Geometries and energies were calculated using the Perdew-Burke-Ernzerhof functional (33) with van der Waals correction proposed by Grimme (i.e., DFT-D2 method) (34), as implemented in the Vienna ab initio simulation package (VASP 5.4.1) (35). The projector augmented wave method was used to describe the interaction between the ions and the electrons (36). The energy cutoff of the plane wave was set to 400 eV. A (2 × 4) supercell of the anatase (101) surface with two stoichiometric TiO2 layers (about 11 Å × 15 Å × 6 Å; see fig. S7, D and E) was used as the substrate. A vacuum gap of 12 Å was used to avoid the periodic image interaction normal to the surface. During the geometrical optimization, the bottom TiO2 layer was fixed at its bulk position, while all other atoms were allowed to relax, until the forces on each atom were smaller than 0.02 eV Å−1. Only the Γ point of the Brillouin zone was sampled. The Gaussian smearing method with a smearing width of 0.2 eV was used to accelerate the convergence of integration at the Brillouin zone. The reaction pathways and transition states were traced by the climbing image nudged elastic band method with a spring constant of 5.0 eV Å−2 (37, 38). No obvious differences in the structural properties (<0.002 Å) and energy barriers (<0.002 eV) were observed when k points were increased from the Γ point only to a 2 × 1 × 1 k-point mesh and the energy cutoff from 400 to 600 eV (table S6), indicating that the calculations had converged, and the computational settings used here were reliable for describing the studied reactions. The coupling effect between two adjacent vanadyl species was generally applicable in dimeric and higher-order polymeric vanadia structures. During the NH3-SCR of NO over the polymeric vanadyl species, the coupling effect between the two adjacent vanadyl species (i.e., within a dimer unit of vanadia) at the reaction site accelerated the whole catalytic cycle, and hence, it could be expected that dimeric and higher-order polymeric vanadia would have similar effects on the SCR reaction. The dimeric vanadyl species is the basic structural unit of various polymeric vanadia structures and can reasonably represent the coupling effect in polymeric vanadia structures; hence, it was used as the model in our DFT calculations.

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