Computation of Electrode Potentials and Alignment of Electronic Energy

Home , Galvani potential, Volta potential

Computation of Electrode Potentials and Alignment of Electronic Energy Levels Jun Cheng, Marialore Sulpizi, Michiel Sprik ([email protected]) University of Cambridge Contents

1. Reversible and ideally polarizable electrodes 2. Absolute electrode potentials and workfunctions 3. Alignment of vertical and adiabatic energy levels

Key source textbooks:

F Liquids, Solutions, and Interfaces, W .R. Fawcett (Oxford University Press). BF Electrochemical Methods, A. J .Bard and L R. Faulkner (Wiley) B Fundamentals of Electrochemistry, V. S. Bagotsky (Wiley) G Physical Electrochemistry, E. Gileadi (Wiley). 1.1: Pt(111)/water interface at potential of zero charge

-6 Three kind of levels just outside vacuum 0 Electronic levels that are -4 electrode potentials CBM -2 Electronic levels that are -2 not electrode potentials

Electrode potentials that -4 0 Fermi are not electronic levels level − -6 2 OH/OH OH/H O potential vs SHE [V] SHE vs potential 2 -8 4 energy relative to vacuum [eV] energy relative to VBM -10 6 Pt (111) water

All experimental data 1.2: Pt(111)/water interface at potential of zero charge

-6 More electronic levels just outside vacuum 0 IP: Vertical ionization -4 potential of the OH− ion. CBM -2 EA: Vertical electron -2 aﬃnity of OH• radical. -EA -4 0 Fermi λ level − -6 2 OH/OH potential vs SHE [V] SHE vs potential λ -8 4

-IP vacuum [eV] energy relative to VBM -10 6 Pt (111) water

All experimental data 1.3: The two modes of an electrochemical cell

load source I

U U anode e e cathode cathode e e anode

I I K K K K e e e e Cl Cl ½H2 H ½ 2 H ½H2 Cl ½Cl2

Oxidation Reduction Reduction Oxidation

Generating electricity Driving reactions UUrev 1.4: The reversible potential is the potential at open circuit

Cl2 g H2 g o 2Cl aq 2H (aq)

U Electrode reactions in equilibrium o 'G k T p H2 pCl2 U B ln 2 2 2e 2e >@H >Cl @

Nernst relation

K K Nature of electrodes irrelevant e e

½H2 H ½Cl2 Cl

No current

U = Urev 1.5: Thermodynamics of electrolytic solutions

Chemical potential of species i in phase α ∂Gα α = μi α ∂ni

Separation in a standard chemical potentials and activity α α,◦ α μi = μi + kBT ln ai (1)

α with activity related to concentration ρi of species i in phase as ρα aα = γα i i i c◦

Due to the neutrality condition (for a 1:1 electrolyte) α − α ρ+ ρ− =0 α α Only the mean activity coeﬃcient γ+γ− can be measured 1.6: Thermodynamics of ﬁnite conducting phases

Neutrality condition relaxed. Electrochemical potential of species i in phase α ∂G˜α ˜α = (2) μi α ∂ni

Separation in a chemical and electric component α α α μ˜i = μi + qiφ (3)

• α μi is the chemical potential of species i in phase α

• qi is the charge of species i • φα is the inner (or Galvani) potential of phase α

Excess charge in conductors accumulates at the boundaries

Inner potentials account for the charge inbalance at the boundaries only

α Electrostatic solvation interactions are included in μi 1.7: Cell potential as inner potential diﬀerence of electrodes

Chlorine reduction (slide 1.3) − − Cl2 +2e → 2Cl can be carried out at the cathode of a cell Pt , H2 | H2SO4, H2O || H2O, NaCl | Cl2, Pt with H2 oxidation at the anode + − H2 → 2H +2e Cell potential = potential cathode - potential anode

U = φPt − φPt 1.8: Overpotential and electrochemical equilibrium

Electrochemical reaction free energy at the cathode S Pt S Pt,◦ S Pt Δ ˜ =2˜ − − 2˜ − =2 − − − 2 − 2 − Gc μCl μe μCl2 μCl μCl2 μe e0 φ φ

Similarly for the electrochemical reaction free energy at the anode ˜ S Pt − S − Pt,◦ S − Pt ΔGa =2˜μH+ +2˜μe μH2 =2μH+ μH2 +2μe +2e0 φ φ

The salt bridge eliminates the electrical potential diﬀerence between S and S

φS = φS Adding we obtain the total electrochemical reaction free energy S S Pt Pt Δ ˜ =Δ˜ +Δ˜ =2 − +2 + − − +2 − G Gc Ga μCl μH μCl2 μH2 e0 φ φ

= −2e0Urev +2eU

The overpotential is the driving force for the electrochemical cell reaction

η = U − Urev 1 S S = − − +2 + − − Urev μCl μH μCl2 μH2 2e0 ΔG˜ =2e0η 1.9: Open circuit voltage of cells under redox control

Cl2 g H2 g o 2Cl aq 2H (aq)

U Electrode reactions in equilibrium o 'G k T p H2 pCl2 U B ln 2 2 2e 2e >@H >Cl @

Nernst relation

K K Nature of electrodes irrelevant e e

½H2 H ½Cl2 Cl

No current 1.10: Electro-active species is adsorbed on surface

Example: Volmer reaction H aq e m o H

U Step in Hydrogen Evolution Reaction

Nature of electrode matters.

K K e

½H2 H H *H

Redox reference Electrosorption 1.11: Open circuit voltage without redox control

Electrode acts as a capacitor Q U M pzc C(Q) U Mpzc v Hfermi

Point of zero charge for metals

Nature of electrode matters.

K e K

½H2 H

Redox reference Double layer 1.12: Non-polarizable and polarizable interfaces

Non-polarizable n • Potential remains constant over a wide U current window

• Rest potential Urest equal to reversible Urest potential Urev of redox couple 0 j o • Urest determined by bulk activities

n Polarizable • Current remains zero over a wide U potential window • Urest Rest potential Urest dependent on 0 j o electrode surface and composition electrolytic solution (no redox control) 1.13: Equivalent circuit representation

Modelling the response to small perturbations in potential (steps or harmonic)

Cdl Cdl: Double layer capacitance, varying with charge (potential)

RF: Interface (Faradaic) resistance, exponentially dependent on potential Rs Rs: Solution (Ohmic) resistance, independent potential RF

• RF large: polarizable interface Only alternating current (AC) response

• RF small: non-polarizable interface: Sustains direct current (DC)

Note: This model is an oversimpliﬁcation 1.14: Computation of electrode potentials: Outline

Should apply to both reversible and ideally polarizable electrodes

• We cannot use Nernst law (no electron exchange between metal and solution) • Instead potentials will be formulated in terms of transfer to vacuum

Potentials vs SHE of M|S interface separated in absolute potentials ◦ U | (she) = U | (abs) − U + (abs) M S M S H /H2

Absolute potentials can be identiﬁed with workfunctions

1. S. Trasatti, Pure & Appl. Chem., 1986, 58, 955–966 2. S. Trasatti, Electrochim. Acta, 1990, 35, 269–271

See also Fawcett(F) 2.1: Inner, outer and surface potentials

The inner potential is separated in outer potential ψα and surface potential χα φα = ψα + χα

The outer or Volta potential is

measured at a point in vacuum just outside the phase boundary

Coming from inﬁnity this point is

• close enough so that all work againts surface charge has been carried out. • far enough so that image and dipole forces have not yet caught on.

The outer potential accounts for the work against surface charge 2.2: Separate potential for work againts surface dipoles

Deﬁnition of the real potential α α α αi = μi + qiχ (4) relation to the electrochemical potential α α α μ˜i = αi + qiψ α α α μ˜i , αi and ψ can be determined by thermodynamic experiment

• α ∞ μ˜i is the work to insert a particle i from in phase α (Eq. 2). α • qiψ is the work to bring a particle i from ∞ to “just outside” phase α. • α αi is therefore the work to insert it from there in phase α

The implication is that α α μi , φα and χ cannot be determined by experiment without further “extra-thermodynamical” assumptions 2.3: Workfunctions and chemical potentials Minus the real potential with the gas-phase chemical potential as reference α − α,◦ − g,◦ Wi = αi μi deﬁnes the work function to be distinguished from the solvation free energy α,◦ α,◦ − g,◦ ΔsGi = μi μi (5) which compares chemical potentials. The diﬀerence is the surface potential α,◦ − α α ΔsGi = (Wi + qiχ )

Rearranging the standard real potential can be written as α,◦ − α g,◦ αi = Wi + μi (6) Example: aqueous proton ◦ − g,◦ αH+ = WH+ +Δf GH+ (7)

g,◦ g,◦ where μH+ =Δf GH+ is the formation free energy of the gas-phase proton

1 + − H2(g) → H (g) + e (vac) 2 Note that the gas-phase reference state for the electron is an electron at rest g,◦ ⇒ α,◦ − α μe =0 αe = We (8) 2.4: Interface and contact potentials

Special notation for the diﬀerence in inner potential accross an interface β α αΔβφ = φ − φ Similarly for the diﬀerence in outer potential (contact potential) β α αΔβψ = ψ − ψ

D 'E \ Triple interface potential loop E D β α \ \ αΔβφ = αΔβψ + χ − χ (9)

Only αΔβψ can be measured F E FD

D I E I D 'E I 2.5: Electron exchange between two metals Copper wire welded to a platinum electrode:

e\ Pt Electrochemical equilibrium Cu 'Pt \ Cu Pt μ˜e =˜μe e\ Cu Pt Substituting Eq. 3 Cu We We Cu − Cu Pt − Pt μe e0φ = μe e0φ ~Cu ~Pt Pe Pe and Eq. 1 with ae =1 Cu,◦ − Cu Pt,◦ − Pt μe e0φ = μe e0φ

We ﬁnd for the Galvani potential diﬀerence Pt,◦ − Cu,◦ e0CuΔ Ptφ = μe μe (10) Standard chemical potentials are instrinsic (but unmeasurable) quantities ⇒ The interface potential at a metal-metal contact is a constant and so is the contact potential (using Eqs. 4,6,8 and 9) Pt,◦ − Cu,◦ − Pt Cu Cu − Pt e0CuΔ Ptψ = μe μe χ + χ = We We which can be measured using a Kelvin probe (see F) 2.6: Two similar metals not in equilibrium

Two copper wires connected to two electrodes, Cu and Cu, at diﬀerent potential Cu Cu − Cu Cu Cu − Cu μ˜e = μe e0φ , μ˜e = μe e0φ

Cu Cu Then, because μe = μe we have

e0CuΔ Cu φ = CuΔ Cu μ˜e

The chemical work for exchanging an electron between electrode terminals equals the electrical work for exchange of a unit negative test charge provided the terminals are made of the same metal 2.7: Electrochemical cell for X•/X− couple

load Cu Cu' Cell reaction

• 1 − + U X (s)+ H2(g) → X (s)+H (s) 2 M anode Pt cathode Potential vs SHE e e ΔrG UX•/X− (she) = − e0

S S' Assuming standard conditions at anode K K e e

• ½H2 H X X

Oxidation Reduction

Electrochemcial speciﬁcation of cell .. • Cu | Pt | H2(g) | HCl, H2O ..H. 2O, NaX, X | M | Cu (11) 2.8: Formal deﬁnition of potential for cell Eq. 11

The cell potential U is the potential between positive and negative electrode

U = CuΔ Cu φ

Interface potentials along an electrochemical circuit add to zero (Volta’s law)

CuΔ Ptφ + PtΔ Sφ + SΔ S φ + S Δ Mφ + MΔ Cu φ + Cu Δ Cuφ =0

Pt,◦ Cu,◦ • e0CuΔ Ptφ = μe − μe using Eq. 10:

• SΔ S φ = 0 assuming the salt bridge provides adequate electrical contact.

Cu,◦ M,◦ • e0MΔ Cu φ = μe − μe using Eq. 10:

Substituting and rearranging gives μM,◦ μPt,◦ U = SΔMφ − − SΔPtφ − (12) e0 e0

This expression is completely general making no assumptions about reversibility of the electrodes + 2.9: The reference H /H2 electrode is reversible 1 → + − Equilibrium condition for reference electrode (2H2 H + e )

1 g s Pt μ˜ =˜μH+ +˜μe 2 H2

Separating in chemical and electric component according to Eq. 3 1 g s s Pt − Pt μ = μH+ + eφ + μe e0φ 2 H2

Separating in standard and activity term (Eq. 1) + ◦ ◦ Pt,◦ 1 ◦ H p e Δ φ = μ + + μ − μ + k T ln 0 S Pt H e H2 B ◦ 2 c pH2

Substituting in Eq. 12 assuming standard conditions for the reference electrode

M,◦ ◦ 1 ◦ μ + − μ μe H 2 H2 UM|S(she) = SΔMφ − − (13) e0 e0 gives the potential of the M|S interface vs the standard hydrogen electrode (SHE)

Pt,◦ μe has been eliminated from the ﬁnal expression for UM|S(she) 2.10: Separation in absolute single electrode potentials

Expression Eq. 13 for UM|S(she) is already separated in single electrode terms

M,◦ ◦ 1 ◦ μ + − μ μe H 2 H2 UM|S(she) = SΔMφ − − e0 e0 but these are measurable potentials. Adding the solvent surface potential ◦ U | (she) = U | (abs) − U + (abs) (14) M S M S H /H2 M,◦ μ S UM|S(abs) = SΔMφ − + χ (15) e0 ◦ 1 ◦ μ + − μ ◦ H 2 H2 S U + (abs) = + χ H /H2 e0

◦ S ◦ μH+ and χ adduptorealpotentialαH+ andwehavefortheSHEterm 1 ◦ ◦ − ◦ eU + (abs) = αH+ μH (16) H /H2 2 2

◦ Inserting Eq. 7 which uses H2(g) as reference state for the proton (μ =0) H2 ◦ g,◦ e U + (abs) = Δ G + − W + (17) 0 H /H2 f H H

◦ WH+ is an ionic workfunction and U + (abs) is therefore measurable H /H2 2.11: Measuring ionic work functions

Using an electrochemical cell with an air gap (Kenrick cell)

Cell conﬁguration Cu | Hg | air | HCl | Pt, H2 | Cu

Potential accross airgap

SΔHgψ =0

Cell potential

s Hg 1 ◦ U = α + + α − − μ H e 2 H2 See Fawcett[F,3] 2.11: Absolute potential of the X•/X− cathode reaction

• − − SΔMφ in Eq. 15 ﬁxed by Nernst equation for X (S) + e (M) → X (S) equilibrium

◦ ◦ aX• • − − e0 UM|S(abs) = e0UX /X (abs) = αX• αX− + kBT ln aX−

The diﬀerence in real potentials can be expanded using Eq. 6 ◦ ◦ g,◦ g,◦ − − − − • αX• αX− = μX• μX− + WX WX

Thermodynamic cycle deﬁning the adiabatic ionization potential (AIP) of X−

◦ AIPX− UX•/X− (abs) = e0

Substitution in Eq. 14 using Eq. 17 gives U vs SHE ◦ g,◦ − − e0 UX•/X− (she) =AIPX + WH+ Δf GH+ (18)

− AIPX− can be interpreted as the workfunction for electrons bound in X with the same “ just outside solution reference” as the proton workfunction. 2.12: General expression for absolute electrode potential

To rewrite UM|S(abs) of Eq 15 in terms of work functions we apply Eq. 9

M S SΔMφ = SΔMψ + χ − χ

InsertinginEq15gives

M,◦ μ M UM|S(abs) = SΔMψ − + χ e0

Chemical potential and surface potential are again combined to a work function M We UM|S(abs) = + SΔMψ (19) e0

Eq 19 for UM|S(abs) is the central result for computational electrochemistry and was derived by Trasatti[1,2] 2.13: Absolute Electrode potential as work function Absolute potential of interface M|S eU | (abs) = W (20) e M'S\ M S e M M = W − e0 MΔSψ e\ e\ S e Contact potential relation M|S − M M M S We We We We SΔMψ = e0

M|S P~M We is the reversible work for e transferring an electron from inside M S the metal to outside the solution

These relations are valid whether the electrode is under redox control or not

Potential vs SHE ◦ U | (she) = U | (abs) − U + (abs) M S M S H /H2 2.14: Two ways to estimate the absolute SHE potential

Applying Eq. 14 to the potential at which metals carry no net charge (PZC) pzc pzc ◦ U (she) = U (abs) − U + (abs) Hg|S Hg|S H /H2 for the Hg/water interface (the model inert electrode), gives ◦ pzc pzc U + (abs) = U (abs) − U (she) H /H2 Hg|S Hg|S

Substituting in Eq. 19 yields and expression for the absolute SHE Hg ◦ We pzc U + (abs) = + Δ ψ − U (she) H /H2 S Hg Hg|S e0 in terms of electronic properties of the Hg electrode. Compare to Eq. 17 ◦ g,◦ e U + (abs) = Δ G + − W + 0 H /H2 f H H which is an expressions in terms of ionic properties of the aqueous proton ◦ These rather diﬀerent estimates of U + (abs) are equivalent and should agree H /H2 2.15: Experimental estimation of the absolute SHE “Best” experimental values selected by Trasatti (1986)[1] (See also Fawcett[3])

+ − 1 1 ◦ H (g)+e (vac) → H2(g) μ = −15.81 eV 2 2 H2 + Work function H WH+ =11.36 eV pzc − Potential of zero charge of Hg UHg|S(she) = 0.191 V Hg Electron workfunction of Hg We =4.50 eV

Volta potential diﬀerence SΔHgψ = −0.248 V ◦ Recommended value for absolute SHE U + (abs) = 4.44 V H /H2

Standard state for H+ is ideal gas at 1 mol/dm−3 Standard state for e− is vacuum at rest More recent results using mass spectroscopy methods (Tissandier et al, 1996)[4]

+ ◦ − Solvation free H ΔsGH+ = 11.53 eV

Which would imply for the surface potential of water (Eq. 5) s − ◦ χ = (ΔsGH+ + WH+ ) /e0 =+0.17V Consistent with other estimated of χs =+0.14 V[3] 3.1: Model system for electron work function method

The slab geometry familiar from computational surface science[5,6,7] vacuum | metal(M) | solution(S) | vacuum

MS z perpendicular slab Periodic boundary conditions in x, y (blue lines) VS Red line: x, y S 'M \ averaged Poisson potential. VM

VM,VS asymptotic potential on M side respectively S side

VM = VS

This a consequence of the inﬁnite extension in x, y directions 3.2: Electrode potentials from electron workfunctions

M The gap between the Fermi level ( F =˜μe ) and the vacuum on the S side − − M|S e0VS F = We (21)

The work to transfer an electron from M to the point “just” outside S = = The work for transfer from M to just outside M and from there to S

M|S M − We = We e0 MΔSψ and, therefore, with Eq. 20 M μ˜e e0U | (abs) = −VS − M S e

Similarly the gap between the Fermi level and the vacuum on the M side − − M eVM F = We Substracting from Eq. 21 gives M|S − M We We VM − VS = SΔMψ = e0

This relation, underlying the work function method, is due to Otani et al.[5] 3.3: Potential of zero charge for the Pt(111)/water interface

Computed using the electron work function method

Pt Pt|S − Pt pzc Reference DFT We We We UPt|S Solvent Experimenta -5.93−1.10.4 Otani et al.[5] PBE 5.8 −1.2 0.2 MD (liquid) Tripkovic et al.[6] PBE 5.74 −1.09 0.21 staticb ,, RPBE 5.60 −0.14 1.02 ” Schnur et al.[7] PBE - −0.7 0.6 MD (2ML) Jinouchi et al.[8] RPBE 5.96 −0.39 1.13 continuum ,, RPBE 5.96 −1.1 0.5 ML + cont.

Energy in eV, potentials in V a As quoted in Ref. [8] b Using an “optimal” ordered (ice-like) water conﬁguration selected according certain criteria

pzc ≈ Variation UPt|S in response to solvent ﬂuctuation 1V. Thermal averaging crucial. 3.4: Pt(111)/water interface at potential of zero charge

-6 just outside 0 -4

-2 -2

-4 0 Fermi level - OH/OH -6 2

potential vs SHE [V] vs potential OH/H O 2 -8 4 energy relative to vacuum [eV] energy relative to vacuum [eV] exp. PBE BLYP -10 6 Pt (111) water Pt results from Otani et al[5], Water results from Cheng and Sprik[9] 3.5: Vertical and adiabatic levels

Relation between total (free) energy curves and photo electron energy levels

Non-linear solvent response AO(x) IPR =ΔA + λO (22) EA =ΔA − λ (23) AO* vacuum O R O O Linear solvent response ≡ AO λO = λR λ AR(x)

IPR 'A EAO EAO Substituting 1 OR ΔA = (IPR +EAO) 2 'A AR* 1 λ = (IPR − EAO) OO 2 OR A R IPR Marcus-Hush theory

Free energy curves Electronic levels 3.6: Electrode potential representation

Aligning the adiabiatic energy level:ΔA =AIPR of Eq. 18 − g,◦ e0 UO/R (she) =ΔA + WH+ Δf GH+

IPR and EAO diﬀer from ΔA by a reorganization energy (Eqs. 22 and 23)

Reorganization is a process conserving charge (non-Faradaic)

Aligning the vertical detachment level (not an electrode potential) − g,◦ e0 UR (she) =IPR + WH+ Δf GH+

Aligning the vertical attachment level (not an electrode potential) − g,◦ e0 UO (she) =EAO + WH+ Δf GH+

Vertical energy levels are aligned in the same way as the adiabatic level 3.7: Ionization free energy from thermodynamic integration

Deﬁne an “artiﬁcial” mapping Hamiltonion depending on coupling parameter η.

Hη =(1− η) HR + η HO 0 ≤ η ≤ 1 superimposing reduced (R) and ionized (O) groundstate. At ﬁxed atomic position

∂Hη = EO − ER =ΔoxE ∂η

ΔoxE is the intantaneous vertical ionization energy in the model supercell

Integrating the canonical average ··· η of the energy gap

1 ΔoxA = dη ΔoxE η 0 gives the adiabatic ionization energy ΔA =AIPR oﬀset by a bias potential V0.

AIPR =ΔoxAR + e0V0 (24)

V0 is an unphysical bias in the electrostatic reference due to application of PBC e0V0 will be cancelled by a bias of opposite sign in the proton work function WH+ 3.8: DFTMD model system for hydroxyl

DFTMD Model 1 OH• radical

31 H2O In cubic box of L = 9.86 Å (periodic boundaries) 10-50 ps MD runs

Electronic structure TZV2P basis, 280 Ry (density cutoff)

No explicit counterion 3.9: Thermodynamic gap integral: examples

OH o OH• e - OH Non-linear: must integrate 1 'A dK 'E R N ³ K OH• 0

SH o SH• e

Linear: can use LRA - SH 1 K 'A | 'E 'E 2 0 K 1 SH• 1 VIP VEA 2 red ox 3.10: Vertical and adiabatic energy gaps

Thermodynamic integral for ionization (oxidation) free energy (Eq. 24)

1 ΔoxA = dη ΔoxE η =AIPR − e0V0 0

The initial value (η = 0) is the biased vertical IP of R

ΔoxE η=0 =IPR − e0V0

The ﬁnal value (η = 1) is the biased vertical EA of O

ΔoxE η=1 =EAO − e0V0

The two point quadrature to the integral 1 ΔoxAX− = (IPX− +EAX• ) − e0V0 2 is equivalent to the linear response approximation of Marcus theory 3.11: Half reaction scheme for the SHE[10,11]

e vac 1 2 H2 g H g

'V0 0

− AIPX− e0V0 WH+ + e0V0 adiabatic ionization desolvation

Reversible potential X/X− couple according to Eq. 18 ◦ g,◦ − − e0 UX•/X− (she) =AIPX + WH+ Δf GH+

Condition for ΔV0 = 0: electrostatic reference determined by solvent 3.12: Coupling oxidation and deprotonation

Hess law XH 2.3kBT pKa X H aq ◦ ◦ ΔdhG =2.30kBT pKa + e0Uox(she)

Key objective o $ ' dh G eUox SHE Satisfy Hess law by construction. Method Implement acid dissociation by completely removing the proton x 1 from the model system X 2 H2 g

PCET triangle: Acid dissociation of an hydride XH(s) → X−(s)+H+(s) followed by oxidation of the conjugate base by the product aqueous proton

− + • 1 X (s)+H (s) → X (s)+ H2(g) 2 is equivalent to dehydrogenation of the hydride

• 1 XH(s) → X (s)+ H2(g) 2 Note: dehydrogenation only involves neutral species 3.13: Half reaction scheme for the pKa[10,11]

H g

'V0 0

− ADPXH + e0V0 WH+ e0V0 adiabatic deprotonation solvation

Thermodynamic cycle for the pKa of XH − ◦ 3 2.30 kBT pKaXH =ADPXH WH+ + kBT ln c ΛH+ (25) with the standard chemical potential of H+ approximating the dissociation entropy 3.14: Deprotonation by discharging the acid proton

real proton dummy proton (no charge) Adiabatic discharge of proton XH → Xd− transforming it to a dummy d

Implemented by a mapping Hamiltonian superimposing PES of XH and X−

Hη =(1− η) HXH + η HX− + Vr extented by a harmonic restraining potential Vr to keep the dummy in place.

∂Hη = EX− − EXH =ΔdpE ∂η is the vertical deprotonation gap. The coupling parameter integral

1 ΔdpAXH = dη ΔdpEXH η 0 is a biased adiabatic deprotonation energy (ADP) related to the true ADP as

ADPXH =ΔdpAXH − e0V0 − ΔzpEH(X) (26) with a correction for the proton zero point motion added. 3.15: Computing the proton workfunction and pKa

The workfunction is the proton is aproximated by the ADP od teh hydronium ion

+ + − − + WH =ΔdpAH3O e0V0 ΔzpEH (OH2)

+ where ΔdpAH3O is the deprotonation integral of the hydronium

+ + ΔdpAH3O = dη ΔdpEH3O η 0

Substituting in the expression for the pKa of Eq. 25 − ◦ 3 2.30 kBT pKaXH =ADPXH WH+ + kBT ln c ΛH+ we ﬁnd for the acidity constant of XH ◦ 3 − + − − + + 2.30 kBT pKaXH =ΔdpAXH ΔdpAH3O ΔzpEH(X) ΔzpEH (OH2) + kBT ln c ΛH ◦ 3 The kBT ln c ΛH+ term accounting for the translational dissociation entropy ◦ 3 ln c ΛH+ pKaH O+ = = −3.2 3 2.30 is by construction the pKa of the hydronium ion (should be -1.74) 3.16: DFTMD model system for hydronium

DFTMD Model + 32 H2O r H In cubic box of L = 9.86 Å (periodic boundaries) 20-50 ps MD runs

Electronic structure TZV2P basis, 280 Ry (density cutoff + 3.17: Accumulated averages for deprotonation gap H3O 3.18: pKa for some small acids compared to experiments[11]

Energies eV'dp AXH pKa exp. 'pKa

HCl 15.1 7 70

H3O 15.35 3.2 1.74 1.5

HCOOH 15.8 5 3.75 1

H2S 16.0 8 7.0 1

CH3SH 16.1 11 10.3 1 NH4 16.1 11 10.2 1

Errrors statistical uncertainty < 0.1 eV MUE =1 3.19: Computing redox potentials and dehydrogenation free energy

Our expression fr the redox potential of the X•/X− couple (Eq. 18) ◦ g,◦ − − e0 UX•/X− (she) =AIPX + WH+ Δf GH+

Substituting Eq. 24 and 26 ◦ g,◦ • − (she) = Δ − +Δ + − Δ − Δ + e0 UX /X oxAX dpAH3O f GH+ zpEH (OH2)

1 1 g,◦ = d Δ − + d Δ + − Δ − Δ + η oxEX η η dpEH3O η f GH+ zpEH (OH2) 0 0

From the integral of the gap for simultaneous removal of electron and proton

1 ◦ − g,◦ − ◦ 3 ΔdhGXH = dη ΔdhEXH η Δf GH+ ΔzpEH(X) + kBT ln c ΛH+ 0

The correction terms ◦ 3 + − + kBT ln c ΛH = 0.19eV ΔzpEH (OH2) =0.35eV

are not small at all and add up to −0.54 eV for the dehydrogenation free energy 3.20: Dehydrogenating liquid water using DFTMD[12]

0.8 0.7 H2O OH H H2O OH H exp. BLYP 1.9 1.3 2.7 2.1

OHx 1 H x 1 2 2 OH 2 H2

Energies in eV 0.7 H2O OH H HSE* 1.7 2.5 *Guidon, M.; Hutter, J.; VandeVondele, J. “Auxiliary Density Matrix Methods for x 1 Hartree-Fock Exchange Calculations.” OH 2 H2 J. Chem. Theory Comput.(2010),6, 2348. 3.21: Oxidizing and reducing liquid water

Ionizing water creates a water radical cation which within 0.4 ps transfers a proton +• • + H2O +H2O → HO +H3O (27)

Reducing water creates a solvated electron −• − H2O → H2O+e (aq) (28)

Fully relaxing an optical electronic excitation yields the combined product ∗ • + − H2O +H2O → HO +H3O + e (aq) (29)

Review of experimental data • The vertical IP of reaction 27 is minus the VBM of liquid (9.9 eV) • The vertical EA of reaction 28 is minus the CBM of liquid water (1.5 eV) • the fundamental gap of liquid water is 8.7 eV, the optical gap 6.5 eV • The free energy change of reaction 29 is 5.6 eV. 3.22: Ionizing liquid water using DFTMD

0.7 H2O OH H Energies in eV BLYP

1.8 2.1 1.3

pK x 2 aH2O x x 1 H2O OH H2 0.1 2 0.7 H2O OH H HSE*

2.9 2.5 1.7

pK x 12 aH2O x x 1 H2O OH 2 H2 (Cheng) 0.7 3.23: Band edges of water ligned up with OH/OH− couple

0 -4 CBM -2 -2 -EA -4

SHE [V] SHE 0 λ vs vacuum [eV] -ΔA -6 2 vs

λ -8 potential potential 4 -IP energy VBM -10 6 Exp. BLYP HSE06 3.24: Pt(111)/water interface at potential of zero charge -6 just outside 0 -4

-2 -2 -EA -4 0 Fermi level λ -6 2 -ΔA potential vs SHE [V] vs potential λ -8 4

-IP energy relative to vacuum [eV] exp. BLYP PBE -10 6 Pt (111) water 3.25: Frost diagram for oxygen reduction

(a) Self consistent reaction ﬁeld calculation from reference [13] References

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