practice most approaches involveeither applying potential-dependent thermodynamic corrections to computationsat zero charge72–74 or converting computations using several xed charges toa series of effective electrode potentials.75–77 The addition of a single electron toAg20 raises the potential by nearly 3 V, which makes it difficult to use a model thissmall for electrochemistry applications. If we did, the addition of a charge wouldalso need to be balanced by variation in the double layer properties to properlycapture the electrical environment experienced by molecules near the cluster.Clearly, a more nely tunable approach is needed to capture the effects ofchemically relevant potential changes on the order of tens to hundreds of mV. Insemiempirical models, it has previously been shown that the electrode potential




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can be modeled by applying a xed shi to the parameters corresponding to themetal atomic orbital energies, which we refer to as the orbital energy shiapproximation (OESA).45,78 Indeed, this approach has been used to model theeffect of potential on the adsorption geometries45,79–81 and reaction pathways45,80,82

of small molecules on cluster electrodes. While this approach is a signicantapproximation that likely captures the essence of an electrochemical experimentonly near the point of zero charge of the electrode, this range of potentials is whatis of most interest in modeling the SERS/TERS experiments.

The main focus of this paper is on the development of a new computationalapproach based on the INDO/SCI method to model electrochemical chargetransfer and present results for the Ag20–pyridine model system. Because bothAg20 and pyridine are closed shell, the most stable [Ag20–pyridine] charge-localized state is a closed-shell singlet and the most stable [Ag20

+–pyridine�]charge-transfer state is a triplet. Given this, the approach we have developeddetermines E�0 for each geometry and solvent environment by applying a shi ofthe Ag orbital energies such that the charge-localized singlet and charge-transfertriplet are equal in energy. We illustrate this approach for a series of geometries ofthe pyridine/Ag20 complex and show that this approach predicts signicantdifferences in E�0 among the geometries that are weakly correlated with signi-cant changes in the interaction energies. This demonstration thus indicates thepotential of this approach for providing chemical insight into the electrochemicalproperties of large molecules on complex surfaces such as those studied in SERS/TERS-based electrochemical experiments.

2. Computational methodology

The geometries of the isolated tetrahedral Ag20 cluster and pyridine molecule andof the Ag20–pyridine complex were determined using density functional theorywith the BP86 functional83,84 and a double-zeta (DZ) basis set. For the Ag atoms,a frozen-core approximation was used for all electrons in the 4p and lowerorbitals, designated DZ.4p. Scalar relativistic effects were incorporated using thezero-order regular approximation (ZORA).85 Single-point energy calculations werealso performed at the same level of theory for a series of interaction geometriesconstructed using the optimized geometries of the isolated Ag20 and pyridinemoieties. The single-point energies were computed both in the gas phase andusing the COSMO solvent model66 with dielectric constants 3 of 2, 5, and 80,corresponding roughly to hexane, chloroform, and water, respectively. Thesolvent radius was set to 1.0 A in all cases, and the screening energy was scaled bya factor of f(3)¼ (3� 1)/(3 + 0.5) relative to a perfect conductor for consistency withthe original COSMO algorithm.66 All density functional theory calculations wereperformed using the Amsterdam Density Functional (ADF) 2014 program.86

Semiempirical calculations using the Intermediate Neglect of DifferentialOverlap (INDO) Hamiltonian were performed as single-point calculations usingour recently developed parameters for Ag65,87 and Zerner’s INDO/S parameters88

for all other atoms. For the Ag20–pyridine system, the closed-shell singlet groundstate was rst computed, and then the triplet states were computed as excitedstates of the ground state using a conguration interaction (CI) approach withsingle excitations (SCI). Although the lowest triplet state could be obtainedthrough a separate open-shell SCF calculation, computation of the triplets as




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excited states of a singlet ground state allows states of both multiplicity to beobtained from one calculation and simplies the analysis of orbital contributionsto the transition. All possible one-electron excitations of triplet multiplicity weregenerated, and the lowest 2000 congurations were included in the CI matrix.This matrix was then diagonalized to obtain the lowest 500 triplet states. For theoptimized geometries, the singlet excited states were also computed using ananalogous procedure. Note that none of the singlet and only the lowest of thetriplet excited states is needed for the E�0 calculation, however the higher energystates provide useful physical insight as described later. Calculations were per-formed in the gas phase and using our recent implementation68 of the COSMOsolvent model66,67 with 3 ¼ 2, 5, and 80. This implementation of COSMO includesa perturbative correction to the excited-state energies to account for relaxation ofthe solvent upon excitation.67,68 For comparison with optical experiments, thiscorrection is typically scaled by f(n2) ¼ (n2 � 1)/(n2 + 0.5), where n is the refractiveindex at optical frequencies, to account for solvent relaxation on optical timescales. However, since we are interested here in the relative energies of the groundsinglet and triplet states when the solvent is fully relaxed, we set the dielectricconstant for the excited-state correction equal to the dielectric constant in theground state.

To compute the electrochemical properties, the INDO parameters corre-sponding to the Ag s, p, and d orbital energies were shied by a value corre-sponding to the applied potential, and the lowest singlet and triplet states werecomputed as described above. As has been previously used in semiempiricalmodels,45,78,80 we assume that a 1 eV shi in orbital energies corresponds to a�1 Vchange in potential. We dene the formal potential E�0 as the shi in the Agorbital energies required to obtain equal energies for the rst charge-localizedsinglet and charge-transfer triplet states. To determine E�0, the potential wasvaried using linear interpolation based on the singlet–triplet energy gap until thesinglet and triplet state energies converged to within 0.001 eV. As we have notreferenced these potentials to a standard electrode, and indeed the connection ofthe metal cluster to an externally connected electrode is not included in themodel, the changes in the formal potential between different geometries orsolvent environments are more signicant than the exact values of the formalpotential in a particular system.

3. Results and discussion3.1. Optimized Ag20–pyridine complexes

First, we examine the Ag20–pyridine complexes in the absence of an appliedvoltage. We consider two prototypical geometries where the pyridine is interacting

Fig. 1 Interaction geometries of the Ag20–pyridine complex.




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with (1) one of the triangular surfaces and (2) one of the vertex atoms of thetetrahedral Ag20 cluster as shown in Fig. 1. Since the INDO parameters aredesigned to reproduce spectroscopic parameters and do not accurately reproducegeometries, we consider the geometries and interaction energies obtained at theBP86/DZ level. Consistent with previous computational studies,54 the “Vertex”complex is more stable, with an interaction energy of �0.61 eV and an N–Agdistance of 2.35 A; the “Surface” complex has an interaction energy of �0.31 eVand the smallest N–Ag distance is 2.48 A.

In these complexes, most of the molecular orbitals (MOs) at the INDO level arelocalized on either the Ag20 or the pyridine moiety as shown in Fig. 2; however,wavefunction overlap between the two moieties allows for some mixing. Impor-tantly, the Frontier MOs are localized solely on the Ag20 moiety, and the rst MOswith signicant pyridine character are several eV below the HOMO or above theLUMO.We note that INDO predicts ground-state charge-transfer (CT) of about 0.4electrons from pyridine to Ag20 in both complexes, primarily involving CT fromthe s bonding orbitals on pyridine into the Ag20 sp band. This behavior is similarto what is obtained from DFT calculations, but the magnitude of the CT is largerin INDO. Since the Ag parameters were tuned to reproduce the optical propertiesof Ag clusters,65 but have not been benchmarked for interactions of Ag with otheratoms, further evaluation of these parameters may be needed.

We have also computed the singlet excited states of these complexes at theINDO/SCI level. As has been observed previously for the isolated Ag20 cluster,65

there are many excited states involving local excitations on the Ag20 moiety, withthe main plasmonic absorption peak occurring at around 3.4 eV. The presence ofthe pyridine has little effect on the local excited states and primarily introducesnew excited states with CT character. The small coupling between CT and localexcitations allows some mixing of the CT excitations with local excitations ofcomparable energy but does not signicantly shi the CT excited-state energiesrelative to the pure CT excitations. The rst states with signicant CT characteroccur at 3.60 eV and 3.71 eV for the surface and vertex complexes, respectively,several tenths of an eV higher in energy than the main plasmonic absorption

Fig. 2 (Left) Molecular orbital energies and (right) excited-state energies for optimizedgeometries of the Ag20–pyridine complex at the INDO/SCI level.




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peak. This is signicantly higher in energy than the lowest charge-transfer statespreviously computed using TD-DFT at the BP86/TZP level (2.16 eV and 1.44 eV,respectively, for the surface and vertex complexes),54 providing an indication ofthe importance of self-interaction effects in the DFT results.

We now turn to the electrochemical properties of these complexes; as detailedin the methodology section, each value of the potential applied to the Ag20 elec-trode corresponds to a shi in the INDO parameters for the Ag atomic orbitalenergies (�1 V potential¼ 1 eV energetic shi).45,78 In particular, we are interestedin the range of potentials where the charge-localized [Ag20–pyridine] and charge-transfer [Ag20

+–pyridine�] states are close in energy. Within this range of poten-tials, the most stable charge-localized state is the rst singlet state and the moststable charge-transfer state is the rst triplet state; thus, we compute the formalpotential E�0 for each interaction geometry as the shi in the Ag atomic orbitalenergies such that the charge-localized and charge-transfer states areisoenergetic.

Fig. 3 (Top) Energetic difference between the first singlet and triplet states and (bottom)total charge on the pyridinemoiety as a function of the applied potential, for the optimizedsurface geometry of the Ag20–pyridine complex at the INDO/SCI level.




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To examine the evolution of the charge-transfer and charge-localized states, werst scan the potential within the range of roughly �0.5 V relative to E�0 for thesurface complex (Fig. 3). Across this range, the singlet–triplet energy gap hasa nearly linear variation with potential. Application of a potential signicantlyreduces the ground-state singlet CT from pyridine to Ag20, and in addition, thetransition from the rst singlet state to the rst triplet state involves nearly a fullelectron being transferred from the sp band of Ag20 to the rst p* orbital onpyridine (which we note is opposite in direction to the ground-state CT describedabove). E�0 occurs at �4.57 V in the surface geometry and �5.11 V in the vertexgeometry in the gas phase. Note that these potentials are referenced to the INDOparameters and not to a standard electrode as is typical for experimentally re-ported values. Further work to benchmark these potentials is in progress.

An important insight from the formal potentials, �5.11 V and �4.57 V, is thatthe 0.44 V difference between the two geometries is four times the energeticdifference between the rst CT excited states for the same geometries but at zeroapplied potential (i.e., when there is no shi in the Ag orbital energies). Sincepyridine is 3.2 A further from the geometric center of the Ag20 cluster in the vertexgeometry than in the surface geometry, a larger energy is required to overcome theelectrostatic attraction and induce CT in the vertex geometry. At zero appliedpotential, the substantial mixing between CT and local excitations dilutes thiseffect. This highlights the importance of electrochemical calculations at anapplied potential in determining the variation of formal potential with geometry.Indeed the applied potential signicantly changes the electronic structure of thesinglet and triplet states, which is in contrast with what has been assumed inprevious computational studies that suggested minimal changes in adsorptionproperties with potential.72 Likewise, the difference between the formal potentialsis larger than the 0.30 eV difference in interaction energies between the twogeometries. Although it has recently been proposed that in certain systemschanges in E�0 correlate with interaction energies,89 this correlation is limited tosystems where the interaction energy of either the oxidized or the reduced formcan be treated as a constant, and thus the lack of correlation between the two inour system is not surprising.

The procedure to determine E�0 is relatively straightforward in this systembecause both of the isolated neutral moieties are closed-shell and the rst localtriplet state localized on Ag20 is roughly 1.5 eV higher in energy than the singletground state. Since shiing the Ag orbital energies changes the energy of the CTexcited states without signicantly perturbing the local excited states and thecoupling is weak enough that the CT and local excitations only mix signicantlywhen their energies are within a few tenths of an eV, the rst triplet state atpotentials near E�0 is straightforward to identify as the rst CT state. If a largercluster electrode with a smaller band gap were used, the rst local triplet would belower in energy and amore complex analysis may be required to identify the statesof interest.

3.2. Effect of interaction distance

Since we have shown that our computational approach reveals differences in E�0

between different adsorption sites, we now evaluate the effects of solvent and ofa broader range of interaction geometries on E�0. We rst consider the effect of the




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distance between the Ag20 and pyridine moieties in the surface and vertexcongurations using geometries constructed from the optimized structures of theisolated Ag20 and pyridine moieties. Importantly, for all of the geometries andsolvent environments studied, the singlet–triplet transition at potentials near theformal potential involves nearly a full charge transfer from the Ag20 sp band toa pyridine p* orbital.

Consider rst the effect of distance in the surface conguration (Fig. 4). Sincethe geometries here are not allowed to fully relax, the strongest interaction of�0.25 eV in the gas phase occurs at a distance of 2.6 A, 0.06 eV weaker than in thefully optimized structure described in the previous section. Notably, the inter-action becomes weaker with increasing solvent dielectric constant due to partialde-solvation, with a maximum interaction strength of only �0.09 eV in water (3 ¼80). In polar solvents, de-solvation also leads to a small barrier to adsorption atdistances between 3 and 4 A.

The interaction distance also has a signicant effect on E�0. In the gas phase,Fig. 4 (bottom) shows that E�0 becomes monotonically less negative withincreasing distance. At distances less than roughly 3.5 A, a small increase in

Fig. 4 (Top) Interaction energy at the BP86/DZ level and (bottom) E�0 at the INDO/SCIlevel as a function of interaction distance in the surface configuration for the Ag20–pyri-dine complex.




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distance signicantly reduces the wavefunction overlap between the twomoieties,leading to strong sensitivity of E�0 to distance. In contrast, at larger distanceswhere wavefunction overlap is negligible, the change in E�0 is primarily related tothe energy required to overcome the electrostatic attraction between the positivelyand negatively charged moieties in the charge-transfer state, leading to a weakerdistance dependence. As the solvent dielectric constant increases, E�0 becomesless negative due to screening of the Coulomb interactions, particularly at longerdistances where electrostatic effects dominate. Solvent screening also leads toa weaker distance dependence of E�0 in the long range; in water, E�0 varies by lessthan 0.1 V at distances greater than 3.6 A. We note that E�0 varies over a muchbroader range than the interaction energies: even though the strongest interac-tion energy is�0.09 eV in water, E�0 becomes more negative by 0.54 V between theadsorbed (2.6 A) and 8.0 A structures. This implies that the details of the wave-function overlap between the two moieties have important effects on the elec-trochemical behavior that are not captured by comparing the strengths of theinteractions in various geometries.

Similar results are seen in the vertex conguration (Fig. 5). As seen in theprevious section, the interaction is about twice as strong in the vertex congu-ration as in the surface conguration and is maximized at slightly shorterdistances. We also see that solvation has a smaller effect on the interaction energyin the vertex conguration; this is unsurprising since this conguration involvesa smaller region of spatial interaction between the two moieties, so less de-solvation is required. In the long-distance limit, E�0 is more negative in thevertex conguration than in the surface conguration because, as mentionedpreviously, the pyridine is 3.2 A further from the geometric center of the clusterfor equivalent N–Ag distances. Because of this difference in effective distance, ineach solvent environment E�0 at a distance of 4.8 A in the vertex conguration isquite comparable to E�0 at 8.0 A in the surface conguration. In the short-distanceregime, adsorption at the vertex has a much smaller effect on E�0 than adsorptionto the surface despite the signicantly stronger interaction at the vertex. This isdue to a combination of smaller wavefunction overlap, since the pyridine onlyinteracts signicantly with one Ag atom in the vertex conguration, and the largereffective distance for charge separation.

3.3. Effect of other interaction geometries

To study the effect of adsorption geometry on E�0, we also examine the effect ofother displacements of the pyridine relative to the Ag20 moiety, rst consideringtilting the pyridine moiety relative to the Ag20 with the nitrogen atom at a xeddistance of 2.6 A from the nearest Ag atom. In both the surface and vertexorientations, the interaction is strongest when the pyridine is standing up on theAg20 cluster (q¼ 0�) and becomes weaker with increasing tilt angle (Fig. 6a and b).In the surface orientation, the interaction becomes strongly repulsive at very largetilt angles; this effect is not observed in the vertex orientation where tilting thepyridine has a much smaller effect on the extent of wavefunction overlap betweenthe two moieties. Consideration of implicit solvation somewhat weakens theinteractions in both orientations and leads to a slightly weaker dependence of theinteraction energy on tilt angle, but it does not qualitatively change trends in theinteraction energies.



Fig. 5 (Top) Interaction energy at the BP86/DZ level and (bottom) potential required forelectron transfer at the INDO/SCI level as a function of interaction distance in the vertexconfiguration for the Ag20–pyridine complex.


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In the vertex orientation (Fig. 6d), E�0 is weakly dependent on the tilt angle andis more weakly tilt-dependent than the interaction energy due to the minimalchanges in wavefunction overlap with tilt. As seen for the interaction energies,solvent effects shi E�0 to smaller values but with minimal effects on the geometrydependence. In the surface orientation (Fig. 6c), E�0 becomes less negative at tiltangles larger than 50� because of the smaller effective difference for chargeseparation. In polar solvents, there is little change in E�0 at angles less than 50�; incontrast, in the gas phase there is a peak in E�0 at a tilt angle of 40� (which is alsopresent for the vertex orientation at 70�). This is primarily related to differences inthe contributions of several electron congurations to the triplet CT state. Ag20has several nearly degenerate orbitals near the HOMO as shown in Fig. 2, leadingto several low-energy congurations where an electron is excited from one of theseorbitals into an unoccupied orbital on pyridine. In polar solvents, the differencesin charge distributions lead to stabilization of one of these congurations, andthe triplet state is nearly a pure excitation from the singlet state that variessmoothly in energy across the series. In contrast, in the gas phase several CT



Fig. 6 (a and b) Interaction energies at the BP86/DZ level and (c and d) E�0 at the INDO/SCIlevel as a function of tilting of the pyridine in the (a and c) surface and (b and d) vertexorientations relative to the Ag20 moiety.


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congurations are very close in energy near 0�, and the triplet state is stabilized bymixing of these congurations; at tilt angles near 40�, larger energetic differencesbetween the congurations inhibit mixing, leading to a higher-energy triplet stateand thus a larger magnitude of E�0.

We now consider the effect of displacements of the pyridine across one of thetriangular faces of the Ag20 cluster, maintaining a constant 2.6 A distance betweenpyridine and the Ag20 surface. For displacements of the pyridine either parallel (xaxis; Fig. 7a) or perpendicular (y axis; Fig. 7b) to an Ag20 edge, the interaction isstrongest when the pyridine is directly atop an Ag atom (x ¼ 0.0, �3.0 A; y ¼0.0, +4.8 A). The interaction energies do not vary signicantly between thedifferent atop sites. The interactions are 0.12–0.14 eV weaker at the two-foldbridge sites (x ¼ �1.5 A; y ¼ �2.4 A) and 0.19–0.20 eV weaker at the three-foldhollow sites (y ¼ �1.6, +3.6 A) than in the atop sites. Consideration of implicitsolvation leads to a constant offset in the interaction energies as described earlier,but has a negligible effect on the relative interaction energies in differentgeometries.

The effect of displacements on E�0 is more complex. As seen for the interactionenergies, solvation primarily yields an offset in E�0 and has a less signicant effecton the relative formal potentials in different geometries. For displacements alongthe x axis (Fig. 7c), E�0 is largest in magnitude in the atop sites where the inter-action is strongest and is smallest in magnitude near the bridge sites; thedifference in formal potentials between the bridge and atop sites is around 0.4 V,several times larger than the difference in interaction energies between the twosites. In the bridge sites, a decrease in the HOMO–LUMO gap combined with



Fig. 7 (a and b) Interaction energies at the BP86/DZ level and (c and d) E�0 at the INDO/SCIlevel as a function of displacement of the pyridine (a and c) parallel and (b and d)perpendicular to the edge of the Ag20 triangular face.


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greater Coulomb stabilization of the triplet state lead to a decrease in themagnitude of E�0. Along the y axis (Fig. 7d), there are slight maxima in themagnitude of E�0 near the atop sites; however, the primary trend is a generalincrease in the magnitude of E�0 with increasing y displacement from the straightedge toward the vertex of the triangle. When pyridine is displaced toward the edgeof the Ag20 face (y < 0), the HOMO–LUMO gap is reduced, stabilizing the tripletstate and decreasing E�0. In contrast, when the pyridine is displaced toward theAg20 vertex (y > 0), the orbital energies have a smaller effect on E�0, and thedominant effect is a weaker Coulomb stabilization of the triplet leading to anincrease in the magnitude of E�0. This is consistent with the larger magnitude ofthe formal potential seen in the vertex conguration than in the surface cong-uration as described earlier. These results show that the details of the chemicalinteractions in each geometry have an important inuence on the electrochemicalbehavior that is in general weakly correlated with the interaction energies or withthe CT excited-state energies at zero applied potential. Note also that the 0.4 Vrange of formal potentials that we have found in Fig. 6 and 7 is similar to therange of formal potentials seen in the recent experiments in the Van Duynegroup.30

4. Conclusions

Developing quantum mechanical models for electrochemical charge transfer iscritical to understand the results of recent nanoelectrochemistry experiments30,90

that show a large variation in the formal potential E�0 between different adsorbedmolecules. Here, we have developed an approach to compute changes in E�0 with




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adsorption geometry based on the semiempirical INDO Hamiltonian. Usinga metal cluster as a model electrode, we apply a potential by shiing the metalatomic orbital energies to effectively change the Fermi level without perturbingthe electrode electronic structure. Since both the neutral molecule and the neutralmetal cluster in our model system are closed-shell singlets, we compute E�0 as thepotential where the charge-localized singlet and charge-transfer triplet are equalin energy.

Our model predicts signicant variation in E�0 as a function of the Ag20–pyri-dine geometry, with this variation strongly dependent on details of the wave-function overlap and, in many cases, weakly correlated with the interaction energyor differences in the energies of the CT excited states at zero applied potential.Although the results here focus on a simple model system, they highlight the needfor detailed electrochemical calculations for molecules in various adsorbedgeometries to understand the chemical origins of the experimentally observedvariation in E�0. Further development will enable this approach to providechemical insights into the electrochemical behavior of more complex systemsrelevant to the recent nanoelectrochemical experiments.

Acknowledgements

This work was supported by the Air Force Office of Scientic Research MURI(FA9550-14-1-0003). We thank Richard Van Duyne, Bo Fu and Colin Van Dyck forhelpful conversations.

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