Adiabaticity and Rate Constants of Electron Transfer, Proton Transfer, and Proton-Coupled Electron Transfer Quantum Dynamics Zachary Goldsmith and Tyler Lytle Department of Chemistry, University of Illinois at Urbana-Champaign

I. Introduction    Charge transfer reactions are a ubiquitous and prominent subset of chemical processes. A charge transfer reaction can involve transferring electrons (ET), protons (PT), or both protons and electrons (PCET). Important examples of charge transfer are biological processes such as photosynthesis and the conversion of cytochrome to the cytochrome oxidase complex.[1,2] Photosynthesis is driven by an ET across a membrane in a photosynthetic reaction center that introduces a charge imbalance. In an effort to correct this charge imbalance, a proton-pumping process will occur to create ATP.[2] However, these two charge-transfer reactions can couple together to affect the aforementioned proton-pump.[1] The conversion of cytochrome to the cytochrome oxidase complex occurs by four sequential ET reactions, and PT also couples with ET in this conversion.[1,2] Analogous charge transfers are crucial to electrochemical reactions and in solid-state materials as well.[1,2] Since charge transfer reactions are so general to a wide variety of chemistries, a method for describing PCET dynamics is valuable. While three kinetic regimes exist for PCET, this paper will focus on electronically adiabatic PT, diabatic ET, and the dynamics of their subsequent coupling. After an understanding of PCET has been established, there will be a graphical and conceptual discussion of adiabaticity in PCET.  

II. Electron Transfer  a. Qualitative Description  

 

 To begin the discussing ET, a qualitative understanding should be first espablished. Suppose two diatomic molecules exist called q1 and q2. The molecule q1 lacks an electron compared to molecule q2, so q2 will transfer an electron to q1. Initially, q2's bond length is longer than q1's but the bond lengths will become equal as the two molecules start to transfer the electron. Once the electron has been transferred from q2 to q1, the bond length of q1 will

Fig.  1.  The  model  reaction  for  ET,  two  diatomic  molecules  (e.g.  H2  and  H2+)  whose  bond  lengths  change  upon  electron  transfer.[2]

now be longer than the bond length of q2, as illustrated in Figure 1.[2] Another depiction of ET is given by Figure 2. The blue line is the potential of the reactants as a function of the difference in bond lengths, q. The orange line is the potential of the products as a function of the difference in bond lengths, q. The ET takes place where the two curves intersect, so the ET takes place when the bond lengths of the two molecules are the same.[2] The bottom of the reactants' curve is when q2's bond length is longer than q1's bond length, and the bottom of the products' curve is when q1's bond length is longer than q2's bond length..[2] The rate of the ET reaction is the rate of crossing the barrier times its relative population assuming tunneling is negligible.[2] This description of ET will work even if the two molecules are bonded together.[2]

 Fig.  2.  The  potential  energy  as  a  function  of  q.  q=q1-­‐q2.[2]  

 b. Parabola Model  

               !! =

!!! ! + !! ! (IIb.1)  

             !! =

!!! ! − !! !    (IIb.2)  

 The potential energy curves shown in Figure 2 are approximated by equations IIb.1 and IIb.2 where f is the force constant, VR is the potential of the reactants, Vp is the potential of the reactants, q is the difference in bond length, and qo is the value of q where the potential reaches a minimum.[2] The difference in the potential energy of the reactants with the potential energy of the products is    !! − !! = 2!!!!          (IIb.3).[2]    Equation IIb.3 can be rearranged to yield:    

! = !!!!!

!! − !!        (IIb.4)  This is important because the x-axis in Figure 2 can be expressed in terms of the difference in potential energies as opposed to the difference in bond lengths.[2] This becomes important when the reaction is carried out in solution because the number of degrees of freedom is large so the difference in potential energies is easier to calculate than q. Figure 2 assumes the free energy change, ΔG, is 0.[2] However, a more general representation of ET is shown in Figure 3.  

  Figure 3 is the ET reaction for a reactant, D-A, going to the product, D+-A-.[2] In this reaction, D stands for the electron donor and A stands for the electron acceptor.[2] The differences in Figure 2 and Figure 3 are that ΔG is non-zero, the y-axis is ΔG, and x-axis indexes the combined polarization and vibrational energies.[2]  c. Classical Method for Electron Transfer Rates  

 In order to determine the ET rate using classical transition-state theory, the solvent will be treated as a continuum dielectric and quantum mechanical effects will be momentarily neglected.[2] The free energies discussed will ignore entropy, so the free energy is the potential energy.[2] Referring to Figure 3, if qc is the crossing point, the free energy of the reactants has to be equal to the free energy of the products at this point,    !! !! = !! !! (IIc.1), and    !!! !! − !! ! = !!! + !

!! !! − !! ! (IIc.2).[2]  

 

Fig.  2.  The  generalized  ET  reaction  in  solution.[2]

IIc.2 is obtained using IIc.1 and the fact that the free energy of the products is lower in energy than the free energy of the reactants by ΔG0, the free energy of the reaction. IIc.2 can be solved for qc, yielding    !! =

!!!

! !!!!!+ !!!!!

! (IIc.3).[2]  

 The quantity, λ, is known as the reorganization energy and defined as the difference in potential energy between the reactants and products as    ! = !

!! !! − !! ! (IIc.4).[2]  

 ΔG‡, the barrier height, is the difference in free energy of the crossing point and the bottom of the reactant free energy parabola defined as    !"‡ = !! !! − !! !! = !! !! − !! !! = !! !! = !

!! !! − !! !

(IIc.5).[2]    The VR(qR) term vanishes because at qR the free energy is 0. If IIc.3 is substituted for qc and IIc.4 is substituted for qP-qR and qP+qR, IIc.5 becomes    !"‡ = !

!!! + !!! ! (IIc.6).[2]  

 IIc.6 gives the height of the free energy barrier as a function of the reorganization energy and the free energy of reaction.[2] The rate for an ET reaction is thus obtained by using Arrhenius kinetics,    

! = !!!!!!!! = !!

! !!!!!!

!!!!! (IIc.7)    and substituting the barrier height for Ea since the free energy has no entropic contributions in this treatment.[2] A is a factor that depends on the frequency of crossing the barrier’s peak, and IIc.7 is the rate constant for an ET reaction treated classically.[2]

 d. Quantum Mechanical Method for Electron Transfer Rates   While equation IIc.7 is useful in some circumstances, it fails similarly to most classical formulations. One flaw is the temperature dependence of the ET rate. The classical result predicts the ET rate to go to 0 as the temperature goes to 0K, which is experimentally known to be unphysical.[2] Experimental results show that at low temperature the rate becomes independent of temperature, but at high temperatures the classical result is followed fairly closely.[2] Also, the classical rate assumes the barrier has to be crossed and neglects tunneling.[2] The rate has to be calculated using quantum mechanics if the

spacing between vibrational energy levels is large compared kBT. The criterion used to determine if the quantum or classical treatment is necessary is !!

!!!!!≫ 1    !"#$%"&  

 !!

!!!!!≪ 1    !"#$$%!#" (IId.1).  

 A graphical view of this is given in Figure 4. The vibrational wavefunctions have been superimposed on Figure 3, and, where the wavefunctions overlap, tunneling needs to be taken into account.[2]

 

The Hamiltonian used to describe this system is known as the spin-boson or polaron model, which couples two electronic states to a large number of vibrational states that can be either quantum mechanical or classical, given by (IId.2):  

 

   where |R> is the vibrational wavefunction of the reactant and |P> is the vibrational wavefunction of the product. The first term in square brackets represents the energy from the free energy curve of the reactants and the second term in square brackets represents the energy from the free energy curve of the products.[2] ER

o and EPo are the ground state energies, and qR,l and

qP,l are the origins of the displacements of the reactants and products.[2]

Fermi's Golden Rule gives the rate of ET to be:    ! = !!!

!| < !|!|! > |! !"#$ = !!!

!!!"! !"#$ (IId.3)  

 

Fig.  3.  The  overlap  of  vibrational  wavefunction.[2]

in which HRP is the donor/acceptor electronic mixing matrix element, and DFWC is the density of states-weighted Franck-Condon Factor.[2] If a specific coupled vibrational state is given a frequency, w, and the equilibrium displacement is defined as Δqe, the reorganization energy is given as  

 !! =

! !!! !

! (IId.4).[2]  

 The rate of the ET reaction thus becomes  

! = !!!

!!!"! 4!"!!!

!!! !

!!!!!!!

!!!!

!!!!!!!

!!!!!! !!!!

!!!!!!          (IId.5)    where λ0 is    !! = !" ! !

!!!+ !

!!!− !

!!!!− !

!! (IId.6).[2]  

 In equation IId.6, Δe is the amount of charge transferred, rD the radius of the donor, rA is the radius of the acceptor, R is the separation between the centers of the donor and acceptor, εoo is the optical frequency dielectric constant, and εo is the 0 frequency dielectric constant.[2] In equation IId.5 the terms in front of the summation are the frequency for the ET without a barrier present, (4!!!!!)!!/! is the classical density of states and the summation is the Franck-Condon factor, which is the sum over all of the possible overlap integrals between some initial vibrational state, v, and some final vibrational state, v'.[2]

 III. Proton Transfer  

Similarly, the quantum dynamics of proton (H nucleus) transfer can be modeled. While nuclei are typically considered to exhibit classical dynamics in the context of Born-Oppenheimer approximation, it is known that many biological and chemical systems prominently feature PT by tunneling[3]. The classical position of H, such as in a molecular dynamics (MD) simulation, is replaced by a vibrational-like wavefunction in a harmonic eigenbasis. For the purposes of this paper, only PT with electronically adiabatic dynamics will be considered. This stipulation allows for the PT to be governed by a single Born-Oppenheimer potential energy hypersurface. The derivation of rate constants for adiabatic (semi-classical) and non-adiabatic (quantum/non—Born-Oppenheimer) PT proceeds highly analogously to that of the constants, k, in Section II. In order to avoid redundancy and introduce a novel concept, this derivation will be left out in favor of a more rigorous treatment of PCET dynamics.

FIG. 5. One-dimensional illustration of a potential surface for H motion in which the two lowest eigenstates have nearly equivalent reactant and product character. In this configuration, the energy difference between the two depicted adiabatic eigenstates is proportional to the tunneling rate between the reactant and product configurations.  

IV. Proton-Coupled Electron Transfer PCET is most basically represented by the following four-state model with electronic states defined as[4]:

D and A represent the donors and acceptors of electrons, e, and protons, p, respectively. H is the transferring proton. For the sake of brevity and cohesion with earlier parts of this paper, the construction of the multiconfigurational VB proton vibrational and electronic free energy surfaces and subsequent wavefunctions will be omitted. This part of the derivation is highly dependent on the response of the electron density outside of the PCET cavity. For solution chemistry this is often represented by a dielectric polarizable continuum. In accordance with section IId, the rate expression for non-adiabatic PCET will be constructed from Fermi’s Golden Rule. Figure 4 illustrates this non-radiative transition between reactant and product diabatic vibronic manifolds. In this derivation, ! will represent the coordinates of the modes of solvent polarization; the proton coordinate is q, and other solute modes are Q. So, the mixed electronic/proton vibrational states for the reactants and products have energies !!! (!, !) and !!!!(!, !) with corresponding wavefunctions !! !!; !, ! !!! (!, !) and !!! !!; !, ! !!!!(!, !) . Now, we begin by calculating separable vibronic states of the solute and solvent modes. We assume the solute modes are harmonic independent of the proton vibrational state !.

!!! !, ! = !!

! ! + !!! ! ,          ! = !, !!

Therefore, the solute mode wavefunctions, !!! ! , are eigenfunctions of the

harmonic Hamiltonian !!! = !! + !!

! ! , with eigenvalues !!! and kinetic

energy !! . Similarly, the solvent mode wavefunctions, !!"! ! , are

eigenfunctions of the Hamiltonian !!! = !! + !!

! ! with eigenvalues !!! + !!"

! . Thus the full reactant and product wavefunctions can be represented as

!!"#! !! , !,!, ! = !! !!; !, ! !!! (!, !)!!"

! ! !!! !

The corresponding energies are ℇ!"#! = !!

! + !!"! + !!

! . Three assumptions must further be made before implementing the golden rule. First, that the coupling !(!, !) is independent of the solvent coordinate in the region of interest and !!" is constant. The coupling is expressed as

!!"!,!"# = !!" !!! |!!!! !!"! |!!"!! !!" = !!! |!(!, !)|!!!!

Second, that the PCET is nonadiabatic i.e. the coupling above is small compared to kBT. Third, that the solvent relaxtion time is short enough such that there is always thermal equilibrium in the reactant states. Then, finally, the nonadiabatic rate constant for PCET between !!"#

! and !!"#! is

! =2!ℏ !!"#

!

!"#

!!"#,!"#!!(

!"#

ℇ!"#! − ℇ!"#!! )

!!"#! = !!(

ℇ!"#!

!!!)

!!

ℇ!"#!

!!!!"#

The P values are Boltzmann distribution functions for the reactant states. Considering two limits allows us to recover an expression very similar to the result of section IId. First consider the high-temperature limit, i.e. ℏ! ≪ !!!. This encourages vibrational excitation to achieve the reorganization energy and behave purely diabatically (as in IId). In addition, we may neglect intramolecular forces to recover the rate expression

! =2!ℏ !!

!

!

!!"!(4!!!"!!!)

!!!

!

exp   −!!!"! + !!"

!

4!!"!!!

V. Graphical Representation: Adiabaticity

Adiabatic and diabatic (nonadiabatic) regimes in fact are the extremes in terms of kinetic regimes[5]. A lot of real PCET chemistry falls somewhere between these two regimes—making their dynamics difficult characterize analytically. However, quantum chemistry methods that generate the pertinent vibronic wavefunctions and free energy surfaces have enabled a semiclassical measure of a charge transfer’s adiabaticity[5]. This formulation

is based on expressions for the proton and electronic tunneling times, !! and !!.

!!~!!"

|!"|!!  ,      !!~

ℏ!!"

In these expressions, !!" is the electronic coupling between the diabatic states, !" is the difference between the slopes of the potential energy surfaces at the

crossing point, !! =!(!!!!)

!, the tunneling velocity at the crossing point,

with Vc the potential energy at the crossing point and E the tunneling energy. The adiabaticity parameter, p, is thusly

! =!!!!  

When p << 1, the electron tunneling time is much greater than that of the proton and the overall process is non-adiabatic. When p >> 1, however, the process is adiabatic. To illustrate this difference, consider Figure 6.

FIG. 6[6]. (a) Diabatic potential curves (black) correspond to the two localized states on the donor and acceptor, respectively. The donor and acceptor ground state proton vibrational wavefunctions are shown in blue and red, respectively. They represent the localization at the potential minima on the donor and acceptor. The vibronic coupling for this transfer is the product of the electronic coupling !!" and the overlap of the reactant and product proton vibrational wavefunctions, !!! |!!!! (b) Electronically adiabatic potential well along with the two lowest proton vibrational wavefunctions. The vibronic coupling for this system is a function of the energy splitting !. Both proton vibrational wavefunctions have significant density on both sides of the transfer—representative of the adiabaticity condition !! ≫ !!

VI. References

1. S. Hammes-Schiffer, Acc. Chem. Res., 2001, 34, 273-281. 2. P.F. Barbara, T.J. Myer, M.A. Ratner, J. Phys. Chem., 1996, 100, 13148-

13168. 3. S. Hammes-Schiffer, J. C. Tully, J. Chem. Phys., 1994, 101, 4657-4667 4. A. Soudackov, S. Hammes-Schiffer, J. Chem. Phys., 2000, 113, 2385-

2396 5. Y. Georgievskii, A. A. Stuchebrukhov, J. Chem. Phys., 2000, 113, 10438 6. S. Hammes-Schiffer, A. Soudackov, J. Phys. Chem. B, 2008, 112, 14108