Fig. 5: Schematic and quantitative description of the reaction kinetics.

a The traditional Butler–Volmer reaction kinetics. Starting from product (blue curves), at zero applied potential, (U₀) the reaction between reactant, R and product, P₀ shows a transition state (TS₀) with R₀ spatial distance along the reaction path. The transition state (TS₁) shifts toward the reactant with lower in free energy gives R₁ spatial distance at applied potential U₁ between reactant R and product P₁ (red curves). The product forms with a single electron transfer from electrode to product, leading an electron transfer jump along the reaction pathway. b, c Grand canonical potential kinetics methodology, illustrating the relationship between the TS geometry and charge as the reaction changes continuously with applied potential, leading to a continuous potential dependent reaction path. The blue curve represents the reaction pathways for cis-COOH to CO conversion on Ni–N₄ site at zero (0 V) potential while the red curve is for −0.8 V applied potential. At 0 V, the reaction between reactant, R or R_0V and product, P_0V produces a transition state (TS₀) with OC–OH = 2.77 Å (blue dotted line) and an energy barrier of 16.89 kcal mol⁻¹. At −0.8 V applied potential the energy barrier for the transition state TS_−0.8V decreases to 2.98 kcal mol⁻¹ with OC–OH = 1.46 Å (red dotted line). The black line connecting the transition states at 0 V (TS₀) and −0.8 V (TS_−0.8V) shows the linear continuous relation between the applied potential and species charge within the system. The dotted circles indicate the transition states at different applied potentials. d, e The geometry of the transition state at zero (TS₀) and −0.8 V (TS_−0.8V) applied potential, resulting a charge difference of 0.6 e. The arrow represents the TS direction on applied potential. (brown: carbon, blue: nitrogen, green: nickel, red: oxygen, off white: hydrogen atom).