Fig. 4: Results analysis of the PdH_x system with different H concentrations (x = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1).

a The mixing energy of PdH_x with different H concentrations. The mixing energy of each structure is calculated by referencing to the energy of the Pd slab and PdH₁ with the lowest adsorption energy. Therefore, only one sample in H concentrations of 0 and 1, while each other concentration contains 5000 samples. b The occupation ratio of H in different layers of Pd along with the total H concentration. The Pd slab consists of four layers of Pd atoms, providing five possible adsorption/embedding layers for H. The black, red, and blue lines represent H adsorption on the top and bottom surfaces of Pd, the second and fourth adsorption layers, and the third adsorption layer, respectively. Due to the symmetry of the slab model, the black and red lines correspond to two equivalent layers. c The average Pd-Pd coordination numbers of surface Pd atoms, calculated for the lowest energy configurations at each concentration gradient. The structures of pure Pd, PdH_0.5, and PdH₁ are shown. The red dot, error bars and violin plot show the average coordination number, the standard deviation of the average coordination number and the distribution of different Pd atom coordination numbers, respectively. d The concentration of hydrogen in Pd as a function of potentials. The dashed line shows the potential at −0.1 V vs RHE, corresponding to the PdH_0.6 structure. e The adsorption energies of *COOH and *CO on the lowest energy PdH_x structures at different hydrogen concentrations. f The free energy diagram of CO₂ reduction to CO with pure Pd and PdH_0.6. The insets show the *CO adsorption configurations on Pd and PdH_0.6 substrates. The free energy curves are obtained using the computational hydrogen electrode model. The hydrogen concentration in (a–e) is expressed as the unitless ratio of the number of hydrogen atoms to the number of Pd atoms. Color code: silver: Pd; yellow: H; red: O; black: C.