In the optimisation of WOCs, computational techniques are increasingly employed to test proposed WOC design principles.1–8 When evaluating proposed
catalytic cycles, the free energies of the various catalytic intermediates are compared.1–6 This comparison often uses correction factors to approximate the
energetic contributions of the protons and electrons transferred during the catalytic step. 1,3,5,9–12 Here the pitfalls of the commonly employed approach are
highlighted, and an improved framework is suggested based on the CSA.
Figure 4.1 The four PCET steps between the catalytic intermediates I1 – I0. Vertical lines denote electron transfer, horizontal lines proton transfer. Stable intermediates are shown in black. The ligand exchange I0 + 2H2O → I1 + O2 is also shown.
Consider the PCET reaction
shown in Figure 4.1, which has a change in free energy
Δ𝐺(𝐼𝑖⟶ 𝐼𝑖+1) = 𝐺(𝐼𝑖+1) − 𝐺(𝐼𝑖) + Δ𝐺(𝐻+) + Δ𝐺(𝑒−). (4.2) As mentioned in Chapter One, G has traditionally been approximated by
𝐺 = (𝐸𝑣𝑎𝑐+ 𝑍𝑃𝐸𝑣𝑎𝑐− 𝑇𝑆𝑣𝑎𝑐) + 𝛿𝐸𝑠𝑜𝑙𝑣, (4.3)
𝛿𝐸𝑠𝑜𝑙𝑣= 𝐸𝑠𝑜𝑙𝑣− 𝐸𝑣𝑎𝑐. (4.4)
This becomes problematic because it calls for the solvation energy of a proton 𝛥𝐺𝑠𝑜𝑙𝑣(𝐻+) =-11.45 eV,13 which will always dominate 𝛥G completely: 𝛥G is usually of the order of 1 eV. In the past,14,15 the energetic contributions of the
proton and electron have often been approximated by combining the proton and electron into ½H2. Upon closer inspection the validity of the ½H2 approximation
is somewhat questionable, as in the PCET reactions considered H+ is always
bonded to another ion. The transfer of H+ is a combination of the formation of
one O – H bond and the breaking of another, with an intermediate step where H is coordinated in between. As noted by Nachimuthu et al. previously,16“modelling
proton transfer reactions is often challenging because of the complexity of processes involving H-bond network rearrangement.”
And yet, it is this complexity of the network rearrangement processes that is ignored when using the ½ H2 approximation. This may have been sufficient when
considering heterogeneous catalysts, but as we move into molecular catalysts the how and why of the PCET process needs to be addressed. Two intermediates can no longer be seen as isolated from each other, instead we must also seek to optimise the processes between them. To do so, explicit water molecules and a metal ion (𝑀𝑒) are included within the simulation box, to act as proton and electron acceptors respectively. In this way the charge carriers and the processes via which they are transferred from the catalyst can be considered explicitly. For the PCET reaction shown in Eqn. (4.1), the equivalent equation in the simulation box is
𝐼𝑖+ 𝑀𝑒3+⟶ 𝐼𝑖+1+ 𝐻𝑠𝑜𝑙𝑣+ + 𝑀𝑒2+, (4.5) where 𝐻𝑠𝑜𝑙𝑣+ denotes the solvated proton, which is often part of a more complicated structure.17–19 We can then decouple the PCET reaction into an
electron- and proton-transfer process. For the electron transfer step
𝐼𝑖+ 𝑀𝑒3+⟶ 𝐼𝑖++ 𝑀𝑒2+. (4.6)
In the context of the CSA methodology, the energy needed to transfer an electron from the catalytic intermediate to the electron acceptor can be calculated by
Δ𝐸𝑒−= 〈𝐸𝐾𝑆(𝐼𝑖++ 𝑀𝑒2+)〉 − 〈𝐸𝐾𝑆(𝐼𝑖+ 𝑀𝑒3+)〉, (4.7)
where 〈𝐸𝐾𝑆〉 is the time-averaged KS energy at 300 K. One should note that Δ𝐸𝑒− also includes the reorganisation energetic contributions resulting from the electron transfer.20 This includes contributions from internal vibrational and
external solvent rearrangement. Because the number of particles, charges, bonding patterns, and conformations of the reactants and products in Eqn. (4.6) remains the same, the change in entropy and zero point energy will be negligible, i.e. Δ𝐸𝑒−≈ Δ𝐺𝑒−.
The proton-transfer process
𝐼𝑖+⟶ 𝐼𝑖+1+ 𝐻𝑠𝑜𝑙𝑣+ (4.8)
may be investigated using constrained CPMD along the reaction coordinate of proton solvation. Δ𝐺𝐻+ can then be calculated for each reaction step according to Eqn. (1.24) in Chapter 1, where the respective 𝑥 = d(O→H) (see also Figure 4.2). So practically, in CSA the total change in energy for the 𝐼𝑖⟶ 𝐼𝑖+1 PCET reaction simulation is given by
Δ𝐺𝐶𝑆𝐴= Δ𝐺𝐻++ Δ𝐸𝑒−. (4.9)
Δ𝐸𝑒− includes energetic contributions from both the oxidation of the catalyst, as well as the reduction of 𝑀𝑒3+. If these events are to be considered independently from each other,
Δ𝐸𝑒−= Δ𝐸𝑒−(𝐼𝑖⟶ 𝐼𝑖+) + Δ𝐸𝑒−(𝑀𝑒3+⟶ 𝑀𝑒2+), (4.10) then it is of crucial importance that Δ𝐸𝑒−(𝑀𝑒3+⟶ 𝑀𝑒2+) remains constant
throughout the catalytic cycle. If this is the case, then the change in free energy for the catalyst would be given by
Δ𝐺(𝐼𝑖⟶ 𝐼𝑖+1) = Δ𝐺𝐻+(𝐼𝑖⟶ 𝐼𝑖+1) + Δ𝐺𝑒−(𝐼𝑖⟶ 𝐼𝑖+1)
= Δ𝐺𝐻+(𝐼𝑖⟶ 𝐼𝑖+1)
+ (Δ𝐸𝑒−− Δ𝐸𝑒−(𝑀𝑒3+⟶ 𝑀𝑒2+)).
(4.11)
Via this formalism, the way has been paved for an energetic consideration of the
process of a reaction step which includes both electron and proton transfer.
Although this transcends the static consideration which uses the correction term ½ H2, it does introduce the extra complication of the energetic contribution due
to the electron acceptor. We propose that if this contribution is kept constant throughout the catalytic cycle, it will provide less of a complication than one may initially expect.
WOC Ru-bpy is used, whose catalytic cycle is described in Chapter Two (see also Scheme 4.1). Ru-bpy provides an excellent text case as its catalytic cycle has been explored both experimentally and computationally using static methods.21
Furthermore, it has a relatively small number of atoms, which for a test case is an attractive property computationally. Here the first and second catalytic PCET steps of this WOC are examined within CSA. The results of this are then compared to experimental data, as well as computational data obtained using static methods.