Reaction mechanism and kinetics for CO2 reduction on nickel single atom catalysts from quantum mechanics

Experiments have shown that graphene-supported Ni-single atom catalysts (Ni-SACs) provide a promising strategy for the electrochemical reduction of CO2 to CO, but the nature of the Ni sites (Ni-N2C2, Ni-N3C1, Ni-N4) in Ni-SACs has not been determined experimentally. Here, we apply the recently developed grand canonical potential kinetics (GCP-K) formulation of quantum mechanics to predict the kinetics as a function of applied potential (U) to determine faradic efficiency, turn over frequency, and Tafel slope for CO and H2 production for all three sites. We predict an onset potential (at 10 mA cm−2) Uonset = −0.84 V (vs. RHE) for Ni-N2C2 site and Uonset = −0.92 V for Ni-N3C1 site in agreement with experiments, and Uonset = −1.03 V for Ni-N4. We predict that the highest current is for Ni-N4, leading to 700 mA cm−2 at U = −1.12 V. To help determine the actual sites in the experiments, we predict the XPS binding energy shift and CO vibrational frequency for each site.


Derivation of GCP (U) equation
In the GCP-K formulation, the free energy F(n) and GCP (U) can be obtained from either constant charge or constant potential calculations. In this study, we first calculated F(n) by QM method at different charges, and then fitted the quadratic curve by using the equation of

S3 Supplementary Note 2 Spin effect evaluation
To study the possibility of spin polarization for hybrid DFT, we simplified the system to the finite Ni doped graphene-like system below and carried out B3LYP-D3 hybrid functional calculations using Jaguar. This used the Los Alamos core-valence effective core potential (describing Ni with 18 explicit electrons) with the LACV3P**++ basis set. For the Ni-N4 system without ligand we found that the ground state is a singlet with no spin polarization. The lowest triplet state is higher by 0.95 eV; it has pz singly occupied orbitals. This suggests that the configuration on the Ni is closed shell d8 with doubly occupied xy, yz, xz, and z 2 d-orbitals and empty x 2 -y 2 overlapping the 4 N sp 2 lone pairs pointing at the Ni. Here the triplet corresponds to a graphene  to * transition.
We also carried out the PBE-D3 calculation with Jaguar and found a ground state singlet with the triplet 0.91 eV higher.
In the VASP PBE-D3 calculations, the ground state was closed shell even though we allowed spin polarization. To explore further the possibility of open shell character, we required an Ms=1 state (triplet) and found an energy 0.25 eV higher with the unpaired spins on the graphene. Thus we conclude that the ground state is correctly described in VASP PBE-D3.
For the most important intermediate (cis-COOH), B3LYP-D3 produces a doublet ground state with the unpaired spin in the x 2 -y 2 orbital. The lowest quartet state is 0.28 eV higher with unpaired spins also in the  and * orbitals. We interpret this as the radical C of (HO)C=O forming a covalent bond to the Ni triplet excited state that starts with a hole in the dz 2 and an electron in the x 2 -y 2 . This leads to a covalent C-Ni  bond leaving the unpaired spin in the x 2 -y 2 orbital. For PBE-D3 we found the doublet ground state with the quartet 0.67 eV higher. Thus for the Ni-N4 system with and without ligand, PBE-D3 and B3LYP-D3 predict the same trend.
In the VASP PBE-D3 calculations on the (cis-COOH) system, we found a closed shell description, with partial occupation of the x 2 -y 2 orbital. Thus we conclude that the PBE-D3 description is adequate.

Supplementary Flow Chart 1
The procedure shows how to obtain the grand canonical potential as a function of fixed potential.
Geometry optimization using VASPsol as a function of charge ( − 0 ) = 1 , 2 , 3 , … Single point calculation using same charge in jDFTx with CANDLE solvation method to get the Free energies (F1, F2, F3 ……) as a function of charge ( 1 , 2 , 3 , …) respectively Finally, GCP (U) obtain by the minimization of grand canonical free energies, G (n; U) via Legendre transformation which converts fixed charge free energy, F (n), to fixed potential grand canonical potential, GCP(U) according to following equation Here we show a simple demonstration of our new grand canonical potential calculation by applying constant potential method. We initially obtain the free energy as a function of constant charge method, then we minimize the free energy quadratic equation as a function of applied potential (U1 vs RHE). Then, we obtain the direct dependence of applied potential (U1 vs RHE) on GCP (U) value. As the applied potential (U1) is changed, it leads to changes in the charges within the system as in the relation of U and nmin.

S6
Supplementary  and (e, f) side view of our used models for the system of COOH to CO conversion. We compare the energy barrier for the system with more explicit water (six) molecules with our model (three water molecules). We found that both systems have same energy barrier, but more waters make the system computationally expensive. charges on the TS as the potential is applied to initiate the reduction process. The transition state at zero potential is close to the product (OC-OH2 is 3.62 Å) while with applied potential it moves towards the reactant. The initial bond distance at 0 V (2.19 Å) between OC-OH in the trans-COOH TS decreases linearly with applied potential, reaching 1.44 Å at -0.5 V (b). In contrast the distance between O(COOH)-H(H2O) gradually increases with potential (c). Compared to the cis-COOH to CO path, the trans-COOH path has a lower energy barrier, requiring less overpotential to overcome the barrier because of the extra charge initially in the trans-COOH system. The charges within the TS species vary linearly with potential as reaction progresses in the forward direction.
So, normalized current density for 3.14 × 10 16 active sites is, j (mA/cm 2 ) = Current obtained from theoritical calculation × Total number of Ni active sites (exp. calc. value) Total number of Ni active sites obtained theoriticallly