Expansion to an Electron Coupled Proton Transfer

To consider the coupling of the electron transfer to a proton transfer the larger masses in these quantum transfer processes have to be considered, as done early by Marcus et al.²³⁰ The two-dimensional ET coordinate becomes one cross section in the PCET energy landscape. The second proton transfer coordinate X is the third dimension as illustrated in Figure 2-19 A (p.59). Two coordinates can be seen in the square scheme introduced above (Figure 1-9, p.13). In this case, the proton transfer occurs along coordinate Xp, whereas the two possible electron transfer steps are along coordinate Q^e.

For the PCET in α especially the Soudackov-Hammes-Schiffer (SHS) theory is of interest. SHS has been applied to discriminate between HAT and a CPET.^{109, 231} These two similar cases are proposed for di-tyrosine peptides by theorists.106, 107, 232 They especially discussed in the α subunit the PCET step between Y⁷³¹ and Y⁷³⁰. Therefore, this section will focus on the SHS theory. However, many diverse theories will give rise to a statistically Arrhenius dependence either multiplied of summed with a dynamic prefactor.⁹⁷ Nevertheless, they differ in the actual realization as reviewed in recent reviews.^{95-97, 233}

The SHS theory is based on a VB description of the four possible steps in the consecutive processes of ET/PT and PT/ET. It uses for the environment a multistate continuum model instead of atomistic models for considering solvent effects. The proton donor-acceptor motion has been incorporated. For this motion linear response theory in combination with Fermi’s golden rule formulation were used here. In most of the modern models the vibronic coupling is taken into account by summation of the Boltzmann populations Pk of the initial state. Sμν is the overlap of the vibrational wave functions for the μ and the ν state. This is fully analog to the description of absorbance and fluorescence probabilities by the Frank Condon theory.^234-236 A rate constant for an equilibrated system at each X value can be obtained from these approximations (Eq. 2-36).²³⁷

Proton Coupled Electron Transfer

In the high temperature and or low frequency regime, the Eq. (2-36) for the X mode it is further simplified. Taken an equilibrium position at ΔX = 0, the simplest form can be derived as Eq. (2-37).^{238, 239}

Here the exponential decay is dependent indirectly on X. Mp and ωp are the X-mode effective mass and frequency, respectively. α^μν is the decay parameter of the vibrational overlap. S^p is here the vibronic overlap in the equilibrium state (ΔX = 0).

The model of Dogonadze, Kuznetsov and Levich has not only separated the electron from the proton movement (BO approximation), but also considered a second case where the proton movement is adiabatic to the solvent (frequency =0S). To illustrate possible relative effects the following magnitudes were given: 0S≈₁₀¹¹ _Hz≪_n≈10¹⁴ Hz≪_e≈ 10¹⁵ Hz. Here n describes the frequency of the bound reactive proton (I and F state) and

ethe electron frequency bound to the proton acceptor in an ionic PT step.

Both hydrogen atom transfer (HAT) and concerted proton coupled electron transfer (CPET) are usually vibronically non-adiabatic due to the small proton wave function overlap that produces vibronic couplings ≪k^BT.¹⁰⁹ Many biological PCETs are electronically non-adiabatic. For CPET reactions within these non-adiabatic reactions, the Eq. (2-37) is valid.²⁴⁰

Figure 2-19: Extension from ET to a PCET A) The extension to a second coordinate X renders the ET to a two dimensional diabatic electron proton free energy surface connecting the vibronic states µ and v as functions of two collective solvent coordinates. One coordinate is strictly related to ET (Qe) and the other associated with PT (Xp). The equilibrium coordinates, the reaction free energy ΔGR° and reorganization energy λμν are indicated similarly to Figure 2-17. Adapted from ref 241. B) Free energy along the reaction coordinate represented by the dashed line in the nuclear coordinate plane of panel A. Qualitative potential energy surfaces (PESs) and pertinent ground state proton vibrational functions are shown in correspondence to the reactant minimum, transition state and product minimum. ref. 242 C) Vibrational mode overlap in the diabatic PESs for the initial and final ET states and vibrational function: initial ψD(I) (blue) and final state ψ^D(II) (red). Small 𝑉𝑉_{𝐼𝐼𝐼𝐼}^el case is depicted. D) Large electronic coupling 𝑉𝑉_{𝐼𝐼𝐼𝐼}^el in an adiabatic ground PES. For an adiabatic system the vibronic coupling is half of the splitting between the energies of the symmetric (cyan) and antisymmetric (magenta) vibrational states of the proton.

The excited vibrational state of the antisymmetric state is shifted up by 0.8 kcal/mol for a better visualization. Adapted from ref. 109.

2.4.2.1 CPET versus HAT

The comparison between HAT and CPET is difficult. Already the definitions are essential, whereas the HAT and the CPET is known to account for a single site and a multisite acceptor, respectively. This definition is fragile. Quantum effects hamper the knowledge of an exact position at a given time, thus superposition of different acceptors has to be treated.^{242, 243} Thus, especially in the transition between a tyrosine radical stacked to a tyrosine, the electron acceptor orbital is not exactly defined.^{106, 107}

Proton Coupled Electron Transfer

A more vigorous definition follows from the nature of the transferred particle. For an HAT an electron neutral particle is transferred, leading to minimal reorganization energies. Thus, the electron is moving stringent to the adiabatic Born Oppenheimer approximation concomitant with the proton. In the CPET case a non-adiabatic transfer is present. By the comparison of a benzyl/toluene and a phenyl/phenoxy system it was revealed that the first case is an HAT and the later a CPET.²⁴⁴ A strong difference between the proton transfer p and the electron transfer speed e could be shown in the two cases.

The ratio between p _and _eis the adiabaticty degree parameter p. Thus p≪1 are PCET reactions and p≫1 are HAT reactions. The transfer in the phenoxy/phenol couple occurred over a π-complex (proton donor-acceptor distance: 2.4 Å) with electrons 80 times faster than the proton movement.

In the benzyl/phenol case a σ complex (proton donor-acceptor distance: 2.72 Å) was formed here the proton movement was calculated to be 3.5 times faster than the electron, thus the electron can respond instantaneously to the proton motion. Further analysis revealed that the electronic coupling V_IF^el is significantly different with 700 cm^-1 to 14300 cm^-1 (CASSCF calculations) between cases Figure 2-20 A and B, respectively. Figure 2-20 demonstrates the effect and clearly illustrates the differences between both cases. In general, the adiabaticity of a PCET reaction can be taken as a good indicator to discriminate PCET and HAT.¹⁰⁹

Figure 2-20: Adiabatic potential along the transferring hydrogen coordinate. Two cases the phenoxy/phenol (A) and the benzyl/toluene (B) system are shown. The ground state is depicted in blue dots and the excited state in red dots. The black dashed lines represent the initial I and the final state II. The mixing of these states with the electronic coupling 𝑉𝑉_{𝐼𝐼𝐼𝐼}^{𝑒𝑒𝑒𝑒} leads to the adiabatic ground and excited states as shown by the calculation (red and blue dots). In case A, the adiabatic and diadiabatic states are virtually identical due to the small electronic coupling.

Picture modified from ref. 109.

2.4.2.2 CPET between a di-Tyrosine Model

The interaction between two backbone connected Y groups has been also studied in the biological context for Y⁷³⁰ and Y⁷³¹ in α by Kalia and Hummer.¹⁰⁶ They could show that in the “π-stacked” arrangement in contrast to the linear geometry of Figure 2-20A, the ground state potential decreases and the electronic coupling increases. Nevertheless, they came to the conclusion that a PCET takes place between this geometry. Water participation is possible but the energy barrier increases for water molecule mediated PCET from 8.5 to 14.1 kcal/mol. Notably, their model revealed that by exchanging the Y with a NO²Y (cf.

Figure 1-6, p.10) with a higher redox potential, this bias favors water mediated PCET. In this thesis, we will also apply bias on the natural di-tyrosine, by the introduction of a 3-amino tyrosine. The change from vacuum to water in a conductor like screening model increased the barrier by up to ≈4 kcal/mol.

Over all, this also demonstrates the necessity for high level calculations. In general, it highlights that QM and MM calculations are important to understand the basics and common principles of PCET reactions.

Proton Coupled Electron Transfer