The standard hydrogen electrode and potential of zero charge in density functional

calculations

Vladimir Tripkovic,1 Marten E. Bjorketun,1 Egill Skulason,2 and Jan Rossmeisl1, ∗

1Center for Atomic-scale Materials Design, Department of Physics,

Technical University of Denmark, DK-2800 Lyngby, Denmark2Science Institute, VR-III, University of Iceland, IS-107 Reykjavik, Iceland

(Dated: May 25, 2011)

Methods to explicitly account for half-cell electrode potentials have recently appeared within theframework of density functional theory. The potential of the electrode relative to the standard hydro-gen electrode is typically determined by subtracting the experimental value of the absolute standardhydrogen electrode potential (ASHEP) from the calculated work function. Although conceptuallycorrect, this procedure introduces two sources of errors i) the experimental estimate of the ASHEPvaries from 4.28 to 4.85 V and, as we show here, ii) the calculated work function strongly dependson the structure of the water film covering the metal surface. In this work, we first identify the mostaccurate experimental reference for the ASHEP by revisiting the up-to-date literature and validatethe choice of electron reference level in single electrode density functional setups. By analyzing adozen different water structures, built up from water hexamers, in their uncharged (potential of zerocharge - PZC) states on Pt(111), we determine three different criteria − no charge transfer, no netdipole and high water flexibility − that a water structure should possess in order for its computedASHEP to closely match the experimental benchmark. We capture and quantify these effects bycalculating trends in ASHEP and PZC on eight close-packed transition metals considering the fourmost simple and representative water models. In addition, we demonstrate how the work functionchanges with exchange correlation functional. Finally, it is shown that the ASHEP’s dependence onthe water structure and metal surface does not play a decisive role when evaluating the energetics ofcharge transfer reactions, if an internal reference scale that links work function scale to the thermo-chemical scale of the standard hydrogen electrode is used instead of the experimentally determinedvalues [Chem. Phys. Lett. 466, 68 (2008)].

PACS numbers: 68.08.-p,73.30.+y,71.15.Nc,71.15.Mb

I. INTRODUCTION

In electrochemistry all half-cell electrode potentials aregiven relative to a chosen reference electrode of somewell-known reaction, for example the standard calomelor the standard hydrogen electrode (SHE). In many elec-trochemistry experiments, taking the difference betweentwo electrode potentials alleviates the problem of deter-mining the potential on an absolute scale. Nevertheless,the absolute potential scale is still of great importancewhen comparing electrochemical and ultra-high vacuum(UHV) experiments and when trying to match semicon-ductor and solution energy levels1 in for instance photo-electrochemical devices.

Density functional theory (DFT) based methods formodeling electrochemical systems and, in particular, fortreating charge transfer reactions at the electrochemicalinterface have just started to appear.2–12 In these simu-lations the electrochemical cell is customarily split intotwo half-cells and the redox reactions taking place at thetwo electrodes are studied separately. Here the abso-lute potential enters as a key parameter because it isno longer possible to measure potential differences be-tween the two electrodes. We would, however, like to dis-tinguish between three different categories of theoreticalelectrochemistry studies. In the first category we find in-vestigations of the thermodynamics of electrochemical re-

actions and calculations of redox potentials.13,14 In thesestudies the potential does not have to be taken explic-itly into account in the DFT simulation, but can ratherbe added posteriori. The second category comprises theenergetics of charge transfer reactions. To study thesetypes of problems it is necessary to explicitly include thepotential in the DFT model. At the same time it is pos-sible to introduce an internal potential reference, whichmeans the accuracy of the absolute potential is of lessimportance.15,16 Finally, in the third category we findproblems like the matching of semiconductor and solu-tion energy levels that demand highly accurate estimatesof the absolute potential.

In general, the DFT based methods use the work func-tion (WF) of the water-covered metal electrode as a mea-sure of the absolute electrode potential. In order to ob-tain a relative potential scale, the WF scale needs to becoupled to a normal thermo-electrochemical scale, for in-stance of the SHE. This matter can be approached intwo conceptually different ways. The most straightfor-ward and most common is to use an experimental valueof the absolute standard hydrogen electrode potential(ASHEP), i.e. the experimental WF that correspondsto SHE conditions, as reference. However, there is alarge uncertainty in the ASHEP reported in the litera-ture; they vary from 4.28 to 4.85 V.17–26 Many prop-erties − including energy barriers, and hence rates, ofcharge transfer reactions − are strongly potential depen-

2

dent, which renders the use of the most correct exper-imental estimate essential. Moreover, since the calcu-lated WF depends strongly on the structure of the waterat the metal/aqueous interface,27 employing an experi-mental reference is a sound approach only as long as thereal system is faithfully mimicked in the simulations. Itshould be noted, though, that it is usually difficult tocapture properties of bulk water in these types of first-principles simulations as computational constraints lim-its the number of water molecules that can be explic-itly treated.6,28 Alternatively, one can use a theoreticalvalue of the ASHEP, internal to the system and perhapsdifferent for different metal/water/vacuum setups. Thisapproach has the advantage that it requires a less re-alistic representation of the simulated metal/water in-terface, but in turn a scheme for determining the the-oretical ASHEP is needed. We have recently devisedsuch a scheme15 and subsequently successfully imple-mented it to model the hydrogen evolution and oxidationreactions.16

Although, as we have already discussed, many prob-lems can be solved using an internal “low quality”ASHEP reference, there are cases like the previouslymentioned matching of energy levels when a good theo-retical estimate of the ASHEP is vital. Therefore, in thecurrent paper we present an extensive investigation of thetheoretical ASHEP. Particular attention is given to howthe choice of water model, used to emulate the interfacialwater, affects the calculated ASHEP and makes it deviatefrom the experimental counterpart. Much effort is alsoput into evaluating the somehow related quantity poten-tial of zero charge (PZC). By thoroughly examining theavailable literature, we first determine the most reliableexperimental benchmark value for the ASHEP. Invok-ing the fundamental concepts of the absolute potentialscale, developed by Trasatti and others,1,29–34 we thenpinpoint the most appropriate choice for the referencelevel of the computed WF. Subsequently, by briefly revis-iting our earlier works, we show how an internal ASHEPreference can be readily established.15,16 Next, we per-form a comprehensive investigation of Pt(111)/water sys-tems. We demonstrate how the WF, which correspondsto the PZC when the water film contains no ions, varieswith the structure of the interfacial water and identifythe different contributions to the WF. Guided partly bythis study we identify three properties the model waterfilm must possess in order to reproduce the experimen-tal ASHEP. We then evaluate the ASHEP for a set ofrepresentative electrode/water systems and quantify thedeviations from the experimental value owing to the useof unrealistic water structures.

II. EXPERIMENTAL ESTIMATES OF THE

ASHEP

Experimental values of the ASHEP reported in the ear-lier literature resided mostly on measurements of changes

in the WF upon water adsorption. The WF was mea-sured by means of immersed electrode setups20–22, UHVstudies35,36 or, similarly, through measurements of thepotential difference across the Hg|air|H+|Pt,H2 cell inthe absence of any specific adsorption or charge trans-fer (so-called streaming Hg jet method)17,18. There isa large scatter in reported values obtained using thesetechniques (4.44 to 4.85 V). The discrepancies have beenattributed to, among other factores, surface contamina-tion, the orientation of water in UHV experiments be-ing different from the one in bulk water, and to partialcharge transfer.29,31 Here, it is worth noticing that thelower value has been identified as the most reliable one byTrasatti and also the value recommended by the Interna-tional Union of Pure and Applied Chemistry (IUPAC).1

FIG. 1: (a) Born-Haber cycle for the standard hydrogen elec-trode. (b) Energies in (a) shown schematically on a step dia-gram. The ASHEP is equal to µe− .

A second and more direct approach to measure theASHEP is through the Born-Haber cycle shown in Fig. 1,where ∆Gd, ∆GI, αS(H+) and µe− are dissociation andionization free energies, the real potential of the proton insolution and the chemical potential of the electron withrespect to its reference state (a measure of the ASHEP).As indicated in Fig. 1, H+(S)+e−(M) will be in equilib-rium with 1

2H2(g) at the SHE potential under standardconditions (pH = 0, pH2

= 1 bar, T = 298 K). Since thefree energies of H2(g) and H+(S) are independent of themetal used as electrode as long as H+ is solvated in thebulk of the solution, the equilibrium of the reaction

H+(S) + e−(M) ↔1

2H2(g) (1)

uniquely defines the Fermi level EF(M) (which makesit metal independent). In other words, the amount ofcharge transferred on different metal surfaces will alwaysbe such that the metal Fermi levels will become alignedat SHE conditions.

3

The ASHEP can thus be determined directly fromthe Born-Haber cycle, without involving the WF, if theproton solvation energy is accurately measured. Re-cently, through very precise determination of αS(H+),the ASHEP has been estimated to be 4.42 V.25,37 Thisvalue is very close to the 4.44 V recommended byTrasatti. In fact, it is highly unlikely that this strikingagreement between two completely different approachescan be the result of a pure coincidence. It should benoted, however, that in an even more recent study, aslightly lower value, ASHEP = 4.28 V, was suggested.26

This value was obtained by using the electron in vacuumat 0 K as the zero level for the electron energy and thestandard hydration free energy of the proton, ∆G0

S(H+),

instead of αS(H+). Nevertheless, as we shall see in nextsection, the most natural and convenient choice of zerolevel for the electron energy in our DFT setup is theelectron in vacuum close to the solution surface, whichis also the zero level assumed by IUPAC.1 In this workwe will therefore use 4.44 V as the experimental value tobenchmark our calculated ASHEPs against.

III. FREE ELECTRON REFERENCE IN

EXPERIMENTS AND DFT

As Trasatti has pointed out, the ASHEP depends onthe chosen reference state for an electron at rest.1,29,32

He has showed that there exist three such physically con-ceivable levels: at rest in vacuum at infinity, in the bulkof the solution, or in vacuum close to the surface of thesolution. From his analysis, he concluded that the bestreference is the near-solution-surface vacuum because itestablishes a direct link between surface science and elec-trochemical experiments and furthermore it is amenableto experimental determination.1,32 As we shall see in thefollowing it is also the most natural reference in DFTcalculations.

A typical electrochemical cell is illustrated in Fig. 2a.Appropriate half-cells are obtained when the electro-chemical cell is vacuum cleaved at a point in the solutionwhere the electrostatic potential (EP) is no longer af-fected by the metal surface. Four WFs required to bringan electron from the Fermi level of the metal Mi to a cho-sen reference state are indicated, where labels Φ∞

i , ΦSi ,

Φi and Φ′

istand for far (or infinite) vacuum, bulk so-

lution, near vacuum, and near-solution-surface vacuumWFs. Three of them − Φ∞

i, ΦS

iand Φ′

i− are potentially

relevant for determination of the ASHEP.In conventional periodic DFT calculations Φ∞

iis not

defined because of infinite surfaces imposed by periodicboundary conditions. As a consequence of the infiniteextension of the surface even a point in vacuum infinitelyfar from the surface will feel the presence of the sur-face dipoles. The second plausible reference is the pointin bulk solution. Obtaining a solution reference pointis straightforward given that enough water is includedin the cell. This, however, reflects on the system size

FIG. 2: (Color online) (a) Schematics of an electrochemicalcell showing a set of work functions, {Φk

i }, measuring thework needed to bring an electron from the Fermi level of themetal to four different free electron reference states. Φ∞

i , ΦS

i ,Φi and Φ′

i denote infinite vacuum, bulk solution, near vac-uum, and near-solution-surface vacuum WFs. (b) DFT modelof an electrochemical cell comprised of Pt(111) and Pd(111)electrodes immersed in a common ion-free aqueous solution,vacuum cleaved half-way between the electrodes. The figureillustrates the convergence of the electrostatic potential withrespect to the number of water layers. The water layers aremirror imaged on the two electrodes.

and the computational cost needed to perform such arelaxation is staggering. Additionally, a portion of thewater must be fixed because any wiggling of the wa-ter molecules can shift the reference level.6 In contrast,the near-solution-surface vacuum level can be readily ob-tained by the WF in metal/water/vacuum setups and,moreover, the measurement does not entail any confine-ments. Hence, it is the most natural free electron refer-ence state in DFT calculations and Φ′

iis thus the most

suitable WF.

The DFT counterpart to an electrochemical cell isshown in Fig. 2b. In ordinary DFT simulations of suchsetups the Fermi levels of the two electrodes (M1 and

4

M2) will always be aligned irrespective of the absolutepotential and the electrode material. Once the Fermilevels of M1 and M2 are aligned, Φ′

1 and Φ′

2 should alsobecome equal (since alignment of the Fermi levels meansthat the electrodes have assumed the same potential),given that the solution phase is thick and polarizableenough to screen the fields from the metals. However,this is usually not the case in DFT calculations due to thelimited amount of water that one can afford to includein the simulations. Convergence of the EP profile fora Pt(111)|water|vacuum|water|Pd(111) cell with respectto the number of water layers is displayed in Fig. 2b. Agradual improvement of the near-solution-surface refer-ence point with increasing thickness of the water is clearlyobserved. The mid-vacuum EP discontinuity (obtainedby electrostatic decoupling of periodically repeated su-percells) is fairly large in the cell containing only a singlewater layer. However, it becomes much smaller after ad-dition of a second layer and it essentially vanishes whena third water layer has been added. In Sec. V we willdemonstrate how inadequate water structures introduce,sometimes substantial, errors in the calculated ASHEPand PZC. It turns out that efficient screening is one ofthe keys to accurately predicting these properties.

IV. ESTABLISHING A THEORETICAL,

INTERNAL, REFERENCE FOR THE ASHEP

In this section we describe how an internal referencefor the ASHEP can be established in a DFT-based elec-trochemistry study. The methodology is summarizedhere because of its relevance for the following study. Formore details about the procedure we refer to our previousworks.15,16

We start with an atomic setup consisting of a metalslab and an electrolyte represented by water layers out-side the surface (cf. Fig. 2b). The metal/electrolyte in-terface is charged by adding hydrogen atoms to the firstwater layer outside the metal surface. These hydrogenatoms spontaneously separate into protons that becomesolvated in the water bilayer and electrons that are trans-ferred to the surface of the metal slab. The charge sepa-ration, in turn, creates an EP drop across the interface.The surface charge, and hence the potential, can be var-ied in steps by changing the concentration of protons inthe water layer.

A link between the thermo-electrochemical scale of theSHE and the WF Φ′ can be established by focusing on thefree energy of the solid/liquid interface as it is chargedwith protons and electrons. The total or integral freeenergy per surface metal atom (or surface area) relativeto H2 for a system with n protons and N surface atomsis given by

Gint = (G(N, n) − G(N, 0) −n

2µH2

)/N (2)

where µH2is the reference chemical potential of hydro-

gen. Gint corresponds to the free energy stored in the

interface set up by the protons in the water layer andtheir counter charge in the metal. Gint will be quadraticin potential if the interface behaves as a perfect capac-itor. The derivative of Gint with respect to the protonconcentration is the chemical potential of protons andelectrons

dGint

d(n/N)= µ(H++e−) −

1

2µH2

. (3)

The role of the µH2term is to define the reference. Hence,

if we choose µH2to equal the free energy of H2(g) at stan-

dard conditions, the WF corresponding to the minimumof Gint will define the potential of the SHE (reaction 1 isin equilibrium) on an absolute scale.

V. RESULTS AND DISCUSSION

In the following we first (Sec. VA) calculate the PZCof Pt(111) using a large number of different water mod-els and both Perdew-Burke-Ernzerhof (PBE) and Re-vised Perdew-Burke-Ernzerhof (RPBE) exchange corre-lation (XC) functionals (for computational details seeAppendix A). We define the PZC as the WF Φ′

Pt(111)

of the metal covered with an ion free (i.e. in our case un-protonated) water film.1,30 The calculated PZC is foundto depend strongly on the structure of the water film andthe choice of XC functional turns out to be of importancetoo. We disclose the physical origin of the large scatterin calculated values and discuss its consequences. Subse-quently (Sec. VB), the ASHEP and PZC are computedon the most close-packed surfaces of eight transition met-als, M = {Ru, Pd, Pt, Au, Ag, Re, Rh, Ir}, using fourqualitatively different water structures. These results arethen used to discern metal (or WF) dependent trends inthe PZC and to illustrate the importance of a set of phys-ical properties of the water film for accurate estimationof the ASHEP. Finally, in Sec. VC, we comment brieflyon the use and applicability of external (universal) andinternal ASHEP references.

Before presenting the PZC and ASHEP results we willbriefly introduce the water models used in this study(more specific details are given in the subsequent sec-tions). The water films are formed from bilayers of dif-ferent net dipole orientations, stacked in a layer-by-layerfashion. From UHV and DFT studies it has been in-ferred that on many close-packed transition metal sur-faces, such as Pt(111)38–43, Ru(0001)43–45, Pd(111)43,46

and Rh(111)43,47, the low-temperature structure consistsof water molecules adsorbed in the form of hexagonalrings, which gives rise to the well known honeycomb pat-tern. Furthermore, it has been shown that the diversityof the hexagon based structures is rich. For example,the layer forming on Ru(0001) is half dissociated44 andtriangular depressions are found in the layer forming onPt(111)48. The hexagonally structured water layer hasbeen named bilayer structure because it consists of twodifferently oriented water molecules, located at slightly

5

different distances from the surface; one is lying flat withthe molecular dipole plane nearly parallel to the surfaceand the other one has a dangling hydrogen bond directedeither away or toward the surface. To distinguish be-tween these two orientations we will henceforth refer tothem as Up and Down structures. We note that thesestructures have previously been successfully employed inmodeling of hydrogen oxidation/evolution and oxygen re-duction reactions.16,49

Although there is still little experimental evidence asto whether these structures will be preserved at ambientconditions, a recent molecular dynamics study showedthat a mixture of Up and Down structures is likely toexist at T = 300 K on Pt(111) and Ru(0001).27 Al-beit in the same study, the authors found that on someother close-packed transition metal surfaces (Au, Ag andPd) water molecules acquire random orientations and thehexagonal pattern becomes disrupted. These metals willalso be investigated in this work, but since our aim isto systematically study how the PZC and ASHEP varywith water model rather than to identify the most stablewater structure under a certain set of environmental con-ditions, the uncertainty about the actual structure willnot affect our conclusions.

A. Sensitivity of the PZC to water structure and

XC functional

In this section we carry out a detailed analysis of thePZC of Pt(111) covered with thin water films. The PZCis evaluated for water films of single and double bilayerthickness, for almost all conceivable combinations of Up,Down and Neutral (a mix of molecular dipoles pointingup and down) bilayers. Additionally, we look at thickerfilms, containing up to five Neutral layers. Some PZCdata for the other seven transition metals will be pre-sented in next section, Sec. VB. The results for Pt(111)are summarized in Table I. The systems are grouped ac-cording to the nominal net dipole moment (DWL = xd)of the adsorbed water film (we use the convention thatx > 0 for net dipoles pointing away from the metal sur-face, x < 0 for net dipoles pointing toward the surfaceand x = 0 for neutral films). Schematics of the films areshown in the first column of the table, with arrows in-dicating the dipole moments of individual water layers.Two arrows pointing in the same direction indicates afinite dipole moment, whereas oppositely aligned arrowsindicates a nominally neutral water layer that is expectedto have no net dipole.

According to DFT calculations with the RPBE (PBE)XC functional, the WF ΦPt(111) of pristine Pt(111) is 5.60(5.74) eV. As a comparison, the experimental value forPt(111) currently given in CRC Handbook of Chemistryand Physics is 5.93 eV.50 Upon adsorption of a waterfilm the metal WF changes by ∆Φ = Φ′

Pt(111) −ΦPt(111).

The magnitude and sign of ∆Φ depends on the struc-ture of the water film and the distance between the

metal surface and the first water bilayer. As shown byMaterzanini et al. and by Jinnouchi and Anderson, ∆Φcan, to a good approximation, be divided into two dis-tinct contributions.11,51 The first, ∆Φorient, stems fromthe orientation of the water molecules and is obtained bycalculating the dipole moment of the water film, isolatedand frozen in the geometry it adopts at the metal surface.The second contribution, the polarization part, ∆Φpolar,is caused by charge redistribution at the metal/waterinterface upon adsorption of the water film. It can bedetermined by analyzing the laterally averaged electro-static potential difference, perpendicular to the surface,between the full metal-water system on one hand and theisolated metal slab and water film on the other. As seenin Table I, the two contributions (∆Φorient + ∆Φpolar)add up to almost exactly ∆Φ.

When taking a closer look at Table I we first notice thatthe net dipole moment of the water film is of paramountimportance for the magnitude of the WF. As a rule ofthumb, a net dipole moment pointing away from the sur-face lowers Φ′

Pt(111), whereas the opposite effect is ob-

served for a net dipole moment pointing toward the sur-face. This result is what you would intuitively expectto see and is fully consistent with earlier findings.27 Wealso see that when the net dipole starts building up inone direction − that is, when the dipoles of the indi-vidual water molecules become more and more orientedin one direction − Φ′

Pt(111) will gradually approach the

saturation limit. This effect is clearly demonstrated alsoin Fig. 2b where up to three Down bilayers have beenstacked upon each other. Further inspection of Table I re-veals that the magnitude of ∆Φpolar is strongly correlatedwith the distance dPt−O between the Pt surface and theO closest to the metal surface in the first water layer, es-pecially for systems with moderate charge redistribution(|∆Φpolar| ≤ 1). Moreover, the RPBE and PBE function-als often give substantially different Φ′

Pt(111). This dif-

ference can also be traced back to variations in ∆Φpolar.PBE usually predicts a smaller metal-water separation,resulting in a larger charge redistribution. If, on the otherhand, the electronic structure is calculated for a fixednuclear geometry, the two functionals yield very similarΦ′

Pt(111).

It is worth noticing that even when performing cal-culations at the PBE level, we generally overestimatethe metal-water separation for Down (3.12 A) and Up(3.49 A) structures compared to corresponding theoret-ical values found in the literature. Other authors typ-ically report dPt−Os of approximately 2.7 A for thesestructures.11,27,42,43,48 As a result, compared to others,we underestimate the charge transfer and thus get a lessnegative ∆Φpolar and a larger Φ′

Pt(111). Most experi-

mental values reported for the PZC of Pt are of the or-der 0.4 V vs. SHE,30,52,53 which would correspond toa Φ′

Pt(111) of about 4.8 eV. Obviously, among our sys-

tems only a few of the neutral water films, treated atthe PBE level, match these experimental values. We

6

TABLE I: Work function Φ′

Pt(111) of Pt(111) in different Pt/water/vacuum setups. Upon adsorption of water on pristine

Pt(111), the work function changes with ∆Φ = Φ′

Pt(111) − ΦPt(111). ∆Φ is conveniently separated into two contributions: i)∆Φorient, given by the static dipole of the isolated water film, corresponding to an expected nominal dipole moment DWL = xd;and ii) ∆Φpolar, a polarization contribution, arising due to charge transfer between the water film and the metal surface whenthey are brought in close contact. ∆Φpolar is strongly dependent on the distance dPt−O between the Pt surface and the Oclosest to the metal surface in the first water layer.

Pt/water DWL Φ′

Pt(111) ∆Φ ∆Φorient ∆Φpolar dPt−O

model (eV) (eV) (eV) (eV) (A)

2d 4.01a/4.12b -1.59a/-1.62b -1.49a/-1.35b -0.11ad/-0.26bd 3.77a/3.49b

2d 3.58a/2.91b -2.02a/-2.83b -2.08a/-2.06b +0.08ad/-0.76bc 3.62a/2.72b

0d 5.31a/4.43b -0.29a/-1.31b -0.12a/-0.34b -0.16ad/-0.96bc 3.94a/2.33b

0d 5.52a/5.41b -0.08a/-0.33b -0.02a/-0.25b -0.04ad/-0.08bd 4.33a/4.17b

0d 5.43a/4.98b -0.17a/-0.76b -0.05a/-0.29b -0.11ad/-0.47bc 3.74a/3.18b

0d 5.55a/4.54b -0.05a/-1.20b -0.01a/-0.22b -0.05ad/-0.97bc 3.90a/2.39b

0d 5.46a/4.65b -0.14a/-1.09b -0.05a/-0.15b -0.08ad/-0.94bc 3.83a/2.44b

0d 5.55a/5.78b -0.05a/+0.04b -0.01a/+0.05b -0.06ad/-0.05bd 3.88a/4.01b

0d 5.44a/5.64b -0.16a/-0.10b -0.03a/-0.04b -0.12ad/-0.07bd 3.72a/3.97b

-2d 6.73a/5.96b +1.13a/+0.22b +1.25a/+0.98b -0.06ad/-0.77bc 4.10a/3.12b

-2d 6.54a/6.71b +0.94a/+0.97b +2.07a/+2.36b -1.12ac/-1.40bc 4.09a/4.08b

-4d 7.42a/7.63b +1.82a/+1.89b +4.24a/+4.65b -2.41ac/-2.77bc 4.06a/3.80b

aRPBE exchange-correlation functional.bPBE exchange-correlation functional.cConsiderable charge transfer between the Pt surface and the wa-

ter film.dSmall charge transfer between the Pt surface and the water film.

note, however, that a really rigorous treatment of thePZC would require statistical averaging − either overa complete set of static low-temperature water models,weighted by their total energy (stability), or of a longroom temperature molecular dynamics simulation.

To gain further insight into the difference betweenthe two XC functionals and the relationship betweenthe metal-water separation and the WF, we analyze thestacks of Neutral bilayers more carefully. Special atten-tion is paid to how the geometry and the WF change asthe thickness of the water film is increased. Fig. 3 showsthe 0 K structures of Pt(111) with water films consist-ing of two and four Neutral layers, optimized with PBEand RPBE, respectively. For the thinner film, PBE pre-dicts a much smaller Pt-water distance (dPt−O = 2.39 A)than RPBE (dPt−O = 3.90 A). The smaller separationresults in a much larger charge transfer (c.f. Table Iand Fig. 3) and hence a much larger contribution from∆Φpolar to Φ′

Pt(111). For this particular system PBE pre-

dicts a 1 eV smaller Φ′

Pt(111) than RPBE, mainly due

to the charge transfer effect. When more water layersare added little happens to the equilibrium distance inthe RPBE simulation (dPt−O = 3.88 A for the four layerthick film) and hence Φ′

Pt(111) remains fairly constant.

However, with PBE the separation suddenly increases todPt−O = 4.01 A as the fourth layer is added. Accordingly,the amount of charge transfer is now similar to that pre-

FIG. 3: (Color online) Optimized structures of two (a, b) andfour (c, d) Neutral water layers adsorbed on Pt(111). Thecalculations were performed with the PBE (a, c) and RPBE(b, d) XC functionals. The solid black lines indicate laterallyaveraged redistribution of electronic charge upon adsorptionof the water films.

dicted by RPBE (c.f. Fig. 3) and Φ′

Pt(111) = 5.78 eV also

agrees well with the Φ′

Pt(111) = 5.55 eV given by RPBE.

7

The abrupt change in dPt−O, in the case of PBE, is amanifestation of a competition between hydrogen-bond-mediated inter-layer interactions in the water film on onehand and the film’s desire to minimize its surface en-ergy at the interface on the other.54 The two competingcontributions assume optimal values at different pointsin geometry space. When the fourth layer is added theinter-layer interaction starts to dominate, resulting in achange in dPt−O. In the case of RPBE the metal-waterinteraction is much weaker and the inter-layer interac-tion will dominate already from the start. Whether therelatively large dPt−O observed for thicker films repre-sents a true physical property of the interface or if it is acomputational artifact is uncertain given current densityfunctionals’ limited ability to accurately estimate the en-ergy of hydrogen bonds. Nevertheless, in connection tothis it is worth mentioning that careful analysis of lit-erature data has indicated that the magnitude of ∆Φdetermined from water adsorption data is most likelyhigher than the corresponding ∆Φ at the electrochem-ical interface, where the thickness of the water reachesmacroscopic dimensions.30 If this observation is true, itis consistent with our finding of an increase in dPt−O andreduction in charge transfer for thicker films.

B. Trends in the PZC and sensitivity of the

ASHEP to water structure

FIG. 4: (Color online) Structures of four different water mod-els containing a single additional hydrogen atom. a) model1: Neutral-Neutral structure, b) model 2: Down structure, c)model 3: Up-Down structure and d) model 4: Down-Neutralstructure.

For high-quality calculations of the ASHEP one shoulddemand from the water film representing the electrolytei) that it exhibits no net dipole moment when uncharged,ii) that it exchanges no or very little charge with the ad-jacent metal under PZC conditions, and iii) that its wa-ter network is flexible. The first criterion simply ensuresthat the model reproduces the expected zero average netdipole of a thick finite temperature water film. Imposingthe second criterion guarantees that additional hydrogen

TABLE II: Checkboard showing to what extent each of thefour tested water models satisfy the criteria that have to befulfilled under PZC conditions in order to ensure a correctestimate of the ASHEP.

Water No net No charge Flexibility of

model dipole transfer water network

Model 1 X X X

Model 2 X X

Model 3 X X

Model 4 X

atoms added to the water film will donate electrons tothe metal, which results in a decrease of the electrodepotential. If the charge transfer would be significant al-ready at the PZC, one could in the extreme case end upin a situation where additional hydrogen atoms will notdonate any charge to the metal surface. Finally, the thirdcriterion is to assure that the film, just like bulk waterwith its high dielectric constant, is efficient at screeningelectric fields.

Out of the manifold structures used in the analysis ofthe PZC we have selected four (see Fig. 4), that fulfill avarying number of the above criteria, for further inves-tigation. Their properties are qualitatively different interms of net dipole, charge transfer and ability to screenelectric fields. We calculate the ASHEP and PZC, at theRPBE level, on the eight close-packed transition metalsurfaces using these water structures. The subsequentanalysis of the results then provides a clear indicationhow and to what degree the various criteria influencethe theoretical ASHEP. More precisely, the water mod-els chosen include the Neutral-Neutral structure (model1), a flexible water structure which, at the RPBE level,is predicted to exhibit negligible charge transfer and netdipole at the PZC; the Down structure (model 2) with nocharge transfer but finite dipole moment; the Up-Downstructure (model 3), a more rigid water structure withessentially zero charge transfer and net dipole; and, fi-nally, the Down-Neutral structure (model 4) with finitedipole and substantial charge transfer. How well the fourwater models satisfy the suggested physical criteria canbe seen in Table. II. Besides the aspiration for diversityin electronic properties, the choice of water models wasstipulated by computational cost (the water films shouldcontain as few water molecules as possible) and the pos-sibility of adding hydrogen atoms to the water structurewithout disrupting it (i.e. without rearranging the wa-ter dipoles). The latter constraint is the reason why outof the three possible mono-bilayer structures (Down, Upand Neutral) only the Down structure was selected; thedipoles pointing up in the other two models have a strongtendency to reorient toward the surface after addition ofextra hydrogen atoms.

To determine the ASHEP, we use the scheme for cou-pling the work function scale to the thermo-chemical

8

scale, outlined in Sec. IV and described in detail inour previous works.15,16 Each individual Gint-vs.-Φ

′

parabola, used to establish a link between the two scalesfor one specific system, contains a point correspondingto the PZC (cf. Fig. 6, Appendix B). Hence, for eachmetal and water model we automatically obtain the PZCas part of the ASHEP calculation.

In Fig. 5 we have plotted the absolute values of the po-tential of zero charge and standard hydrogen electrode(UPZC and USHE, obtained from the parabolas in Ap-pendix B), for the eight close-packed transition metals,against the work function Φ of the corresponding pris-tine metal surfaces. From these plots it is evident thatthe choice of water model will affect the calculated valueof both potentials. The PZC results are consistent withthe previous finding for Pt(111) that the value of UPZC isdictated by the water structure, which complicates com-parison with experimental UPZC data and limits the con-clusions that can be drawn from calculations on one spe-cific metal/water system. Yet, since we apply each watermodel to a range of metals, it is still possible to dis-cern some general trends. For instance, UPZC is found tovary linearly with Φ. Such linear relations between UPZC

and Φ have been observed before, in measurements ondifferent facets of Au and Ag,29 and was theoreticallypredicted by Bockris33 who emphasized that the linear-ity will depend upon whether the water dipoles or chargetransfer is independent of the nature of the metals. Inother words, the linear trend is expected to hold as longas the water structure of the interface is fairly constant,i.e. if the dipoles assume the same orientation on allmetals, which is exactly the case for our artificially con-structed water films. When the charge transfer is negli-gible, as in models 1 − 3, the slope will be close to 1 (cf.Fig. 5a-c). On the other hand, when the charge transferis substantial as in model 4 the slope becomes essentiallyzero. This is due to the fact that the water at the in-terface then acts as a perfect screening medium. Accord-ingly, the potential drop at the interface vanishes and theWF will be solely determined by the surface dipole at thewater-vacuum interface.

All the differences pertaining to UPZC should, in prin-ciple, be eliminated under SHE conditions, since SHEis the universal reference point and as such independentof the electrode material used in the measurements (seethe discussion in Sec. II). Therefore, ideally, the slopeof USHE(Φ) should be zero. A USHE(Φ) slope other thanzero thus reflects the imperfect screening of the waterused in the simulation. As evident from Fig. 5, model 4possesses the best and model 3 the worst screening prop-erties. This result can be rationalized in terms of theability of water layers to adjust their position and theamount of charge transferred. When water moleculesin the first layer point toward the surface they will bemore effective in screening than in the case when theyare unphysically constrained and form a rather rigid wa-ter structure as in model 3.

One can further conclude from Fig. 5 that the magni-

tude of the ASHEP and its deviation from the experimen-tal reference (4.44 V) depends mainly on the net dipoleof the water film, but also to some extent on the film’sability to facilitate charge transfer. The ASHEP is com-monly measured in bulk solution, not at the metal/waterinterface as in our approximate models. Therefore, a wa-ter film featuring no net dipole is a better representationof the experimental situation and consequently models 1and 3 exhibit ASHEP values closest to the experimentalbenchmark. The fact that the average ASHEP valuespredicted by models 1 and 3 (4.25 (4.30 without Ag) and4.68 V) are in rather good agreement with the experi-mental value may seem surprising given the significantoverestimation of PZC (c.f. Table I and the discussion inSec. VA). However, the additional protons present in thefirst water bilayer at potentials cathodic of PZC will helpreduce the metal-water separation significantly. Hence,as we start charging the surface, the separation will soondecrease to a value that would be more consistent withthe distance in the real system under PZC conditions.As the surface is further charged the separation does notchange much. Accordingly, the points further to the lefton the Gint-vs.-Φ

′ parabolae may in some sense corre-spond to more accurately described interfaces; hence thesurprisingly good estimates.

To conclude, in the beginning of this section it wassuggested that a water film employed in a calculation ofthe ASHEP should satisfy three physical criteria. Wecould see in Table. II that only model 1 satisfies all threeconditions. The following calculations and analysis havethen shown that this model also gives the best estimateof the ASHEP.

C. Internal versus external potential reference

Finally, we would like to briefly comment on the use ofexternal and internal ASHEP references. By external ref-erence we mean a universal reference, like the one we havetried to establish in this study, that is valid for all metalsand in good agreement with the experimental counter-part. By internal reference we instead mean a referencethat is valid only for one particular metal/water systemand can be obtained from a free energy parabola like oneof those found in Fig. 6 in Appendix B. For some prob-lems, such as the matching of semiconductor and solutionenergy levels mentioned in the introduction, the use of anexternal reference is a necessity. However, for tacklingother problems, notably the energetics of charge transferreactions, using an internal reference is sufficient. Em-ploying an internal reference often significantly reducesthe computational burden because it allows the use ofsimpler water models. It also ensures that errors intro-duced by the use of approximate water structures anddifferent electrode materials will be born out. The lat-ter is a consequence of error cancellations between theUSHE point and any other point on the same Gint-vs.-Φ

′

parabola. Hence for studies of charge transfer reactions

9

FIG. 5: (Color online) Dependence of the potential of zero charge (red, triangles) and the potential of the standard hydrogenelectrode (green, circles) on the pristine metal work function for a) model 1, b) model 2, c) model 3 and d) model 4. Thehorizontal dashed line indicates the experimental reference value (4.44 V) for the absolute standard hydrogen electrode potential.Notice that the point with the lowest work function (Ag) is off in models 1 and 4, because of the instability of the water filmin these systems. The dashed and solid lines are the fits with and without this point.

we do not expect that the use of the exact water model,capable of accurately screening and solvating the pro-ton, will significantly improve results obtained with sim-pler models. We have previously ascertained this pointby showing that the energetics of the hydrogen evolu-tion/oxidation reactions are preserved regardless of theinterfacial water structure.16

VI. SUMMARY AND CONCLUSIONS

In density functional theory (DFT) based models ofelectrochemical systems the work function (WF) of theelectrode, placed in an electrode/water/vacuum environ-ment, is usually used as a measure of the absolute poten-tial. Determining the WF that corresponds to the ab-solute standard hydrogen electrode potential (ASHEP)with this kind of setup constitutes a great challenge. Dueto present limitations in computer power it is not practi-cally feasible to emulate bulk water in a large-scale elec-trochemistry study. Instead, one is usually limited tomuch less sophisticated water models. The near-vacuumreference level for the electrons in such models is directlydependent on the structure of the water layer, thus affect-ing the calculated WF. This uncertainty translates intoan arbitrariness not only in the calculated ASHEP butalso in the theoretical estimate of the potential of zerocharge (PZC) of an electrode. In this paper we have madean attempt to shed some light on these issues. The sensi-tivity of the PZC to water structure has been quantified

through detailed analysis of a large set of Pt(111)/watersystems. Likewise, by systematically analyzing four dif-ferent water structures on a series of close-packed transi-tion metal surfaces, we have demonstrated how the choiceof water model affects the calculated ASHEP. However,to properly assess and compare the qualities of ASHEPsobtained with different water models one needs a well-grounded experimental benchmark. After careful exam-ination of the available literature it was concluded thatthe most reliable value, of those measured using experi-mental setups matching our DFT model, is 4.44 V.

To quantify the effect of water structure on the calcu-lated PZC, we split the change in WF of Pt(111) result-ing from adsorption of a water film into two terms − astatic dipole term, determined by the orientation of thewater molecules, and a charge polarization term, reflect-ing the amount of charge transferred between the waterand the metal. In general, both terms contributed sig-nificantly to the total change in WF and the size of thelatter was found, not surprisingly, to correlate stronglywith the distance between the metal and the first waterbilayer. Furthermore, we investigated what impact thechoice of exchange correlation (XC) functional − PBE orRPBE − has on the results. A significant difference wasobserved between PBE and RPBE for systems containingonly a few water bilayers, when the first bilayer containedmolecular dipoles oriented toward the surface. PBE con-sistently gave lower WF estimates. This corresponds toa lower PZC and implicitly influences also the ASHEP,though probably to a somewhat lesser extent. The origin

10

of the lower WFs in the PBE calculations could be tracedback to smaller metal-water separations, something thatpromotes charge transfer. However, as more water lay-ers were added, the metal-water separation suddenly in-creased in the PBE calculation to a distance similar tothat predicted by RPBE. As a consequence the differencein WF vanished at this point.

Finally, based partly on the PZC study we recognizedthree properties the model water film must possess in or-der to yield an ASHEP value that is close to the experi-mental benchmark and independent of the metal surface;under PZC conditions the film: i) should have no netdipole moment; ii) should not facilitate charge transferto the metal surface; iii) should consist of a water net-work that is flexible enough to allow efficient screening.We then demonstrated the importance of these criteriaby evaluating the ASHEP for four simple, but qualita-tively different, water structures. A water film consistingof two neutral water bilayers stacked upon each other wasidentified as the best choice. It appeared to possess allthree properties and produced the ASHEP closest to theexperimental benchmark.

Acknowledgments

CAMD is funded by the Lundbeck foundation. TheCatalysis for Sustainable Energy initiative is funded bythe Danish Ministry of Science, Technology and Innova-tion. Support from the Icelandic Research Foundation,the Danish Center for Scientific Computing, the DanishCouncil for Technology and Innovation’s FTP programand the Strategic Electrochemistry Research Center isgratefully acknowledged.

Appendix A: Computational details

The electronic structure calculations have been carriedout using density functional theory with the Perdew-Burke-Ernzerhof (PBE)55 and Revised Perdew-Burke-Ernzerhof (RPBE)56 functionals for exchange and corre-lation. Lattice constants were optimized for bulk metalsusing the RPBE fuctional and were then used in all calcu-lations, including those with the PBE functional. Metalelectrodes were represented by periodically repeated 3layer slabs, separated by at least 12 A of vacuum in thedirection perpendicular to the surface. This amount ofvacuum ensured convergence of work functions and ener-gies. Inclusion of a 4th layer had negligible influence onthe presented results. Surface unit cells of various sizes −(3×2), (3×3), (3×4), (6×3) and (6×4) − sampled with(4 × 6), (4 × 4), (4 × 3), (2 × 4) and (2 × 3) Monkhorst-Pack k-point sampling grids57 were used to account fordifferent proton concentrations (potentials). In all casessymmetry was applied to further reduce the number ofk-points. The dipole correction was used in all cases to

decouple the electrostatic interaction between the period-ically repeated slabs.58 The Kohn-Sham equations weresolved using a plane wave basis set with a plane waveand density cutoff of 26 Ry. Ionic cores were describedwith Vanderbilt ultrasoft pseudopotentials59. A Fermismearing of 0.1 eV was used and energies were extrapo-lated to an electronic temperature of 0 K. The two bot-tom layers of the slab were fixed in their bulk positions,while all other atoms were relaxed until the magnitudeof the forces acting on them were less than 0.01 eV/A.All calculations were performed using the Dacapo code60,integrated with the Atomic Simulation Environment61.

Appendix B: Integral free energy

FIG. 6: (Color online) Dependence of the integral free en-ergy, Gint, on the work function of the metal in contact withwater for the eight investigated transition metals, shown fora) model 1, b) model 2, c) model 3 and d) model 4. Themean average values and standard deviations of the ASHEPfor model 1, 2, 3 and 4 are (4.25/0.17 with Ag, 4.30/0.10without Ag), (5.08/0.11), (4.68/0.17) and (5.27/0.20 with Ag,5.34/0.06 without Ag) V respectively.

In Fig. 6 the integral free energy Gint obtained withmodels 1, 2, 3 and 4, respectively, has been plotted ver-sus the WF Φ′ calculated for the metal/water/vacuumsetup. The points Gint = 0 to the far right in the graphsare obtained for the uncharged systems, without any ad-ditional hydrogen in the water, and thus correspond tothe UPZC of the different metals. The minima of theparabolae, on the other hand, correspond to the USHEs(cf. Eq. 3). Notice that only three points have been usedto define the Au and Ag parabolae in model 4. Includingmore points would improve the accuracy but unfortu-nately points corresponding to large surface cells couldnot be obtained due to substantial reconstruction of thewater.

11

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