Grand canonical electronic density-functional theory: Algorithms and applicationsto electrochemistryRavishankar Sundararaman, William A. GoddardIII, and Tomas A. Arias

Citation: The Journal of Chemical Physics 146, 114104 (2017); doi: 10.1063/1.4978411View online: http://dx.doi.org/10.1063/1.4978411View Table of Contents: http://aip.scitation.org/toc/jcp/146/11Published by the American Institute of Physics

THE JOURNAL OF CHEMICAL PHYSICS 146, 114104 (2017)

Grand canonical electronic density-functional theory:Algorithms and applications to electrochemistry

Ravishankar Sundararaman,1,2,a) William A. Goddard III,2,3,b) and Tomas A. Arias4,c)1Department of Materials Science and Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180,USA2Joint Center for Artificial Photosynthesis, California Institute of Technology, Pasadena, California 91125, USA3Materials and Process Simulation Center, California Institute of Technology, Pasadena, California 91125, USA4Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA

(Received 16 January 2017; accepted 27 February 2017; published online 16 March 2017)

First-principles calculations combining density-functional theory and continuum solvation mod-els enable realistic theoretical modeling and design of electrochemical systems. When a reactionproceeds in such systems, the number of electrons in the portion of the system treated quantummechanically changes continuously, with a balancing charge appearing in the continuum electrolyte.A grand-canonical ensemble of electrons at a chemical potential set by the electrode potential istherefore the ideal description of such systems that directly mimics the experimental condition.We present two distinct algorithms: a self-consistent field method and a direct variational freeenergy minimization method using auxiliary Hamiltonians (GC-AuxH), to solve the Kohn-Shamequations of electronic density-functional theory directly in the grand canonical ensemble at fixedpotential. Both methods substantially improve performance compared to a sequence of conventionalfixed-number calculations targeting the desired potential, with the GC-AuxH method additionallyexhibiting reliable and smooth exponential convergence of the grand free energy. Finally, we applygrand-canonical density-functional theory to the under-potential deposition of copper on platinumfrom chloride-containing electrolytes and show that chloride desorption, not partial copper mono-layer formation, is responsible for the second voltammetric peak. Published by AIP Publishing.[http://dx.doi.org/10.1063/1.4978411]

I. INTRODUCTION

Density-functional theory (DFT) enables theoretical elu-cidation of reaction mechanisms at complex catalyst surfaces,making it now possible to design efficient heterogeneouscatalysts for various industrial applications from first princi-ples, for example, for high-temperature gas-phase transfor-mation of hydrocarbons to a variety of valuable chemicalproducts.1,2 The extension of this predictive power to electro-catalysis would be highly valuable for an even broader classof technological problems, including a cornerstone of futuretechnology for renewable energy: converting solar energy tochemical fuels by electrochemical water splitting and carbondioxide reduction.3 Accurately describing electrochemicalphenomena, however, presents two additional challenges.

First, the electrolyte, typically consisting of ions in a liquidsolvent, strongly affects the energetics of structures and reac-tions at the interface. Treating liquids directly in DFT requiresexpensive molecular dynamics to sample the thermodynamicphase space of atomic configurations. Historically, a numberof continuum solvation models that empirically capture liquideffects have enabled theoretical design of liquid-phase cat-alysts.4,5 More recently, empirical solvation models suitable

a)Electronic mail: sundar@rpi.edub)Electronic mail: wag@wag.caltech.educ)Electronic mail: taa2@cornell.edu

for solid-liquid interfaces,6–8 joint density-functional theory(JDFT) for efficiently treating liquids with an atomic-scalestructure,9 and minimally empirical solvation models derivedfrom JDFT10,11 have made great strides towards reliable yetefficient treatment of electrochemical systems.

Second, electrons can flow in and out of the electrodeas electrochemical reactions proceed. Changes in electroniccharge of electrode surfaces and adsorbates can be especiallyimportant because the electrolyte stabilizes charged config-urations with a counter charge from the ionic response. Forexample, reduction of formic acid on platinum at experimen-tally relevant potentials is dominated by formate ions ratherthan neutral molecules at the surface.12 Proton adsorptionon stepped and polycrystalline surfaces involves displacingoxidatively adsorbed water at relevant potentials, resulting innon-integer charge transfers and an anomalous pH dependencedeviating from the Nernst equation.13

Accounting for the electrolyte response using our solva-tion models,8,11 and adjusting the electron number to matchexperimentally relevant electrode potentials, realistic pre-dictions of electrochemical reaction mechanisms have nowbecome possible.14 In particular, application of this methodol-ogy to the reduction of CO on Cu(111) predicts onset potentialsfor methane and ethene formation with 0.05 V accuracy incomparison to experiment, for a wide range of pH varying from1 to 12.15 However, conventional DFT software and algorithmsare optimized for solving the quantum-mechanical problem at

0021-9606/2017/146(11)/114104/14/$30.00 146, 114104-1 Published by AIP Publishing.

114104-2 Sundararaman, Goddard III, and Arias J. Chem. Phys. 146, 114104 (2017)

a fixed electron number, requiring extra work (both manual andcomputational) to calculate properties for a specified electrodepotential.

Electric potentials and fields play an important rolein fields besides electrochemistry. Density-functional theoryapproaches accounting for electric potential have been devel-oped in special cases for field emission from metal sur-faces using a jellium model16 and for calculating capaci-tance in metal-insulator-metal17,18 and carbon nanotube sys-tems.19 Calculating non-equilibrium transport of electronsin nanoscale systems also requires accounting for potentialdifference between reservoirs in a DFT calculation.20 First-principles molecular dynamics approaches have been devel-oped to emulate fixed potential using fluctuating numbers ofelectrons between time steps.21 However, in all these cases,each involved self-consistent DFT calculation contains a fixednumber of electrons and is carried out using a conventionalcanonical-ensemble algorithm.

This paper introduces algorithms for grand canonicalDFT, where electron number adjusts automatically to tar-get a specified electron chemical potential (related to elec-trode potential), thereby enabling efficient and intuitive first-principles treatment of electrochemical phenomena. Section IIsummarizes the theoretical background of first-principles elec-trochemistry using JDFT and continuum solvation models andsets up the fundamental basis of grand-canonical DFT. Then,Section III introduces the modifications necessary to maketwo distinct classes of DFT algorithms, the self-consistentfield (GC-SCF) method and the variational free energy min-imization using auxiliary Hamiltonians (GC-AuxH), directlyconverge the grand free energy of electrons at fixed potential.Sections IV B and IV C establish the algorithm parameter(s)that optimize the iterative convergence of the GC-SCF and GC-AuxH methods, respectively, while Section IV D compares theperformance of these algorithms for a number of prototypicalelectrochemical systems. Finally, Section IV E demonstratesthe utility of grand canonical DFT by solving an electrochem-ical mystery: the identity of the second voltammetric peak inthe under-potential deposition (UPD) of copper on platinumin chloride-containing electrolytes.

II. THEORYA. Background: Electronic density functional theory

The exact Helmholtz free energy A of a system of inter-acting electrons in an external potential V (r) at a finite tem-perature T satisfies the Hohenberg-Kohn-Mermin variationaltheorem,22,23

A = minn(r)

(AT

HKM[n(r)] +∫

drV (r)n(r)

), (1)

where ATHKM is a universal functional that depends only on the

electron density n(r) (and temperature) and not on the exter-nal potential. However, constructing approximations for thisunknown universal functional that accurately capture the ener-gies and geometries of chemical bonds in terms of the densityalone is extremely challenging, partly because the quantummechanics of the electrons is completely implicit in AHKM[n]

(dropping the T labels here onward for notational simplicity;all the functionals below depend on temperature).

Most practical approximations in electronic density-functional theory follow the Kohn-Sham approach24 thatincludes the exact free energy of a non-interacting system ofelectrons with the same density n(r). The universal functionalis typically split as

AHKM[n] = Ani[n] +∫

dr∫

dr′n(r)n(r′)2|r − r′ |︸ ︷︷ ︸

EH [n]

+EXC[n], (2)

where Ani[n] is the non-interacting free energy (which wedescribe in detail below), the second “Hartree” term EH [n]is the mean-field Coulomb interaction between electrons(using atomic units e, me, ~, kB = 1 throughout), and the final“exchange-correlation” term EXC[n] captures the remainderwhich is not known and must hence be approximated. Theexact free energy of non-interacting electrons is

Ani[n] = minψi(r), fi→ n(r)

∑i

( fi2

∫dr|∇ψi(r)|2︸ ︷︷ ︸Kinetic

−T S(fi)︸︷︷︸Entropy

), (3)

which includes the kinetic energy and entropy contributions ofa set of orthonormal single-particle orbitals ψi(r) with occu-pation factors fi ∈ [0, 1]. Above, the single-particle entropyfunction is S(f ) = −f log f − (1 − f ) log(1 − f ). Note that welet the orbital index i include spin degrees of freedom as welland therefore do not introduce factors of 2 for spin degener-acy. For notational convenience, we also let i include Blochwave-vectors in the Brillouin zone (in addition to spin andband indices) for periodic systems.

The minimization in (3) is constrained such that thedensity of the non-interacting system∑

i

fi |ψi(r)|2 = n(r), (4)

the density of the real interacting system. Performing this min-imization with Lagrange multipliers VKS(r) (the Kohn-Shampotential) to enforce the density constraint and εi (Kohn-Sham eigenvalues) to enforce orbital normalization constraintsresults in a set of single-particle Schrodinger-like equations

−∇2

2ψi(r) + VKS(r)ψi(r) = εiψi(r) (5)

for stationarity with respect to ψi(r), and the Fermi occupationcondition

fi =1

1 + exp εi−µT

(6)

for stationarity with respect to f i. Here, the electron chemicalpotential µ appears as a Lagrange multiplier to enforce theelectron number constraint and is chosen so that

∑fi = N , the

number of electrons in the system.Optimizing the total free energy functional (1) with

AHKM[n] implemented by (2, 3) then yields the stationaritycondition with respect to electron density,

VKS[n](r) = V (r) +δ

δn(r)(EH [n] + EXC[n]) . (7)

Conventional density-functional theory calculations thenamount to self-consistently solving the Kohn-Sham equation

114104-3 Sundararaman, Goddard III, and Arias J. Chem. Phys. 146, 114104 (2017)

(5) along with (7), coupled via the electron density con-straint (4).

The Kohn-Sham potential is arbitrary up to an overalladditive constant: changing this constant introduces a rigidshift in the eigenvalues ε i and the electron chemical potentialµ, but does not affect the occupations f i, electron density, orfree energy. For finite systems (of any charge) and for neutralsystems that are infinitely periodic in one or two directions, thisarbitrariness can be eliminated by requiring that the potentialvanishes infinitely far away from the system, giving mean-ing to the absolute values of µ and ε i as being referenced to“zero at infinity.” However, for materials that are infinite in allthree dimensions, such as periodic crystalline solids, there isno analogous natural choice for the zero of potential, makingthe absolute reference for µ and ε i meaningless.

More importantly, for systems that are periodic in one,two, or all three directions, the net charge per unit cell mustbe zero, otherwise the energy per unit cell becomes infinite.In terms of a finite system size L and then taking the limitL → ∞, the energy per unit cell of a system with net chargeper unit cell diverges ∝ ln L for one periodic direction, ∝L fortwo periodic directions, and ∝L2 for three periodic directions.Therefore, the number of electrons per unit cell is physicallyconstrained to keep the unit cell neutral in systems with anyperiodicity, and only finite systems (like molecules and ions)have number of electrons as a degree of freedom.

B. Electrochemistry with joint density-functionaltheory

For describing electrochemical systems and electrocat-alytic mechanisms, we are typically interested in adsorbedspecies in a solid-electrolyte interface which exchange elec-trons with the solid (electrode). In these systems, the solidsurface, which we would describe in a density-functional cal-culation as a slab periodic in two directions, does have a netcharge per unit cell that depends on the electrode potential.In contrast to the discussion at the end of Sec. II A, this isphysically possible (i.e., has a finite energy) because the elec-trolyte contains mobile ions that respond by locally increasingthe concentration of ions of opposite charge near the surface,thereby neutralizing the unit cell. (The charge per unit areasof the electrode and electrolyte are equal and opposite.)

Next to an electrolyte, the charge of a partially periodicsystem (one-dimensional “wires” or two-dimensional “slabs”)is no longer constrained, allowing the number of electrons tovary. Now, the absolute reference for the electron chemicalpotential µ does become physically meaningful, and µ nowcorresponds to the electrode potential that controls the numberof electrons in the electrode.

However, treating electrochemical systems using elec-tronic density-functional theory alone is extremely challeng-ing for a variety of reasons. First, treating liquids requires astatistical average over a large number of atomic configura-tions to integrate over thermodynamic phase space. For this,techniques such as molecular dynamics typically require cal-culations of at least 104–105 atomic configurations (instead ofjust one for a solid). Second, such calculations require a largenumber of liquid molecules to minimize finite size errors in the

molecular dynamics. For example, for electrolytes with a real-istic ionic concentration of 0.1 M, there is on average one ionfor a few hundred solvent molecules. Making statistical errorsin the ion number manageable in such calculations thereforerequires >103 solvent molecules with >104 electrons, con-trasted with typical 10–100 atoms with 100–103 electrons inthe electrode slab + adsorbate of interest. Combined, thesefactors make density-functional molecular dynamics simu-lations of electrochemical systems prohibitively expensivecomputationally, in additional to being difficult to set up andanalyze.

A viable alternative to the above direct approach is toemploy joint density-functional theory9 (JDFT), a variationaltheorem akin to the Hohenberg-Kohn theorem that makes itpossible to describe the free energy of a solvated system interms of the electron density n(r) for the solute and in termsof a set of nuclear densities Nα(r) (where α indexes nuclearspecies) of the solvent (or electrolyte). Specifically, the equi-librium free energy of the combined solute and solvent systemsminimizes

A = minn(r), Nα (r)

(AJDFT[n(r), Nα(r)]

+∫

drV (r)n(r) +∑α

∫drVα(r)Nα(r)

), (8)

where V (r) is the external electron potential, Vα(r) is the exter-nal potential on the liquid nuclei, and AJDFT is a universalfunctional independent of these external potentials. Separatingout the Hohenberg-Kohn electronic density functional AHK[n]for the solute, the total free energy is

AJDFT[n, Nα] = AHKM[n]︸ ︷︷ ︸electronic

+ Adiel[n, Nα]︸ ︷︷ ︸solvation

. (9)

In practice, the functional Adiel, much like AHKM, is unknownand needs to be approximated. Importantly, the liquid isnow described directly in terms of its average density ratherthan individual configurations, and the expensive quantum-mechanical theory of the electrons is restricted to the solutealone, thereby addressing both the sampling and system-sizeproblems that make density-functional molecular dynamicsprohibitively expensive.

Typically, we are interested in a situation where theexternal potentials on the liquid are zero, and the liq-uid only interacts with itself, and with the electronsand nuclei of the solute. In this situation, we can per-form the optimization over liquid densities and define theimplicit functional AJDFT[n] = AHKM[n] + Adiel[n], where wedefine AX [n]≡ min Nα (r)AX [n(r), Nα(r)] for X = JDFT andX = diel. We will work with these reduced functionals belowfor simplicity.

The framework of joint density-functional theory encom-passes an entire hierarchy of solvation theories. Further sepa-rating the solvation term Adiel into a classical density functionalfor the liquid25–27 and an electron-liquid interaction functionalunlocks the full potential of JDFT to describe an atomic-scale structure in the liquid without statistical sampling. Start-ing from “full-JDFT” and performing perturbation theory forlinear-response of the liquid result in the non-empirical SaLSA

114104-4 Sundararaman, Goddard III, and Arias J. Chem. Phys. 146, 114104 (2017)

solvation model10 which continues to capture the atomic-scale nonlocality in the liquid response and introduces no fitparameters for the electric response of the solvent.

At the simplest end of the JDFT hierarchy, there are con-tinuum solvation models that neglect the nonlocality of theliquid response and replace it by that of an empirically deter-mined dielectric cavity (optionally with Debye screening dueto electrolytes). This includes our recent CANDLE solvationmodel11 that builds on the stability of SaLSA for highly polarsystems, and earlier solvation models suitable for moleculesand less-polar systems such as our GLSSA13 model7 (or itsequivalent, VASPsol8) and the comparable Self-ConsistentContinuum Solvation (SCCS) model.6,28 Even the traditionalquantum-chemistry finite-system solvation models such as thePCM series5 and the SMx series4 can be mapped on to this classof solvation models.

For simplicity, we will work here with this simplest classof continuum solvation models. The theoretical considerationsand algorithms in this work focus primarily on the electronicdensity-functional theory component, are largely agnostic tothe internals of the solvation model, and therefore straightfor-wardly generalize up the JDFT hierarchy to the more detailedand complex solvation theories. Essentially, all the simpleelectron-density-based continuum solvation models6–8,11 canbe summarized abstractly as7,10

Adiel[n(r)] =∫

drρel(r)(K−1 − χ)

−1− K

2ρel(r) + Acav[s].

(10)Here, the first term is the electrostatic solute-solvent inter-action energy given by the difference between the solvent-screened Coulomb interaction and the bare Coulomb inter-action K of the total solute charge density, ρel(r) = n(r)+ ρnuc(r), where ρnuc(r) is the solute nuclear charge density.The screened Coulomb interaction term above is expressed interms of the bare Coulomb interaction K and the solvent sus-ceptibility operator χ, defined for the local-response modelsby

χ · φ(r) ≡ ∇ ·

(εb − 1

4πs(r)∇φ(r)

)︸ ︷︷ ︸

Dielectric

−κ2

4πs(r)φ(r)︸ ︷︷ ︸Ionic

. (11)

Here, εb is the bulk dielectric constant of the solvent and

κ =√

4π∑

i NiZ2i /T is the inverse Debye screening length in

vacuum of a set of ionic species of charge Z i with concentra-tions N i (the net inverse Debye screening length in the presenceof the background dielectric is κ/

√εb). The cavity shape func-

tion s(r) modulates the dielectric and ion response and variesfrom zero in the solute region (no solvent present) to unityin the solvent at full bulk density. Finally, the second termof (10), Acav, empirically captures effects beyond mean-fieldelectrostatics, such as the free energy of cavity formation inthe liquid and dispersion interactions between the solute andthe solvent.29 Different solvation models at this level of thehierarchy only differ in the details of how s(r) is determinedfrom the electron density (or even from atomic positions andfit radii for the PCM5 and SMx4 solvation models) and in thedetails of Acav.

In practice, evaluating the first term of (10) requires the

calculation of φ(r) = (K−1 − χ)−1ρel(r), which is the net

electrostatic potential including screening by the solvent (andelectrolyte). The second part of the first term in (10) repre-sents the electrostatic interactions between electrons and thenuclei in vacuum which cancels corresponding terms in thevacuum DFT functional (specifically the Hartree, Ewald, andlong-range part of the local pseudopotential terms), so thatall long-range terms that contribute to the net free energyare present in the first term of (10). Finally, substituting (11)and K−1 = −∇2/(4π) into the definition of φ(r) results in thelinearized Poisson-Boltzmann (modified Helmholtz) equation,

− ∇ · (εbs(r)∇φ(r)) + κ2s(r)φ(r) = 4πρel(r). (12)

Without ionic screening (second term of (12)), the abso-lute reference for φ(r) is undetermined and does not contributeto the bound charge induced in the liquid, χφ(r) (given by thefirst term of (11) alone), because ∇(constant) = 0. However,with ionic screening, a constant shift of φ(r) does affect thesecond terms of (11) and (12), producing a charge responsein the liquid and making the absolute reference of the electro-static potential φ(r) meaningful. In particular, integrating (12)over space yields∫

drκ2

4πs(r)φ(r) =

∫ρel(r). (13)

From (11), the left hand side is the negative of the total boundcharge in the liquid, while the right hand side is the total chargeof the solute system. Therefore, the continuum electrolyteautomatically compensates for any net charge in the soluteand makes the complete system neutral. As a side effect, theabsolute reference of the electrostatic potential is meaningfuland automatically corresponds to “zero at infinity.” This hap-pens because the Green’s function of (12) is exp(−κr/

√εb)/r

in the bulk liquid with finite κ (instead of 1/r), causing φ(r) toexponentially decay to zero outside the solute region (whereρel = 0).

Reference 30 gives a detailed version of the above dis-cussion including rigorous proofs of the absoluteness of thereference for φ(r), and numerical details and algorithms forefficiently solving (12) with periodic boundary conditions inplane-wave basis DFT calculations. The result that the elec-trolyte neutralizes charges in the solute is true more generally,even for solvation models with nonlinear7 and/or nonlocalresponse.10 However, as Ref. 7 describes in detail, for the gen-eral nonlinear case, unit cell neutralization is not automaticand must be imposed using a Lagrange multiplier constraint;this Lagrange multiplier then fixes the absolute value of φ(r)such that it is zero at infinity.

The key conclusion of this discussion is that continuumelectrolytes present two advantages. First, automatic chargeneutralization implies that we can perform well-defined cal-culations with net charge per unit cell in the solute. Second,meaningful absolute reference values for potentials impliesthat the electron chemical potential µ (which determines theelectron occupations in (6)) is also referenced to zero at infin-ity. Consequently, µ is related directly to the potential U ofthe electrode providing the electron reservoir in experiments.

114104-5 Sundararaman, Goddard III, and Arias J. Chem. Phys. 146, 114104 (2017)

This potential is typically referenced to the standard hydrogenelectrode (SHE), so that

µ = µSHE − U, (14)

where µSHE is the absolute position of the standard hydrogenelectrode relative to the vacuum level (i.e., the zero at infinityreference). Calculations can employ either the experimentalestimate µSHE = −4.44 eV to relate the absolute and rela-tive potential scales31,32 or a theoretical calibration based onthe calculated and measured potentials of zero charge of sol-vated metal surfaces.30 The latter approach has the advantageof minimizing systematic errors in the solvation model sincethey cancel between the calculations used for calibration andprediction. For example, with the CANDLE solvation modelwe use below, the calibrated µSHE = −4.66 eV.11

C. Grand-canonical density-functional theory

Using joint density-functional theory of continuum solva-tion models to treat electrolytes as described above, we can cal-culate the Helmholtz free energy and electrochemical potential(µ, and hence U) of specific microscopic configurations ofadsorbates on electrode surfaces at a fixed number of electronsN in the solute subsystem (electrode + adsorbates). However,in electrochemical systems, N is an artificial constraint becauseelectrons can freely exchange between the electrode and anexternal circuit. Instead, experiments set the electrode poten-tial U, and hence the electron chemical potential µ, and Nadjusts accordingly as a dependent variable. Thermodynam-ically, this corresponds to switching the electrons from thefinite-temperature, fixed-number canonical ensemble to thefinite-temperature, fixed-potential grand-canonical ensemble.Correspondingly, the relevant free energy minimized at equi-librium is the grand free energy Φ = A − µN , instead of theHelmholtz free energy A.

The most straightforward approach to fixed-potential DFTfor electrochemistry is to repeat conventional fixed-chargeDFT calculations at various electron numbers N to reach atarget chemical potential µ. Using a steepest descent/secantmethod to optimize N results in convergence of µ typicallywithin 10 iterations.14 However this approach is inefficientsince it requires multiple DFT calculations to calculate thegrand free energy and charge of a single configuration, typi-cally taking 3 times the time of a single fixed-charge calcu-lation (not 10 times because subsequent calculations have abetter starting point and converge quicker).14

A more efficient approach would be to directly optimizethe Kohn-Sham functional in the grand-canonical ensemble.For a solvated system, this corresponds to a small modificationof the minimization problem (8) with AJDFT given by (9) andwith AHKM evaluated using the Kohn-Sham approach ((2) and(3)). The only differences are that the Lagrange multiplier term−µ(

∑i fi − N) that enforced the electron number constraint is

replaced by −µ∑

i fi which implements the Legendre trans-form of the Helmholtz free energy to the grand free energy,and that the fixed-N constraint is removed. The electron occu-pation factors are still Fermi functions (6), but µ is specifiedas an input (instead of being adjusted to match a specified N).Operationally, this amounts again to solving the Kohn-Sham

eigenvalue problem (5) self-consistently with

VKS[n](r) = V (r) +δ

δn(r)(EH [n] + EXC[n] + Adiel[n]) , (15)

where there is now a single extra contribution to the potentialdue to the solvent (electrolyte) and with the aforementionedchanges to the Lagrange multipliers and constraints.

This conceptually simple modification, however, presentsnumerical challenges to the algorithms commonly used forsolving the Kohn-Sham problem, such as the self-consistentfield (SCF) method, which solves the Kohn-Sham eigen-value problem from an input density (or Kohn-Sham poten-tial), adjusting this density (or potential) iteratively until self-consistency is achieved. A common instability for metallicsystems in the SCF method is “charge sloshing”: the elec-tron density oscillates spatially between iterations instead ofconverging. Switching to the grand-canonical ensemble cansignificantly exacerbate this problem: electrons can now addi-tionally slosh between the system and the electron reservoir.Here, we present modifications to the SCF method and an alter-nate algorithm that allow reliable and efficient convergence forgrand-canonical Kohn-Sham DFT.

III. ALGORITHMSA. Self-consistent field method: Pulay mixing

Given electron density n(i)in (r), the Kohn-Sham equation

(5) with potential VKS given by (15) defines orbitals ψj andeigenvalues εj. At a given electron chemical potential µ,these in turn define the occupations f j given by (6) and anew electron density n(i)

out(r) given by (4). The Self-Consistent

Field (SCF) method attempts to find n(i)(r) such that n(i)out(r)

= n(i)in (r).There are two algorithmic ingredients to this method:

solution of the Kohn-Sham eigenvalue equations and optimiza-tion of the electron density. The eigenvalue equations remainunchanged between conventional and fixed-potential calcu-lations. To solve these, we use the standard Davidson algo-rithm.33 The difficulty in fixed-potential calculations arises inthe electron density optimization, which we discuss below.

A robust and frequently used algorithm for charge-densityoptimization is Pulay mixing34 with Kerker precondition-ing.35 Briefly, Pulay mixing assumes that the residual R[nin(r)]≡ nout(r) − nin(r) is approximately linear in the input elec-tron density nin(r) and calculates the optimum input electrondensity as a linear combination of previous iterations,

noptin (r) =

∑i

αin(i)in (r). (16)

Minimization of the norm of the corresponding residual

F(αi) =∑

ij

αiαj

∫drR[n(i)

in (r)]MR[n(j)in (r)] (17)

with the constraint∑

i αi = 1 yields a set of linear equationsdetermining the coefficients αi, where M is the metric fordefining the norm of the residual. Finally, the next input den-sity is obtained by mixing the optimum input density with itscorresponding output density,

n(i+1)in (r) = nopt

in (r) + KR[noptin (r)], (18)

114104-6 Sundararaman, Goddard III, and Arias J. Chem. Phys. 146, 114104 (2017)

where K is the Kerker preconditioning operator.The metric M and preconditioner K serve to balance

the influence of different components of the electron densityto the optimization procedure. These are usually defined inreciprocal space, expanding n(r) =

∑G n(G)eiG ·r, where G

are reciprocal lattice vectors. In reciprocal space, the Hartreepotential in (15) takes the form VH (G) = n(G)4π/G2, causingsmall G (long-wavelength) variations of the electron densityto produce larger changes in the Kohn-Sham potential thanlarge G ones. Uncompensated, this makes the electron densityoptimization unstable against long-wavelength perturbations(the charge-sloshing instability). The Kerker preconditioner

K(G) = AG2

G2 + q2K

(19)

mitigates this problem by suppressing the contribution of theproblematic small G components in determining the next inputdensity. The metric

M(G) =G2 + q2

M

G2(20)

enhances the contribution of small G components in the resid-ual norm, thereby prioritizing the optimization of these com-ponents when determining nopt

in . The wavevectors qK and qM ,which control the balance between short and long-wavelengthcomponents, and the prefactor A, which sets the maximumfraction of nout that contributes to the next nin, can be adjustedto optimize the convergence of the SCF method. See Ref. 36for a detailed discussion of the Pulay-Kerker SCF approachand its performance for conventional fixed electron number(canonical) DFT calculations.

The G = 0 component of the electron density, whichequals N/Ω, where Ω is the unit cell volume, remains fixedin canonical DFT calculations and is therefore excluded fromthe metric and the preconditioner. This is no longer true infixed-potential DFT, where the number of electrons changes.With the above prescriptions, the Kerker preconditioner (19)→ 0 as G → 0 which will prevent the electron number fromchanging between SCF iterations. Likewise, the G→ 0 diver-gence of the Pulay metric (20) causes the residual norm tobecome undefined when the electron number changes betweeniterations.

In order to generalize the Pulay-Kerker SCF approachfor fixed-potential DFT, we therefore need to fix the G → 0behavior of both the preconditioner and the metric. The needfor the preconditioner and the metric arose from the recipro-cal space Coulomb operator 4π/G2 in the Hartree potential.In a uniform electrolyte with Debye screening, the reciprocalspace Coulomb operator instead takes the form 4π/(εbG2 + κ2)(from (12) in reciprocal space with s = 1). Since the elec-trolyte is responsible for fixing the indeterminacy of the G = 0component of the potential, a reasonable ansatz for extendingPulay-Kerker to fixed potential calculations is replacing G2

with G2 + q2κ , where

qκ =κ√εb

. (21)

In Section IV B, we show that setting

K(G) = AG2 + q2

κ

G2 + q2κ + q2

K

(22)

and

M(G) =G2 + q2

κ + q2M

G2 + q2κ

(23)

indeed makes the grand canonical self-consistent field (GC-SCF) method function efficiently, with optimum convergencefor qκ given by (21).

B. Variational minimization: Auxiliary Hamiltonianmethod

An alternate approach to solving the Kohn-Sham equa-tions is to directly minimize the total (free-)energy functional(1), with AHKM given by (2, 3), in terms of the Kohn-Shamorbitals as independent variables. For joint density-functionaltheory, this amounts to

A = minψi(r), f i

∑i

( fi2

∫dr |∇ψi(r)|2 − TS(fi)

)+ EH [n] + EXC[n] + Adiel[n] +

∫drV (r)n(r)

], (24)

where n(r) is now derived from ψi and f i as given by (4).In the above minimization, the orbitals ψi must be orthonor-mal, the occupation factors must satisfy 0 ≤ fi ≤ 1, andoptionally,

∑i fi = N for the canonical fixed electron number

case.For insulators at T = 0, the occupations f i are known

in advance. The constrained optimization over orthonormalψi is most efficiently carried out using a preconditionedconjugate-gradients (CG) algorithm on unconstrained orbitalsusing the analytically continued energy functional approach.37

Briefly, this approach evaluates the energy functional (24) ona set of orthonormal orbitals, which are a functional of theunconstrained orbitals used for minimization (see Ref. 37 forfurther details).

The general case of metallic systems and/or finite T addi-tionally requires the optimization of f i, which is challengingfor non-linear optimization algorithms because of the inequal-ity constraints 0 ≤ fi ≤ 1. One possibility is to update the fill-ings from the Kohn-Sham eigenvalues using (6) after every fewsteps of the CG algorithm,38 but this hinders the convergenceof CG because the functional effectively changes each time thefillings are altered. The “Ensemble DFT” approach39 rectifiesthis convergence issue by optimizing the occupation factors atfixed orbitals in an inner loop and performing the optimiza-tion of orbitals in an outer loop using the CG method, but thisincreases the computational cost compared to the case of theinsulators. The SCF approach is typically much more compu-tational efficient than these variants of the direct variationalminimization algorithm with variable occupations.36

An alternate strategy for direct variational minimizationwith variable occupations is to introduce an auxiliary subspaceHamiltonian matrix Haux as an independent variable40 andsetting the occupations fi = f (ηi) in terms of the eigenval-ues ηi of Haux instead of the Kohn-Sham eigenvalues εi.(Here, f (η) is the Fermi function given by (6), and the electronchemical potential µ is chosen so as to satisfy the electron num-ber constraint

∑i f (ηi) = N .) This eliminates the problematic

inequality constraints on the occupations, and upon mini-mization, the auxiliary subspace Hamiltonian H ij

aux approaches

114104-7 Sundararaman, Goddard III, and Arias J. Chem. Phys. 146, 114104 (2017)

the true subspace Hamiltonian, H ijsub = 〈ψi |HKS |ψj〉, where

HKS = −∇2i /2 + VKS(r) is the Kohn-Sham Hamiltonian. By

choosing the undetermined unitary rotations of the orbitalsψi to diagonalize Haux, Ref. 40 further shows that the gradi-ent of the free energy with respect to the independent variablessimplifies to

δAδψi(r)

= f (ηi)(HKSψi(r) −

∑j

ψj(r)H jisub

)(25)

and

∂A

∂H ijaux

= δij(Hiisub − ηi)

∂f (ηi)∂ηi

− δij∂µ

∂ηi

∑k

(Hkksub − ηk)

∂f (ηk)∂ηk

+ (1 − δij)Hijsub

f (ηi) − f (ηj)

ηi − ηj. (26)

These gradients are used to perform line minimization alonga search direction in the space of independent variables. Withevery update of the independent variables, the orbitals are re-orthonormalized and the unitary rotations of the orbitals areupdated to keep the auxiliary Hamiltonian diagonal.

In the preconditioned CG algorithm, the next search direc-tion is obtained as a linear combination of the current searchdirection and the preconditioned gradients given by

Kψi(r) = TinvδA

δψi(r)(27)

and

KH ijaux= −K(H ij

sub − ηiδij), (28)

where Tinv and K are preconditioners. The role of precondi-tioning is to balance the weight of different directions in theminimization space in the explored search directions, and ide-ally the preconditioner equals the inverse of the Hessian (whichis difficult to compute exactly). For the orbital directions, thestandard preconditioner Tinv resembles the inverse of the domi-nant kinetic energy operator in the Kohn-Sham Hamiltonian.37

For the auxiliary Hamiltonian direction, the preconditionerremoves the Fermi function derivatives and finite differencefactors from (26) in order to more equitably weight all com-ponents of Haux. The preconditioning factor K controls theoverall contribution of the Haux components relative to thatof the orbital components and is adjusted to achieve optimumconvergence.

In this auxiliary Hamiltonian (AuxH) approach, theorbitals and occupations are continuously and simultaneouslyoptimized to minimize the total free energy, resulting in bet-ter convergence and computational efficiency comparable tothe fixed-occupations insulator case, and competitive with theSCF method even for metallic systems. See Ref. 40 for fur-ther details on the algorithms and performance comparisonsfor conventional fixed electron number calculations.

Now, for fixed potential calculations, we set the occupa-tions to Fermi functions of the auxiliary Hamiltonian eigen-values at a specified µ, instead of selecting µ based on theelectron number constraint. Correspondingly, the second termof the auxiliary Hamiltonian gradient (26), which arises fromthis constraint, drops out, and the algorithm requires no furthermodification.

The convergence rate of this algorithm, however, is sensi-tive to the preconditioning factor K and we propose a modifiedheuristic to update K automatically and continuously. At theend of each line minimization, the derivative of the optimizedfree energy Amin with respect to K can be evaluated fromthe overlap between the auxiliary Hamiltonian gradient andsearch direction.40 The line-minimized energy is optimum for∂Amin/∂K = 0. Therefore, if variation of Amin with respect toK is convex, we should increase K if we find ∂Amin/∂K < 0and vice versa. However convexity is often lost if K is ini-tialized at too high a value. Therefore, our heuristic tries tozero ∂Amin/∂K while limiting the contribution of the auxiliaryHamiltonian gradient. In particular, we update

K ← K ×max

[exp

(fsat

(−∂Amin/∂K

gtot

)),

gtot

2gaux

](29)

at the end of each line minimization, where fsat(x) ≡ x/√

1 + x2

to saturate the factor by which K can change in one iteration,gaux is the overlap of the Haux components of the gradientand preconditioned gradient, and gtot is the total overlap of thegradient and preconditioned gradients (orbital + Haux). Finally,we reset the conjugate-gradient algorithm (i.e., set the searchdirection to the negative of the preconditioned gradient direc-tion) after K has increased or decreased by a factor greater thane2. We do this because dynamically changing the precondi-tioner technically invalidates the strict orthogonality of the CGsearch direction with previous directions. Section IV C showsthat this heuristic exceeds the convergence obtained with fixedK, while Section IV D shows that the grand-canonical auxiliaryHamiltonian (GC-AuxH) algorithm consistently outperformsGC-SCF for fixed-potential calculations.

IV. RESULTSA. Computational details

We implement all algorithms and perform all calculationsusing the open-source plane-wave density-functional theorysoftware, JDFTx.41 Below, we specify computational and con-vergence parameters in atomic units (distances in bohrs a0

≈ 0.529 Å and energies in hartrees Eh ≈ 27.2 eV), but presentany physically relevant properties in conventional units (Å,eV). All calculations in this work employ the PBE42 exchange-correlation functional with GBRV ultrasoft pseudopotentials43

at a kinetic energy cutoff of 20 Eh for Kohn-Sham orbitalsand 100 Eh for the charge density. The metal surface calcu-lations use inversion-symmetric slabs of at least five layers,with at least 15 Å vacuum separation and truncated Coulombpotentials44 to minimize interactions with periodic images. ForBrillouin zone integration, we use a Fermi smearing of 0.01Eh and a Monkhorst-Pack k-point mesh along the periodicdirections with the number of k-points chosen such that theeffective supercell is larger than 30 Å in each direction. Weuse the CANDLE solvation model to describe the effect of liq-uid water and Debye screening due to 1M electrolyte, whichwe showed recently to most accurately capture the solvationof highly charged negative and positive solutes.11 We empha-size that the methods and algorithms described above do not

114104-8 Sundararaman, Goddard III, and Arias J. Chem. Phys. 146, 114104 (2017)

rely on specific choices for the pseudopotential, exchange-correlation functional, k-mesh, or solvation model; we keepthese computational parameters constant here for consistency.

B. Convergence of the GC-SCF method

The self-consistent field (GC-SCF) algorithm summa-rized in Section III A depends on several parameters thatcontrol its iterative convergence. All these parameters are com-mon to the conventional fixed-charge version (SCF) and thefixed-potential variant (GC-SCF) introduced here, except forthe low-frequency cutoff wavevector qκ that is necessary toallow the net electron number to change in the fixed-potentialcase. Figure 1 compares the dependence of GC-SCF conver-gence on qκ for a prototypical calculation of an electrochemicalsystem: a Cu(111) surface treated using a five-layer inversion-symmetric slab surrounded by 1M aqueous non-adsorbingelectrolyte treated using the CANDLE solvation model,11 ata fixed potential of µ = −0.208Eh (1 V SHE). (The remain-ing GC-SCF parameters are set to their default values whichwe discuss below.) The best convergence is obtained withqκ = 0.17 which corresponds to the Debye screening length ofthe electrolyte (21). For small qκ , including the conventionalcase of qκ = 0, the number of electrons does not respondsufficiently quickly, stalling at about 0.1 electrons from theconverged value, correspondingly with the free energy stallingat about 0.1 Eh (≈2.7 eV) away from the converged value.Larger qκ causes the electron number to change too rapidly,hindering convergence and eventually leading to a divergenceas seen for the case of qκ = 0.7. An issue remains in theconvergence independent of qκ : after initial convergence, thefree energy oscillates at the 10−6 Eh level, while the electronnumber oscillates at the 103 level.

Keeping qκ at this optimum value given by (21), wenext examine the dependence of GC-SCF convergence on theremaining algorithm parameters for the same example system.Figure 2 shows the dependence on the Kerker mixing wavevec-tor qK , which helps stabilize the GC-SCF algorithm against

FIG. 1. Dependence of GC-SCF convergence on low-frequency cutoffwavevector qκ. The upper panel shows the convergence of the grand freeenergy Φ (towards its final value Φ0) on a logarithmic scale, and the lowerpanel shows that of the electron number N (towards its final value N0). Thebest convergence is obtained with qκ = 0.17a−1

0 , the inverse Debye-screeninglength, which we use as the default value henceforth. Results shown here arefor a five-layer (5ML) Cu(111) slab solvated in 1M CANDLE aqueous elec-trolyte, with potential fixed to 1 V SHE ( µ = −0.208Eh), starting from aconverged neutral calculation of the same slab in vacuum.

FIG. 2. Dependence of GC-SCF convergence on Kerker-mixing wavevectorqK . Good convergence is observed near the typical recommended value qK= 0.8 a−1

0 (≈1.5 Å1). Convergence is relatively insensitive to qK near thisvalue, but becomes unstable for small qK approaching the low-frequencycutoff qκ . System and remaining details are identical to Figure 1.

long-wavelength charge oscillations. Optimal convergence isobtained for the typical recommended value36 of 0.8 a−1

0(≈1.5 Å1). As expected, convergence is relatively insensi-tive to the exact choice of qK , as long as qK does not becomeso small that convergence is ruined by charge sloshing. Noticethat the final convergence beyond the 10−6 Eh and 103 elec-tron level remains an issue that is not resolved for any choiceof qK .

Next, Figure 3 shows the variation of GC-SCF conver-gence with the wavevector qM controlling the reciprocal-space metric used by the Pulay algorithm. The convergenceis entirely insensitive to this choice, and we henceforth setqM = qK = 0.8 a−1

0 (the recommended value36). Again, thefinal convergence issue remains unaffected by the choice ofqM .

Finally, Figure 4 compares the dependence of GC-SCFconvergence on the maximum Kerker mixing fraction A, whicheffectively controls what fraction of the new electron densityis mixed into the current value. We find nominally the bestconvergence for A = 0.5, but the performance of other valuesis not much worse. Smaller values of A lead to greater stabilityinitially, but marginally slower convergence later on, whilelarger values of A lead to greater oscillations initially, but fasterconvergence later on. Regardless, as before, good convergence

FIG. 3. Dependence of GC-SCF convergence on Pulay-metric wavevectorqM . Convergence is relatively insensitive to qM , and we set qM = qK = 0.8 a−1

0henceforth. System and remaining details are identical to Figure 1.

114104-9 Sundararaman, Goddard III, and Arias J. Chem. Phys. 146, 114104 (2017)

FIG. 4. Dependence of GC-SCF convergence on maximum Kerker-mixingfraction A. Convergence is relatively insensitive to A, except that it slowsdown for small A. We henceforth set A = 0.5 which nominally exhibits thebest convergence. System and remaining details are identical to Figure 1.

is obtained until the free energy reaches the 10−6 Eh level, butcontinues to oscillate at that level beyond that point.

This final convergence issue may not affect practical cal-culations where relevant energy differences are at the 10−3 Eh

level or higher. However, smooth exponential convergence tothe final answer is desirable as this makes it easier to determinewhen the target accuracy has been reached. Unfortunately,no combination of GC-SCF parameters achieves uniformlysmooth convergence for the fixed-potential case. On the otherhand, with our qκ modification, the GC-SCF algorithm at leastconverges to the 10−6 Eh level independent of system size(Figure 5), with the number of cycles required for conver-gence remaining mostly unchanged with increasing number ofCu(111) layers, and with increasing thickness of the solventregion.

Note that the standard fixed-charge SCF method con-verges the energy smoothly and exponentially in most cases,including in our implementation in JDFTx. Our implemen-tation of the GC-SCF method in JDFTx uses exactly thesame code, except for the preconditioner and metric mod-ifications (due to qκ) presented here. Therefore we believe

FIG. 5. Variation of GC-SCF convergence using the default parameters deter-mined above with system size: both with number of slab layers ranging from5ML to 9ML with fixed vacuum spacing 15 a0 and with vacuum spacing rang-ing from 15 a0 to 35 a0 with the 5ML slab. Convergence slows only marginallywith increasing system size, with either vacuum spacing or layer count. Eachcalculation is for solvated Cu(111) charged to 1 V SHE in CANDLE elec-trolyte, starting from the state of the corresponding converged neutral vacuumcalculation.

that the final convergence difficulty in GC-SCF is a prop-erty of the algorithm itself, rather than an implementationissue.

C. Convergence of the GC-AuxH method

The grand-canonical auxiliary Hamiltonian (GC-AuxH)approach discussed in Section III B directly minimizes the totalfree energy of the system without assuming any models forphysical properties of the system (such as dielectric responsemodels that are built into the SCF mixing schemes). This algo-rithm contains a single parameter K, which weights the relativecontributions of the Kohn-Sham orbital and subspace Hamil-tonian degrees of freedom in the conjugate-gradients searchdirection for free energy minimization.

Figure 6 shows the dependence of iterative convergenceof the GC-AuxH algorithm on this preconditioning parame-ter K for the same Cu(111) test problem considered above.If the preconditioning factor K is held fixed, the rate of con-vergence is sensitive to the choice of K, with the optimumchoice being K ≈ 0.3 for this system. With the preconditionerauto-adjusted using the heuristic given by (29), we find thatindeed the convergence picks up from that of the sub-optimalK = 1 towards that of the optimal value. More importantly,we observe smooth exponential convergence of both the freeenergy and the electron number, in contrast to our experienceswith the GC-SCF method.

Figure 7 further shows that this smooth convergence sus-tains with changing system size. In particular, the convergenceis virtually unchanged with the thickness of the solvent regions,but slows down slightly with increasing number of copperlayers in the surface slabs.

D. Comparison of algorithms

Having analyzed and optimized the convergence of theGC-SCF and GC-AuxH methods, we now compare the per-formance of these algorithms for a few different cases. In thiscomparison, we also include the present state of the art: the

FIG. 6. Dependence of the convergence of the GC-AuxH variational-minimize method on K, the preconditioning scale factor for subspace rotationsgenerated by the auxiliary Hamiltonian. Near-optimal convergence is obtainedwhen K is automatically adjusted using the heuristic given by (29), whichwe use by default henceforth. Calculations are for solvated 5ML Cu(111) at1 V SHE, starting from the corresponding neutral vacuum calculation, exactlyas in Figures 1–4. Note the smooth exponential convergence (without oscil-lations in electron number and free energy) here, in contrast to the GC-SCFcase.

114104-10 Sundararaman, Goddard III, and Arias J. Chem. Phys. 146, 114104 (2017)

FIG. 7. Variation of convergence of the GC-AuxH variational-minimizemethod with Cu(111) system size (exactly analogous to Figure 5 for the GC-SCF method). Convergence is invariant with vacuum spacing and slows downonly marginally with number of layers.

“Loop” method which uses a secant method to adjust thenumber of electrons in an outer loop to match the specifiedelectron chemical potential.14 For a fair comparison, we usethe fixed-charge SCF method in the inner loops, because itachieves the fastest convergence; this algorithm works equallywell with the AuxH method in the inner loop, but is thenmarginally slower. In each test case, we start all three algo-rithms from the same starting point: the converged state ofthe corresponding neutral (fixed-charge) calculation. Differ-ent test cases effectively perturb the potential by differentamounts, thereby testing the algorithms for a range of prox-imities between initial and final states. Also, we now com-pare the wall time between algorithms, because there is nostraightforward correspondence between GC-SCF cycles andconjugate-gradient iterations of the GC-AuxH method. Therelative wall-time performance of these fairly distinct algo-rithms will depend to an extent on details of code optimizationfor each. However, there is no perfect metric for comparingthese algorithms and wall time suffices for a rough qualitativecomparison.

FIG. 8. Performance comparison of the new GC-SCF and GC-AuxH methodswith the previous state of the art: a “Loop” over fixed-charge calculations(using the secant method to adjust the charge to match the potential). Bothnew methods reach 10−7 Eh free energy accuracy in half the wall time ofthe Loop method, but the GC-AuxH method is the clear winner with smoothexponential convergence. Calculations here are for the 5ML Cu(111) slab at1 V SHE, as before. Timings are measured on a single 32-core NERSC Corinode in all cases.

First, Figure 8 compares the performance of the algo-rithms for the 5-layer Cu(111) slab used in all the tests sofar. The spikes in the free energy seen in the Loop method arethe points where the electron number changes in the outer loopand a new SCF convergence at fixed charge begins. Both theGC-SCF and GC-AuxH methods are quite competitive, cuttingthe time to convergence within 10−6 Eh in half compared to theLoop method. Given the smooth convergence beyond 10−6 Eh

however, the GC-AuxH method is preferable over GC-SCFfor fixed potential calculations. Note that in the fixed-chargecase, when SCF converges smoothly, it often outperforms theAuxH method as mentioned above. The advantage of the AuxHand GC-AuxH methods is their stability on account of beingvariational methods: the free energy is guaranteed to decreaseat every step. Consequently, the convergence difficulties ofthe GC-SCF method make the variationally stable GC-AuxHmethod relatively more attractive.

Next, we compare these algorithms for more complextext cases. Figure 9 compares the convergence for a five-layerPt(111) slab fixed to a potential of µ = −0.171 Eh (0 V SHE).At this potential, the surface of Pt(111) charges negatively,and the CANDLE solvation model brings the cavity closer tothe electrons to capture the more effective solvation of negativecharges by liquid water. Additionally, the dielectric response ofplatinum is more complex than copper due to the partially filledd shell. Both of these factors make this system harder to con-verge than the previous test case, and therefore the convergenceof the GC-AuxH method is no longer clearly a single expo-nential. Despite this, both the direct grand-canonical methodsconverge faster than the Loop method, with the GC-AuxHmethod eking out an advantage in final convergence as before.

Finally, Figure 10 compares the convergence for chlorideanions adsorbed at one-third monolayer coverage, in a

√3×√

3supercell of a five-layer Pt(111) slab. Despite the increasedcomplexity, the direct grand-canonical methods exhibit thebest convergence, edging out the Loop method by a factorof four in wall time now, again with the smoothest con-vergence for the GC-AuxH method. Due to the systematic

FIG. 9. Similar to Figure 8, but for a 5ML Pt(111) slab at 0 V SHE(µ = −0.171 Eh). Platinum is neutral at ≈0.8 V SHE, so this corresponds tonegatively charging the slab, which activates CANDLE’s asymmetry correc-tion. Additionally, platinum has d bands crossing the Fermi level in contrast tocopper which has occupied d bands. This calculation therefore explores a morecomplicated charge vs. potential landscape, causing deviations from expo-nential convergence, but the relative performance of the algorithms remainssimilar.

114104-11 Sundararaman, Goddard III, and Arias J. Chem. Phys. 146, 114104 (2017)

FIG. 10. Similar to Figures 8 and 9, but for a 5ML Pt(111) slab decoratedwith a

√3×√

3 partial monolayer of adsorbed chloride anions (1/3 coverage)at 0 V SHE. The increased complexity slows down the convergence of theGC-SCF method, which also slows down the Loop over fixed-charge SCFcalculations, while the GC-AuxH method continues to exhibit rapid near-exponential convergence with no charge oscillations, beating the Loop methodby a factor of 4 in time. All subsequent calculations of complex metal surfaceswith adsorbates therefore use the GC-AuxH method.

convergence advantage of the GC-AuxH method, we use itfor all remaining calculations in this work and recommendit as the default general purpose algorithm for convergingelectrochemical calculations.

In Sec. II and all calculations so far, we used Fermi smear-ing where the electron occupations are given by the Fermifunction (6). Practical k-point meshes typically require the useof a temperature T substantially higher than room temperature(we used 0.01 Eh which is approximately ten times higher),which could result in inaccurate free energies. Such errors canbe reduced substantially by changing the functional form of theoccupations and the electronic entropy, with the caveat that thesmearing width T no longer corresponds to an electron tem-perature. Common modifications include Gaussian smearing,where the Fermi functions are replaced with error functions,and Cold smearing,45 where the functional form is chosen tocancel the lowest order variation of the free energy with T (seeRef. 45 for details).

FIG. 11. Comparison of the convergence of the GC-AuxH method for the5ML Pt(111) slab at 0 V SHE using different smearing functions. Insetsshow the variation of converged properties with smearing width. Cold smear-ing45 substantially reduces the finite-width error in the free energy, and toa lesser extent in the electron number, compared to Gaussian and Fermismearing. At the same width of 0.01 Eh, the GC-AuxH method with coldsmearing converges marginally slower than Gaussian or Fermi smearing, butstill faster than when using any of the smearing methods with a smaller width of0.001 Eh.

Figure 11 compares the performance of the preferred GC-AuxH method for various smearing methods. The insets showthe variation of the converged free energy and electron num-ber with smearing width T. Gaussian smearing reduces thecoefficient of the quadratic T dependence compared to Fermismearing, while Cold smearing cancels the quadratic depen-dence altogether, by design. The variation of electron numberwith smearing width is also reduced by Cold smearing, but to alesser extent. Notice that the use of Cold smearing marginallyslows down the iterative convergence of the GC-AuxH method.However, using Cold smearing at a high width of 0.01 Eh is stillfaster than using any smearing method with the lower width0.001 Eh that is close to room temperature (because of the fardenser Brillouin zone sampling required for smaller widths).Therefore, it is still advantageous to use Cold smearing at ele-vated smearing widths, and so we use Cold smearing with awidth of 0.01 Eh for the final demonstration below.

E. Under-potential deposition of Cu on Pt(111)

The application of sufficiently negative (reductive) poten-tials on an electrode immersed in a solution containing metalions reduces those ions and results in bulk electro-depositionof metal on the surface. Additionally, for many pairs of met-als, a single monolayer of one metal deposits on a surfaceof the other at an under potential, that is, at a potential lessfavorable than for bulk deposition. This phenomenon of under-potential deposition (UPD) has several technological applica-tions since it enables precise synthesis of heterogeneous metalinterfaces. It also serves as an archetype for fundamental stud-ies of electrochemical processes (see Ref. 46 for an extensivereview), which makes it a perfect example for demonstratingour grand-canonical density-functional theory method.

The basic reason for underpotential deposition is that theheterogeneous binding between the two metals is stronger thanthe homogeneous binding of the depositing metal to itself.Indeed, metal pairs that exhibit underpotential deposition alsodisplay analogous phenomena in vapor adsorption.47 How-ever, the process in solution is far more complicated and highlysensitive to the composition of the solution because of com-peting adsorbates,48 as well as to the structure of the electrodesurface.49

The UPD of copper on Pt(111) in the presence of chlorideanions is particularly interesting and the subject of consider-able debate in the literature. Voltammetry for this system50

exhibits two well-separated under-potential peaks, as shownin the background of Figure 12. Certain LEED and in situX-ray scattering studies of this system51 find evidence of a2 × 2 bilayer of copper and chloride ions co-adsorbed on thesurface at potentials between the two peaks, suggesting thatone peak corresponds to the formation of a partial layer, andthe second peak to the formation of the full monolayer. In con-trast, other studies50,52,53 do not find this signature and proposethat the additional peak arises from adsorption and desorptionof chloride ions alone.

To address this debate, we perform grand-canonical den-sity functional theory calculations of various configurationsof copper and chlorine adsorbed on a 5-layer Pt(111) slab in√

3 ×√

3 and 2 × 2 supercells. We determine the most stableconfigurations at each potential (µ) and the potentials at which

114104-12 Sundararaman, Goddard III, and Arias J. Chem. Phys. 146, 114104 (2017)

FIG. 12. Underpotential deposition of Cu on Pt(111) from an aqueous solu-tion containing 103 mol/l Cu2+ ions and 104 mol/l Cl ions. Calculated freeenergies for various adsorbate configurations as a function of electrode poten-tial are shown as calculated by explicit fixed-potential solvated calculationsin (a), and from vacuum calculations in (b), with the experimental voltam-mogram50 shown for comparison. Solid red lines indicate no copper, purpledotted lines indicate 2 × 2 partial copper monolayer, and blue dotted-dashedlines indicate full copper monolayer. Thickness of the lines indicates Cl cov-erage, ranging from no Cl (thinnest), 2 × 2 (1/4) coverage (intermediate) to√

3 ×√

3 (1/3) coverage (thickest). Prominent configurations are sketched in(c)-(f). Vacuum calculations in (b) predict only a single voltammetric peakin disagreement with experiment. Explicit potential-dependent solvated cal-culations in (a) predict two peaks in qualitative agreement with experiment(∼0.1 V accuracy), with the second peak due to chloride desorption.50,52,53

The partial Cu monolayer (d) proposed by some51 is not predicted to be themost stable configuration at any relevant potential.

transitions between configurations occur by comparing theirgrand free energies. The relevant grand free energy of a config-uration α containing Nα

Pt platinum atoms in the slab with NαCu

copper and NαCl chlorine atoms adsorbed at the surface within

the calculation cell is

Φα(µ) =

Φα(µ) − µPtNαPt − µCuNα

Cu − µClNαCl

N surfPt

, (30)

normalized by the number of surface platinum atoms N surfPt in

order to correctly compare energies of calculations in differentsupercells. (N surf

Pt = 2 for the unit cell, 6 for the√

3 ×√

3supercell, and 8 for the 2 × 2 supercell, accounting for thetop and bottom surfaces in the inversion-symmetric setup.)Since no light atoms are present, we safely neglect changes invibrational contributions to the free energy between adsorbateconfigurations.

Above, Φα(µ) is the free energy of adsorbate configu-ration α calculated by fixed-potential DFT, which is grandcanonical with respect to the electrons at chemical potentialµ (related to the electrode potential by (14) as discussed atthe end of Section II B). Then, (30) calculates the free energyΦα(µ) which is additionally grand canonical with respect to allrelevant atoms with chemical potentials µPt, µCu and µCl. Sev-eral conventions are possible in defining the electron-grand-canonical free energy Φ, depending on what electron numberwe subtract: change from neutral value, total electron number,

or number of valence electrons in pseudopotential DFT calcu-lations. The atom chemical potentials would then respectivelycorrespond to neutral atoms, bare nuclei, or pseudo-nuclei(nuclei + core electrons in pseudopotential). The full grandcanonical free energy Φ does not depend on this choice. In ourJDFTx implementation, we choose the last option above (num-ber of valence electrons and correspondingly atom chemicalpotentials of the pseudo-nuclei).

The bulk of the platinum electrode sets the Pt chemicalpotential, µPt = EPt(s)− µNe

Pt(s), where EPt(s) is the DFT energyof a bulk fcc Pt calculation with a single atom in the unit cell,and Ne

Pt(s) is the number of valence electrons in that calcula-tion. The second term here implements the electron countingconvention discussed above. Next, copper ions in solution setµCu, but directly calculating the free energy of such ions usingsolvation models is error-prone.11 So, instead, we use the DFTcalculated energy, ECu(s), of a bulk fcc Cu calculation (contain-ing Ne

Cu(s) valence electrons) and relate it to the free energy ofthe ion via the experimentally determined standard reductionpotential UCu2+→Cu(s) = 0.342 V SHE.54 This yields

µCu =(ECu(s) − µNe

Cu(s)

)+ 2

(µ − µSHE + eUCu2+→Cu(s)

)+ kBT ln[Cu2+], (31)

where the second term accounts for the change from Cu(s)to Cu2+ ions, and the final term accounts for change in ionicconcentration from the standard value of 1 mol/l to the currentvalue of [Cu2+] (in mol/l). Similarly, chlorine ions in solu-tion set µCl, but to minimize DFT errors, we connect to theDFT calculated energy, ECl(at), of an isolated chlorine atom(containing Ne

Cl(at) valence electrons), via the experimentally

determined atomization energy ECl2→2Cl(at) = 242.6 kJ/mol,55

gas-phase entropy SCl2(g) = 223.1 J/mol K,54 and reductionpotential UCl2(g)→2Cl− = 1.358 V SHE.54 Specifically,

µCl =(ECl(at) − µNe

Cl(at)

)−

12

(ECl2→2Cl(at) + TSCl2(g)

)−

(µ − µSHE + eUCl2(g)→2Cl−

)+ kBT ln[Cl−], (32)

where the second term accounts for the change from atomicto gas-phase chlorine, the third term for the change to chlo-ride ions, and the final term for the change in chloride ionconcentration to [Cl] (in mol/l).

Figure 12(a) shows the calculated grand free energies asa function of electrode potential for a number of Cu and Cladsorbate configurations on the surface of Pt(111). At highpotentials, the most stable (lowest free energy) configura-tion is 1/4 Cl coverage (Figure 12(f)), which transitions toa clean Pt surface (Figure 12(e)) at a potential of 0.55 VSHE. Upon further lowering the potential, the stable config-uration transitions to a full monolayer of copper with 1/3Cl coverage (Figure 12(c)) at a potential of 0.46 V SHE.Experimentally, the two voltammogram peaks are at approxi-mately (0.63± 0.04) and (0.51± 0.02) V SHE, averaging overthe forward and reverse direction sweeps. Therefore chlorinedesorption and full-copper-monolayer formation are plausibleexplanations50,52,53 of the two peaks, with our first-principlespredictions reproducing well the peak spacing (0.09 eV ver-sus 0.12 eV in experiment), and placing the absolute loca-tions of the peaks to within 0.07 eV. Similar accuracy has

114104-13 Sundararaman, Goddard III, and Arias J. Chem. Phys. 146, 114104 (2017)

been achieved in comparison to experiment for onset poten-tials and product selectivity in CO reduction on Cu(111)15,56

and the oxygen evolution reaction on IrO2(110)57 in con-current work using exactly the same calculation protocol ashere: fixed-potential DFT in JDFTx using the PBE exchange-correlation functional and the CANDLE solvation model. Thepartial 2 × 2 monolayer of copper (Figure 12(d)) proposedby others51 as the reason for the second peak is not themost stable configuration in our calculations at any potential,lying a significant 0.3 eV above the other phases at relevantpotentials.

For comparison, Figure 12(b) shows the analogous resultsthat would be obtained using only conventional vacuum calcu-lations. In the above formalism, this corresponds to assumingΦα(µ) ≈ Aα − µNα

e , where Aα is the Helmholtz energy from aneutral vacuum DFT calculation of configuration α containingNα

e valence electrons. This approximation results in a singletransition directly from the Cl-covered Pt surface to the onewith a copper monolayer, predicting a single voltammogrampeak in disagreement with experiment. Accurate predictionsfor electrochemical systems therefore require treating chargedconfigurations stabilized by the electrolyte at relevant elec-tron potentials, now easily accomplished with the methodsand algorithms introduced in this work.

V. CONCLUSIONS

This work introduces algorithms for directly convergingDFT calculations in the grand-canonical ensemble of elec-trons, where the number of electrons adjusts to maintain thesystem at constant electron chemical potential, while ionicresponse in a continuum solvation model of electrolyte keepsthe system neutral. We show that, with appropriate modifi-cations, grand canonical versions of both the self-consistentfield (GC-SCF) method and direct free-energy minimizationwith auxiliary Hamiltonians (GC-AuxH) method are able torapidly converge the grand free energy of electrons. This sub-stantially improves upon the current state of the art of runningan outer loop over conventional fixed-charge DFT calcula-tions. With detailed tests of the convergence of all these algo-rithms, we show that the GC-AuxH method is the most suitabledefault choice exhibiting smooth exponential convergence tothe minimum.

Grand-canonical DFT directly mimics the experimentalcondition in electrochemical systems, where electrode poten-tial sets the chemical potential of electrons, and the number ofelectrons at the electrode surface (including adsorbates in theelectrochemical interface) changes continuously in response.Describing this change in charge at the surface plays animportant role in accurately modeling several electrochemicalphenomena.12–15 Here, we showcase the new algorithms byanalyzing the under-potential deposition (UPD) of copper onplatinum in an electrolyte containing chloride ions. We resolvean old debate about the identity of a second under-potentialpeak, showing that partial copper monolayers are not plausibleand that the second peak is due to desorption of chloride ions.We expect the new methods presented here to substantiallyadvance the realistic treatment of electrochemical phenomenain first principles calculations.

ACKNOWLEDGMENTS

R.S. and W.A.G. acknowledge support from the JointCenter for Artificial Photosynthesis (JCAP), a DOE EnergyInnovation Hub, supported through the Office of Scienceof the U.S. Department of Energy under Award No. DE-SC0004993. R.S. and T.A.A. acknowledge support from theEnergy Materials Center at Cornell (EMC2), an Energy Fron-tier Research Center funded by the U.S. Department of Energy,Office of Science, Office of Basic Energy Sciences underAward No. DE-SC0001086. Calculations in this work usedthe National Energy Research Scientific Computing Center(NERSC), a DOE Office of Science User Facility supportedby the Office of Science of the U.S. Department of Energyunder Contract No. DE-AC02-05CH11231. We thank KendraLetchworth-Weaver, Kathleen Schwarz, Yuan Ping, Hai Xiao,Tao Cheng, Robert J. Nielsen, and Jason Goodpaster forinsightful discussions.

1M. J. Cheng and W. A. Goddard III, J. Am. Chem. Soc. 137, 13224 (2015).2M. J. Cheng, W. A. Goddard III, and R. Fu, Top. Catal. 57, 1171 (2014).3M. G. Walter, E. L. Warren, J. R. McKone, S. W. Boettcher, Q. Mi, E.A. Santori, and N. S. Lewis, Chem. Rev. 110, 6446 (2010).

4A. V. Marenich, C. J. Cramer, and D. G. Truhlar, J. Phys. Chem. B 113,6378 (2009).

5J. Tomasi, B. Mennucci, and R. Cammi, Chem. Rev. 105, 2999 (2005).6O. Andreussi, I. Dabo, and N. Marzari, J. Chem. Phys. 136, 064102 (2012).7D. Gunceler, K. Letchworth-Weaver, R. Sundararaman, K. A. Schwarz, andT. Arias, Modell. Simul. Mater. Sci. Eng. 21, 074005 (2013).

8K. Matthew, R. Sundararaman, K. Letchworth-Weaver, T. A. Arias, andR. G. Hennig, J. Chem. Phys. 140, 084106 (2014).

9S. A. Petrosyan, J.-F. Briere, D. Roundy, and T. A. Arias, Phys. Rev. B 75,205105 (2007).

10R. Sundararaman, K. Schwarz, K. Letchworth-Weaver, and T. A. Arias, J.Chem. Phys. 142, 054102 (2015).

11R. Sundararaman and W. A. Goddard III, J. Chem. Phys. 142, 064107 (2015).12K. A. Schwarz, R. Sundararaman, T. P. Moffat, and T. Allison, Phys. Chem.

Chem. Phys. 17, 20805 (2015).13K. Schwarz, B. Xu, Y. Yan, and R. Sundararaman, Phys. Chem. Chem. Phys.

18, 16216 (2016).14J. D. Goodpaster, A. T. Bell, and M. Head-Gordon, J. Phys. Chem. Lett. 7,

1471 (2016).15H. Xiao, T. Cheng, W. A. Goddard III, and R. Sundararaman, J. Am. Chem.

Soc. 138, 483 (2016).16Y. Gohda, Y. Nakamura, K. Watanabe, and S. Watanabe, Phys. Rev. Lett.

85, 1750 (2000).17M. Stengel and N. A. Spaldin, Phys. Rev. B 75, 205121 (2007).18S. Kasamatsu, S. Watanabe, and S. Han, Phys. Rev. B 84, 085120 (2011).19K. Uchida, S. Okada, K. Shiraishi, and A. Oshiyama, Phys. Rev. B 76,

155436 (2007).20M. Brandbyge, J.-L. Mozos, P. Ordejon, J. Taylor, and K. Stokbro, Phys.

Rev. B 65, 165401 (2002).21N. Bonnet, T. Morishita, O. Sugino, and M. Otani, Phys. Rev. Lett. 109,

266101 (2012).22P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).23N. D. Mermin, Phys. Rev. 137, A1441 (1965).24W. Kohn and L. Sham, Phys. Rev. 140, A1133 (1965).25R. Sundararaman, K. Letchworth-Weaver, and T. A. Arias, J. Chem. Phys.

137, 044107 (2012).26R. Sundararaman and T. Arias, Comput. Phys. Commun. 185, 818 (2014).27R. Sundararaman, K. Letchworth-Weaver, and T. A. Arias, J. Chem. Phys.

140, 144504 (2014).28C. Dupont, O. Andreussi, and N. Marzari, J. Chem. Phys. 139, 214110

(2013).29R. Sundararaman, D. Gunceler, and T. A. Arias, J. Chem. Phys. 141, 134105

(2014).30K. Letchworth-Weaver and T. A. Arias, Phys. Rev. B 86, 075140 (2012).31S. Trasatti, Pure Appl. Chem. 58, 955 (1986).32J. Cheng and M. Sprik, Phys. Chem. Chem. Phys. 14, 11245 (2012).33E. R. Davidson, J. Comput. Phys. 17, 87 (1975).

114104-14 Sundararaman, Goddard III, and Arias J. Chem. Phys. 146, 114104 (2017)

34P. Pulay, Mol. Phys. 17, 197 (1969).35G. P. Kerker, Phys. Rev. B 23, 3082 (1981).36G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169 (1996).37T. A. Arias, M. C. Payne, and J. D. Joannopoulos, Phys. Rev. Lett. 69, 1077

(1992).38S. Ismail-Beigi and T. A. Arias, Comput. Phys. Commun. 128, 1 (2000).39N. Marzari, D. Vanderbilt, and M. C. Payne, Phys. Rev. Lett. 79, 1337

(1997).40C. Freysoldt, S. Boeck, and J. Neugebauer, Phys. Rev. B 79, 241103(R)

(2009).41R. Sundararaman, D. Gunceler, K. Letchworth-Weaver, K. A. Schwarz, and

T. A. Arias, JDFTx, 2012, http://jdftx.sourceforge.net.42J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).43K. F. Garrity, J. W. Bennett, K. Rabe, and D. Vanderbilt, Comput. Mater.

Sci. 81, 446 (2014).44R. Sundararaman and T. Arias, Phys. Rev. B 87, 165122 (2013).45N. Marzari, D. Vanderbilt, A. D. Vita, and M. C. Payne, Phys. Rev. Lett. 82,

3296 (1999).46E. Herrero, L. J. Buller, and H. D. Abruna, Chem. Rev. 101, 1897 (2001).47M. T. Paffett, C. T. Campbell, T. N. Taylor, and S. Srinivasan, Surf. Sci. 154,

284 (1985).

48J. H. White and H. D. Abruna, J. Electroanal. Chem. Interfacial Electrochem.300, 521 (1991).

49H. D. Abruna, J. M. Feliu, J. D. Brock, L. J. Buller, E. Herrero,J. Li, R. Gomez, and A. Finnefrock, Electrochim. Acta 43, 2899(1998).

50H. Bludau, K. Wu, M. Zei, M. Eiswirth, H. Over, and G. Ertl, Surf. Sci. 402,786 (1998).

51N. Markovic, H. A. Gasteiger, C. A. Lucas, I. M. Tidswell, and P. N. Ross,Surf. Sci. 335, 91 (1995).

52Y. Soldo, E. Sibert, G. Tourillon, J. L. Hazemann, J. P. Levy, D. Aberdam,R. Faure, and R. Durand, Electrochim. Acta 47, 3081 (2002).

53M. S. Zei, K. Wu, M. Eiswirth, and G. Ertl, Electrochim. Acta 45, 809(1999).

54CRC Handbook of Physics and Chemistry, 93rd ed., edited by W. M. Haynes(Taylor and Francis, 2012).

55See http://cccbdb.nist.gov/ for NIST Computational Chemistry Comparisonand Benchmark Database.

56H. Xiao, T. Cheng, and W. A. Goddard III, J. Am. Chem. Soc. 139, 130(2017).

57Y. Ping, R. J. Nielsen, and W. A. Goddard III, J. Am. Chem. Soc. 139, 149(2017).