Improving Molecule–Metal Surface Reaction Networks Using the Meta-Generalized Gradient Approximation: CO2 Hydrogenation

Density functional theory is widely used to gain insights into molecule–metal surface reaction networks, which is important for a better understanding of catalysis. However, it is well-known that generalized gradient approximation (GGA) density functionals (DFs), most often used for the study of reaction networks, struggle to correctly describe both gas-phase molecules and metal surfaces. Also, GGA DFs typically underestimate reaction barriers due to an underestimation of the self-interaction energy. Screened hybrid GGA DFs have been shown to reduce this problem but are currently intractable for wide usage. In this work, we use a more affordable meta-GGA (mGGA) DF in combination with a nonlocal correlation DF for the first time to study and gain new insights into a catalytically important surface reaction network, namely, CO2 hydrogenation on Cu. We show that the mGGA DF used, namely, rMS-RPBEl-rVV10, outperforms typical GGA DFs by providing similar or better predictions for metals and molecules, as well as molecule–metal surface adsorption and activation energies. Hence, it is a better choice for constructing molecule–metal surface reaction networks.


S.2.2 List of formulas for Gibbs free energy calculation
The wavenumber of imaginary modes and modes with a wavenumber lower than 50 cm -1 is taken to be 50 cm -1 in all calculations to avoid that     B  − 1 approaches 0, which would cause U vib and S vib to blow up.3N-6 becomes 3N-5 for linear molecules in the formulas below.Table S1 provides a comprehensive overview of the adsorption configurations and adsorption energies of all reaction intermediates involved in the reaction network.The sites indicated in the table are depicted in Figure S5.

S.3 Adsorption energy and site of the intermediates
CO 2 does not show a specific adsorption morphology on the Cu(211) or Cu(111) surface.Both in the gaseous phase and physisorbed, CO 2 molecules maintain their linear structure, with C-O bond lengths of 1.18 Å, similar to the experimental measurement of 1.16 Å 1 , which further strengthens the argument for physisorption of CO 2 .
To the best of our knowledge, no experimental studies exist on H 2 O adsorption on the Cu(211) surface.However, Brosseau et al. 2 studied H 2 O adsorption on defect-rich Cu(100) surfaces, reporting an adsorption energy of approximately -0.52 eV, which is similar to the adsorption energy we find on the stepped Cu(211) surface.
CO adsorbs to the bridge site on the step, displaying an adsorption energy of -0.96 eV.Radnik et al. 3 , employing electron energy loss spectroscopy techniques, studied the adsorption position and orientation of CO molecules on the Cu(211) surface, demonstrating that CO tends to adsorb on the top or bridge sites of the Cu(211) step surface, presenting an adsorption energy of -0.605 ± 0.015 eV.
Methanol synthesis might also proceed through the hydrogenation of intermediates like COHOH* and its derivatives.Schreiner et al. 4 identified three possible isomeric structures of COHOH, namely t,t-COHOH, and t,c-COHOH, through infrared spectroscopy and high-level ab initio coupled cluster theory calculations.Surface adsorption, however, was not included in this study.We find the most stable COHOH* to be the t,t-COHOH* configuration, which binds to the top site of the step on Cu(211).Both variants of the carboxyl pathway on Cu(111) are depicted in Figure S10.From this figure, it is again clear that the comparison between both variants of the carboxyl pathway for Cu(111) is the same as for Cu(211).The most favourable carboxyl pathway and rate-controlling step are the same for both facets.

S.6.2 Carboxyl pathway
The barriers for the most favourable carboxyl pathway, i.e., through CO*, on Cu(111) are higher or similar to the barriers on Cu(211), with a rate-controlling step that is 0.18 eV lower on Cu(211).For the pathway through COHOH*, the comparison is less unambiguous.E.g., the barrier for COHOH* dissociation is 0.16 eV higher on Cu(211), while the barrier for HCOH* hydrogenation is 0.08 eV lower on Cu(211).From the presence of CO* in the carboxyl pathway it becomes clear that the exploration of the CO 2 dissociation pathway, the last pathway discussed in the introduction of the main paper, is important.This pathway is depicted in Figure S11 for Cu(211) and in Figure S12   The CO 2 * dissociation barrier on the flat Cu(111) surface is 0.22 eV higher than the dissociation barrier on the stepped Cu(211) surface.In previous experimental studies, the CO 2 dissociation barrier on the Cu(110) surface was found to be 0.69 eV 5 , while on the relatively flat Cu(100) surface, the barrier was 0.96 eV 6 .

S.6.3 CO 2 dissociation pathway
The dissociation of CO 2 on stepped and kinked surfaces is significantly easier on stepped surfaces than on flat surfaces, which is consistent with our calculations.The highest barriers on Cu(111) are those for CO

Figure S1 :
Figure S1: Convergence of the adsorption energy of HCOOH on Cu(111) as a function of the cutoff for rMS-RPBEl-rVV10.

Figure S2 :
Figure S2: Convergence of the adsorption energy of HCOOH on Cu(211) as a function of the cutoff for rMS-RPBEl-rVV10.

Figure S3 :
Figure S3: Convergence of the adsorption energy of HCOOH on Cu(111) as a function of the k-point grid for rMS-RPBEl-rVV10.

Figure S6 .S. 6 S. 6 . 1
Figure S6.2D (Z, r) cuts of the PES for O2/Cu(111) of O 2 dissociation on Cu(111) bridge site calculated with RPBE-D3, BEEF-vdW and rMS-RPBEl-rVV10, respectively.O 2 molecule is parallel to the surface.Z is the distance between O 2 molecule and the surface, r is the bond length between two O atoms.The solid lines represent the 0 eV reference value corresponding to O 2 and the surface in equilibrium and far away from each other.The energy difference between consecutive equipotential lines is 0.02 eV.

Figure S8 :
Figure S8: Potential energy diagram of both variants of the formate pathway on Cu(111), the path through H 2 COO* is indicated in blue, the path via HCOOH* in red.Activation energy barriers are in eV and intermediate states are depicted in the figure.Both variants of the formate pathway on Cu(111) are depicted in Figure S8.From this figure, it is clear that the comparison between both variants of the formate pathway for Cu(111) is the same as for Cu(211).The most favourable formate pathway is the same for both facets, the rate-controlling step on Cu(111) is CH 3 O* hydrogenation.The barriers for the formate pathway through HCOOH* on Cu(111) are higher or similar to the barriers on Cu(211), except for HCOO* hydrogenation, which is 0.26 eV lower on Cu(111).The barriers for the pathway through H 2 COO* are lower on Cu(111).

Figure S9 :
Figure S9: Potential energy diagram of both variants of the carboxyl pathway on Cu(211), the path through COHOH* is indicated in blue, the path via CO* in red.Activation energy barriers are in eV and intermediate states are depicted in the figure.

Figure S10 :
Figure S10: Potential energy diagram of both variants of the carboxyl pathway on Cu(111), the path through COHOH* is indicated in blue, the path via CO* in red.Activation energy barriers are in eV and intermediate states are depicted in the figure.

Figure S11 :
Figure S11: Potential energy diagram of the CO 2 dissociation path on Cu(211).Activation energy barriers are in eV and intermediate states are depicted in the figure.
for Cu(111).The potential energy diagram for Cu(211) highlights that the direct dissociation of adsorbed CO 2 * to yield CO* is thermodynamically unfavourable, with a reaction energy of 0.18 eV, and is associated with an energy barrier of 1.07 eV.Subsequently, CO* undergoes hydrogenation to produce CH 3 OH via the same reactions as in the carboxyl pathway.The highest barrier and rate-controlling step is CO 2 * dissociation with a barrier of 1.07 eV.This pathway thus resembles the carboxyl pathway and can be summarized as CO 2 * → CO* → HCO* → CH 2 O* → CH 3 O* → CH 3 OH*.

Figure S12 :
Figure S12: Potential energy diagram of the direct CO 2 dissociation path on Cu(111).Activation energy barriers are in eV and intermediate states are depicted on the figure.

S. 6 . 4
Figure S13: Potential energy diagram of the formate path on Cu(211), in red, and Cu(111), in blue .Activation energy barriers are in eV and intermediate states are depicted in the figure.

Table S1 :
Preferred binding sites and adsorption energies (E ads ) of the species involved in hydrogenation of CO 2 on Cu(211) and Cu(211) calculated with rMS-RPBEl-rVV10.Site H, Hs, s, Bs, Ts, fcc, hcp and Phy stand for 3-fold hollow, hollow step, step, bridge step, top step, face-centred cubic, hexagonal close-packed and physisorption.These sites are indicated on FigureS5on the Cu(111) and Cu(211) surface.

Table S3 :
List of transition states on Cu(211) and Cu(111) surface with corresponding total transition state energies calculated with rMS-RPBEl-rVV10.

Table S4 :
List of gas phase species with energies, zero-point energy, entropy and integrated heath capacity at 500 K calculated with rMS-RPBEl-rVV10.

Table S5 :
List of gas phase species with energies, zero-point energy, entropy and integrated heath capacity at 500 K calculated with BEEF-vdW.

Table S6 :
List of gas phase species with energies, zero-point energy, entropy and integrated heath capacity at 500 K calculated with RPBE-D3.

Table S7 :
List of adsorbates and transition states at Cu(111) surface with corresponding total energies calculated with BEEF-vdW and RPBE-D3.