## 2 Reaction Mechanisms and Kinetics of Electrocatalytic Nitrate Reduction
### DFT Calculation as an Advanced Tool for eNO\({}_{3}\)R
#### 2.2.4 eNO\({}_{3}\)R vs. HER  
Generally, an ideal catalyst for eNO\({}_{3}\)R should maintain high stability and catalytic activity while should effectively suppress the Hydrogen Evolution Reaction (HER). For this reason, Strong competition from HER presents a significant challenge in aqueous systems [72]. Notably, [PERSON] et al. [51] found that HER competes strongly with nitrate reduction; at pH = 0, the formation of H\({}_{2}\) requires only 0.25 eV, lower than the 0.37 eV needed for nitrate reduction on Cu(111), making eNO\({}_{3}\)R less favorable in acidic conditions. In neutral environments (pH = 7), eNO\({}_{3}\)R outcompetes HER, while in alkaline conditions (pH = 14), HER predominates [51]. This indicates that pH significantly influences the competitive dynamics between HER and eNO\({}_{3}\)R. However, it is not desirable to fully suppress HER, as the Adsorption strengths of the H atom or active hydrogen species are essential for the next step--nitrogen hydrogenation to produce ammonia, as discussed in the previous section of nitrate to ammonium. Typically, the binding energy of *H serves as a useful descriptor of hydrogen evolution activity across a variety of catalyst materials. To analyze selectivity trends for the NO3 RR among transition metals, [PERSON] et al. [66] used an approach similar to that used for CO\({}_{2}\) reduction. Accordingly, they plotted the difference U\({}_{\rm L}\)(eNO\({}_{3}\)R) \(-\) U\({}_{\rm L}\)(H\({}_{2}\)) against the limiting potential U\({}_{\rm L}\)(eNO\({}_{3}\)R) for various transition metals as illustrated in Figure 7a. A more positive value of U\({}_{\rm L}\)(eNO\({}_{3}\)R) \(-\) U\({}_{\rm L}\)(H\({}_{2}\)) correlates with increased  
Figure 6: Schematic illustration of the possible hydrogenation of nitrous oxide including O-end, O-side, N-end, and N-side pathway to NH\({}_{3}\).  
selectivity for eNO\({}_{3}\)R over HER, suggesting that catalysts in the upper right region exhibit both high activity and selectivity for eNO\({}_{3}\)R. Importantly, this analysis focuses on selectivity for the eNO\({}_{3}\)R relative to HER rather than specifically towards NH\({}_{3}\) or N\({}_{2}\). The results suggest that Cu emerges as the most active and selective catalyst for the eNO\({}_{3}\)R, aligning with previous studies identifying Cu as a leading candidate. Additionally, while Ir shows higher catalytic activity than Cu, its lower selectivity towards eNO\({}_{3}\)R products leads to H\({}_{2}\) being the dominant product under eNO\({}_{3}\)R conditions. Furthermore, it is critical to develop strategies to effectively manage this competition. Therefore, active sites should preferentially bind NO\({}_{3}\) over H atoms, with \(\Delta\)G\({}_{\text{NO3}}>\Delta\)G\({}_{\text{H}}\), and the formation of HER byproducts can be minimized with U\({}_{\text{L}}\)(H\({}_{2}\)) \(>\) U\({}_{\text{L}}\)(NH\({}_{3}\)) [66]. In this context, [PERSON] et al. [68] considered TM\({}_{2}\)/N\({}_{6}-\) G (TM\({}_{2}\) = Sc\({}_{2}\), T\({}_{2}\), V\({}_{2}\), Cr\({}_{2}\), Mn\({}_{2}\), Fe\({}_{2}\), Co\({}_{2}\), Cu\({}_{2}\), and Zn\({}_{2}\)) as potential catalysts. The adsorption capabilities of NO\({}_{3}\) and the suppression of HER (illustrated in Figure 7b) was assessed. Among the nine candidates, Cr\({}_{2}\)/N\({}_{6}-\) G, Mn\({}_{2}\)/N\({}_{6}-\) G, and Cu\({}_{2}\)/N\({}_{6}-\) G met the selectivity criteria: \(\Delta\)G\({}_{\text{NO3}}>\Delta\)G\({}_{\text{H}}\) and U\({}_{\text{L}}\)(H\({}_{2}\)) \(>\) U\({}_{\text{L}}\)(NH\({}_{3}\)). The free energy diagrams for HER indicated uphill \(\Delta\)G\({}_{\text{PDS}}\) values of 0.59, 0.24, and 0.37 eV for Cr\({}_{2}\)/N\({}_{6}-\) G, Mn\({}_{2}\)/N\({}_{6}-\) G, and Cu\({}_{2}\)/N\({}_{6}-\) G, respectively, at their corresponding limiting potentials of \(-\)0.46 V, \(-\)0.45 V, and \(-\)0.36 V. Thus, these three systems were identified as promising candidates for nitrate reduction to ammonia [68].