## 2 Experimental Section
### DFT Calculation  
The CASTEP program was employed to simulate how STG interacts with the molybdenumite (001) surface, both with and without the presence of copper ions. This program offers an effective way to implement the DFT plane-wave pseudopotential approach [22; 23; 24; 25], and is applied to optimize the geometry of adsorbate-surface systems. The PBE/GGA functional was adopted to treat exchange-correlation interactions [26]. An ultrasoft pseudopotential was employed to depict the electron-ion interactions [27; 28; 29]. In all calculations, the cutoff energy was selected to 500 eV, and the Monkhorst-Pack scheme was applied for k-space sampling. For electronic minimizer, the density mixing method was applied with a convergence tolerance of 2.0 \(\times\) 10\({}^{-6}\) eV/atom. The criteria for structure optimization and total energy calculations were set as follows: an energy tolerance of 2 \(\times\) 10\({}^{-5}\) eV/atom (a), a maximum force tolerance of 0.05 eV/A (b), and a maximum displacement tolerance of 0.002 A (c).  
As shown in the distribution diagrams in Figure 2, the speciation calculations for 8 mg/L Cu\({}^{2+}\) solution, as a function of pH, indicate that Cu\({}^{2+}\) dominates under acidic conditions (pH < 6), while CuOH\({}^{+}\) becomes the major species at approximately pH 8. At higher pH values, species such as Cu(OH)\({}_{2}\)(aq), Cu(OH)\({}_{3}\)\({}^{-}\), and Cu(OH)\({}_{4}\)\({}^{2-}\) gradually become dominant. Based on this trend, CuOH+ was selected as the representative hydrolyzed copper species under near-neutral conditions for subsequent DFT modeling.  
The optimized cell structure of molybdenite and the surface structure of molybdenite are shown in Figure 3. The molybdenite surface was modeled using a (7 \(\times\) 3) MoS\({}_{2}\) (001) surface supercell. This model included six atomic layers and a 30 A vacuum region, which was sufficient to minimize any interaction between adjacent slabs. During the calculation, the three bottom atomic layers of the MoS\({}_{2}\) (001) surface were fixed in place. The molecular structures of H\({}_{2}\)O, STG, and CuOH\({}^{+}\) were pre-optimized and are shown in Figure 4. The hydrated structure of the molybdenite (001) surface, the STG adsorption structure on the hydrated molybdenite (001) surface, and the STG adsorption structure on the CuOH\({}^{+}\)-modifiedmolybdenum (001) surface were further optimized on a one-by-one basis. The interaction energy between the adsorbate and the molybdenite (001) surface is expressed by the following equation:  
\[\text{E}_{\text{ads}}=\text{E}_{\text{surface + adsorbate}}-\text{E}_{\text{surface}}-\text{E}_{\text{adsorbate}} \tag{1}\]  
Here, \(\text{E}_{\text{surface + adsorbate}}\) is the total energy before and after the interaction with the adsorbate, \(\text{E}_{\text{surface}}\) represents the surface, \(\text{E}_{\text{adsorbate}}\) is the adsorbate energy; and \(\text{E}_{\text{ads}}\) is the adsorption energy between adsorbate and the molybdenite (001) surface. A more negative Eads implies a stronger adsorption tendency.  
Figure 3: Cell structure of molybdenite (**a**) and the (001) surface structure (**b**) of molybdenite.  
Figure 2: Distribution diagrams of 8 mg/L Cu\({}^{2+}\) solution as a function of pH.