# Noble Metal-Based Heterogeneous Catalysts for Electrochemical Hydrogen Evolution Reaction
## 3 Catalytic Mechanism of the HER
### D-Band Center Theory  
The classical volcanic curve describes the semi-quantitative relationship between the HER rate and \(\Delta\)G\({}_{\text{H*}}\) of M-H*, which is essentially closely related to the electronic structure of the metal surface. The DFT calculations also provide greater insight into the electronic structure, particularly how the d-orbital electrons affect the adsorption strength of H* on a given electrode surface.  
The d-band model established by [PERSON] et al. [29; 57] is an approximate description of bonding between adsorbent atoms on a given metal surface. As shown in Figure 4c [29], the valence space orbitals of adsorbent atoms are first coupled with the s-band on the metal surface to form a low-bonding band with energy close to the s-band of the metal, named the reforming band. The reforming band is coupled with the metal d-band, and the energy is further reduced to form a bonding band, with energy lower than the reforming band, and an anti-bonding band with energy higher than the d-orbital, which is called the final bonding band. If the anti-bonding band is less filled with electrons, then the larger the part is above the Fermi level, indicating that the bonding is more stable. If the antibonding band is more filled with electrons, then the smaller the part is above the Fermi level, demonstrating that the bonding is less stable. The stability of the bonding between the adsorbent and the metal surface is influenced by the coupling of the s-band with the valence orbital of the adsorbent and the d-band with the reforming band. [PERSON] believed that the energy range and shape of the reforming orbit formed by the valence shell orbit of adsorbates and the s-band of metals in the first step are not very different for all transition  
Figure 4: (**a**) Volcano plot between the logarithm of exchange current density of a hydrogen electrode and the Gibbs adsorption free energy of a hydrogen atom based on the DFT calculation [8]. (**b**) “Volcano plot” of Pt-modified with different metal hydroxide clusters (the inset is a partial enlargement corresponding to the top of the volcano) [43]. (**c**) Schematic illustration of the formation of a chemical bond between an adsorbate valence level and the s– and d–states of a transition-metal surface [29]. Copyright 2020, 2021 [PERSON] Wiley and Sons; 2005 Springer Nature.  
metals. It can be considered that the reforming band formed by all transition metals in the first step is similar. Therefore, the d-band of the second step plays an important role. If the d-band center determines the energy range of the anti-bonding band formed in the second step, then the energy level of the d-band center determines the degree of the anti-bonding band being filled by electrons. It further determines the stability and strength of the adsorption bond, which is the d-band center (\(\varepsilon_{\text{d}}\)) theory. The interaction between H* orbitals and metal d-orbitals will produce fully filled bonding orbitals with low energy and partially filled anti-bonding orbitals with high energy located above the metal d-band. The strength of the M-H bond depends on the relative occupation of the anti-bonding H 1s-M d. The higher the occupation of anti-bonding orbitals is, the weaker the bonding strength will be. Therefore, the adsorption energy of H* and \(\Delta\)G\({}_{\text{H}^{*}}\) of transition metal-based catalysts can be qualitatively predicted and verified by comparing the calculated d-band states density on the metal surface with the Fermi energy levels. Each metal system can adjust its d-band center by alloying and constructing heterogeneous interfaces and optimizing the H* adsorption energy, thus improving the HER performance.