Computational Details

2-1. The computation method by Truhlar

We used density functional theory (DFT) with the M11-L functional in Gaussian 09.

The 6-311+G-(2df,2p) basis set was used for H, C, N, O, the optimization and frequency calculations were done using the LanL08 pseudopotential for the Ru metal, and the single-point and solvent model MG3S was used for the metal (Ru). The solvent effect in ethanol was considered using the solvation model (SMD) ³⁶.

2-2. The computation method

Gaussian software, whose name originates from the Gaussian basis sets in the software, is the most widely used software in quantum chemical calculations. It uses the principles of quantum mechanics to generate wave functions and calculate energy.

It can predict gas and liquid conditions, various chemical reactions and properties of molecules, such as molecular structure and energy, vibrational frequencies of molecular systems, infrared and Raman spectroscopy, etc. It can be used to observe intramolecular and intermolecular reactions, the intermediate product, and even the molecular structure of the transition state. We computed the single-point energy, vibrational frequencies, solvent effects, and performed geometry optimization and NPA and population calculations. Some of the calculation results can be used to evaluate reaction free energy potentials and pKa values.

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2-2-1. Single-Point Energy

Calculating single-point energy involves calculating wave function and associated charge density in terms of molecular coordinates. The purpose of a single point energy calculation is to find the charge density for which the total energy functional is minimized.

2-2-2. Geometry optimization

The properties of a molecule are obtained from the representative minimum-energy geometry of the molecule, which is obtained by geometry optimization based on the changing interatomic distances in the molecules and the corresponding energy changes. The energies generated by the changes in molecular geometry are called potential energy surfaces (PES). In terms of mapping coordinates and energy of processes, structures are adjusted in accordance with the PESs until a stable minimum value is obtained, which represents the geometry of a stable molecule.

Figure 2-2-2. Potential Energy Surface

As shown in Figure 2-2-2, there are several significant points in PES: global maximum, global minimum, local maximum, local minimum, and saddle point. Global maximum is the highest point of the entire PES, while global minimum is the lowest point, indicating the most stable structure. Local maximum and local minimum, respectively, are the highest and lowest points in a region; the saddle point is crossing point of two high points and two low points, signifying the transition state of these two

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low points.

In the optimization process, the first derivative of the energy with respect to displacement of the atoms is zero for the global minimum, local minimum, or the saddle point. A second derivative is used to determine whether the obtained result is a saddle point or not. If the obtained frequency is positive, it implies that the associated energy is located on a local minimum; if the frequency value is negative, it implies that the associated energy lies on the saddle point. In addition, when the difference in energy falls in the preset criteria, the program considers that the optimization process has converged.³⁸

2-2-3. Frequency

Single-point energy calculations and geometry optimization ignore the vibrational states of each atom in the molecule, frequencies associated with these vibrational states were analyzed by obtaining the second derivatives of the energy.

Frequency analysis for the system also includes thermodynamic analysis in terms of enthalpy and entropy; in the default case, the system calculates the value for a temperature of 298.15 K and pressure of 1 atm using the formula for Gibbs free energy:

G = H − TS

The resulting of Gibbs free energy plays an important role in this article, and calculated E⁰ and pKa values are important for the analysis.

2-2-4. Solvation model

The transition state and molecular properties are different between the gas phase and the liquid phase, because the electrostatic field generated by the solution will affect the properties of the molecule. To obtain the molecular properties in liquid phase, a solvation model must be used.

The theoretical models of systems in non-aqueous solution are referred to as

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consistent reaction field (SCRF) models, where the solvent is described as a continuous reaction field, constituted by the dielectric constant of the solvent. A solute molecule in solvent space will be recalculated in liquid phase in terms of intermolecular forces and reaction field forces.

The algorithmic efficiency and results are related to the solvation model, so it is important to choose a suitable solvation model. The following Figure. 2-2-4 displays four types of solvation model.

Figure 2-2-4. Four type of solvation model

1. Onsager (dipole & sphere) Model: This is the simplest reaction field, where the solute occupies a fixed spherical cavity of a particular radius within the solvent field. An external dipole will induce a dipole in the medium of the molecule, and the solvent dipole produced by the applied electric field will in turn interact with the molecular dipole, leading to net stabilization.

2. Tomasi’s Polarized Continuum Model (PCM): The cavity is defined by a series of interconnected atomic spheres, which induce a dipole in the cavity of the solvent reaction field to generate a dipole field according to each atom, leading to electric polarization. This is the most representative model.

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3. Isodensity Polarized Continuum Model (IPCM): The cavity is defined as an isodensity surface of the molecule, which is based on the wave functions calculated in the gas phase. The cavity shape is then calculated until it attains a stable value.

4. Self-Consistent Isodensity Polarized Continuum Model (SCI-PCM): This procedure solves for the electron density that minimizes the energy, including solvation energy. In contrast to IPCM, SCI-PCM accounts for full coupling between the cavity and the charge density and includes coupling terms.³⁸

2-2-5. The calculation of potential, pK

, and free energy in solvent

In the 1 M standard state, a proton dissociation free energy change in aqueous solution is given by ∆G_aq^∗ (superscript * denotes under 1 M standard state), Which is calculated using the following thermodynamic cycle:

∆G_aq^∗ ＝∆G_𝑔^° + ∆∆G_s^∗+ ∆G^°→∗

∆G_𝑔^° = G_𝑔^°(𝐴⁻) + G_𝑔^°(𝐻⁺) − G_𝑔^°(𝐻𝐴)

∆∆G_s^∗ = ∆G_s^∗(𝐴⁻) + ∆G_s^∗(𝐻⁺) − ∆G_s^∗(𝐻𝐴)

Scheme 2-2-5-1. The thermodynamic cycle of dissociation in gas and solute phase

∆G_g^° can be calculated at 1 atm standard state by initial count or DFT, ∆G_s^∗ is the solvation free energy, G_g^°(𝐻⁺) is the gas-phase proton free energy, whose value at 298 K is −6.28 kcal/mol as calculated by the Sachur-Tetrode equation^39,40,41,

∆G_s^∗(𝐻⁺) value of the aqueous solution at 298 K (−263.98 kcal/mol) was proposed by

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Tissandier and coworkers⁴⁰, ∆G^°→∗ is the free energy difference (1.89 kcal/mol) between the standard states in the gas and liquid phases.

Due to differences in the thermal correction between gas phase and liquid phase, convergence of the geometry optimization in the liquid phase to obtain ∆G_s^∗ is difficult.

It cannot be calculated by vibration frequency analysis directly in a continuum solvation model. We do not expect the geometry to change much before and after proton and electron transfer; therefore, the error in the thermal correction term is ignored. The thermal correction term that can be neglected because the differences in thermodynamic cycle are small.

The ∆G_aq^∗ value as calculated above, can be used to calculate 𝑝𝐾_𝑎values of the various intermediates as per the following formula:

𝑝𝐾_𝑎 = ∆G_aq^∗ 2.303𝑅𝑇

where R is the gas constant and T denotes the temperature in Kelvin.

Scheme 2-2-5-2. Thermodynamic cycle for the proton-coupled reduction of species

Reduction reaction can be represented by a thermodynamic cycle, as shown in

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Scheme 2-2-5-2. R is a reduced acidic species and O is an alkaline oxidized species.

The energy of the gas-phase free electrons is −0.006 kcal/mol; if m > 0, it indicates that the reduction reactions involved electron transfer coupled with proton transfer, i.e. the reductions are PCET processes. The free energy change associated with the liquid phase can be obtained from the thermodynamic cycle.

Standard potential can be obtained by the following equation:

𝐸_𝑂|𝑅^∗ = −∆𝐺₀^∗

𝑛𝐹 − 𝐸_𝑆𝐶𝐸^∗ = −(∆𝐺_𝑂|𝑅^∗ − 𝑛∆𝐺_𝑁𝐻𝐸^∗ )

𝑛𝐹 − 𝐸_𝑆𝐶𝐸^∗

Here, ∆𝐺₀^∗ is the relative free energy of the standard hydrogen electrode (NHE), and

∆𝐺_𝑁𝐻𝐸^∗ values are 4.28V, and the potential of the standard reference electrode is given by 𝐸_𝑆𝐶𝐸^∗ = +0.24 V vs. NHE.^39,43 For m> 0, the standard reduction potential of PCET at pH = 0 is obtained using the Nernst Equation expressed in terms of pH as follows:

E = 𝐸_𝑂|𝑅^∗ +𝑅𝑇 𝑛𝐹ln (𝑎_𝑂

𝑎_𝑅) −𝑚

𝑛 ∙ 0.0591 ∙ 𝑝𝐻

where m and n are the numbers of protons and electrons, respectively, R is the gas constant, T is the temperature in Kelvin, F is the Faraday constant, 𝑎_𝑖 is a chemical activity (i = O or R); in the reduction half-reaction, the second term is zero.

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