Grand canonical rate theory for electrochemical and ...

doi.org/10.26434/chemrxiv.8068193.v5

Grand canonical rate theory for electrochemical and electrocatalyticsystems I: General formulation and proton-coupled electron transferreactionsMarko Melander

Submitted date: 10/06/2020 • Posted date: 12/06/2020Licence: CC BY-NC-ND 4.0Citation information: Melander, Marko (2019): Grand canonical rate theory for electrochemical andelectrocatalytic systems I: General formulation and proton-coupled electron transfer reactions. ChemRxiv.Preprint. https://doi.org/10.26434/chemrxiv.8068193.v5

A generally valid rate theory at fixed potentials is developed to treat electrochemical and electrocatalyticpotential-dependent electron, proton, and proton-coupled electron reactions. Both classical and quantumreactions in adiabatic and non-adiabatic limits are treated. The applicability and new information obtainedfrom the theory is demonstrated for the gold catalyzed acidic Volmer reaction.

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Appendix to Grand canonical rate theory for

electrochemical and electrocatalytic systems I:

General formulation and proton-coupled electron

transfer reactions

Marko M. Melander

Nanoscience Center, P.O. Box 35 (YN) FI-40014, Department of Chemistry,

University of Jyvaskyla, Finland

E-mail: [email protected]

10 June 2020

Abstract. In this appendix the rate constants for non-adiabatic reactions within the

grand canonical rate theory are presented.

1. Non-adiabatic ET and PCET reaction rates within GCE

As shown in the main paper to this appendix, computation of fixed potential rates for of

electronically adiabatic reactions does not yield any fundamental difficulties as compared

to the canonical case; after finding the barrier, one can simply use a simple TST-like

expression to compute the reaction rate using grand free energies. Besides the reaction

barrier, rates depend on the prefactor which is crucial for modelling ET, PT, and PCET

reactions which often feature (non-)adiabatic tunneling. Given the importance of ET,

PT, and PCET in both applications and fundamental studies, the prefactors are treated

here for consistency of the general framework while the computational applications will

be presented separately[1].

Including the prefactor beyond TST presents some difficulties and care is needed.

In particular, the treatment of non-adiabatic processes is difficult; electronic transition

matrix elements are not defined for states with different number of electrons when

only particle conserving operators are used. In other words, the transition needs to be

particle conserving. The non-adiabatic flux-side correlation function utilizes a projection

operator which is explicitly depends on the particle number.[2] Hence, developing a

transition probability which is independent of the particle number is not straight-

forward and therefore one cannot directly use the effective GCE-EVB states to compute

the electronically non-adiabatic rate. Instead, the electronic transition matrix element

needs to be computed separately for each canonical transition which preserve particle

Grand canonical rate theory 2

number upon transition. Afterwards, a summation over the canonical rates is performed

to express the non-adiabatic ET/PCET rate as a GCE expectation value.

To obtain the non-adiabatic TST rate, the Golden-rule approach is used herein.

In the canonical ensemble, the Golden-rule formulations are well established.[3, 4, 5, 6]

Below the theory of non-adiabatic ET and PCET rates within GCE is developed. It

is stressed that the non-adiabatic approach is inherently quantum mechanical and ET,

PT, and PCET reactions describe quantum mechanical tunneling processes.

1.1. Non-adiabatic ET rate

To start with, localized electronic states |iN〉 are specified as eigenstates corresponding

to the electronic Hamiltonian HelN . Electronic states are defined for initial (i) and final

(f) states with a fixed number of particles (N). Then the electronic energies for the

initial and final states at fixed particle number at nuclear geometry Q are

〈iN |HelN |iN〉 = εiN(Q) and 〈fN |Hel

N |fN〉 = εfN(Q) (1)

Within the Born-Oppenheimer approximation (BOA), the nuclear wave functions

and their energies ε in the initial (|mN〉) and final (|nN〉) electronic states are obtained

from

[TQ + εiN(Q)] |mN〉 = εmN |mN〉 and

[TQ + εfN(Q)] |nN〉 = εnN |nN〉(2)

where TQ is the nuclear kinetic energy. Within BOA, the total vibronic wave

function and the corresponding energy factorize as

|imN〉 = |iN〉 |mN〉 and EimN = εiN + εmN (3a)

|fnN〉 = |fN〉 |nN〉 and EfnN = εfN + εnN (3b)

As the different energy contributions are additive, the canonical partition functions

can be factorized

QNi = exp[−βεiN ]

∑m

exp[−βεmN ] and QNf = exp[−βεfN ]

∑n

exp[−βεnN ] (4)

At this point all relevant canonical quantities have been defined and the focus turns

to the GCE formulation of the Golden-rule rate. The GCE partition function for the

initial state is

Ξi =∑N

exp[βµN ]QNi (5)

This equation is inserted in the general GCE rate expression. For the non-adiabatic

limit, the Golden rule rate expression is used. The Golden rule expression is consistent


with the general rate theory based on the flux approach if a non-adiabatic Hamiltonian

and suitable flux operator are utilized. The GCE-NATST rate constant is then

kGCE−NATST =2π

hΞi

∑N

e−β(εiN−µN)∑m,n

e−βεmN

∣∣∣ 〈Nnf |VN |imN〉∣∣∣2δ(EimN − EfnN)

=2π

h

∑N

∑m,n

pimN

∣∣∣ 〈Nnf |VN |imN〉∣∣∣2δ(EimN − EfnN)(6)

where pimN is the population of the vibronic state |imN〉. Next, a significant

simplification is made; it is assumed that the vibrational part of the canonical partition

function does not depend on the number of electrons in the system. This assumption

directly implies that the reorganization energy is potential independent which should

be a reasonable assumption for electronically non-adiabatic reactions. As a result

QNi = exp[−βεiN ]

∑m exp[−βεmN ] ≈ exp[−βεiN ]

∑m exp[−βεm] = exp[−βεiN ]Qm and

the GCE partition function becomes

Ξi ≈ Qm

∑N

exp[−β(εiN − µN)] = QmΞi (7)

Inserting this approximation in the GCE-NATST rate expression gives

kGCE−NATST ≈2π

hΞi

∑N

e−β(εiN−µN)∑m,n

e−βεmN

Qm


=2π

h

∑N

piN∑m,n

pmN

∣∣∣ 〈Nnf |V |imN〉∣∣∣2δ(EimN − EfnN)

(8)

where piN,el = exp[−β(εiN − µN)]/Ξi,el and pmN = exp[−βεmN)]/Qm.

This equation has the structure of the canonical Golden rule rate weighted by the

GCE probability of being in the initial electronic state iN . To simplify the notation, one

can momentarily concentrate only on the canonical part of the above rate expression.

As shown in Section 2.2, using the Fourier transform presentation of the delta function,

gives

kGCE−NATST ≈∑N

V 2N,fi

2h2 piN

∫dtC(t) (9)

where C(t) is an energy autocorrelation function (see Section 2.2). The

autocorrelation function maybe extracted from time-dependent quantum or classical

dynamics. However, to obtain a closed form for the rate equation, herein the

autocorrelation function is expressed using a cumulant expansion[7]. Using the second

order cumulant expansion, assuming that all solvent degrees of freedom are classical

and taking the short time approximation[8] to the correlation function results in (see

Section 2.2):


kGCE−NATST ≈∑N

piNV 2N,if

h√

4πkBTλexp

[−

(∆ENfi + λ)2

4kBTλ

](10)

The reorganization and reaction energies are defined as λ = Eim(QF ) − Ein(QI)

and ENfi = EN

fn(QF )− ENim(QI). While the above development rests on the assumption

of constant reorganization energy, the solvent structure depends on the potential and

charge state of the electrode and reorganization energies for reactions near the electrode

may not be constant. To account for this, the reorganization energy can be further

separated to inner and outer sphere components as discussed in Section 2.4. If this

separation is invoked, one can alleviate the assumption that the total reorganization

is independent of the particle number and instead assume that only bulk solvent

(outer sphere) reorganization is a constant while the inner-sphere reorganization energy

depends on the particle number i.e. is potential-dependent.

1.2. PCET kinetics within GCE

The PCET kinetics is based on the PCET rate theory of Soudackov and Hammes-

Schiffer. Within the canonical ensemble the relevant rate expressions were derived in

Refs. [9, 10, 11, 12] and here this treatment is extended to the GCE yielding PCET

rate constants at fixed electrode potentials. The PCET rate constant derivation follows

a similar procedure as the one used above for the ET rates. In the case of PCET, an

additional geometric variable q for the transferring proton is introduced. Within BOA,

the total vibronic wave function is then

|iumN〉 = |iN(q,Q)〉 |uN(Q)〉 |mN〉 (11)

where it is explicitly written that the electronic wave function |iN〉 depends

explicitly on the proton q and system coordinate Q while the proton wave function

|uN(Q)〉 depends on the system coordinate Q. The wave functions and corresponding

energies are solved using equations similar to the ET case

〈iN |HelN |iN〉 = εiN(q,Q) and

〈fN |HelN |fN〉 = εfN(q,Q)

(12a)

[Tq + εiN(q,Q)] |iuN〉 = εiuN |iuN〉 and

[Tq + εfN(q,Q)] |fvN〉 = εivN |fvN〉(12b)

[TQ + εiuN ] |mN〉 = EmN |mN〉 and

[TQ + εfvN ] |nN〉 = EnN |nN〉(12c)

where Tq and TQ are the kinetic energy operators for the proton and other nuclei,

respectively. Within BOA, the total energy of the at fixed N is written as a simple sum

of the three contributions:


EiumN = εiN + εiuN + EmN (13)

and similarly for the final diabatic state. Furthermore, coupling constant is given

as

〈Nnvf |V (R)N |iumN〉 ≈〈Nvf |V (R)N |iuN〉q 〈Nn|mN〉Q = V (R)NuvS

Nnm

(14)

The SHS treatment of PCET rates is valid for reactions ranging from vibronically

non-adiabatic to vibronically adiabatic scenarios[13] and rate expressions for various

well-defined limits have been achieved. The SHS PCET rate theories are derived

following a path analogous to the derivation of ET rates and extension to the GCE

is rather straightforward. As done by SHS, the Golden rule formulation is used. The

details of this derivation are presented in the Section 2.1 . The simplest GCE-PCET

rate is given for the short time approximation of the energy gap correlation is valid in

the high-temperature limit and static proton donor-acceptor R distance as

k =∑N,u

piu∑v

∣∣V (R)Nuv∣∣2

h√

4πkBTλuvexp

[−(∆EN

uv + λuv)2

4kBTλuv

](15)

where the reaction energy between vibrational states iuN and fvN is ENuv =

EfvnN(qF , QF ) − EiumN(qI , QI). The state-dependent reorganization energy λuv =

Eium(qF , QF ) − Eivn(qI , QI) is assumed independent of the particle number. If some

vibrational modes (besides the R mode) are sensitive to changes in the particle number,

they can be separated from the total reorganization energy by decomposing the total

reorganization energy to inner- and outer-sphere components as shown in Section

2.5. Depending on the form of the prefactor, both electronically and vibronically

adiabatic and non-adiabatic limits of PCET can be reached within the semiclassical

treatment[14, 15, 16] of the prefactor.

1.3. Hybrid GCE-NA-EVB model

Bridging the the NA and adiabatic GCE-EVB models developed above and in the main

paper, vibronic or electronic diabatic states along the reorganization are considered as

shown in Figure 1. As in the NA model, the hybrid model takes the reorganization

of the unreactive nuclei Q as the reorganization and reaction coordinate. However,

unlike in the NA model, the hybrid GCE-NA-EVB model treats the reorganization

coordinate at a fixed electrode potential, as done in the GCE-EVB model. The effective

barrier is then ∝ exp[−β(Λ + ∆Ω)2/4Λ]. At the crossing point of the initial and

final states along Q coordinate the vibronic/electronic diabatic states are brought in

to resonance so that nuclear/electron tunneling can take place. The contribution of

tunneling between different canonical diabatic states depends on the applied potential.

The GCE prefactor can be approximated as a GCE expectation value of canonical


Figure 1. Schematic picture for electrochemical PCET. The pink (green) line depicts

an initial (final) vibronic diabatic state as a function of the environment reaction

coordinate. The insets show two types of proton curves along the proton transfer

coordinate at the initial, transition and final solvent coordinates. The top (bottom)

inset shows the electronically non-adiabatic (adiabatic) proton curves. The dashed

solid (dashed) electronically non-adiabatic curve corresponds to electron localized at

the initial (final) state. In each inset the black and orange curves correspond to two

different electrode potentials or states with different number of electrons.

prefactors ∝ V 2µ,if ≈

∑N piNV

2N,if =

⟨V 2N,if

⟩µ. Combining the the barrier and prefactor

leads to the NA-GCE-EVB rate constant

kNA−GCE−EV B ≈

⟨V 2N,if

⟩µ√

4kBTΛexp

[−β (Λ + ∆Ω)2

4Λ

](16)

Unlike the NA-TST rates in the previous section, the hybrid model makes of the

GCE free energies. Also, unlike the GCE-EVB model, the hybrid includes a well-defined

way to compute the prefactor as detailed in Section 3 of the SI. Using the Landau-Zener

approach[17], the hybrid model offers a transparent way to interpolate between adiabatic

and non-adiabatic reactions in a single framework. This hybrid model also provides a

tempting way to include and estimate non-adiabatic and tunneling effects in large scale

DFT studies and material screening with kinetics.


1.4. Analysis of the non-adiabatic GCE rates

In this section the computation of prefactors was considered to go beyond the

TST limit. In particular (non-adiabatic) tunneling in ET and PCET was treated.

The main difficulty in the GCE non-adiabatic rate theory is the treatment of the

electronic/vibronic coupling constant; this term is defined only for particle conserving

transitions. This precludes the straightforward use of initial and final GCE diabatic

states which have different number of electrons at the same geometry. The NA-GCE-

EVB hybrid offers a well-defined way to approximate the non-adiabatic effects using

GCE diabatic states but the prefactors still need to be computed using canonical states

and GCE averaging. Only at the thermodynamic limit when the particle number

fluctuation approaches zero can the GCE diabatic states be used for computing the

coupling constant. However, at this limit the GCE-NATST is equal to the canonical

NATST as only a single particle number state is populated i.e. pi becomes a delta

function at some particle number. At the thermodynamic limit either using fixed

potential GCE states or fixed particle number canonical states will give equivalent

results, as they should.

Even at the thermodynamic limit the present treatments differ from the traditional

Dogonadze-Kutzetnotsov-Levich[5], Schmikler-Newns-Anderson[18, 19, 20], and SHS

approaches. The crucial difference is that the present formulation does not rely on the

separation of the total interacting wave function to non-interacting or weakly interacting

fragments. Also, in the present approach, the applied electrode potential does not only

affect the electrode potential is self-consistently treated to affect all electrode, reagent,

and solvent species. This way the inherent complexity of the electrochemical interface

is naturally included in the Hamiltonian and the wave functions in a self-consistent

manner. For instance, the work terms entering Marcus[21] or other electrochemical

rate theories[22, 23] do not need to be computed in the present formalism. Another

crucial difference is that the charge transfer kinetics are not decomposed into single

electron orbital contributions. Instead, the work herein formulates the kinetics in terms

of many-body diabatic wave functions. In the canonical ensemble, such an approach has

been shown[24] to provide accurate barriers, prefactors, and overall kinetics for electron

transfer reaction in battery materials.

For small systems where particle number fluctuations are pronounced the

summation over particle numbers need to be performed. While straightforward in

principle, the amount of calculations can seem daunting at first. However, as the

populations depend exponentially on the energy and target chemical potential, piN ∼exp[−β(EiN − µN)], only a limited number of states will contribute to the summation.

It is expected that the infinite summation can be safely reduced to summation over a

small number (5–10) of different charge states covering the electrode potential range of

interest. Again, at the thermodynamic limit only a single calculation per potential is

needed.


2. Supporting information for the appendix

2.1. PCET kinetics within GCE

Continuing the PCET scheme set up in the paper section 1.2., the PCET rate constant

is written using the Golden Rule formulation. This gives

kGCE−PCET =2π

hΞi

∑N,u,v,m,n

e−β(EiumN−µN)∣∣∣ 〈Nnvf |VN |iumN〉∣∣∣2δ(EiumN − EfvnN)

=2π

h

∑N

∑u,v

∑m,n

piumN

∣∣∣ 〈Nnvf |VN |iumN〉∣∣∣2δ(EiumN − EfvnN)(17)

The obtained form is analogous to the GCE-ET theory developed herein and shares

the structure of the canonical PCET rate of SHS. As assumed for ET part, it is

expected that the vibrational part of the system does not depend on the number of

particles. However, no such assumption is made for the transferring proton i.e. the

proton potential depends on the charge state. This is written as

Ξi =∑N,u,m

e−β(EiumN−µN) ≈ Qm

∑N,u

e−β(εiN+εiuN−µN) = QmΞiu (18)

At this point it is important to stress that the vibronic coupling depends sensitively

on the proton donor-acceptor distance R which is included in the rate expression. It is

assumed that the coupling can be decomposed as

〈Nnvf |V (R)N |iumN〉 ≈ 〈Nvf |V (R)N |iuN〉q 〈Nn|mN〉Q = V (R)NuvSNnm (19)

Inserting these two approximations result in PCET rate constant of the form

kGCE−PCET ≈2π

h

∑N,u,v

e−β(εiN+εiuN−µN)

Ξiu

∑m,n

e−βEmN

Qm

∣∣V (R)Nuv∣∣2∣∣SNmn∣∣2δ(EiumN − EfvnN)

=2π

h

∑N,u,v

piuN∑m,n

pm∣∣V (R)Nuv

∣∣2∣∣SNmn∣∣2δ(EiumN − EfvnN)

(20)

This form is amenable to the direct treatment as performed by SHS. Depending

on the treatment of the R coordinate, several appropriate limits may be considered

each yielding a different canonical rate constant. The derivations for the R-dependent

PCET rates follow a similar (but more complex [11]) cumulant expansion as performed

above for ET. Hence, the GCE-PCET rate can be obtained by extending the approach

presented above for the ET. The extension of PCET in GCE is straight-forward and here

I present only the most simple result valid under the same conditions as the Marcus-

like expression derived above for ET. Specifically, one assumes that[25] i)the short time


approximation of the energy gap correlation is valid, ii) high-temperature limit is taken,

and iii) that the R coordinate gives Eq. 15.

2.2. Fourier transform and cumulant expansion for ET reactions

Here it shown how the non-adiabatic rate constants are obtained from cumulant

expansion to the energy autocorrelation function. First, the autocorrelation function is

related to the delta function as

∑m,n

pimN


=1

2πh

∑m,n

pimN

∣∣∣ 〈Nnf |VN |imN〉∣∣∣2 ∫ dteit(EimN−EfnN )/h

=1

2πh

∑m,n

pimN 〈fmN |VN |inN〉〈inN |VN |fmN〉∫dteit(EimN−EfnN )/h

≈ 1

2πh

∑m,n

pimN

∣∣∣ 〈fN |VN |iN〉∣∣∣2 ∫ dt 〈mN |nN〉〈nN |mN〉 eit(EimN−EfnN )/h

=1

2πh

∑m,n

pimNV2N,if

∫dt∣∣∣〈nN |mN〉q∣∣∣2eit(EimN−EfnN )/h

=V 2N,if

2πh

∫dt⟨eit(EimN/he−it(EfnN )/h

⟩q

=V 2N,if

2πh

∫dtC(t)

(21)

where C(t) is an energy autocorrelation function. The last two equations are

amenable to two different ways of computing the rate constant. The last can be used

with a cumulant expansion approach, while the second last has the form of a thermally

averaged Franck-Condon treatment is presented in Section 2.3 for completeness.

While nuclear quantum effects maybe important and can be included in the

computation of C(t)[26], in the present work, nuclear degrees of freedom are treated

classically. Following either Geva[26] or Marcus[27], the autocorrelation function can be

expressed using a cumulant expansion[7]. Use of the second order cumulant expansion

results in

〈exp[iEfnN t/h] exp[iEimN t/h]〉i ≈

exp

[−ith

⟨∆EN

fi

⟩− 1

h2

∫ t

0

dτ1

∫ τ1

0

dτ2C(τ1 − τ2)

](22)

where⟨∆EN

fi

⟩is the average free energy gap between the final and initial electronic

diabatic states. Also C(τ1−τ2) =⟨δ∆EN

fi(τ)δ∆Efi(0)⟩

where δ∆ENfi = ∆EN

fi−⟨∆EN

fi

⟩.

C(τ1−τ2) is directly linked to the vibrational spectral density of the system[27, 28, 11, 6].

To obtain a manageable expression for the rate, the short time approximation or slow


fluctuation limit[8] to the correlation function is used: C(τ1− τ2) ≈ C(0) =⟨δ(∆EN

fi)2⟩.

Inserting this in Eq. (22) yields

exp

[− 1

h2

∫ t

0

dτ1

∫ τ1

0

dτ2C(τ1 − τ2)

]≈ exp

[− t

2

h2 (⟨δ(∆EN

fi)2⟩]

(23)

This is inserted in Eq. (21) to give

∑m,n

pimN


≈V 2N,if

2πh

∫ ∞−∞

dt exp

[it

h

⟨∆EN

fi

⟩− t2

h2 (⟨δ(∆EN

fi))2⟩]

=V 2N,if

2πh

√2πh2⟨

δ(∆ENfi)

2⟩ exp

[−⟨∆EN

fi

⟩2

2⟨δ(∆EN

fi)2⟩]

≈V 2N,if

2π

√π

kBTλexp

[−

(∆ENfi + λ)2

4kBTλ

](24)

where on the last line it has been assumed that the free energy surfaces are quadratic

along the energy gap coordinate. The reorganization and reaction energies are defined as

λ = Eim(QF )−Ein(QI) and ENfi = EN

fn(QF )−ENim(QI). A generalization to asymmetric

GCE-diabatic energy curves can be made following Mattiat and Richardson[29].

Furthermore, it is assumed that the curvature of the quadratic surfaces is the same

for all particle numbers N in which case the reorganization energy does not depend on

N . This should be to a rather good approximation as the reorganization is related to

the reorientation of the surrounding medium which is expected be rather insensitive to

the number of electrons in the system. For example, in the spin-boson model, which in

the canonical ensemble yields the Marcus rate, the reorganization energy is only related

to the bath frequencies in thermal equilibrium.[6] If the spin-boson model is applied

to the present GCE case, the vibrational, bosonic Hamiltonian would be assumed to

be independent of the number of electrons and yield directly the reorganization energy

which is indepenedent of the number of particle for the GCE. The assumption that

the reorganization energy is independent on the particle number can also be reinforced

by doing a re-derivation of the rate using the thermalized Franck-Condon approach as

shown in Section 2.3.

2.3. Franck-Condon derivation of the non-adiabatic rate

The Franck-Condon treatment starts from the second-last line of Eq. (21) by noticing

that


1

2πh

∑m,n

pimNV2N,if

∫dt|〈nN |mN〉|2eit(EimN−EfnN )/h =

V 2N,if

2πFC(∆E)i

(25)

where FC(∆E)i is the thermalized Franck-Condon factor. In general case, the

thermalized Franck-Condon factor can be computed by Fourier transforming it and

using generating functions.[30] As shown in Ref. [8] chapter 6,the FC-factor can be

written using the spectral density function Jfi(ω) to give

FC(∆E)i =1

2πhexp[G(0)]

∫ ∞−∞

dt exp[it∆EN

fi/h+G(t)]

≈∫ ∞−∞

dt

2πhexp

[it

∆ENfi − λh

]exp

[− λt

2

βh2

]

=

√1

4πkBTλexp

[−

(∆ENfi + λ)2

4kBTλ

]

where G(t) =

∫ ∞0

dω cos(ωt)(1 + 2n(ω))JIF (ω)− sin(ωt)JIF (ω)

≈∫ ∞

0

dω(ωt)2

βhωJIF (ω)− i

∫ ∞0

dωtωJIF (ω)

(26)

using the high-temperature approximation (1 + 2n(ω) ≈ 2kBT >> 1), slow-

fluctuating Debye solvent assumptions, and∫∞

0dωωJIF (ω) = λ/2. Hence, if the spectral

density not sensitive to the number of electrons, the reorganization energy is independent

on the number of electrons in the systems. For practical purposes this is expected to be

a good approximation. When the approximate FC factor is introduced, Eq. (25) gives

the Marcus rate in the GCE.

2.4. Decomposition of the reorganization energy to inner- and outer-sphere

contributions

The total reorganization energy is often[9, 31, 21, 32] modified differentiate between

inner- and outer-sphere contributions. This is achieved by partitioning the surrounding

molecules to tightly bound ligands or inner-solvent solvent molecules and the bulk

solvent. While this is not necessary in the approach taken in this work, separating the

effect the nearby atoms or molecules and the solvent might be useful for a understanding

the role of different constituents on the overall reaction. In both computational and

theoretical studies this separation occurs naturally if the bulk solvent is presented as a

continuum as in the work of Dogonadze et.al.[5, 4] for ET and SHS[9] for PCET.

To single out the solvent reorganization energy, a solvent polarization coordinate

Q is introduced. As detailed in Ref. [9] this coordinate introduces a new parametric


dependence to the electron, proton, and vibrational Hamiltonians, wave functions and

energies. Here it is shown how an additional solvent coordinate modifies the ET reactions

and the PCET kinetics can be treated analogously.

First, a solvent coordinate Q is introduced. The solvent coordinate is

orthogonal to other coordinates which allows writing the wave function as |imaN〉 =

|iN(q,Q)〉 |mN(Q)〉 |aN〉 where |aN〉 is the wave function related to solvent

polarization. All other quantities obtain a parametric dependence on Q. The initial

state solvent wave functions are eigenfunctions obtained from

[TQ + εmN ] |aN〉 = EaN |aN〉 (27)

and similarly for the final state. Above, TQ is the kinetic energy operator for the

outer-sphere species. Then the total energy is given by

EimaN = εiN + εimN + EaN (28)

and the total coupling between the initial and final states is

VimaN,fnbN = 〈fmbN |VN |imaN〉≈ 〈fN |VN |iN〉〈nN |mN〉q 〈bN |aN〉Q= Vif,NSnm,NSab,N

(29)

Assuming that the outer-sphere free energy related to the solvent reorganization

is independent of the particle number allows separating its contribution from the total

grand partition function

Ξi =∑m,a,N

exp[−β(EimaN − µN)]

≈ Qa

∑m,N

exp[−β(EimN − µN)] = QaΞim

(30)

Note that inner-sphere energies and partition function explicitly depend on the

particle number. Inserting the last two equations in the golden rule expression yields

k =2π

hΞi

∑Nabmn

e−β(εiN−µN+βEiaN+εmN )

∣∣∣ 〈Nnvf |VN |iumN〉∣∣∣2δ(EimaN − EfnbN)

≈ 2π

h

∑N

∑m,n

pimN∑a,b

paNV2if,NS

2nm,NS2

ab,Nδ(EimaN − EfnbN)(31)

where pimN = exp[−β(εiN + εmN − µN)]/Ξim and paN = exp[−βEaN/Qa]. As

done above, representing the delta function as a Fourier transform allows writing


k =∑N

V 2if,N

h2

∫dt⟨eit(εmN/he−it(εnN )/h

⟩q×⟨eit(EaN/he−it(EbN )/h

⟩Q

=∑N

V 2if,N

h2

∫dtGmn,N(t)gab,N(t)

(32)

where auxiliary correlation functions Gmn,N(t) and gab,N(t) are introduced providing

a connection to the work of SHS[9, 11]. To be specific, Gmn,N(t) characterizes the inner-

sphere contributions while gab,N(t) is related to the outer-sphere solvent polarization.

Different approximations for the correlation functions presented by SHS in Ref. [9, 11]

can be readily used here as well to derive various well-defined limits of the rate equation.

For example, assuming that the intra-molecular modes can be neglected leads to Eq.(21)

with a/b replacing the m/n indices. Within this assumption and repeating the steps

leading to Eq. 10 shows that resulting reorganization energy is the solvent reorganization

energy and the inner-sphere interactions contribute only to the reaction energy.

If the intra-sphere contributions cannot be neglected, the rate equations become

rather cumbersome in general. However, the case Gab,N(t) ≈ Gab(t) i.e. that the

outer-sphere contribution to rate is independent of the particle number, deserves some

attention. For this, the inner- and outer-sphere components are separated by rewriting

Eq.(31) using a convolution[32]

k =∑N

piN2πV 2

if,N

h

∫dEf(x)F (∆EN

fi − x) (33)

with f(x) =∑

mn pmNS2nm,Nδ(ε

imN − εinN + E) and

F (ENfi − x) =

∑ab paNS2

ab,Nδ(EaN − EbN + ∆ENfi − x) as shown for single N in

Ref.[32]. f(x) and F (ENfi − x) represent inner- and outer-sphere contributions to the

activation energy. Again various forms for both terms can be derived[32]. To retain

consistency, a high-temperature approximation for quadratic solvent modes is used.

This gives[32, 9, 11]

F (ENfi − x) =

1

h√

4πkBTλNoexp

[−

(∆ENfi + λNo )2

4kBTλNo

](34a)

f(x) = FC(∆E − x)i (34b)

where FC(∆E − x)i is a modified Franck-Condon factor given in (26) and λNo is

recognized as the outer-sphere reorganization energy. Making the high-temperature and

slow-fluctuating Debye solvent approximations as done in Eq (26) allows performing the

convolution integral. This yields [32]

k =∑N

piN2πV 2

if,N

h

1

h√

4πkBT (λNo + λNi )exp

[−

(∆ENfi + λNo + λNi )2

4kBT (λNo + λNi )

](35)


Finally the assumption that the outer-sphere contributions do not depend on the

particle number can be applied to give

k =∑N

piN2πV 2

if,N

h

1

h√

4πkBT (λo + λNi )exp

[−

(∆ENfi + λo + λNi )2

4kBT (λo + λNi )

](36)

From this form it can be seen that the total reorganization energy can be separated

to a particle number independent solvent contribution λo and a reorganization energy

of the inner sphere component λNi which depends explicitly on the particle number.

3. Declaration of interest

Declarations of interest: none

4. References

[1] Melander M M 2020 Grand canonical rate theory ii: Addressing non-adiabaticity and tunneling

in gold-catalyzed Volmer reaction from density functional theory in preparation

[2] Richardson J O and Thoss M 2014 The Journal of Chemical Physics 141 074106

[3] Hammes-Schiffer S and Stuchebrukhov A A 2010 Chemical Reviews 110 6939–6960

[4] Dogonadze R and Kuznetsov A 1975 Progress in Surface Science 6 1 – 41

[5] Dogonadze R 1971 3. theory of molecular electrode kinetics Reactions of Molecules at Electrodes

ed Hush N (Wiley-Intersciences) pp 135–228

[6] Nitzan A 2006 Chemical Dynamics in Condendsed Phases: Relaxation, Transfer, and Reactions

in Condensed Molecular Systems (Oxford University Press)

[7] Kubo R 1962 Journal of the Physical Society of Japan 17 1100–1120

[8] May V and Kuhn O 2011 Charge and Energy Transfer Dynamics in Molecular Systems vol 3rd

(WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

[9] Soudackov A and Hammes-Schiffer S 2000 The Journal of Chemical Physics 113 2385–2396

[10] Soudackov A and Hammes-Schiffer S 1999 The Journal of Chemical Physics 111 4672–4687

[11] Soudackov A, Hatcher E and Hammes-Schiffer S 2005 The Journal of Chemical Physics 122 014505

[12] Venkataraman C, Soudackov A V and Hammes-Schiffer S 2008 The Journal of Physical Chemistry

C 112 12386–12397

[13] Hammes-Schiffer S 2012 Energy Environ. Sci. 5(7) 7696–7703

[14] Goldsmith Z K, Lam Y C, Soudackov A V and Hammes-Schiffer S 2019 Journal of the American

Chemical Society 141 1084–1090

[15] Georgievskii Y and Stuchebrukhov A A 2000 The Journal of Chemical Physics 113 10438–10450

[16] Skone J H, Soudackov A V and Hammes-Schiffer S 2006 Journal of the American Chemical Society

128 16655–16663

[17] Newton M D 1991 Chemical Reviews 91 767–792 (Preprint

https://doi.org/10.1021/cr00005a007) URL https://doi.org/10.1021/cr00005a007

[18] Schmickler W 1986 Journal of Electroanalytical Chemistry and Interfacial Electrochemistry 204

31 – 43 ISSN 0022-0728

[19] Schmickler W 2017 Russian Journal of Electrochemistry 53 1182–1188

[20] Santos E, Lundin A, Potting K, Quaino P and Schmickler W 2009 Phys. Rev. B 79(23) 235436

[21] Marcus R A 1965 The Journal of Chemical Physics 43 679–701

[22] Lam Y C, Soudackov A V, Goldsmith Z K and Hammes-Schiffer S 2019 The Journal of Physical

Chemistry C 123 12335–12345


[23] Nazmutdinov R R, Bronshtein M D and Santos E 2019 The Journal of Physical Chemistry C 123

12346–12354

[24] Park H, Kumar N, Melander M, Vegge T, Garcia Lastra J M and Siegel D J 2018 Chemistry of

Materials 30 915–928

[25] Hammes-Schiffer S, Hatcher E, Ishikita H, Skone J H and Soudackov A V 2008 Coordination

Chemistry Reviews 252 384 – 394 the Role of Manganese in Photosystem II

[26] Sun X and Geva E 2016 The Journal of Physical Chemistry A 120 2976–2990

[27] Georgievskii Y, Hsu C P and Marcus R A 1999 The Journal of Chemical Physics 110 5307–5317

[28] Hynes J T 1989 Chemical Physics Letters 162 19 – 26

[29] Mattiat J and Richardson J O 2018 The Journal of Chemical Physics 148 102311

[30] Englman R and Jortner J 1970 Molecular Physics 18 145–164

[31] JOM Bockris S K 1979 Quantum Electrochemistry (Plenum Press)

[32] Kestner N R, Logan J and Jortner J 1974 The Journal of Physical Chemistry 78 2148–2166

download fileview on ChemRxivappendix.pdf (649.78 KiB)



Grand canonical rate theory for electrochemical andelectrocatalytic systems I: General formulation andproton-coupled electron transfer reactions

Marko M. Melander

Nanoscience Center, P.O. Box 35 (YN) FI-40014, Department of Chemistry,University of Jyvaskyla, Finland


10 June 2020

Abstract. Electrochemical interfaces present a serious challenge for atomisticmodelling. Electrochemical thermodynamics are naturally addressed within thegrand canonical ensemble (GCE) but the lack of fixed potential rate theoryimpedes fundamental understanding and computation of electrochemical rateconstants. Herein, a generally valid electrochemical rate theory is developed byextending equilibrium canonical rate theory to the GCE. The extension providesa rigorous framework for addressing classical reactions, nuclear tunneling andother quantum effects, non-adiabaticity etc. from a single unified theoreticalframework. The rate expressions can be parametrized directly with self-consistentGCE-DFT methods. These features enable a well-defined first principles routeto address reaction barriers and prefactors (proton-coupled) electron transferreactions at fixed potentials. Specific rate equations are derived for adiabaticclassical transition state theory and adiabatic GCE empirical valence bond(GCE-EVB) theory resulting in a Marcus-like expression within GCE. FromGCE-EVB general free energy relations for electrochemical systems are derived.The GCE-EVB theory is demonstrated by predicting the PCET rates andtransition state geometries for the adiabatic Au-catalyzed acidic Volmer reactionusing (constrained) GCE-DFT. The work herein provides the theoretical basisand practical computational approaches to electrochemical rates with numerousapplications in physical and computational electrochemistry.

Keywords: electrochemical kinetics, grand canonical, free energy relations, Volmerreaction, constrained DFTSubmitted to: J. El. Chem. Soc.


1. Introduction

Electrochemical reactions and especially electrocatal-ysis are at the forefront of current green technologiesto mitigate climate change. To realize and utilize thefull potential of electrocatalysis, selective and activecatalysts are needed for various applications and re-actions including e.g. oxygen and hydrogen reduc-tion/evolution reactions, nitrogen reduction to ammo-nia and CO2 reduction.[1] These and other electrocat-alytic/electrochemical reactions are based on succes-sive proton-coupled electron transfer (PCET), electrontransfer (ET), and proton transfer (PT) reactions; theunique aspect of electrochemistry is the ability to di-rectly control PCET, ET, and PT kinetics and ther-modynamics by the electrode potential.[2]

Besides the catalyst material, electrocatalytic per-formance is controlled by the electrolyte compositionand electrode potential. To translate these to mi-croscopic, computationally treatable quantities, it isthe combination of the electrolyte and electron elec-trochemical potentials which determine and controlthe (thermodynamic) state of electrochemical systems.Therefore, an atomic-level computational model needsto provide an explicit control and description of thesechemical potentials as depicted in Figure 1. In ther-modynamics fixing the chemical potentials is achievedthrough a Legendre transformation from a canonicalensemble to a grand-canonical ensemble (GCE) forboth electrons and nuclei.[3] This calls for theoreti-cal and computational methods to treat systems wherethe particle numbers are allowed to fluctuate and thechemical potentials are fixed.

The theoretical basis for fixed potential electronicstructure calculations was developed by Mermin whoformulated electronic density functional theory (DFT)within GCE.[4, 5]. Later, GCE-DFT has been gener-alized for treating nuclear species either classically orquantum mechanically [3, 6, 7, 8, 9]. The GCE-DFTprovides a fully DFT, atomistic approach for comput-ing free energies of electrochemical and electrocatalyticsystems at fixed electrode and ionic/nuclear chemicalpotentials.[3] Importantly, the free energy from a GCE-DFT calculation is in theory exact and unique to agiven external potential. In practice, the (exchange-)correlation effects in both quantum and classical sys-tems need to be approximated. The thermodynamicGCE framework has already been adopted by the elec-tronic structure community to model electrocatalytic

thermodynamics at fixed electrode[10, 11, 12, 13, 14,15, 16, 17, 18, 19, 20, 3] and ion potentials[3, 14, 12].Based on the large number of theoretical and compu-tational works utilizing GCE-DFT, the computationalframework for thermodynamics within GCE seems gen-erally accepted. The thermodynamic approach hasprovided fundamental atomic level insight on reactionsat complex electrochemical interfaces and enabled com-putational catalyst screening using free energy rela-tions, Volcano curves, and scaling relations.[1]

Figure 1. Pictorial description of a proper electrochemicalinterface at fixed electron µ and solvent/electrolyte µ± chemicalpotentials.

However, it has been shown that a purely thermo-dynamic perspective on electrocatalysis is not sufficientfor understanding and predicting activity, selectivity,or catalytic trends.[21, 22, 23, 24] Besides applicationsin catalysis and material science, electrochemical ki-netics are fundamentally important and provide a wayto understand complex solvent effects, electron andnuclear tunneling, and non-adiabatic reactions. Ide-ally both fundamental and applied kinetic computa-tional/theoretical studies should make use of generaland self-consistent first principles Hamiltonians withinGCE. This has, unfortunately, remained unattainabledue to theoretical and methodological difficulties andomissions.[25] Surprisingly, a general GCE rate theoryhas not yet been established; mending this deficiencyis the central goal of the present work.

Before diving to the development of the GCErate theory, it is worth considering what new andimportant information can be obtained from a general


electrochemical rate theory. First and foremost, thetheory needs to accurately capture the intricacies ofET, PT, and PCET reactions as function of theelectrode potential. Therefore, a general treatment ofelectrochemical reaction rates needs to be applicableto 1) both inner-sphere and adiabatic as well asouter-sphere and non-adiabatic reactions, 2) sequentialET/PT or decoupled reactions as well as simultaneousPCET reactions, 3) tunneling of both electrons andnuclei, and 4) be combined with general first principlesGCE Hamiltonians. The motivation for including eachof these four requirements is discussed next.

First, adiabatic inner-sphere reactions present alarge and important class of electrocatalytic reactionsas demonstrated by a large body of computationalworks aiming to evaluate rate constants for this classof reactions[11, 12, 20, 19, 21, 26, 27, 28, 29, 30,31, 32]. However important adiabatic reactions are,all electrochemical reactions are certainly not inner-sphere nor adiabatic. In particular, both vibronicand electronic non-adiabatic effects are frequentlyencountered in outer-sphere and long-range ET, PT,and PCET reactions.[25, 33] Even for electrocatalyticreactions, non-adiabaticity may be present and theimportance and contribution of non-adiabaticity maydepend on the electrode potential.[34, 35] As a concreteexample, it has been shown that only the inclusion ofvibronic non-adiabaticity in electrochemical hydrogenevolution reaction can explain experimentally observedTafel slopes and kinetic isotope effects.[34]

Second, there are several reactions where thePT and ET are decoupled for kinetic reasons.For example, in alkaline ORR pure ET has beenproposed as the rate determining step[36, 37, 38, 39].Recent experiments of ORR on carbon-based materialsconclusively demonstrate that ET is the rate- andpotential-determining step.[40, 41]. Also solution pHcan alter the reaction mechanism and e.g. CO2

reduction can proceed through simultaneous PCETin acidic and through decoupled PCET (ET-PT) inalkaline solutions[42, 43]. In general, decoupled ETand PT are expected to play an important role onweakly bonding electrode surfaces in oxygen, CO2,CO, alcohol etc. reduction reactions.[44] In suchreaction-catalyst combinations long-range ET/PT maytake place warranting the inclusion non-adiabaticityeffects. From an applied perspective, decoupledsteps may enable circumvention of thermodynamicscaling relations and lead to identification of novelelectrocatalysts.[45]

Third, ET, PT, and PCET include the transferof very light particles and therefore quantum effectsmay be very important. Especially nuclear tunnelinghas a long tradition in electrochemistry[46] andexperiments have conclusively demonstrated that

room-temperature hydrogen tunneling takes placeduring ORR on Pt, and at low over-potentialstunneling is the prevalent reaction pathway.[47]Tunneling contributions are rarely considered inthe field of computational electrocatalysis which ismainly due to tradition and methodological difficulties;the computational electrochemistry community hasadopted tools and classical transition state theory(TST) from computational heterogenous catalysiswhere reactions take place at high temperatures andquantum effects are considered negligible. On the otherhand, the theoretical electrochemistry community hastraditionally considered ET, PT, and PCET in thenon-adiabatic, tunneling framework[36, 48, 49, 50,51, 52, 53, 34, 54, 55, 56, 57, 58, 59, 60, 61, 62].The computational community has been slow inadopting the language and approaches developed inthe theory community which has resulted scarcity offirst principles study of tunneling in electrochemicalenvironments.

Fourth, theoretical electrochemistry has a longtradition of using model Hamiltonian formulationsto understand reaction kinetics. For instance,Marcus[62], Dogonadze-Kutzetnotsov-Levich[48, 49],Schmickler-Newns-Anderson[63, 64], or Soudackov-Hammes-Schiffer[33, 34, 53, 60, 61, 65] theories haveprovided the basis for understanding electrochemicalkinetics. The main drawback of these methods is thatthey are difficult to parametrize in a self-consistentmanner and require effective parameters obtainedfrom either experiments, simple DFT calculations,or a mixture of these. Yet, widely differentparametrizations for the same reaction can resultin similar rates. For instance, differences as largesas ∼ 3-4 eV in reorganization energy and thecoupling matrix elements[66, 67, 35, 68] lead topractically identical reaction rates; it is clear thatsome unphysical error or parameter cancellation takesplace. The difficulty of parameter estimation anderror cancellation limits the physical/chemical insightobtained from model Hamiltonians. Furthermore,model Hamiltonians are static and (usually) not self-consistent. Typically, the electrode potential servesto role of changing the Fermi-level in an otherwisestatic electronic structure. Even when potential-dependent electrostatic interactions and work termsare included [35, 51, 67, 69], most parameters suchas the solvent reorganization energy, chemical bondingcharacterized by Morse potentials, electrode structure,tunneling matrix elements etc. remain unchanged bythe electrode potential. As such, it is unlikely thatmodel Hamiltonians can quantitatively capture thecomplexity of electrochemical reactions. Besides issuesrelated to self-consistency, model Hamiltonians studiesof non-adiabatic reactions implicitly rely on the single


orbital picture which is highly problematic for firstprinciples Hamiltonians as discussed in the SupportingInformation Section 1. Instead, modern fixed potentialfirst-principles methods explicitly incorporate theeffect of electrode potentials on the interfacialproperties and bonding. Especially the GCE-DFThas proven to provide a well balanced and rigorousdescription electrochemical interfaces. However,using general first principles methods for addressingET/PCET kinetics in general have remained largelyelusive thus far.

The above discussion highlights how differentreactions and phenomena have been and can beaddressed in the theoretical and computationalcommunities. Computational works utilize high-quality ab initio Hamiltonians but rate constants arebased on tools derived from heterogeneous catalysisand electrocatalytic reactions have been studiedonly using classical adiabatic TST theory. Thesecomputational studies describe the electrochemicalinterfaces in a self-consistent way and there is noneed for empirical parametrization of the TST rateequation. Thus far, these methods have only givenaccess to the reaction barrier but not the prefactorbeyond the TST approximation. Estimates onimportance of the prefactor has relied on perturbativerate theories with model Hamiltonians at the non-adiabatic limit to describe electron/proton tunneling.Other theoretical works extend the Newns-Anderson-Schmickler model Hamiltonian to study both classicaladiabatic TST and non-adiabatic tunneling reactions.While both barriers and prefactor have been computed,the models are evaluated using non-self-consistentparametrization. Therefore researchers have beenbe faced with a difficult choice: Should the studyinclude all the complexity addressed in a self-consistentmanner using an ab initio approach but with therestriction of classical TST approximation withoutgeneral prefactors? Or should the studies includeprefactors to reflect non-adiabaticity or tunneling butwith a empirically-parametrized model Hamiltonian?

In this work this difficulty is resolved bydeveloping a generally valid electrochemical rate theorywhich can be directly combined with fixed-potentialab initio methods. This is achieved by deriving agrand canonical rate theory building on Miller’s generalequilibrium (micro)canonical rate theory [70, 71, 72].As Miller’s theoretical framework is equally validfor adiabatic and non-adiabatic as well as quantum,semiclassical, and classical rate expressions[73] and canutilize both model or first principles Hamiltonians[33,57, 58, 59, 60, 61, 74, 75] the presented novel GCEextension provides a generally valid electrochemicalrate theory; the developed GCE rate theory enablesusing all canonical rate theories in constant potential

simulations. In particular, the work herein provides aunified rate theory for computing reaction barriers aswell as the prefactors making the theory applicable totreat adiabatic and non-adiabatic reactions, classicaland tunneling reactions, and PT, ET, and PCET onequal footing using GCE-DFT methods.

Besides developing a general and exact GCE ratetheory, approximate techniques for adiabatic reactionsare developed; non-adiabatic reactions are treatedusing the same formalism in a future publications.First, for adiabatic ET, PT and PCET reactionsa generalized GCE transition state theory (TST) isderived. Second, adiabatic Marcus-like[62] empiricalvalence bond theories (GCE-EVB) are developed.These lead to well-defined non-linear free energyrelationships ideally suited for materials’ screeningpurposes with kinetic information as demonstrated forthe acidic Volmer reaction on Au(111) in Section 4.Crucially, the developed rate theories can be seamlesslycombined with modern computational methods basedon (GCE-)DFT to facilitate self-consistent evaluationof rate constants without experimental parameters.The fixed potential rate theory will expand the type ofsystems, conditions, and phenomena in electrocatalysisamenable for first principles modelling.

The paper is organized as follows. In Section2 a general rate theory and TST within GCE aredeveloped. Rest of the paper focuses on ET andPCET kinetics within GCE. Section 3 shows howthe adiabatic barrier and rate of ET and PCETreactions are computed using GCE-EVB and freeenergy perturbation theory to developed a fixedpotential version of Marcus theory. Tafel slopes andother useful quantities as extracted from GCE-EVBare analyzed. A simple computational demonstrationof the GCE-EVB for Au-catalyzed Volmer reaction ispresented in Section 4. Next, additional computationalaspects for evaluating the rate constants are discussedin 5. Finally, the advances and results are summarized.

2. Rate theory in the grand canonicalensemble

2.1. Ensemble considerations

The GCE is open and the system exchanges matterwith its surroundings. The thermodynamics ofGCE are well understood[76] and we have recentlyshown that both electrochemical thermodynamicquantities for both classical and quantum particlescan obtained rigorously from GCE multi-componentDFT[3]. GCE provides a rigorous and natural wayto compute all thermodynamic expectation values atfixed electrode potentials by including the electrodepotential explicitly in the ab initio Hamiltonian. Thisis also the case for rate constants and fixed potential


rate constants are GCE expectation values of canonicalrate constants as shown below.

To address the GCE rate constants, one needsto consider the dynamics of open (quantum) systemswhich is still an active area of research.[77, 78]. Thetreatment of open system dynamics directly affects theGCE rate theory. First, GCE phase space volume isnot globally conserved and the Liouville theorem doesnot hold in general and computed ensemble propertieswill depend on time if the system is not in equilibriumor is non-stationary.[79, 78, 80]. For the presentwork it is important that equilibrium and short-timeproperties are unique and time-independent in theGCE[79, 81]. At other times the expectation valuesdepend sensitively on the coupling between the systemwith the particle reservoir and introduces the reservoirtime scales.[80] As a result, time-dependent quantitiessuch as particle fluxes and correlation functionsentering the general flux formulation of rate theory(see below) would require extensive sampling andcareful computation.[78, 80] To avoid the treatmentof explicitly time-dependent quantities, the GCEtheory developed herein only utilizes equilibrium andinstantaneous quantities. Therefore, non-equilibriumprocesses cannot be treated using the approach takenhere. Neglecting the bath time-scale and coupling alsomeans that electron transfer kinetics from the electron”bath” to system (see Figure 1) are assumed fast,a condition satisfied by well-conducting electrodes.Neither of the the above restrictions on treating thebath coupling and time scale are expected to greatlyaffect the use or validity of the developed GCE ratetheory in electrochemical and electrocatalytic systems.

A related consideration from on the treatmentof open systems is particle conservation. If a quan-tum system is characterized by particle conserv-ing operators (H Hamiltonian, S entropy, and Nparticle number), even time-dependent observablesare obtained as ensemble weighted (pn) expecta-

tion values from O(t) = Tr[ρU(t0, t)O(t)U(t, t0)

]=∑

n pn 〈ψn|U(t0, t)O(t)U(t, t0)|ψn〉. Note, that changesbetween states with different number of particles arenot included in the propagator when both the propa-gator U and the operator O are particle conserving.[82]Hence, even explicit propagation of the wave func-tion does not allow sudden jumps in particle numbers.Therefore, in the extension of (micro)canonical ratetheory to the GCE, only particle conserving reactionsare considered. Then, all equilibrium quantities arealways well-defined but jumps between states with un-equal number of particles are suppressed. While thisis not an issue for adiabatic reactions with smoothchanges in the number of particles, the prefactors en-tering e.g. non-adiabatic rate constants need to beformulated so that particle conservation is respected.

Therefore, all rate expressions derived herein will onlyutilize particle conserving operators.

2.2. General grand canonical rate theory

After establishing the particle conserving and equilib-rium nature of the rate constants, the GCE rate con-stants can be formulated. To allow various types of re-actions to be described, the exact equilibrium canonicalrate expression due to Miller[70, 71, 72, 83] is adopted:

k(T, V,N)QI =

∫dEP (E) exp[−βE] = lim

t→∞Cfs(t)

(1)where QI is the canonical partition function of

the initial state, and β = (kBT )−1. The firstexpression is written in terms of transition probabilityat a given energy P (E). The second expressionutilizes a canonical flux-side correlation function

Cfs(t) =1

(2πh)f∫dpfdqf exp(−βH)δ[f(q)]qh[f(qt)]

for f degrees of freedom. δ[f(q)] constrains thetrajectories to start from the dividing surface, q is theinitial flux along the reaction coordinate, and h[f(qt)]is the side function which includes the dynamicinformation whether a trajectory is reactive or not.

Based on the discussion in Section 2.1 on thedynamics of open systems, only the t → 0+

and t → ∞ should be considered for the flux-side correlation function in the equilibrium rateexpressions. Depending on the choice of P (E) orH and h[f ] non-adiabatic and adiabatic (nuclear)quantum effects are included in the rate.[84, 85, 86, 87].It is noteworthy that P (E) and Cfs are computedusing only particle conserving operators[71] and theconditions discussed above are satisfied when (1) isused as the starting point for formulating GCE rateconstants.

To compute reaction rates at fixed potentials astraight-forward, yet novel, extension of the canonicalrate theory to the GCE is made:

k(µ, V, T )ΞI =1

2π

∞∑N=0

exp[βµN ]

∫ ∞−∞

dE exp[−βEN ]P (EN )

=

∞∑N=0

exp[βµN ]k(T, V,N)Q0 = limt→∞

Cµfs(t)

(2)

where ΞI = exp[βµN ]QI is the initial state grandpartition function and k(T, V,N) was introduced in(1). Above, N is the number of species (nuclearor electronic) in the system and Cµfs is the GCEflux-side function. The previous equation showsthat all canonical rate equations can be applied to


electrochemistry within GCE approach and that fixedpotential electrochemical rate constants are GCEaveraged canonical rates constants.

The above equations are completely general andvarious flavors of rate theories can be extractedby invoking different Hamiltonians and transitionprobabilities, but they are somewhat cumbersomefor computational purposes. Indeed, it would beconvenient if the GCE rates could be directly evaluatedwithout explicitly summing over different particlenumbers. One way to achieve this is to makethe transition state theory (TST) assumption[72, 71,70] but generalized to GCE herein. In canonicalTST, the instantaneous limt→0+ Cfs(t) is consideredcorresponding to the assumption that there areno recrossings of the dividing surface. Bothquantum/classical and adiabatic/non-adiabatic TSTsare written as [88, 89, 90, 91]

kTST (T, V,N)QI(T, V,N) = limt→0+

Cfs(t) (3)

and the exact rate is recovered by introducing acorrection

k(T, V,N) = limt→∞

κ(t)kTST (T, V,N)

with κ(t) =Cfs(t)

Cfs(t→ 0+)

(4)

where κ(t) is the time-dependent transmis-sion coefficient which at long times is κ =k(T, V,N)/kTST (T, V,N).[92] Inserting this equationin (2) results in the most general grand canonical rateconstant. Significantly simplified rate constants are ob-tained when focusing on classical nuclei and using TST.As derived in the SI Section 2, for classical nuclei theTST result is [71, 72]:

k(T, V, µ)ΞI =

∞∑N=0

exp[βµN ]

∫dEPcl(E) exp[−βE]

≈∑N

exp[βµN ]kBT

hQ† ≡ kBT

hΞ†

(5)

where Pcl(E) denotes transition probability forclassical nuclei but the electrons are of course quantummechanical[75, 93] with details given in [72] and theSI Section 2. The previous equation shows that thestructure of GCE-TST and canonical TST are similarwhich is true for open system in general if memoryeffects are neglected[94]. To obtain the GCE rateconstant without invoking the TST approximation, onecan use the transmission coefficient κ to write

k(T, V, µ) =

∑∞N=0 exp[βµN ]κ(T, V,N)

kBT

hQ‡

ΞI

≈ 〈κµ〉kBT

h

Ξ‡

ΞI= 〈κµ〉

kBT

hexp[−β∆Ω‡

](6)

where it is assumed that an effective transitionprobability 〈κµ〉 can be used. To complete thederivation for the classical GCE rate constant, therate is expressed in terms of grand energies with thedefinition Ωi = − ln(Ξi)/β and ∆Ω‡ = Ω‡−ΩI for theGCE barrier. Above the only new assumption besidesgrand canonical equilibrium distribution and TST, isthat the flux out of the transition state 〈κµ〉 can betreated as an expectation value and separated from thebarrier. For large enough systems and small variationsin the particle number this is a justified assumption.

The above development establishes the generalfixed chemical potential rate theory. For classical,adiabatic reactions the rate constants in GCE areessentially the same as in the canonical ensemble.Within TST approximation the rate constant isdetermined by the grand free energy barrier andeffective prefactor. The transmission coefficient needsto be approximated but this depends on the case athand; examples for the adiabatic and non-adiabaticharmonic GCE-TSTs expression valid for fully opensystem are derived in Supporting Information section3. A more thorough treatment on the theoryand computation of non-adiabatic and tunnelingcorrections will be presented in forthcoming work.

2.3. Semi-grand canonical ensemble

The above development is valid when both nuclearand electronic subsystems are open. A significantsimplification results if one assumes that the reactionrate does not explicitly depend on the number of somenuclei in the system. In a typical first principlescalculation this simplification is often exploited whenthe system can be divided to two subsystem: 1)classical electrolyte species consisting of nuclei andelectrons and 2) electrode + reactants treated eitherclassically or quantum mechanically. Typically thenumber of nuclei constituting the electrode andreactant are fixed while the electrolyte and electronchemical potential are fixed. Fixing only theelectron and electrolyte chemical potentials definesa semi-grand canonical ensemble used for derivingthe thermodynamics of electrocatalytic systems withinGCE-DFT[3]. In this treatment is often utilizedin e.g. Poisson-Boltzmann type models where theelectrolyte is at a fixed chemical potential but theenergetics do not explicitly depend on the number of


electrolyte species. Then, summation over the numberof electrode/reactant nuclei or the electrolyte speciesis not needed.

Herein the semi-GCE is applied to derive rateconstants as a function of the electrode potential.From now on, I assume that the reaction rates dependexplicitly only on the number and/or chemical potentialof electrons in the system. Then, the state of thesystem is determined by T , V , number of nuclei ofthe electrode+reactant NN , chemical potential of theelectrolyte, chemical potential of the electrons µn, andnumber of electrons in the system N unless explicitlyspecified otherwise. Electroneutrality is maintainedby the electrolyte. A widely utilized harmonic TSTrate for constant number of nuclei and constantelectrochemical potentials are derived in section 3 ofthe Supporting Information.

3. Adiabatic barriers and rates fromGCE-EVB

To compute the GCE-TST rate at a given electrodepotential, the grand energy barrier in (6) needs tobe obtained. For electronically adiabatic reactionsmethods like the constant-potential[20] nudged elasticband[95] can be used. An alternative method forcomputing the grand energy barrier is to formulatea Marcus-like[62] approach or empirical valence bond(EVB) theory[96] within GCE. Such models arecommonly utilized in electron[62] and proton transfertheories.[65, 96, 97, 98, 99] Here, the treatmentis based novel development of GCE diabatic statesand the extension of the canonical thermodynamicperturbation theory to the GCE to facilitate derivationof a GCE-EVB rate theory (see SI sections 3 and 4).The GCE-EVB theory provides a theoretically well-justified and computationally affordable way to obtainfixed potential barriers at various electrode potentials;the adiabatic barrier needs to be explicitly computedonly at a single electrode potential while barriers atother potentials can be obtained using well-definedextrapolation of (17). The utility of the GCE-EVBtheory is demonstrated in Section 4.

In canonical EVB and Marcus theories usediabatic states, effective wave functions and freeenergies[62]. This can be extended to GCE byusing two fixed potential, diabatic ground statesurfaces which represent a GCE-statistical mixtureof states with probabilities given by the densityoperator in GCE[3]. Importantly, the diabatic statesobtained using the GCE density operator naturallyinclude many-body effects of the coupled electrode-reactant-solvent system and the complexity of theelectrochemical interface is explicitly included in themodel. Also, there is no need to decompose the

rate constants to orbital dependent quantities(seeSection 1 in the Supporting Information for additionaldiscussion). Then, two grand canonical diabatic all-electron wave functions are used to form an effectivediabatic GCE Hamiltonian. This is analogous tomolecular Marcus theory utilizing a canonical diabaticHamiltonian containing an initial (oxidized) I andfinal(reduced) molecule F .

Following the treatment in the SupportingInformation Section 3, a diabatic 2×2 grand canonicalHamiltonian in (7) can be formed from two diabaticGCE states. The resulting form is analogous to thecanonical EVB methods[96], electron[62], proton[98,99] and proton-coupled electron[65] theories. Thepresent form is, however, crucially different from itspredecessors; based on the approach developed in thiswork, all quantities are defined and computed at fixedelectrode potentials. In the basis two GCE diabaticstates the GCE Hamiltonian is

HGCE−dia =

[ΩII ΩIFΩFI ΩFF

](7)

as derived in Supporting Information Section 3.Here the diagonal elements are the grand energies ofthe initial (II) and final (FF) systems. The off-diagonalelements account for the interaction and mixingbetween the initial and final states. They can becomputed as GCE expectation values of contributionsfrom different N-electron states ψNi as ΩFI =∑N pN (µ)

⟨ψNF∣∣HN

∣∣ψNI ⟩. This is rather straight-forward for ET reactions using e.g. constrainedDFT discussed in Sections 4 and 5. For PT andPCET reactions computing these matrix elementswould require computing the vibronic matrix elementsusing e.g. the semiclassical approach of Georgievskiiand Stuchebrukhov[100] This direct computation isparticularly useful for non-adiabatic rate constantswhich are investigated in future work.

For adiabatic reactions, the direct calculation canbe replaced by the diagonalization of the 2×2 diabaticHamiltonian in Eq. (7). This diagonalizating producesthe adiabatic ground and first excited states as

Ω±ad =1

2

(ΩII + ΩFF ±

√(ΩII − ΩFF )2 + 4Ω2

IF

)(8)

As shown below, the diabatic states cross (ΩII =ΩFF ) at the transition state. This makes it possible tocompute the coupling matrix element as the differencebetween the diabatic states and the adiabatic states.For the ground state one has ΩIF = ΩII − Ω−ad. Theadiabatic ground transition state grand free energycan be computed using e.g. NEB calculations andthe coupling matrix element is simply the difference


between the diabatic transition state grand energy andthe adiabatic one as shown below in Eq. (18).

Finally, the (diabatic) grand canonical statescorrespond to a single electron density which isguaranteed by the Hohenberg-Kohn-Mermin[4, 3] tobe unique for a given electrode potential. Bydefinition, GCE diabatic states are unique groundstates. Such diabatic statas also include the interactionand exchange between all the electrons in the systemand for adiabatic, ground state reactions there is noneed to include addition excited states despite thecontinuum of (single-electron) states of the electrode.In principle it is possible to add other, possibly excitedstates as basis states but here the focus in on treatingadiabatic reaction and excited states beyond the firstexcited state are neglected. If a general quantummechanical Hamiltonian is used, bond breaking isnaturally included in the GCE-EVB model. Theonly disambiguity is the definition of diabatic states.In practice the GCE diabatic energies, (ΩII andΩFF ), can be computed directly by applying usinge.g. cDFT[101, 102, 103] with fixed potential DFTas discussed in Section 5 and shown in Section 4.

3.1. Computation of diabatic GCE energy surfacesand barriers

An approach often used in molecular simulationsfor constructing the diabatic free energy curves isto sample the diabatic potentials along a suitablereaction coordinate. For canonical ET, PT, andPCET reactions the reaction coordinate is the en-ergy gap between the two diabatic states as shownby Zusman[104] and Warshel[105]: ∆Egap(R) =EF (R) − EI(R).[76, 106] From the sampled energygap, free energy curves are obtained as A(R) =−kBT ln(p(Egap(R))) + c. If the distribution is Gaus-sian

(p(Egap(R)) = c exp

[−(∆Egap − 〈∆Egap〉)2/2σ2

]),

the resulting free energy curves a parabolic. The dia-batic barrier in EVB or Marcus theory is then obtainedfrom the intersection of the initial and final diabaticcurves[106, 107, 108, 109].

Within GCE, the energy gap is simply Egap(R;µ) =∑N,i pN,iEgap(Ri, N). As shown in the SI section

4, the gap distributions can be formulated and com-puted by generalizing Zwanzig’s[110] canonical free en-ergy perturbation theory to the GCE. This provides arigorous way to derive the reaction barrier in termsof diabatic states and energies as presented in theSupporting Information Section 4. The reaction en-ergy rate can be computed from the initial-final stateenergy gap distribution functions using a well-knownformula[105, 111, 112, 113, 114, 115, 116]

kIF = κexp[−βgI(∆E‡)

]∫d∆E exp[−βgI(∆E)]

= κpI(∆E‡) (9)

where gi(∆E) is the free energy curve in statei as a function of the energy gap, pI(∆E

‡) isthe gap distribution at the transitions state, and κdenotes an effective prefactor. The reaction rate isdetermined by the energy gap distribution functionpI(∆E) = 〈δ(∆E(R)−∆E)〉I from equation (S30) ofthe Supporting information.

While the approach is general and valid forcomplex reactions, assuming that Egap(R;µ) isGaussian leads to a closed form equation. In thiscase the GCE-diabatic states are parabolic and theMarcus barrier in GCE is given by (13). As shownin the Sections 4 of the SI, the (Gaussian) gapdistribution may be derived using a second ordercumulant expansion resulting in

pI(∆E) =1√

2πσIexp

[−

(∆E − 〈∆E〉I)2

2σ2I

](10)

where 〈∆E〉I is the energy gap expectationvalue in the initial state obtained from equa-tion (S27) in the Supporting Information andσI =

⟨(∆EI − 〈∆EI〉)2

⟩Iis the gap variance. The

Marcus relation is then obtain after standardmanipulations[106, 112] yielding

pI(∆E‡) =

1√4kBTΛ

exp

[−β (∆Ω + Λ)2

4Λ

](11)

where σ2I = σ2

F = 2kBTΛ = kBT (〈∆E〉I −〈∆E〉F ), Λ is the fixed potential reorganization energyand ∆Ω = (〈∆E〉I + 〈∆E〉F )/2 is the reactiongrand energy as depicted in Figure 2. These gapidentities are valid for symmetric reactions and havebeen previously established well for the canonicalensemble[112] and generalized here to the GCE. Inpractice, the reorganization energy is computed asan average of the reorganization energies which aredifferences of the diabatic free energy at the finalgeometries

Λ =1

2[ΩII(RF )− ΩII(RF )I + ΩFF (RI)− ΩFF (RI)]

=1

2[ΛI + ΛF ]

(12)

shown in Figure 2. Finally, the GCE-EVB rateequation using the above assumptions results in anexpression analogous to Marcus equation


k =κ√

4kBTΛexp

[−β (∆Ω + Λ)2

4Λ

](13)

Figure 2. Schematic depiction of the important GCE-EVBquantities. The blue (orange) dashed lines is initial (final)diabatic surface while the black solid line is the adiabatic surface.

The energy barrier of (13) is the diabatic energybarrier. The adiabatic barrier is estimated from (7)using the methods discussed in Section 3.2 below.One caveat to keep in mind is the more involvedcomputation of κ within the GCE as discussed inSection 2 and forthcoming work for non-adiabaticreactions. The above result may safely be used whenκ ≈ 1 for all particle numbers meaning that thereaction is always fully adiabatic and classical.

3.2. Implications of the canonical GCE-EVB ratetheory

If the diabatic grand energy surfaces are symmetricand quadratic they have the same curvature andreorganization energy. In this case, the diabatic grandenergy barrier is estimated from (13). The assumptionon equal curvature can be relaxed[117] (see also SIsection 5). One easy approach to realize this is toutilize an asymmetry parameter αas as[118]

αas =ΛI − ΛFΛI + ΛF

(14)

in terms of the reorganization energies for boththe initial and final states ΛI and ΛF , respectively.The transition state is located at the crossing point

x‡/ξ = − 1

αas+

1

αas

√1− αas

(αas −

4∆Ω

ΛI + ΛF

)(15)

With these definitions the asymmetric diabaticMarcus barrier and rate become

∆Ω‡ =1

4ΛI(x‡/ξ − 1

)2(16a)

k ≈ κ√4kBTΛI

1 + αas1 + αasx‡/ξ

exp[−β∆Ω‡

](16b)

When αas → 0, the regular Marcus barrier andcrossing point are obtained. In Figure 3 the effect ofasymmetry and reaction energy to the reaction barrierand location of the transition state are compared. Itcan be seen that both the barrier heights and itslocation are affected by the asymmetry and reactionenergy.

Figure 3. Left: EVB curves at different different asymmetriesαas. The final state reorganization energy is ΛF = 40 while theinitial state reorganization energy ΛI ∈ [10, 80]. The reactionenergy is ∆Ω = 0 for all curves. Right: EVB curves as afunction of the reaction energy: ∆Ω ∈ [−15, 15] and ΛF =40.The blue (red) curve corresponds to ΛI = 40 (ΛI=60). Both:The dashed line at x = 0 indicates the position of the transitionstate when ΛI = ΛF and ∆Ω = 0. The curve crossing point

equals ∆Ω‡dia.

While the Marcus-like equation results in adiabatic barrier, the adiabatic reaction barrier can beextracted from the diabatic barrier by diagonalizing(7). The adiabatic barrier can also be obtained from(13) using the Hwang-Aqvist-Warshel adiabaticitycorrection[119, 120]

∆Ω‡ad,EV B =(∆Ω + Λ)2

4Λ− ΩIF (x‡) +

(ΩIF (xI))2

∆Ω + Λ

= ∆Ω‡dia − ΩIF (x‡) +(ΩIF (xI))2

∆Ω + Λ(17)

where ΩIF is the off-diagonal matrix of the GCE-EVB Hamiltonian in (7). If the Condon approximationis used, the above equation is greatly simplified asΩIF ≈ ΩIF (x‡) ≈ ΩIF (xI) becomes a geometry-independent constant.

Next changes in the adiabatic GCE-EVB barrieras function of the parameters is analyzed. Fromthe schematics shown in Figures 2 and 3, one canobserve that changes of the minima along the reactioncoordinate correspond to horizontal displacements of


the diabatic states and changes in Λ. Vertical changescorrespond to changes in the reaction grand energy∆Ω. In general, the reorganization energy of inner-sphere reactions taking place on or near the electrodesurface may depend on the electrode potential andinvestigations along this direction are on their way.

Under equilibrium conditions, ∆Ω = 0 and thecorresponding reorganization energy Λ0, the adiabaticbarrier is

∆Ω0,‡ad,EV B =

Λ

4− ΩIF +

(ΩIF )2

Λ≈ Λ0

4− ΩIF (18)

which leads to Λ0 = 4(∆Ω0,‡ad,EV B + ΩIF ) ≈

4∆Ω0,‡dia assuming that ΩIF << Λ0 (this is the case

for e.g. the Au-catalyzed Volmer reaction in Section4). At the equilibrium point, the overpotential is,η = ∆Ω = 0. Assuming for a moment that Λ ≈ Λ0 andreplacing the solution for Λ0 in (17) gives the diabaticbarrier as

∆Ω‡dia = ∆Ω0,‡dia +

∆Ω

2+

(∆Ω)2

16∆Ω0,‡dia

(19)

Inserting (19) in (17) results in the adiabaticreaction barrier as

∆Ω‡ad,EV B = ∆Ω0,‡ad,EV B +

∆Ω

2+

(∆Ω)2

16∆Ω0,‡dia

(20)

This result is well-known in the canonical EVBand Marcus theories. However, in GCE this relationis valid only when constant reorganization energyis assumed. In general, the driving force can bemanipulated easily by changing the electrode potentialwhich is in turn directly related to the absoluteelectron electrochemical potential as EM (abs) ∼ −µn.[3, 121, 122] An experimentally meaningful approach isto study −∂r(T, V, µn)/∂µn as done in a Tafel analysis,for example. Tafel analysis can also be understoodin a more general context of Brønsted-Evans-Polanyi(BEP) and other free energy relations measuring thechange of reaction rate when the reaction energy ischanged[123, 124, 125], as both Tafel and BEP analysesmeasure the reaction rate as a function the reactiondriving force - Tafel analysis focuses on the over-potential and BEP on the free energy. These twoquantities are linked by |∆η| = |∆µn| = |∂∆Ω/∂n|.Defining the rate constant as a function of the electrodepotential E as k(E) = k(E = 0)A(E) exp(−βαE) interms of the prefactor A and the Tafel-BEP coefficientα and the Tafel-BEP coefficient is [2, 123, 124]

d ln k(E)

dE≈ −βα→ α = −β−1 d ln k(E)

dE(21)

where constant α and prefactor are assumed.Within GCE-EVB α is obtained in terms of thereorganization and reaction energies

−βα =d ln k(∆Ω,Λ)

dE

= −∂ ln k

∂∆Ω

∂∆Ω

∂µn

∂µn∂E− ∂ ln k

∂Λ

∂Λ

∂µn

∂µn∂E

= −γ∆Ω′ − c∆Λ′

(22)

where the first term measures how the ratechanges as a function of the reaction energy, γ denotesa BEP coefficient and ∆Ω′ denotes the grand energychange as a function of the over-potential. Thesecond contribution is novel and unique to the GCEformulation. It measures the sensitivity of the rate tochanges in the reorganization energies as a functionalof the potential. This unconventional contribution canbe observed in e.g. the Volmer reaction treated in[67] with a model Hamiltonian and will be discussedin more detail in a future publication.

To facilitate understand the BEP term, onerecognizes that ∆Ω = (AF (〈NF 〉) − AI(〈NI〉) −µn(〈NF 〉 − 〈NI〉). For macroscopic systems, chemicalreactions have NF = NI while simple electrochemicalsteps have NF = NI ± 1. For chemical reactions∆Ω = ∆A and the variation ∆Ω′ is small. Within thecomputational hydrogen electrode (CHE) concept[126]the reaction energy ∆Ω ≈ ∆A0 ∓ η for PCET stepswith ∆A0 computed without any bias potential. Then,αCHE = γ for PCET steps and zero otherwise. Ingeneral such a simple relationship does not hold ingeneral and models such as GCE-DFT can be usedfor computing ∆Ω′ explicitly. Thus far, ∆Ω′ has beenreported in only few studies[20, 21, 127]. In these worksand in Section 4, ∆Ω is found to exhibit a roughlylinear dependence on the applied potential.

Next the BEP γ of Eq (22) is analyzed. Using thediabatic barriers in (19) (obtained assuming constantreorganization energy and constant prefactor), oneobtains

γ =∂ ln k(T, V, µn)

∂∆Ω

∣∣∣∣T,V

≈ −− β 1

2

[1 +

∆Ω

Λ0

](23)

which results in α = −∆Ω′(1/2 + ∆Ω/2Λ0).It is seen that γ is not a simple constant butdepends linearly on the reaction driving force. If thereorganization energy is small the dependence on thereaction grand energy becomes more pronounced asdemonstrated for the Au-Volmer reaction in Section 4.In general, non-linearity of the grand energy barrier hastwo contributions: non-linearity of the diabatic barrierand the potential-dependent reorganization energy.For macroscopic systems non-linearity is established by


including the quadratic part of the diabatic barrier inthe model. Lately[29, 31, 20, 21] this has been observedcomputationally and it is pleasing that the GCE-EVBpicture seems qualitatively correct.

To summarize, the generalized BEP-Tafel rela-tionships have been derived from a microscopic per-spective using grand canonical rate theory. Both vari-ation in the reaction energy barrier and the transitionstate location as a function of the potential can bepredicted using just a few parameters. This is demon-strated for the acidic Volmer reaction in Section 4. Thegeneral form of the BEP-Tafel relation is given in (22).For small over-potentials, the rate is expected to de-pend linearly on the applied potential. For larger over-potentials non-linear dependence is predicted when thereorganization energy is small. The GCE perspectivealso predicts a novel potential-dependent reorganiza-tion energy which is supported by model Hamiltoniancalculations[67] and will be addressed carefully in fu-ture publications.

4. Application of the GCE-EVB theory to theAu-catalyzed Volmer reaction

Here the first demonstration of the GCE ratetheory is provided. I consider the acidic Volmerreaction i.e. proton discharge which is arguablythe simplest and yet relevant electrocatalytic reactionfor hydrogen production and other electrocatalyticreactions. Similarly, gold can be considered as thesimplest electrode material. Yet, the Volmer reaction,even on gold, is not fully understood[128] and thereaction is considered to exhibit nuclear quantumeffects and even vibronic non-adiabaticity.[34, 35] Asa first application of the theory and methodologyderived and developed in this work, I consider mostlyan adiabatic and classical model for the acidic Volmerreaction – quantum effects and a non-adiabaticity arestudied separately in forthcoming publications. Theresults are discussed in the GCE-EVB framework ofSection 3.

The Volmer reaction is modelled as a singlehydronium ion on a 3x3x5 Au(111) surface as shownin Figure 5. The needed free energies were computedusing GCE-DFT as implemented in GPAW[129] withinthe surface-jellium approach[20] with a continuumsolvent model for water[130]. This approach givesall the thermodynamic quantities at a fixed electrodepotential. The potential-dependent minimum energypathways are computed using a nine image nudgedelastic band[95] (NEB) discretization. Geometries andNEB pathways were considered converged when themaximum force was below 0.05 eV/A.

Constant-potential diabatic states and reorganiza-tion energies were computed using constrained DFT

(cDFT) as implemented in GPAW[131]. As in thecanonical case, the constraining potential in GCE-cDFT is introduced as an external potential to theGCE-DFT giving

Ω[n(r), Vc;T, V, µn]cDFT =

Ω[n(r);T, V, µn]DFT + Vc∑σ

(∫drwσc (r)nσ(r)−Nc

)(24)

where Ω[n(r);T, V, µn]DFT is the GCE-DFT energyfunctional[3] and n(r) is the electron density. wσi (r)is the weight function which defines how the charge isto be partitioned, i.e. the regions where charge is to belocalized, Nc is the desired number of excess electronswithin the constrained region, and Vc is the Lagrangemultiplier enforcing the charge/spin localization. Theintroduction of constraining terms in Eq. (24) leads toa new effective potential defined as

vσeff =δΩ[n(r), Vc]

δn(r)=δΩKS [n(r)]

δn(r)+∑c

Vcwσc (r) (25)

Thus, the cDFT potential is just the sum ofthe usual KS potential and the constraining potentialwhich is also used in the self-consistent calculation.The constraint is further enforced by introducing theconvergence criteria

C ≥∣∣∣∣∑σ

∫drwσc (r)nσ(r)−Nc

∣∣∣∣ ,∀ c (26)

The optimize grand canonical free energy underthe specified constraint is[131]

Ω[Vc] = maxVc

minnΩ[n, Vc;T, V, µn]cDFT (27)

To fulfill the cDFT constraints and to performfix potential calculations, the approach in Fig. 4 hasbeen utilized. An example of GPAW scripts used forperforming GCE-cDFT calculations is given in the SIsection 6.

The charge states to define diabatic states arechosen as Nc = +1 state for the hydronium (H3O

+)and neutral Nc = 0 for the final water and adsorbedhydrogen (H2O+H∗). The reorganization energies arecomputed from Eq. (12) at the equilibrium potentialwhich resulted in ΛF = 2.1 eV for the initial stategeometry, and ΛI = 3.2 eV for the final state geometry.The average Λ = 2.65 eV is used in calculating theMarcus barrier of Eq. (13). These cDFT computedreorganization values are in good agreement with the


Figure 4. The two phase optimization loop used for obtainingfixed potential diabatic states. The cDFT optimization is giventhe outer loop of Eq. (27) and the SJM loop is the inneroptimization performed using the algorithm from Ref.[20] tochange the number of electrons N to satisfy the set electrodepotential

.

values used for model Hamiltonian parametrizations ofHuang[68] and Santos[66] but much larger than the oneused by Lam[67]. The reaction asymmetry from (14) is0.2 meaning that the transition state geometry alongthe reorganization coordinate is closer to the initialstate. Analysis of the TST geometries at differentpotentials shows that that reorganization energy is bestpresented either by the Au-O distance or the dihedralangle between the H2O and surface.

In Figure 5 the adiabatic reaction barrier asobtained from NEB calculation is plotted as a functionof the reaction energy corresponding to differentelectrode potentials. First, one observes that thebarriers are very small for all considered electrodepotentials. This is in line with explicit water DFTresults[20] at all electrode potentials and reactionenergies. The figure also shows the adiabatic TSTlocation as a function of the reaction energy fromboth NEB-DFT and extrapolation using (16a). Theextrapolation reproduces the TST geometries verywell, and captures the trends in the TST location.For comparison, explicit solvent calculations exhibit asimilar trend in the TST position as a function of thepotential[20] as the one found here using an implicitsolvent. This example demonstrates the Au-O distanceis good reorganization coordinate and that the TSTlocation is effectively captured by Eq. (16a). UnlikeDFT-NEB calculations, the GCE-EVB requires justone NEB and two reorganization energy calculationsto capture the TST geometry for a range of reactionenergies and electrode potentials.

Figure 5. Above: The reaction pathways at different potentials.The stars show the DFT-NEB computed TST geometry alongreorganization coordinate. The stars correspond to TSTgeometries predicted using equation (14a). Below:The fixedpotential Au(111) Volmer barrier as a function of the reactionenergy. NEB[*] refers to NEB calculations of the present paperwith an implicit solvent while NEB[20] are from [20] withexplicit, ice-like solvent. Also the barrier and reorganizationenergy used in the model Hamiltonian work of [67] are shownand extrapolated using (20).

The results in Figure 5 also show the GCE-EVBbarrier as obtained from Eq. (20) which is evaluatedusing the cDFT computed average reaorganizationenergy. The adiabatic equilibrium barrier (∆Ω0,‡

ad inEq. (20)) is computed from a NEB calculation close tothe equilibrium potential. As the results in Figure 5show, the extrapolation with Eq. (20) provides a veryaccurate way for computing the adiabatic potential-dependent energy barrier. It is also observed thatthe estimate for the reorganization energy used in themodel Hamiltonian work of [67] is very small (∼ 0.3eV) and cannot be used for predicting barriers usingGCE-EVB.

Next, the validity of the adiabatic assumptionis tested using the coupling matrix elements of Eq.


(7). The coupling matrix element is obtained usingEq. (8) at the equilibrium potential transition stateand results in ΩIF = 0.37eV . The coupling constantmaybe used for estimating transmission coefficient 〈κµ〉of Eq. (6) the adiabaticity from the the Landau-Zenerfactor (PLZ) for PCET reaction [132, 35] adopted tothe GCE.

〈κµ〉 =2PLZ

1 + PLZ(28a)

PLZ = 1− exp[−2πγ] (28b)

2πγ =π3/2 |ΩIF |2

hνn√

ΛkBT(28c)

and 〈κµ〉 = 1 is a signature of an electronicallyadiabatic reaction whereas non-adiabatic reactionshave 〈κµ〉 << 1. Eq. (28a) is valid for reactions in thenormal region.[132, 35] νn is the vibrational frequencyalong the reaction coordinate.[133] As discussed above,the reaction coordinate n the Au-catalyzed Volmer isthe water reorientation and νn = 1/τL ≈ 0.5× 109s−1

computed from the water reorientation time τL ≈ 2ps[134]. The expression for γ in Eq. (28c) is validfor quadratic free energy surfaces as derived in Eq.S.15 of the Supporting Information. Evaluating thetransmission coefficient at room temperature usingthe computed average reorganization energy, couplingmatrix element, and water reorganization time gives〈κµ〉 = 1.0 - this shows that the reaction is adiabaticand justifies the treatment in this section.

Besides enabling a reliable prediction of reactionbarriers and TST geometries, the results demonstratethe first GCE-cDFT calculations. Besides showingthat GCE-cDFT is technically possible, the resultsshow that ab initio computed diabatic states offer newinsight to electrocatalytic reactions. In particular, theresults provide a proof-of-principle that GCE-EVB canbe used to accurately estimate barriers using just asingle NEB calculations and a few cDFT calculationswith an expense similar to a standard DFT calculation.

5. Discussion

The distinct advantage of the formalism and theory de-veloped in this paper is that all rate equations can bereadily evaluated with GCE-DFT or other first princi-ples approaches. The presented formalism enables thetreatment electrochemical and electrocatalytic thermo-dynamics and kinetics in terms of the prefactor andbarriers in the same self-consistent framework – theGCE-DFT. Therefore, the same DFT-based tools canbe used to address inner-sphere and outer-sphere kinet-ics and thermodynamics instead of modifying or chang-ing the theoretical and computational framework fordifferent reaction steps[36].

By construction the rate constants include theinterplay between the electronic structure, solvent,electrode potential etc. All quantities can becomputed using self-consistent DFT energies and”wave functions” to include exchange and correlationeffects between all the electrons in the system. TheFermi-Dirac distribution is fulfilled at the DFT leveland, therefore, there is no need to integrate overthe filled/empty orbitals weighted by the Fermi-Diracdistribution in the rate expression as done is traditionalsingle-orbital descriptions (see SI Section 1 and below).Also, the Kohn-Sham-Mermin theorem[3] guaranteesthat GCE-DFT and GCE-EVB states are unique toa given electrode potential and that the GCE-EVBdiabatic inlcude that interactions between all electronsin the metal and the reactants.

The electrode potential is self-consistently treatedand all free energies and prefactors depend explicitlyon the potential. This is in contrast with traditionalmodel Hamiltonian treatments where the electrode po-tential rigidly shifts the Fermi-level without modify-ing any interactions or prefactors [64, 69] or modifiesonly the electrostatic interactions[34, 67]. Also, sep-arate computation of work terms[67, 69, 68, 135] isnot needed because all relevant interaction can be di-rectly included in the general Hamiltonian. Evaluationof chemisorption functions entering adiabatic Newns-Anderson-based models[63, 66, 67, 64, 68, 135] is alsoavoided. Therefore, the current models are free of ap-proximate treatment of semi-elliptic DOSs[67, 66, 135]or fitting the chemisorption functions to a computedDOS[66, 135].

As the developed rate theory utilizes generalab initio Hamiltonians, bond formation/breakingare naturally included. This is again in contrastwith model Hamiltonians which require approximatepotential-independent terms to describe changes inatomic bonding [136, 137, 135]. Instead, asdemonstrated herein, ET, PT, or PCET and bondrupture/formation are naturally captured with GCE-DFT. Bond formation s is also captured by diabaticmodels using cDFT as demonstrated herein forthe Volmer reaction and previously for ET[138],PCET[139] and general chemical reactions[140, 141].

As all necessary terms can be computed fromGCE-(c)DFT, adoption and evaluation of the rate ex-pressions is straight-forward (but potentially labori-ous). While applicability and usefulness of combinedDFT and GCE-EVB was demonstrated for the Volmerreaction, it is worth discussing the additional compu-tational requirements in some detail. First, the simu-lation of charged systems is needed to sample the elec-trode potential. Electroneutrality can be enforced us-ing some variant of the Poisson-Boltzmann model, fordetails see [3]. Fixed potential calculations can be ac-


complished within a single SCF cycle[10], or iteratively[142, 20]. Second, the solvent effects should be in-cluded in the model. In traditional TST-based modelsfor adiabatic reactions the main solvent contributionis thermodynamic and stems from (de)stabilization ofdifferent structures. GCE-EVB models need to involvethe solvent as the reaction barrier is directly related tothe solvent/environment reorganization energy and ne-glecting the solvent contributions will most likely leadto incorrect results.

Given a software capable of handling charged sys-tems and performing constant potential calculations,adiabatic TST rate constants can be readily evaluated.As shown for the Volmer reaction, reaction barriersand adiabatic prefactors at constant potential are ob-tainable using e.g. the NEB[95] method. Evalua-tion of GCE-EVB rate constants requires additionalsoftware capabilities for constructing charge/spin lo-calized diabatic states and to evaluate the electroniccoupling between these states either through directcalculation or diagonalization of the diabatic Hamil-tonian of Eq. (7). Also the reorganization energy,which is an excited state quantity, needs to be com-puted. One widely implemented and available toolfor evaluating the additional parameters is the cDFTmethodology[101, 102, 103] which is implemented inseveral DFT codes[131, 143, 144, 145, 146, 147, 148,149, 150, 151, 152]. The extension of cDFT to GCEwas, for the first time, demonstrated and applied inthis work. GCE-EVB should be accompanied with aconstant potential simulations to compute fixed poten-tial reaction and reorganization energies.

The general framework can also be extendedto treating reactions beyond adiabatic reactions toinclude e.g. non-adiabatic effects, nuclear tunneling,and solvent-dynamics controlled reactions - treatmentof these effects is under current study and willbe published separately. Non-adiabatic effects areexpected for e.g. outer-sphere ET reactions and severalPCET reactions[33]. Several PT and PCET reactionsare also likely include adiabatic or non-adiabaticnuclear tunnelling effects. Also, solvent dynamicsshould be included as these are likely to becomeincreasingly important or even dominant when thereaction is adiabatic and the reaction barrier becomesvery small or vanishes.[153] Under such conditionsthe reorganization will be the slowest process andthe reaction prefactor should reflect this. Last, well-defined interpolation[154, 155, 156] between adiabatic– solvent dynamic –non-adiabatic should be developedor adapted to the fixed-potential rate theory.

Finally, a spectacular feature of canonical Marcusand EVB theory is the observation of an inverted regioni.e. the rate constant starts to decline as the reactionbecomes more exothermic. However, the inverted

region has not been observed for electrochemicalreactions even at large over-potentials. The grandcanonical Marcus rate of (13) seems to predictan inverted region for highly exothermic conditionsand warrants additional discussion. First, theMarcus-like expression is based on linear response orsecond order cumulant treatment[157] which leads toquadratic free energies along the reaction coordinate- the prediction of the inverted region is a directconsequence of the linear response assumption. Togo beyond the quadratic free energy surfaces higherorder cumulants can be added (see SI section 5)to modify the existence of the inverted region.[158]For instance, the inverted region is not predicted forMorse potentials.[159] Secondly, the inverted regionis very sensitive to tunneling effects, excited states,and solvent effects.[160] For example, nuclear quantumeffects are needed to achieve accurate ET rates in theinverted region ET[161] while excited proton vibronicstates dominate PCET reaction rate[159]. Even ifthe inherent approximations on quadratic, classical,and adiabatic nature are acceptable, in GCE-EVBthe reorgnization energy is potential-dependent andthe inverted region is suppressed if ∆Ω′ ≈ 0. Alsothe reorganization energy should saturate at largeoverpotentials.

The prediction of non-existing inverted regionis inherent to several non-adiabatic single orbitalapproaches. In these treatments the inverted regionis avoided by taking the manifold of single-electronstates into account and integrating over the Fermi-Dirac weighted transition rates, as discussed inSupporting Information Section 1. This has resultedin e.g. Marcus-Hush-Chidsey[162], Dogonadze-Levich-Kuztnetsov[49, 48], Soudackov-Hammes-Schiffer[53]models of ET and PCET. On the other hand, adiabaticrate computed using the Newns-Anderson-SchmicklerHamiltonian [63, 64] does not the include the orbital-to-orbital contributions separately as the redox orbitalis coupled with all levels on the metal by definition.In the adiabatic Newns-Anderson-Schmickler model,the free energy surface becomes as single well and thereactant and product become indistinguishable. As thevanishing barrier cannot be treated[63, 154], the barrieris simply extrapolated to zero in the inverted regionand the rate is controlled by the prefactor.

Based on the above discussion, the inverted regionis more complicated than the normal region andrequires careful consideration of excited states andtunneling effect, for example. The vanishing invertedregion is for non-adiabatic single-orbital models andnot for adiabatic models where the exchange betweenall metallic states and the redox molecule are treated.In the GCE-EVB picture the inverted region is inherentto the quadratic grand energy surfaces and simply


extrapolation of the barrier to zero removes theinverted region. However, this is not fully satisfactoryand more studies are needed to study non-quadraticfree energy surfaces, tunneling, and excited states inthe inverted region.

6. Conclusions

In this work a new theoretical formulation forcomputing electrochemical and electrocatalytic rateconstants at a fixed potential is developed. The rateexpressions are obtained by extending the universallyvalid and exact canonical rate theory[70, 71, 72] tothe grand canonical, fixed potential ensemble. Generalconditions and limitations for the fixed potentialrate theory are developed. It is shown that allrate theories developed within the canonical ensemblecan be transferred to the GCE; electrochemical rateconstants are ”just” GCE-weighted canonical rateconstants. This is conceptually important becausethe fixed-potential rate theory enables treating allpotential-driven reactions within a single formalisminstead of relying on separate theories for differentcases.

This provides a unified framework for computingand understanding both the barrier and prefactor froma single formalism towards modelling beyond adiabaticinner-sphere reactions to include e.g. non-adiabaticityand tunneling. Herein, specific rate expressions arederived for typical electrocatalytic adiabatic ET, PT,and PCET reactions. Fixed potential rate expressionhave been derived for i) general electrocatalyticreactions with ((5)) and without ((2)) the TSTapproximation, ii) electronically adiabatic ET, PT andPCET reactions using a grand canonical Marcus-likeGCE-EVB theory in (13).

The GCE-EVB formalism enables computing thegrand energy barrier in terms of fixed potentialreorganization energy and the reaction grand energyin analogy with the canonical EVB or Marcus theory.GCE-EVB can explain and predict the electrocatalytic”Marcus-like” behavior in energy barriers and TSTgeometries as a function of the thermodynamic drivingforce. GCE-EVB enables also the computation ofnon-linear energy relationships and Tafel slopes andgeneral BEP-Tafel relations in goo accuracy using justfew DFT parameters; this has shown for the Volmerreaction using fixed potential (constrained) DFTdeveloped here. Also, quantitatively accurate barriersand TST geometries can be predicted using a few self-consistent GCE-DFT calculations. These features areexpected to make the GCE-EVB approach particularlysuitable for electrocatalyst screening studies.

The developed theory can be directly combinedwith modern, solid-state ab initio methods to capture

the complexity of the electrochemical interface atconstant potential in a self-consistent manner. Inthis sense, the model is fully ab initio and allparameters can be directly computed. A set of widelyimplemented DFT-based tools suffices to compute allthe needed parameters in a self-consistent manner.This should enable the computational communityadopt the theoretical framework and to progress froma thermodynamics-based description of electrocatalysisto addressing also electrocatalytic kinetics underexperimentally realistic conditions.

The advances herein enable further developmentof theory and computational methods to address e.g.tunnelling pathways and non-adiabaticity in electro-chemical systems from first principles. Understand-ing and controlling (non-adiabatic) tunneling can openup new reaction pathways to avoid constraining scal-ing relations[163, 164, 165] encountered for adiabaticPCET reactions. Besides applications, the advancedrate theories will improve fundamental understandingof electrochemical kinetics in e.g. de-coupled and non-adiabatic ET, PT, and PCET. This contributions areespecially important for weakly-binding catalysts, butneglected in computational studies thus far.

7. Acknowledgements

I acknowledge support by the Alfred KordelinFoundation and the Academy of Finland (Project No.307853). Computational resources were provided byCSC – IT CENTER FOR SCIENCE LTD. I alsowish to thank the anonymous reviewer for constructivecriticism and for providing comments to improve themanuscript.

8. Declaration of interest

Declarations of interest: none

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download fileview on ChemRxivmain_article.pdf (1.12 MiB)



Supporting information for ”Grand canonical rate

theory for electrochemical and electrocatalytic

systems I: General formulation and proton-coupled

electron transfer reactions”

Marko M. Melander

Nanoscience Center, P.O. Box 35 (YN) FI-40014, Department of Chemistry,

University of Jyvaskyla, Finland


10 June 2020

Supporting Information 2

1. Problem of combining ”orbital-based” rate theories and first principles

methods: Choosing localized empty and filled orbitals

1.1. Orbital based electron transfer rate theories

There are two commonly used orbital-based approaches for describing charge transfers

rate at electrode surfaces. The first one was developed by Dogonadze, Levich, and

Kutznetsov (DLK)[1], who assumed a weak interaction between the donor and the

acceptor. Their treatment yields an expression similar to Marcus theory and the model

goes by various names including Marcus-Hush-Chidsey[2], Gerischer[3], Marcus-DOS[3]

or just the density-of-states (DOS) model. In the case of a metallic electrode, the

molecular orbitals interact with a continuum of electronic states from the metal and

therefore one needs to integrate over all the metallic bands. An implicit assumption

in the DOS model is that charge transfer takes place between two one-electron orbitals

rather than two many-electron wave functions. Also, the effect of the electrode potential

E is assumed to linearly change the occupation of orbitals without otherwise changing

the one-electron levels. For electrochemical charge transfer reactions require information

only on the molecular orbital ε0 and its DOS ρ0(ε0) and (quasi-continuum) of electrode

bands ε with DOS ρ(ε). In this case the charge transfer is [1, 4] (see also 1)

kDOS(E) =

∫dεW (ε, ε0)f(ε− E)ρ(ε)ρ0(ε0) (1)

where W is the transition probability and f is the Fermi-Dirac distribution.

Originally, the DLK model was derived for the weak interaction limit and harmonic

energy surfaces in which case the result is well-known [1, 4]

kDOS(E) ≈√

β

4πλ

∫ ∞−∞

dε|Hab(ε, ε0)|2f(ε− E)

exp

[−β (λ+ e0(E0 − E)− ε0)2

4λ

] (2)

Hab(ε, ε0) = 〈ψε0a |H|ψεb〉 denotes the Hamiltonian matrix element between the

molecule and electrode orbitals corresponding to energy levels ε0 and ε, respectively,

in the initial a and final b diabatic states. E is the electrode potential and E0 is the

formal equilibrium potential. Depending on the model used for the reactant DOS, the

weakly interacting limit by Dogonadze, Gerischer’s model with a Gaussian dependency

or Schmickler’s model (see below) maybe obtained as shown in Ref. [5]

The other approach is due to Schmickler[6, 7] and has been dubbed as the potential

energy curve (PEC) method[5]. The PEC method applies a modified Newns-Anderson

(N-A) Hamiltonian for building the PECs


Figure 1. Conventional (left) and GCE Marcus theory (right). The conventional

theory is based on transitions between single-electron orbitals while the current GCE

framework utilized general many-electron wave functions.

HN−A = ε0n0 +∑k

εknk +∑k

(vk0c†kcr + vrkc

†0ck)

+1

2

∑i

hωi(p2i + q2

i ) + (n0 − z)∑i

(hωigiqi)(3)

where the terms describe reactant orbital, orbitals of the electrode, electron

exchange terms using coupling matrix elements v, harmonic bath at frequencies ωi,

momenta pi and coordinate qi while the last term couples the reactant at charge state

z to the harmonic bath. Connecting the initial and final states of the redox reaction

along a charge transfer coordinate rq and using N-A Hamiltonian, the PEC is

u(rq, εF ) =r2q

4λ+ (ε0 + rq − εF )〈n(rq)〉

+∆

2πln[(εF − rq − ε0) + ∆2]

(4)

where 〈n(rq)〉 = 1/2 + 1/π tan−1((εF − rq − ε0)/∆) is the charge at rq and

∆(ε′) = π∑

k|v0,k|2δ(ε′−εk) is the effective coupling constant. Then, the charge transfer

barrier is u‡ = u(rq = rmax, εF )− u(rq = 0, εF ) and the rate is

kPEC = κ exp[−βu‡

](5)

At the weak interaction limit, both the DOS and PEC models are in their essence

formulations of Fermi’s Golden rule describing electron transfer between single electron

orbitals.[7]


1.2. Orbital based Fermi Golden rule formulation

Here a modern and general Golden rule is revisited to discuss inherent limitations of the

orbital-based formulation. For this purpose the initial (final) state at the initial (final)

geometry can be approximated by a single diabatic electronic state Ψ(Rinitial) ≈ |ψI〉 |χj〉and using the Hamiltonian in Eq.(6). A similar equation can also be written for adiabatic

states as shown in Ref. [8, 9]. In the diabatic Fermi Golden rule formulation the

Hamiltonian is[10, 11]

Hel =∑i

EIi

∣∣I i⟩ ⟨I i∣∣+∑f

EFf

∣∣F f⟩ ⟨F f∣∣

+∑if

∆if (∣∣I i⟩ ⟨F f

∣∣+∣∣F f⟩ ⟨If∣∣) (6)

where∣∣Kk

⟩= |ψk〉 |χk〉 is a vibronic wave function consisting of |ψk〉, a one

electron orbital and |χj〉 a nuclear wave function. Reaction rates are computed

using the general flux formulation presented in the main article with the following

transition probability and flux[10, 11]: P (E) =1

2(2πh)2 Tr

[F δ(E − HN)F δ(E − HN)

]and F = 1/h∆[|0〉〈1| + |1〉〈0|], respectively. ∆if is a general diabatic coupling term,

which in the Franck-Condon approximation is

∆if =∣∣∣ 〈ψf |V |ψi〉∣∣∣2∑

kl

|〈χk|χl〉|2

= |Vif |2∑kl

|〈χk|χl〉|2(7)

Following standard thermalized Fermi-golden rule derivation[12, 10, 11] for a

transition between two electronic states gives

kI→F =2π

h

∑i∈I,f∈F

ki→f (8a)

ki→f =|Vif |2∑

l exp[−Ei,l]∑lk

exp[−Ei,l]|〈χk|χl〉|2 × δ(Ei − Ef + El − Ek)

= |Vif |2F (Ei − Ef )(8b)

where F (Ei −Ef ) is the thermally averaged Franck-Condon factor (see Section ??

of the SI) . If the nuclear wave functions are taken to be those of a harmonic oscillator,

the Marcus barrier and the rate constant can be obtained from the derivation in Section

2 of the SI. Note that transition between all one electron orbitals are considered here.

To obtain the DOS-model for electron transfer, only a subset of the transition rates

is considered. Intuitively, for an reduction of a molecule, transitions from the localized


occupied metal orbitals to empty orbitals localized at the molecule should be considered.

This leads to

kred =2π

h

∑i∈filled

∑f∈empty

ki→f

=2π

h

∑f∈empty

∫dεiρ(ε)f(ε− εF )ki→f

(9)

where the second equation highlights the close correspondence with the DOS

method (Eq. (1)), ρ(ε) is the DOS and f the Fermi-Dirac distribution. Note also

that the PEC method uses a Hamiltonian similar diabatic Hamiltonian here. In PEC

the total transition probabilities from initial to final state are also computed using

orbital-to-orbital formulation described above.

1.3. Choosing the orbitals

Both DOS and PEC share a fundamental open question: how does one choose the

localized and empty/filled orbitals? This situation is faced in a typical first-principles

calculations, where (canonical) one electron orbitals are highly delocalized even when

charge-localized diabatic states are used, making the choice of active orbitals difficult.

An important result learned from orbital localization methods[13, 14, 15, 16] is that

the energy from a single determinant method such as DFT or Hartree-Fock methods

is invariant to orbital rotation within the occupied molecular orbitals. Thus, occupied

orbitals can be localized using a unitary rotation which leaves the energy, and the total

wave function unchanged ; during this process the spatial shape and spread of filled

one electron orbitals are drastically changed. Also, the empty, virtual orbitals can be

localized separately. However, the filled and empty are not allowed to mix during the

localization to avoid changes in occupation of numbers[17]. As mixing the unoccupied-

occupied subspaces is forbidden, orbital localization is performed separately for the

empty and filled orbitals and consequently two different unitary transforms are required.

A concrete example helps to understand why the orbital localization causes practical

difficulties. Consider for example an outer-sphere ET from an electrode to O2 forming

a superoxide species. Here the initial state wave functions |I〉 would be occupied

orbitals localized on the metal and the final state orbitals |F 〉 would be empty states

localized on O2. After a normal DFT calculation, one performs a unitary transform on

both the initial and final states separately such that the the orbitals are well localized

to the molecule and metal for both states: |I〉 = U∣∣∣IfilledDFT

⟩and |F 〉 = V

∣∣F emptyDFT

⟩, with

U U † = 1 and V V † = 1. Note that nuclear wave function remain unchanged as the

electronic energy is unaffected by the transformation. Thus, the unitary transformation

leaves the thermally averaged Franck-Condon weight unchanged.

However, the electronic coupling elements for a given Vif = 〈i|H|f〉 change

drastically as the electronic orbitals are rotated. This is easily seen from the close


correspondence[18] between the coupling and overlap elements 〈i|V |f〉 ≈ v 〈i|f〉, where

v is a constant. Changing from the localized to delocalized states is written as

〈i|f〉 = 〈idft|U †V |fdft〉 6= 〈idft|fdft〉 when final (empty) and initial (filled) canonical

DFT orbitals are localized separately. Only when U = V is the overlap between the

localized and canonical orbitals the same; this would require of mixing of the filled and

empty canonical orbitals resulting in changes in the total energy and the total wave

function and is therefore discouraged.

As a concrete example consider Eq. (9) where only states below the Fermi-level

contribute to the reduction rate. As potential is changed, some orbitals become empty

or occupied changing the driving force of the reaction which remains unchanged under

the orbital localization. The rate is dictated by transition probability directly related to

the matrix elements in non-adiabatic reactions. Thus, the rate of non-adiabatic electron

transfer reaction depends on how the orbitals are chosen and localized. This can lead

to inconsistent and incorrect interpretation of the electrochemical rate as a function of

the potential if basic single determinant methods are used to parametrize the DOS or

PEC models. Great care is needed when the methods and as emphisized in Ref. [19]

”this is not a failure of the computational methods used but is a consequence of how the

rate constant is defined by the phenomenological equations. It is therefore important to

choose the approach which is equivalent to the experiment or thought experiment that

the theory is attempting to reproduce”.

Based on the above discussion, a very important conclusion is reached: the

rate obtained from the orbital-based Fermi golden rule, DOS or PEC using only

the ”active orbitals” depend on the way the orbitals are localized. Therefore, one

needs to acknowledge that orbital localization needed when the orbital-based models

are parametrized using canonical DFT methods, leads to arbitrary changes in the

rate constant depending on the localization or rotation used scheme. Hence, while

the energy, density, and the total wave function remain unchanged after a unitary

transformation of the orbitals, single orbitals and single orbital overlaps will necessarily

be affected. Therefore, a unitary transformation such as orbital localization will

unphysically affect the rate obtained from methods using one-electron orbitals and

orbital-to-orbital transitions to compute the transition probability. If one electron-based

the DOS or PEC methods are parametrized using first-principles approaches, methods

such as fragment orbital DFT[20] methods might be applicable.

Care is also required when using many-body wave functions for computing the

rates. While unitary transform does not change any observables of a single diabatic

wave function, the off-diagonal matrix elements might be sensitive to orbital rotations.

However, in approaches such constrained DFT employed in this work, the coupling

elements are functionals of the electron densities of the initial and final state[21] and as

such in principle unaffected by orbital localization.


2. Adiabatic and non-adiabatic harmonic TST rates

Here classical harmonic TST (HTST) for adiabatic and non-adiabatic reactions within

GCE are derived.

2.1. GCE-TST with classical nuclei

For classical nuclei, the general rate equation in the GCE is written in terms of the time-

integral of the flux correlation function contains all the dynamic information[22, 11]:

Pr(p,q) = limt→∞ h[f(qt)] =∫∞

0dtd

dth[f(qt)] =

∫∞0dtCff (qt,pt). With this definition

the classical TST rate is

k(T, V, µ)ΞI =∞∑N=0

exp[βµN ]

∫dEPcl(E) exp[−βE]

=∞∑N=0

exp[βµN ]

∫dpdq exp[−βH(p,q)]F (p,q)Pr(p,q)

(2πh)N

=∞∑N=0

exp[βµN ]

∫dpdq exp[−βH(p,q)]F (p,q)

∫∞0dtCff (t)

(2πh)N

≈∞∑N=0

exp[βµN ]kBT

hQ‡∫dtδ(t) =

∑N

exp[βµN ]kBT

hQ†

≡ kBT

hΞ†

(10)

where on the second last line making the short time approximation[22] to Cff (t→

0) =kBT

hQ‡δ(t) leads to the TST expression.

2.2. Adiabatic HTST

The general TST rate equation is shown in Eq. 5 of the main article. First, consider

a general case where potential the number of both nuclei and electrons is allowed to

fluctuate. Usually, for NN classical nuclei the Hamiltonian in mass-weighted coordinates

(xi) and momenta Pi is written as Hcl =∑

i∈NNP 2i + V (xi). V (xi) defines the (Born-

Oppenheimer) potential energy surface.

Then consider a system is open to electrons at a fixed electron chemical potential

while number of nuclei is fixed. Also, the system is assumed adiabatic meaning

that the number and distribution of electrons adjusts instantaneously to the nuclear

configuration. This is the common situation considered in first principles calculations

at fixed electrode potential calculations. For this case, the Kohn-Sham-Mermin theorem

guarantees that electronic energy and distribution are unique to a given electron

chemical potential and external potential (here provided by the nuclei). Hence, the

potential energy V is not only a parametric function of the nuclear positions but also


the chemical potential of the electrons. Furthermore, as shown in Ref. [23], the grand

free energy of the electrons is given by Ωn(T, V,NN , µn; xi). As the nuclei move the

on the effective potential energy surface provided by the electrons, one recognizes that

V (xi, µn) = Ωn(T, V,NN , µn; xi) (see Ref. [23]). Then, for the open electronic system,

the classical Hamiltonian for the nuclei is

H(NN)cl =∑i∈NN

P 2i + V (xi, µn) ≡ Hcl (11)

where I and ‡ denote the initial and transition states. The TST rate is written as

[12]

kTST (NN , V, T )QI =

∫N

dP

∫N

dx exp[−H‡clβ

]δ(f(x))(∇f · PN)h(∇f · PN) (12)

where f is the N − 1 dimensional dividing surface between the reactants and

products, ∇f · PN = Pn‡ is the momentum normal to f identified as the reaction

coordinate, h(∇f · PN) = h(Pn‡) is a step function separating the reactant ad product

basins, and δ(f(xN)) restricts the geometries to lie on the dividing surface. With these

definitions the canonical HTST at fixed electron chemical potentials follows from:

kHTST (T, V,NN) =

∫N

dP

∫N−1

dxPn‡

exp[−H‡clβ

]ZI

=

∫NdP∫N−1

dxPn‡ exp[−β(

∑Ni=0 1/2P 2

i + V (xi, µn)‡)]

∫NdP∫Ndx exp

[−β(

∑Ni=0 1/2P 2

i + V (xi, µn)I)]

=1√2πβ

∫N−1

dx exp[−βV (xi, µn)‡

]∫Ndx exp[−βV (xi, µn)I ]

≈ vN√2π

∏N−1i vi∏N−1i v‡i

exp[−β(Ω‡n − ΩI

n)]

=vN2π

exp[−β(Ω‡N − ΩI

N)]

=vN2π

exp[−∆Ω‡Nβ

]

(13)

where at the second last row the effective potentials are Taylor expanded in terms

of normal mode coordinates with corresponding frequencies vi and vN is the frequency

along the reaction coordinate:V ‡/I = Ω‡/IN + 1/2

∑i viq

2i . The last equality follows from

setting the nuclear vibrational entropy SN = kB ln(∏N−1

i vi/∏N−1

i v‡i

)and setting the

total grand free energy to ΩN = Ωn − TSN . Here the subscript N reminds that the

number of nuclei was kept fixed above. Note that Eq. (13) would be used in

typical first principles calculations at fixed electrode potentials where the

electron chemical potential and number of nuclei are fixed.

The above treatment can also be extended to treat situations in which both the

number of electrons and nuclei are allowed to fluctuate. This is straight-forward and


can be obtained by. Inserting Eq. (13) in Eq. 5 of the main paper and applying Eq. 6

of the main paper leads to

kHTST (T, V, µ) =〈vN〉µ

2πexp[−∆Ω‡β

](14)

where 〈vN〉µ is the effective frequency along the reaction coordinate computed using

effective fixed potential PESs.

2.3. Non-adiabatic HTST

Next, non-adiabatic harmonic transition state theory (NA-HTST) approximation to

the rate is developed. Unlike for the canonical case, only a fixed number of nuclei is

treated. NA-HTST also requires the calculation of matrix elements HAB = 〈ΨA|H|ΨB〉.These HABs are defined only when |ΨA〉 and |ΨB〉 have the same number of both

electron and nuclei. Also, the adiabatic approximation cannot be used and the

electrons do not instantaneously adapt to nuclear positions. Hence, unlike for the

adiabatic case, constant electron number V (x, n) rather that constant electron potential

V (x, µn) is used. The appropriate Hamiltonian is given by Eq. (6), in which Hcl =∑i∈NN

P 2i /2mi + Vi(xi).

Using this Hamiltonian, assuming a quadratic potential V and applying the Golden

rule form the basis for NA-HTST. This derivation can be found in e.g. Ref [24].

Another path, presented below, is to use the classical transitions state theory using

the Landau-Zener transition Pr probability[25, 12] and assuming that the potential

energies are quadratic. Then, the following identities are used: The reorganization

energy and vibrational frequency along the reaction coordinate are related as λ =

2v2N∆q2 = 2mv2

N∆x2, where ∆q and ∆x are the geometric differences of the initial

and final states in mass weighted and cartesian coordinates states, respectively. The

differences of forces can written as gradient of the two parabolas at the transition state

as shown in Ref. [24] to yield |∆F |‡ = λ/∆x. With these definitions, fixed number

(canonical) electronic/nuclear NA-HTST can be derived:


knaHTST (T, V,NN , Nn) =

∫N

dP

∫N−1

dxPrPn‡

exp[−H‡clβ

]ZI

=

∫N

dP

∫N−1

dx

(1− exp

[− 2π|HIF |2

h|Pn‡∇n‡(VI − VF )|

])Pn‡

exp[−H‡clβ

]ZI

linearize exp≈

∫N

dP

∫N−1

dx2π|HIF |2

h|Pn‡∇n‡(VI − VF )|Pn‡

exp[−H‡clβ

]ZI

forces=

∫N

dP

∫N−1

dx2π|HIF |2

hPn‡|∆F |Pn‡

exp[−H‡clβ

]ZI

integrate P=

2π|HIF |2

h|∆F |

∫N−1

dx exp[−V ‡β

]∫Ndx exp[−VIβ]

harmonic TST≈

√2πβ|HIF |2

h|∆F |vN

∏N−1i vi∏N−1i v‡i

exp[−(E‡ − EI)β

]vib. entropy

=√

2πβ|HIF |2

h|∆F |vN exp

[−∆A‡β

]|∆F |‡=λ/∆x≈

√2πβ

√mvN∆x|HIF |2

hλexp[−∆A‡β

]λ=2mv2N∆x2

=

√πβ

h2λ|HIF |2 exp

[−∆A‡β

]Marcus barrier≈

√πβ

h2λ|HIF |2 exp

[−β (∆A0 + λ)2

4λ

](15)

The above rate is derived for fixed number of electrons and nuclei. As done for

the adiabatic case, this fixed particle rate needs to be turned to a fixed potential rate.

In particular, the electronic subsystem needs to be open in order to study kinetics

at a fixed electrode potential. However, generalization of the NA-HTST to GCE is

significantly more difficult compared the the adiabatic as discussed. The electronically

GCE NA-HTST can be accomplished the approach in Section 4 of the main paper.

To gain more insight, it is useful to compare the above derivation to the GCE-EVB

picture used for deriving the GCE equivalent of Marcus barriers and Landau-Zener

transmission probability in Sections 3 and 4 of the main paper. These considerations

directly lead to the hybrid NA-GCE-EVB rate constant

kNA−GCE−EV B(T, V, µn) ≈

⟨√πβ

h2Λ|ΩIF |2

⟩µn

exp

[−β (∆Ω + Λ)2

4Λ

](16)

where the ΩIF is the coupling matrix element, and the prefactor is computed for

either i) some particle number and assumed to independent of the electrode potential or

ii) various particle numbers and weighted according to the grand canonical distribution.


3. Grand canonical diabatic states using density matrices

Here the theoretical and technical details of the GCE-EVB theory are presented. To

form the GCE diabatic states, the work of Reimers[8, 9] on canonical ensembles is

followed. As noted by Reimers, the density matrix ˆρ can be written using either

adiabatic or non-adiabatic states. Especially, when only two electronic states are used,

the connection of the Born-Huang expansion bears striking resemble to the commonly

used 2 × 2 diabatic Hamiltonians used for deriving electron transfer rate theory. In

the canonical ensemble, the diabatic states are φI and φF corresponding to the electron

localized on the initial (I) or final (F) state while the molecular electronic-vibrational

Hamiltonian is

Hdia(N, V, T ) =

[HII HIF

HFI HFF

](17)

with

HII(R) = 〈φI |Hel(R)|φI〉+ Tnuc = EI + Tnuc (18a)

H∗FI = HIF = 〈φI |Htot(R)|φF 〉 (18b)

HFF = 〈φF |Hel(R)|φF 〉+ Tnuc = EF + Tnuc (18c)

where Tnuc is the nuclear kinetic energy operator, H = Hel + Tnuc, and Hel includes

electron kinetic energy and Coulomb energies of the electron-nucleus system. The Born-

Huang, or vibronic, states are

Ψi(R) =∑j

[CIij |ψI〉 |χj〉+ CF

ij |ψF 〉 |χj〉]

=∑k=I,F

|ψk〉∑j

CIki,j |χj〉

(19)

where Ψ, ψ, and χ are the vibronic, electronic, and nuclear wave functions,

respectively. C is the weight of each state. Using these definitions the, the canonical

ensemble density matrix is

ρ(N, V, T ) =

[ρII ρIFρFI ρFF

](20)

with ρAB =∑

j CAjiC

Bji and the total density matrix has dimension (2×Ni)×(2×Ni).

Next the diabatic canonical Hamiltonian is generalized to the grand canonical

ensemble. To simplify the notation, it is assumed that the initial and final can

approximated as a single electronic state and a single vibrational state - extension to

include more vibrational state is straight-forward. Then, the total vibronic state is

written as Ψ(R) ≈ cI |ψI〉 |χI〉 + cF |ψF 〉 |χF 〉. In electron transfer theory the vibronic


states are often assumed to be harmonic but here such an assumption is not needed.

Next, the total number of electrons is allowed to fluctuate while the electron Fermi-

level is fixed. These are effectively introduced by using the equilibrium reduced density

operator within the GCE [23]

ˆρred =∑N

pN∑ij

|ΨNi〉〈ΨNj|

with |Ψi〉 = cI |ψI〉 |χI〉+ cF |ψF 〉 |χF 〉(21)

where pN is the GCE weight for a state with N electrons. The resulting density

matrix will have N -dimensional block-diagonal form with 2 × 2 blocks. Similarly

the Hamiltonian matrix is made of Eq.(17) HNdia blocks. Diagonalizing each block

separately will give canonical adiabatic states whereas Tr[

ˆρredH]

gives the adiabatic

grand canonical free energy. Because the trace is cyclic, both ˆρred and H can be

reorganized which keeps the (diabatic) free energy unchanged as long as diagonal

elements remain at the diagonal. This freedom is utilized can be utilized for reorganizing

the density matrix so that the upper part of ˆρred and H correspond to the initial state and

the lower part to the final diabatic states. Tracing the upper and lower parts separately,

diabatic GC free energies of initial and final states (ΩII and ΩFF ) are obtained. These

states and their effective couplings are used in the main paper to derive the GCE-EVB

model.

4. Grand canonical perturbation theory

Here it is shown how the needed GCE-EVB states rigorously defined and computed

with grand canonical perturbation theory. To keep the present work as general as

possible i.e. allowing both the number of electron and nuclear species to fluctuate, a

simple effective Hamiltonian cannot be specified. Instead, explicit sampling of the GCE

and number of electrons and nuclei is needed. In this case a novel extension of the

canonical thermodynamic perturbation theory[26] to GCE is utilized. Along these lines,

the canonical energy operator H = H0 + V is defined and partitioned to contributions

from the unperturbed H0 part and a perturbation V . The total GC partition function

Ξ and grand energy Ω are given as[23]

Ξ = Tr[H − TS − µN ] and exp[−βΩ] = Ξ (22)

Then, the total grand energy can be multiplied and divided by the unperturbed

grand energy

exp[−βΩ] =exp[−β(Ω− Ω0)]

exp[−βΩ0]=

exp[−β(ΩV )]

exp[−βΩ0]= 〈exp[−βV ]〉0 (23)


where the last identity means that the perturbation part of the grand energy is

obtained with GCE sampling of the perturbation. For electron transfer reactions, the

total Hamiltonian can be written as[27]

H = K + U + Vx (24)

where K is the kinetic energy, U is the interaction energy and Vx is the perturbation

which depends on extent of the reaction: x = 0 and x = 1 correspond to initial and

final states, respectively. A linear switch from the initial to the final state is obtained

using a switching potential Vx = VI − x(VF − VI). This potential defines the initial and

final diabatic states and based on the energies of the initial and final states EI and EF .

Furthermore, one defines the instantaneous energy gap ∆E(R) = EF (R)− EI(R) = X

at geometry R. As noted by Zusman[28] and Warshel[29] (see also Ref. [30] for a

combined discussion), the energy gap coordinate is directly related to the (solvent/bath)

reorganization coordinate and both are often used in deriving electron transfer rates. It

was recently shown by Jeanmairet et.al.[27] that the energy gap coordinate is a valid

reaction coordinate also within GCE.

Combining the two-state GCE diabatic model for the initial I and final F states

with the general perturbation result leads to

exp[−β∆Ω] =〈exp[−βVF ]〉F〈exp[−βVI ]〉I

=

∑N e

βµN∫dPNdRNe−βVF∑

N eβµN

∫dPNdRNe−βVI

=ΞVF

ΞVI

(25)

which results in ∆Ω = −β−1 ln(ΞVF /Ξ

VI

). Next, the sampling is constrained to a

specific region of the energy gap. As recently shown in Ref. [27], a one-to-one mapping

exists between the vertical energy gap 〈∆E〉x, x, the potential Vx, and the probability

(px) of being in microstate sampled from the GCE: x↔ 〈∆E〉x ↔ Vx ↔ px. Introducing

the energy gap coordinate and noting that the energies of I and F are computed from

the same Hamiltonians except for the ”perturbation” part allows writing

∆Ω = −β−1 ln

(∑N e

βµN∫dPNdRNe−β(∆E+V I)∑

N eβµN

∫dPNdRNe−βV I

)=

− β−1 ln⟨e−β∆E

⟩I

= β−1 ln⟨eβ∆E

⟩F

(26)

where ∆E = VF − VI is used. One can also obtain a probability distribution for

the energy gap by performing constrained sampling[12] of the grand energy curves

Ξi(X) =∑N

eβµN∫dPNdRNe−βEiδ(∆E(R)−X) (27a)


pi(X) =Ξi(X(R))

Ξi= 〈δ(∆E(R)−X)〉i =∑

N eβµN

∫dRNdPNδ(∆E(R)−X)e−βEi∑

N eβµN

∫dRNdPNe−βEi

(27b)

so that Ξi =∫dXΞi(X) ≡ e−βΩi and Ωi(X) = −β−1 ln(pi(X)) + Ωi. Above Ωi is

the diabatic grand energy and i = I or F . Using the last identity and observing that

integration over the probability is unity one obtains

Ωi = −β−1 ln

∫dXe−βΩi(X) (28)

An important identity linking the diabatic grand energies to the energy gap is

obtained by using the energy gap as the reaction coordinate. In this case the constrained

grand energy along the energy gap leads to

ΩI(∆E) = −β−1 ln(ΞI(∆E)

)=

− β−1 ln

(∑N

eβµN∫dPNdRNe−βEI(RN )δ(∆E(RN)−∆E)

)

= −β−1 ln

(∑N

eβµN∫dPNdRNe−β(EF (RN )−∆E(R))δ(∆E(RN)−∆E)

)

= −β−1 ln

(eβ∆E

∑N

eβµN∫dPNdRNe−β(EF (RN ))δ(∆E(RN)−∆E)

)

= −∆E − β−1 ln

(∑N

eβµN∫dPNdRNe−βEF (RN )δ(∆E(RN)−∆E)

)= −∆E + ΩF (∆E)

(29)

At this point all relevant free energy identities within the GCE corresponding to

the commonly used identities used for deriving the canonical Marcus theory have been

obtained.[31, 32, 33, 34, 35, 29, 36] Refs. [31, 32, 33, 34, 35, 29, 36] show various ways

to obtain the iconic canonical Marcus rate constant. To arrive at the corresponding rate

constant in the GCE, it is shown that detailed balance is satisfied. At the transition

state the initial and final diabatic grand energies are equal which results in

ΩI(∆E‡) = ΩF (∆E‡)

→ −β−1 ln(pI(∆E

‡))

+ ΩI = −β−1 ln(pF (∆E‡)

)+ ΩF

→ pI(∆E‡)

pF (∆E‡)= exp[−β(ΩF − ΩI)] = exp[−β∆ΩFI ]

(30)

which shows that detailed balance is satisfied. The diabatic grand energy surfaces

are computed from the energy gap distribution[35]


gI(∆E) = −β−1 ln(pI(∆E)) and

gF (∆E) = −β−1 ln(pF (∆E)) + ∆ΩFI

(31)

The transition state can then be identified from the intersection of the relatice grand

energy curcves: gI(∆E‡) = gF (∆E‡). Computing the reaction rate using the standard

transition state theory expression gives

kIF = κexp[−βgI(∆E‡)

]∫d∆E exp[−βgI(∆E)]

= κpI(∆E‡) (32)

showing that the reaction rate is determined by the energy gap distribution function

pI(∆E) = 〈δ(∆E(R)−∆E)〉I from Eq. (27). Note, that microscopic reversibility is

satisfied by construction. To obtain the iconic Marcus rate within GCE, one may follow

the perturbation theory route[26, 35] and perform a cumulant expansion on the energy

gap distribution. It has been shown in several previous studies[35, 37, 32] that the second

order cumulant expansion results a Gaussian form for the energy gap distribution:

pI(∆E) =1√

2πσIexp

[−(∆E − 〈∆E〉I)2

2σ2I

](33)

where 〈∆E〉I is the energy gap expectation value in the initial state obtained from

Eq. (27) and σI = 〈(∆E)2〉I − (〈∆E〉I)2 = 〈(∆EI − 〈∆EI〉)2〉I is the gap variance. The

Marcus relation is then obtain after standard manipulations[32, 25] by inserting these

relations in Eq. (29) result in the GCE Marcus rate of Eq. 12 of the main paper.

5. Cumulant expansion

The cumulant expansion is a widely used and powerful statistical approach for

approximating the reaction barrier. To utilize the cumulant form, the ensemble average

of the energy gap is replaced by its time integral using the Fourier presentation of the

delta function

pI(∆E) =ΞI(∆E)

Ξi= 〈δ(∆E(R)−∆E)〉i =

β

2π

∫ ∞−∞

⟨e−iγβ(∆E(R)−∆E)

⟩Idγ (34)

As shown by e.g. Matyushov and collaborators[38, 39], the exponential form can be

expanded using cumulants to result in non-Gaussian gap distribution functions. In this

treatment, the gap function is written in terms using a cumulant generating function.

One obtains[39] ⟨e−iγβ(∆E(R)

⟩I

= eΩI(γ) (35)

where ΩI(γ) is written in terms of cumulants as


〈(δE(R))n〉a =

(i

β

)n∂nΩI(γ)

∂γn

∣∣∣∣γ=0

(36)

and δE(R) = E(R)−〈E(R)〉 and 〈E(R)n=1〉I = 〈E(R)〉I . For a Gaussian variable,

〈E(R)n>2〉I = 0 which results in quadratic free energy curves along the energy gap

coordinate and the identities shown in the above Section.[38, 39, 40] For more complex

environments and distributions, higher order cumulants contribute and more complex

free energy curves emerge as shown in [38, 39, 40].

6. GCE-cDFT script

from ase import *

from ase.io import write, read

from ase.units import mol, kJ, kcal, Pascal, m, Bohr

from ase.data.vdw import vdw_radii

from gpaw import *

from gpaw.analyse.hirshfeld import HirshfeldPartitioning

from gpaw.analyse.vdwradii import vdWradii

from ase.calculators.vdwcorrection import vdWTkatchenko09prl

from gpaw.cdft.cdft import *

from gpaw.poisson import PoissonSolver

# Import solvation modules

from ase.units import mol, kJ, kcal, Pascal, m

import numpy as np

from gpaw.solvation import (

SolvationGPAW, # the solvation calculator

EffectivePotentialCavity, # cavity using an effective potential

Power12Potential, # a specific effective potential

LinearDielectric, # rule to construct permittivity function from the cavity

GradientSurface, # rule to calculate the surface area from the cavity

SurfaceInteraction # rule to calculate non-electrostatic interactions

)

from gpaw.poisson import PoissonSolver

import sys, os

from gpaw.solvation.sjm import *

# define system

atoms = read('Au_H3O_to_H2O_H_4.46.traj')

######## Potential

E_target_RHE = 0.0 #PZC of the pure gold surface

E_pzc_comp = 5.02 # absolute wrt continuum, computed

E_pzc_exp = 0.559 # vs RHE, Experimental


E_ref_comp_RHE = E_pzc_comp - E_pzc_exp # computational reference potential wrt RHE

E_target = E_ref_comp_RHE + E_target_RHE

# Solvent parameters

u0 = 0.180 # eV

epsinf = 78.36 # Dielectric constant of water at 298 K

gamma = 18.4 * 1e-3 * Pascal * m

T = 298.15 # K

vdw_radii = vdw_radii.copy()

vdw_radii[79] = 2 #Au

atomic_radii = lambda atoms: [vdw_radii[n] for n in atoms.numbers]

# Define SJM calculator

calc = SJM(h=.18,

spinpol = True,

basis = 'dzp',

xc = 'PBE',

maxiter=500,

nbands = -40,

symmetry ='off',

eigensolver = Davidson(3),

kpts = (3,3,1),

poissonsolver='dipolelayer': 'xy',

occupations = FermiDirac(width = 0.05),

mixer = Mixer(beta = 0.05,nmaxold = 5,weight=90.0),

convergence = 'eigenstates': 1.0e-5, # eV^2 / electron

'energy': 1.0e-5, # eV / electron

'density': 1.0e-3,

txt = '1Au111_H3O_to_H2O_H_reorg_CDFT_TS09_%3.2f.txt'%(E_target_RHE),

potential=E_target, #define potential

dpot=0.025,

ne=-0.3,

doublelayer='upper_limit':atoms.get_cell()[2][2]-1,'start':'cavity_like',

cavity=EffectivePotentialCavity(

effective_potential=SJMPower12Potential(atomic_radii, u0,

unsolv_backside=False,

H2O_layer=False),

temperature=T,

surface_calculator=GradientSurface()),

dielectric=LinearDielectric(epsinf=epsinf),

interactions=[SurfaceInteraction(surface_tension=gamma)])

atoms.set_calculator(calc)

atoms.get_potential_energy()

# start cDFT loop with SJM calculator

cdft_a = CDFT(calc = atoms.calc,


atoms=atoms,

charge_regions = [[45,46,47,48]], # atoms for H3O

charges = [0.], # charge of the diabatic state

charge_coefs = [-3.75], #initial guess for Vc

method = 'L-BFGS-B',

txt = '1Au111_H3O_to_H2O_Hreorg_TS09_constant_pot_%3.2f.cdft'%(E_target),

minimizer_options='gtol':0.05) # tolerance for charge constraint

atoms.set_calculator(cdft_a)


# add vdw contributions

# van der Waals parameters

radii = vdWradii(atoms.get_chemical_symbols(), 'PBE')

cc = vdWTkatchenko09prl(HirshfeldPartitioning(cdft_a.calc),radii)

atoms.set_calculator(cc)



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https://doi.org/10.1021/acs.jpcb.8b08865

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