## 2 Reaction Mechanisms and Kinetics of Electrocatalytic Nitrate Reduction
### DFT Calculation as an Advanced Tool for eNO\({}_{3}\)R
#### 2.2.2 Nitrate to Ammonium  
For ammonia generation, research endeavors have shifted their focus towards investigating the subsequent step through DFT studies, specifically examining the hydrogenation of NO\({}_{\text{ads}}\). This pathway is widely acknowledged as the primary route for the synthesis of ammonium; however, it is complex to study due to the addition of hydrogen atoms, which can occur at both the nitrogen (N) and oxygen (O) atoms of NO, two possible products can be formed: HNO (hydrogenated nitrogen) or NOH (hydrogenated oxygen) [69]. Furthermore, recent studies have shown that the hydrogenation process occurs in four distinct steps. These steps correspond to different adsorption configurations of NO, including O-end, O-side, N-end, and N-side pathways (Figure 6) [53]. These hydrogenation steps are also known for their significant energy barriers, which are influenced by several factors, with the electrocataly being the key determinant. [PERSON] et al. [70] showed that the reduction in nitrate by different facets of Pd nanoparticles changes the selectivity and reduction rate. Energetics DFT calculations demonstrated that the reaction energy associated with the dissociation of *NO\({}_{3}\) to *NO\({}_{2}\) + *O was favorable on the Pd(111) facet than on the Pd(100) facet. In contrast, *NO\({}_{2}\) adsorbs more strongly on Pd(100) surfaces which favors strong ammonium formation compared to Pd(111) surfaces. Another study by [PERSON] et al. [55] selected the Pd(111) plane as the Pd facet to build Pd(WO\({}_{3}\))\({}_{3}\) catalyst and they have carried out DFT calculations to investigate the enhanced eNO\({}_{3}\)R activity. The comparison between the calculated adsorption energy of NO\({}_{3}^{-}\) on pure Pd and Pd(WO\({}_{3}\))\({}_{3}\) showed that the energy barrier of the hydrogenation *N to *NH on pure Pd is the largest by 3.38 eV. However, the doping of the surface with (WO\({}_{3}\))\({}_{3}\) reduced this energy barrier. This study also reported that the reaction undergoes a *NO\({}_{3}\)\(\rightarrow\) *NO\({}_{2}\)\(\rightarrow\) *NO\(\rightarrow\) *N \(\rightarrow\) *NH \(\rightarrow\) *NH\({}_{2}\)\(\rightarrow\) *NH\({}_{3}\) pathway remaining the same but the RDS changed from the hydrogenation of *N to *NH on pure Pd to the hydrogenation of *NH\({}_{2}\) to *NH\({}_{3}\) on Pd(WO\({}_{3}\))\({}_{3}\). Furthermore, it is important to note that RDS changes as a function of facet and material composition, as reported by [PERSON] et al. [55]. Figure 5: **(A)** The binding energies of **(a)** *NO\({}_{3}\), **(b)** *NO\({}_{2}\), and **(c)** *NO adsorbates on different transition metals and the horizontal dashed line indicates the chemical potential NO\({}_{3}^{-}\)[66]. **(B)** Volcano plot showing limiting potentials for **(a)** Ni, Rh, Pt, Pd, and Cu based on binding energies of NO and **(b)** Co, Ru, and Ir based on binding energies of OH. The horizontal dashed lines in **(a)** and **(b)** depict the equilibrium potential for the eNO\({}_{3}\)R to ammonia. The vertical dashed line in **(a)** depicts the Gibbs free energy of NO(g) [66]. **(C)** Theoretical volcano plots of the TOF as a function of atomic oxygen (\(\Delta\)E\({}_{\text{O}}\)) and nitrogen (\(\Delta\)E\({}_{\text{N}}\)) adsorption energies for electrocatalytic nitrate reduction on transition metal surfaces based on DFT-based microkinetic simulations at **(a)** \(-\)0.2 V, **(b)** 0 V, **(c)** 0.2 V, and **(d)** 0.4 V vs. RHE. Reaction conditions are T = 300 K with a H\({}^{+}\)/NO\({}_{3}^{-}\) Molar ratio of 1:1. White indicates unphysical regions where the activation energies of some elementary steps are negative. Ag falls within the white region due to errors associated with using linear scaling relationships [67]. **(D)** **(a)** Theoretical limiting potential U\({}_{\text{L}}\) for nine double-atom catalysts supported on N-doped graphene (TM\({}_{2}\)/N\({}_{6}-\) G) screened out by stability analysis.