Theory of Proton-Coupled Electron Transfer
Sharon Hammes-SchifferPennsylvania State University
Note: Much of this information, along with more details, additional rate constant expressions, and full references to the original papers, is available in the following JPC Feature Article:Hammes-Schiffer and Soudackov, JPC B 112, 14108 (2008)
Copyright 2009, Sharon Hammes-Schiffer, Pennsylvania State University
R
AeApDp H
ETPT
De
General Definition of PCET
• Electron and proton transfer reactions are coupled• Electron and proton donors/acceptors can be the same or
different• Electron and proton can transfer in the same direction
or in different directions• Concerted vs. sequential PCET discussed below• Concerted PCET is also denoted CPET and EPT• Hydrogen atom transfer (HAT) is a subset of PCET• Distinction between PCET and HAT discussed below
R
AeApDp H
ETPT
De
Examples of Concerted PCET
ET
PT
Importance of PCET
• Biological processes− photosynthesis− respiration− enzyme reactions− DNA
• Electrochemical processes− fuel cells− solar cells− energy devices
Cytochrome c oxidase4e− + 4H+ + O2 → 2(H2O)
Theoretical Challenges of PCET
• Wide range of timescales− Solute electrons− Transferring proton(s)− Solute modes− Solvent electronic/nuclear polarization
• Quantum behavior of electrons and protons− Hydrogen tunneling− Excited electronic/vibrational states− Adiabatic and nonadiabatic behavior
• Complex coupling among electrons, protons, solvent
Diabatic states:
Single Electron Transfer
e e
e e(2)
(1 D A
D
)
A
−
−
( ) ( )
2 1/ 2 †12
2†
12 coupling between diabatic states
2(4 ) exp ( )
4
:
B Bk V k T G k T
G G
V
π πλ
λ λ
− = − ∆
∆ = ∆ +ℏ
Nonadiabatic ET rate:
Solvent coordinate
2 1 in( ) ( )ez d ρ ρ= − Φ∫ r r
Marcus theory
Inner-Sphere Solute Modes
22 1/ 2 I (1) (2) †
12 1 ,2
(2)(1)vibrational wavefunctions
2(4 ) | exp ( )
,
B Bk V k T P G k Tµ µ υ µ υµ υ
υµ
π π
ϕ ϕ
λ ϕ ϕ− = − ∆ ∑ ∑ℏ
Assumes solute mode is not coupled to solvent →Not directly applicable to PCET because proton strongly coupled to solvent
Single Proton Transfer
p p
p p
( ) D H A
( ) D HA
a
b
+ −
Diabatic states: Solvent coordinate
Proton coordinate: rp (QM)in( ) ( )p b az d ρ ρ= − Φ∫ r r
PT typically electronically adiabatic (occurs on ground electronic state) but can be vibrationally adiabatic or nonadiabatic
• Four diabatic states:
• Free energy surfaces depend on 2 collectivesolvent coordinates zp, ze
• Extend to N charge transfer reactions with 2N states and Ncollective solvent coordinates
Proton-Coupled Electron Transfer
e p p e
e p p e
e p p e
e p p e
(1 ) D D H A A
(1 ) D D HA A
(2 ) D D H A A
(2 ) D D HA A
a
b
a
b
− +
− +
+ −
+ −
⋯⋯
⋯⋯
⋯⋯
⋯⋯
1 1 in
2 1 in
PT (1 ) (1 ): ( ) ( )
ET (1 ) (2 ) : ( ) ( )
p b a
e a a
a b z d
a a z d
ρ ρ
ρ ρ
→ = − Φ
→ = − Φ
∫
∫
r r
r r
Soudackov and Hammes-Schiffer, JCP 111, 4672 (1999)
• Sequential: involves stable intermediate from PT or ETPTET: 1a→ 1b→ 2bETPT: 1a→ 2a→ 2b
• Concerted: does not involve a stable intermediateEPT: 1a→ 2b
• Mechanism is determined by relative energies of diabaticstates and couplings between them
• 1b and 2a much higher in energy → concerted EPT
Sequential vs. Concerted PCETe p p e
e p p e
e p p e
e p p e
(1 ) D D H A A
(1 ) D D HA A
(2 ) D D H A A
(2 ) D D HA A
a
b
a
b
− +
− +
+ −
+ −
⋯⋯
⋯⋯
⋯⋯
⋯⋯
Remaining slides focus on “concerted” PCET:describe in terms of Reactant → Product
• Reactant diabatic state (I)- electron localized on donor De- mixture of 1a and 1b states
• Product diabatic state (II )- electron localized on acceptor Ae- mixture of 2a and 2b states
Typically large coupling between a and b PT states andsmaller coupling between 1 and 2 ET states
Reactant and Product Diabatic States
Diabatic vs. Adiabatic Electronic States
4 diabatic states: 1a, 1b, 2a, 2b4 adiabatic states:Diagonalize 4×4 Hamiltonian matrix in basis of 4 diabatic statesTypically highest 2 states can be neglected
2 pairs of diabatic states: 1a/1b, 2a/2b2 pairs of adiabatic states:Block diagonalize 1a/1b, 2a/2b blocksTypically excited states much higherin energy and can be neglected
2 ground adiabatic states from block diagonalization above:Reactant (I) and Product (II ) diabatic states for overall PCET reaction
H treated quantum mechanicallyCalculate proton vibrational states for electronic states I and II- electronic states: ΨI(re,rp), ΨII(re,rp) - proton vibrational states: ϕIµ(rp), ϕIIν(rp)
Reactant vibronic states: ΦI(re,rp) = ΨI(re,rp) ϕIµ(rp)Product vibronic states: ΦII(re,rp) = ΨII(re,rp) ϕIIν(rp)
Coupling between reactant and product vibronic states typicallymuch smaller than thermal energy because of small overlap →Describe reactions in terms of nonadiabatic transitions between reactant and product vibronic states
Vibronic states depend parametrically on other nuclear coords
Electron-Proton Vibronic States
2D Vibronic Free Energy Surfaces
Reactant (1a/1b) D− A
Product (2a/2b) D A−
• Multistate continuum theory: free energy surfaces depend on 2 collective solvent coordinates, zp (PT) and ze (ET)
• Mixed electronic-proton vibrational (vibronic) surfaces• Two sets of stacked paraboloids corresponding to different proton vibrational states for each electronic state
One-Dimensional Slices
Mechanism: 1.System starts in thermal equilibrium on reactant surface2.Reorganization of solvent environment leads to crossing3.Nonadiabatic transition to product surface occurs with
probability proportional to square of vibronic coupling4. Relaxation to thermal equilibrium on product surface
• Shape of proton potentials not significantly impacted by solventcoordinate in this range
• Relative energies of reactant andproduct proton potentials stronglyimpacted by solvent coordinate
Solvent Coordinate rp
Fundamental Mechanism for PCET
Solvent Coordinate rp
Fundamental Mechanism for PCET
Solvent Coordinate rp
Fundamental Mechanism for PCET
Overview of Theory for PCET
• Solute: 4-state model
• H nucleus: quantum mechanical wavefunction• Solvent/protein: dielectric continuum or explicit molecules• Typically nonadiabatic due to small coupling• Nonadiabatic rate expressions derived from Golden Rule
Hammes-Schiffer, Acc. Chem. Res. 34, 273 (2001)
R
AeApDp H
ETPT
De
e p p e
e p p e
e p p e
e p p e
(1 ) D D H A A
(1 ) D D HA A
(2 )D D H A A
(2 )D D HA A
a
b
a
b
− +
− +
+ −
+ −
⋯⋯
⋯⋯
⋯⋯
⋯⋯
PCET Rate ExpressionSoudackov and Hammes-Schiffer, JCP 113, 2385 (2000)
( )
( ) ( )( ) ( )I
e
1/2 2I †
2†
elIIe pp
24 exp ( )
4
ˆ, ,
B Bk P k T V G k T
G G
V H V S
µ µν µν µνµ ν
µν µν µν µν
µν µν
π πλ
λ λ
− = − ∆
∆
Φ
= ∆ +
≈Φ=
∑ ∑
r rr r
ℏ
Reactant (1a/1b) D− A
Product (2a/2b) D A−
H coordinate
Excited Vibronic States
ETV
Relative contributions from excited vibronic states determined from balance of factors (different for H and D, depends on T)• Boltzmann probability of reactant state• Free energy barrier• Vibronic couplings (overlaps)
( ) 1/ 2 2I †24 exp ( )B Bk P k T V G k Tµ µν µν µν
µ ν
π πλ−
= − ∆ ∑ ∑ℏ
Proton Donor-Acceptor Motion
ETV
De ApDp AeH
R
• R is distance between proton donor and acceptor atoms• R-mode corresponds to the change in the distance R,typically at a hydrogen-bonding interface
• R-mode can be strongly influenced by other solute nuclei, viewed as the “effective” proton donor-acceptor mode
• PCET rate is much more sensitive to R than to electron donor-acceptor distance because of mass and length scales for PT compared to ET
For this PCET reaction, R is distancebetween donor O and acceptor N inPT reaction
Role of H Wavefunction Overlap
• Rate decreases as overlap decreases (as R increases)
• KIE increases as overlap decreases (as R increases)2 2
2 2
( overlap)
( overlap)H H
D D
k V H
k V D∝ ≈
2 2( overlap)H Hk V H∝ ∝
solid: Hdashed: D
(for a pair of vibronic states)
De ApDp AeH
R
• Vibronic coupling (overlap) depends strongly on R• Approximate vibronic coupling as
• Derived dynamical rate constant with quantum R-mode andexplicit solvent
• Derived approximate forms for low- and high-frequency R-modeusing a series of well-defined approximations
Include Proton Donor-Acceptor Motion
ETV
De ApDp AeH
R
( ) ( )el 0eqexpV R V S R Rµν µν µνα ≈ − −
Vel: electronic coupling : proton wavefunction overlap at Req
Req: equilibrium R value
0Sµν
Soudackov, Hatcher, SHS, JCP 122, 014505 (2005)
Dynamical Rate for Molecular Environment
ETV
( )
( ) ( ) ( )
( ) ( ) ( )1 1
2el 0
2
0
1 2 1 2 1 2 1 2 1 22 20 0 0 0
exp
2exp 0
1 1
t
R R R
t t
D R
ij t V S t
iC C t D C d
d d C d d C C
µν
τ τ
αα τ τ
τ τ τ τ τ τ τ τ τ τ
=
× + −
− − − − −
∫
∫ ∫ ∫ ∫
ℏ
ɶ
ℏ
ℏ ℏE
E
( ) ( )( )eq,t R tε ξ= ∆E
( ) ( ) ( ), ,R DC t C t C tE
• Calculate quantities with classical MD on reactant surface• Includes explicit solvent/protein environment• Includes dynamical effects of R-mode and solvent/protein
Soudackov, Hatcher, SHS, JCP 2005
eqR R
DR
ε=
∂∆=∂
ɶ
Time correlation functions:
( )dyn 2
1k j t dt
∞
−∞
= ∫ℏ
Energy gap and its derivative:
Closed Analytical Rate Constant
ETV
Approximations: short-time, high-T limit for solvent and quantum harmonic oscillator R-mode
Parameters depend on T, reorganization energies, reaction free energies, vibronic coupling exponential factor, mass and frequency of R-mode, and difference in product and reactant equilibrium R values
Rate constant expressed in terms of physically meaningfulparameters but requires numerical integration over time
2el 0
I 22
2exp exp 2 (cos 1) ( sin
V Sk P d p i q
µν αµ
µ ν
λ ζ τ χτ τ τ θτ∞
−∞
= − + − + + Ω Ω
∑ ∑ ∫ℏ ℏ
Soudackov, Hatcher, SHS, JCP 2005
High-Frequency R-mode
( )22 0el 0
I
B B
exp4
exp RGV S
k Pk T T
Rk
µαµν
νµνµ
µ ν
λ λ α δλπ
λ λ − −
∆ + = −
Ω ∑ ∑
ℏℏ
2 2
2 2
22
M
R M R
µνα
αλ
δ δ
=
= Ω
ℏ M, Ω: mass and frequency of R-modeα: exponential R-dependence of vibronic couplingδR: difference between product and reactant
equilibrium values of R
Bk TΩ >>
Assumption of derivation (strong-solvation limit): 0Gµνλ > ∆
In this limit, sole effect of R-mode on rate constant is thatvibronic coupling is averaged over ground-state vibrationalwavefunction of R-mode
For very high Ω, use fixed-R rate constant expression
Low-Frequency R-mode
( )( )
( )
22 0el 0 2B
2I
B B
2e expxp
4
GV Sk P
k T k
k T
M Tαµν
α α
µνµνµ
µ ν
λπλ λ λλ
λα ∆ + + = − + +
Ω
∑ ∑ℏ
2 2
2Mµν
α
αλ =
ℏ
M, Ω: mass and frequency of R-modeα: exponential R-dependence of vibronic coupling
( )2
2 2B2 2
2KIE exp
H
D H
D
S k T
MSα α− ≈ − Ω
Approximate KIE(only ground states)
• T-dependence of KIE determined mainly by α and Ω:• KIE decreases with temperature because αD > αH
• Magnitude of KIE determined also by ratio of overlaps:smaller overlap → larger KIE
Bk TΩ <<
Typically λα << λ
Note: this expression assumes δR = 0; a more complete expression is available
• Reorganization energy λ in previous expressions refers tosolvent/protein reorganization energy (outer-sphere)
• Inner-sphere reorganization energy (intramolecular solute modes) can also be included- high-T limit (low-frequency modes): add inner-sphere reorganization energy to solvent reorganization energy
- low-T limit (high-frequency modes): modified rate constantexpression has been derived (Soudackov and Hammes-Schiffer, JCP 2000)
• Calculation of reorganization energies- Outer-sphere: dielectric continuum models or molecular
dynamics simulations- Inner-sphere: quantum mechanical calculations on solute
Reorganization Energies
• Reorganization energies (λ)- outer-sphere (solvent): dielectric continuum model or MD- inner-sphere (solute modes): QM calculations of solute
• Free energy of reaction for ground states (driving force) (∆G0)- QM calculations or estimate from pKa’s and redox potentials
• R-mode mass and frequency (M, Ω) - QM calculation of normal modes or MD - R-mode is dominant mode that changes proton donor-acceptor distance
• Proton vibrational wavefunction overlaps (Sµν , αµν) - approximate proton potentials with harmonic/Morse potentials or generate with QM methods
- numerically calculate H vibrational wavefunctions w/ Fourier grid methods• Electronic coupling (Vel) - QM calculations of electronic matrix element or splittingNote: this is a multiplicative factor that cancels for KIE calculations
Input Quantities
• Experimentally challenging to change only a single parameterExamples: Increasing R often decreases Ω; may impact KIE in opposite wayChanging driving force by altering pKa can also impact R
• Relative contributions from pairs of vibronic states aresensitive to parameters, H vs. D, and temperatureMust perform full calculation (converging number of reactant and productvibronic states) to predict trend
• High-frequency and low-frequency R-mode rate constants are qualitatively differentExample: Low-frequency expression predicts KIE decreases with TFixed-R and high-frequency expressions can lead to either increase or decrease of KIE with T
Warnings about Prediction of TrendsEdwards, Soudackov, SHS, JPC A113, 2117 (2009)
Driving Force Dependence
ETV• Theory predicts inverted region behaviornot experimentally accessible for PCETdue to excited vibronic states withenhanced couplings
• Apparent inverted region behavior could beobserved experimentally if changing driving force also impactsother parameters (e.g., increasing |∆pKa| also increases R)
Free energy vs. Solvent coordinate
0G λ−∆ >0G λ−∆ <
Edwards, Soudackov, SHS, JPC A 2009; JPC B 113, 14545 (2009)
Applications to PCET Reactions• Amidinium-carboxylate salt bridges (Nocera), JACS 1999• Iron bi-imidazoline complexes (Mayer/Roth), JACS 2001• Ruthenium polypyridyl complexes (Meyer/Thorp), JACS 2002• DNA-acrylamide complexes (Sevilla), JPCB 2002• Ruthenium-tyrosine complex (Hammarström), JACS 2003• Soybean lipoxygenase enzyme (Klinman), JACS 2004, 2007• Rhenium-tyrosine complex (Nocera), JACS 2007• Quinol oxidation (Kramer), JACS 2009• Osmium aquo complex/SAM/gold electrode (Finklea), JACS 2010
Experimental groups in parentheses, followed by journal and year of Hammes-Schiffer group application
Theory explained experimental trends in rates, KIEs, T-dependence, pH-dependence
ET
PT
• Overall HAT and PCET usually vibronically nonadiabatic sincevibronic coupling much less than thermal energy: Vµν << kBT
• PT can be electronically nonadiabatic, adiabatic, or in between,
depending on relative timescales of electronic transition (τe) and proton tunneling (τp)electronically adiabatic PT: electrons respond instantaneously
to proton motion, τe << τp
electronically nonadiabatic PT: electrons do not respond
instantaneously, τe >> τp
• HAT ↔ electronically adiabatic PTPCET ↔ electronically nonadiabatic PT
Distinguishing between HAT and PCETSkone, Soudackov, SHS, JACS 128, 16655 (2006)
Quantify Nonadiabaticity: Vibronic Coupling
( )
( )
( )
sc (ad)
ln
2el
el
el
21
2
DA DA
p p p
p
t e
e pt
ct
V V
ep
p
Vp
F v
V
V F v
V Ev
m
κ
κ π
ττ
τ τ
−
=
=Γ +
= =∆
≈ ≈∆
−=
ℏ
ℏ
:
:
:
cV
E
F
V
∆el
energy at crossing point
tunneling energy (vibrational ground state)
:difference of slopes of potential energy curves
electronic coupling
Georgievskii and Stuchebrukhov, JCP 2000; Skone, Soudackov, SHS, JACS 2006
( ) (1) (2)
2 , 1,
|DA D A
p p
V V
κ π
ϕ ϕ
≈ <<
=na el
Electronically nonadiabatic PT:
( )
1, 1
/ 2ad
Electronically adiabatic PT:
DA
p
V
κ ≈ >>= ∆
∆
p eτ τ<
e pτ τ<
Representative Chemical Examples
Phenoxyl/Phenol and Benzyl/Toluene self-exchange reactionsDFT calculations and orbital analysis:
Mayer, Hrovat, Thomas, Borden, JACS 2002
phenoxyl/phenolO---H---O
benzyl/tolueneC---H---C
PCETHAT
SOMO
DOMO
ET and PT between same orbitals
ET and PT betweendifferent orbitals
PCET vs. HAT: Adiabaticity Parameter
el -1
4, 4
14,000cm
p ep
V
τ τ≈ ≈
=(sc) (ad) 2DA DAV V≈ = ∆
(sc) (na) el (1) (2)|DA DA D AV V V ϕ ϕ≈ =
el 1
0.01, 80
700cm
e pp
V
τ τ−
≈ ≈
=
Benzyl-toluene: C---H---C, electronically adiabatic PT, HAT
Phenoxyl-phenol: O---H---O, electronically nonadiabatic PT, PCET
Skone, Soudackov, SHS, JACS 2006
Electrochemical PCET Theory
Derived expressions for current densities j(η)• Current densities obtained by explicit integration over x
• Gouy-Chapman-Stern model for double layer effects
( ) ( )SCH
a axj F dxC x k x
∞= ∫
Venkataraman, Soudackov, SHS, JPC C 112, 12386 (2008)
Rate Constants for Electrochemical PCET
• Nonadiabatic transitions between electron-proton vibronic states
• Integrate transition probability over ε, weighting by Fermidistribution and density of states for metal electrode
• Similar transition probabilities with modified reaction free energy:
( ) ( )00, sG x U U e e xµν µνε ε η φ∆ ≈ ∆ − ∆ + − +
( ) ( ) ( ) ( )1 ,a ak x d f W xε ε ρ ε ε= − ∫
Characteristics of Electrochemical PCET• pH dependence: buffer titration, kinetic complexity, H-bonding• Kinetic isotope effects• Non-Arrhenius behavior at high T• Asymmetries in Tafel plots, αΤ ≠ 0.5
at η=0 (observed experimentally)δReq = 0δReq = 0.05 Å
De ApDp H
Req
Effective activation energy contains T-dependent termsdue to change in Req upon ET; different sign for cathodic and anodic processes →asymmetries in Tafel plots
Cathodic transfer coefficient:
eq B2 R k Tµνα δ±
T 00 eq B 00( 0) 0.5 R k Tα η α δ= ≈ − ΛVenkataraman, Soudackov, SHS, JPC C 2008
Photoinduced PCET
• Developed model Hamiltonian• Derived equations of motion for reduced density matrixelements in electron-proton vibronic basis
• Enables study of ultrafast dynamics in photoinduced processes
Homogeneous Interfacial: molecule-semiconductor interface
Venkataraman, Soudackov, SHS, JCP 131, 154502; JPC C 114, 487 (2009)
Beyond the Golden RuleNavrotskaya and Hammes-Schiffer, JCP 131, 024112 (2009)
• Derived rate constant expressions that interpolate betweengolden rule and solvent-controlled limits
• Includes effects of solvent dynamics• Golden rule limit
- weak vibronic coupling, fast solvent relaxation- rate constant proportional to square of vibronic coupling, independent of solvent relaxation time
• Solvent-controlled limit- strong vibronic coupling, slow solvent relaxation- rate constant independent of vibronic coupling, increases as solvent relaxation time decreases
• Interconvert between limits by altering physical parameters• KIE behaves differently in two limits, provides unique probe
webPCEThttp://webpcet.chem.psu.edu
• Interactive Java applets allow users to perform calculations on model PCET systems and visualize results• Harmonic, Morse, or generalproton potentials• “Exact”, fixed R, low-frequencyor high-frequency R-mode rateconstant expressions• Plot dependence of rates and KIEs as function of temperature and driving force• Analyze contributions of vibronic states• Access via free registration
