CO₂ electroreduction on single atom catalysts: the role of the DFT functional

Abstract

One key process involving single atom catalysts (SACs) is the electroreduction of CO₂ to fuels. The chemistry of SACs differs largely from that of extended catalytic surfaces, presenting an opportunity to improve the ability to activate very stable molecules, such as CO₂. In this work, we performed a density functional theory (DFT) study of CO₂ activation on a series of SACs, focusing on the role played by the adopted functional in activity predictions. The role of the exchange–correlation functional has been widely investigated in heterogenous catalysts, but it is less explored in SACs. We tested the widely used PBE and the PBE+U corrected functionals against the more robust hybrid PBE0 functional. The results show that PBE is reliable if one is interested in qualitative predictions, but it leads to some inaccuracies in other cases. A possible way to attenuate this effect is by adopting the PBE+U framework, as it gives results that are very similar to PBE0 at an acceptable computational cost. The results of this study further underline the importance of the computational framework adopted in predicting the activity of SACs. The work suggests that one needs to go beyond PBE for quantitative estimates, an important consideration when performing screening and high-throughput calculations.

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1. Introduction

The capture of CO₂ and its conversion into valuable chemicals, including fuels, is involved in key processes in the ongoing energy transition.^1–4 The conversion of CO₂ to fuels would create, in principle, a carbon neutral cycle that can mitigate the impact of the greenhouse effect. The main issue associated with this objective is the strong stability of CO₂, which in turn makes its activation hard.⁵ The process can be activated by working under harsh conditions with the help of a catalyst.^6–8 The process can be conducted either under thermal or electrochemical conditions, depending on the way energy is supplied.^9,10

Electrocatalysis offers a viable method for electroreducing CO₂ under ambient conditions.^11–13 A significant amount of research is dedicated to the discovery of novel catalytic materials and gaining a fundamental understanding of the physical chemistry behind this reaction. Typical catalytic materials are metal particles,^14–16 based on critical and noble metals.

Single atom catalysis is a relatively new frontier with the potential to reduce the amount of metal loading in catalytic materials.^17–20 A single atom catalyst (SAC) is made by depositing transition metal atoms with atomic precision on a given support. In addition, metal atom SACs exhibit a specific local coordination that strongly affects their reactivity.^21–24 This characteristic makes SACs analogous to coordination chemistry compounds,²⁵ as they can form molecular adducts that do not form on conventional extended surfaces.^26–30 This includes not only unconventional reaction intermediates, but also complexes with the solvent, that can compete with the classical reaction intermediates identified on the surface of metal electrodes. If the solvent can also act as a ligand (such as water), it can play a twofold role in SACs, serving both as a solvent and as a ligand. We have recently observed this effect in CO₂ electroreduction on a set of SACs supported on nitrogen-doped graphene by means of DFT calculations.³¹

Quantum chemical calculations provide a valuable contribution to the fundamental understanding of the chemistry of SACs.^32,33 Calculations enable access to a SAC with atomistic precision, which would otherwise be difficult to achieve experimentally. It is however important to underline that quantitative predictions can only be made if the simulated model corresponds to the real one. To access this information, it is often mandatory to combine theory with spectroscopy techniques.^34,35

Besides the quality of the model, the way in which the electronic structure of the system is described plays a primary role.³⁶ The most widely used methodology for simulating a SAC is DFT, which relies heavily on the parametrization of the exchange and correlation functional. Although there is an extensive literature dedicated to the assessment of the accuracy and reliability of different functionals on extended catalytic surfaces,^37–41 this aspect is less explored in SACs. The number of screening and high-throughput studies aimed at the discovery of new SACs is increasing steadily, but it largely depends on the standard parametrization of the functional known as generalized gradient approximation (GGA).^33,42–52 However, it is well known that self-interaction error corrected functionals such as the DFT+U approach or hybrid functionals can provide a more accurate description of the chemistry, in particular when transition metals with open shell structures are involved.

In this work, we assessed the accuracy of different DFT functionals in simulating the CO₂ electroreduction on SACs. We tested two popular choices against a reference hybrid functional: the widely adopted PBE functional (GGA),⁵³ and the PBE+U correction⁵⁴ (DFT+U) vs the PBE0 functional (hybrid).^55,56 Previous studies showed that PBE0 can be considered a good benchmark in the framework of DFT.³⁶ We investigated the role of the functional in (i) the CO₂ activation process, (ii) the nature and stability of the first electrochemical intermediates, and (iii) the competing action of the solvent. CO₂ activation and the first electrochemical step (addition of the first H⁺/e⁻ couple) are two key processes in CO₂ electroreduction. The main purpose of this work is to investigate the effect of the adopted functional on the predictions of the activation of CO₂ on SACs. To address this point, we studied a specific model system, comprising a graphene nanosheet doped with nitrogen exhibiting pyridinic coordination, which is a likely active site, although it is not the only one.⁵⁷ Indeed, N-doped graphene is widely adopted as a support for SACs, due to its capability to stably bind metal atoms.^58,59 A distribution of different active sites can be present in real samples. There are basically three most common configurations; a nitrogen atom replacing a carbon one (graphitic defect), four nitrogen atoms having a porphyrin-like arrangement close to a carbon divacancy (pyridinic defect), and a nitrogen atom terminating 5-membered rings instead of the lattice hexagonal ones (pyrrolic defect).^60,61 The catalytic activity may be affected by the local coordination. Some insight into the nature of the species can be evinced from XPS measurements.⁶² Also, given the complexity of catalytic pathways leading to fuel formation, in this study we do not proceed further with the reaction mechanism since we are not focused on a specific product, but rather on assessing the accuracy of the DFT functional for describing CO₂ activation. The results show that the use of PBE leads to significant deviations when compared to PBE0, while the use of PBE+U strongly attenuates the discrepancy. This study suggests that despite PBE being reliable for establishing general trends, it may lead to inaccurate results when trying to be quantitative. This problem can be attenuated by adopting the PBE+U approach.

2. Computational details

The calculations have been performed at the level of density functional theory (DFT) as implemented in the Vienna Ab Initio Simulation Package (VASP).^63–65 Simulations were performed including spin polarization and employing the projector-augmented wave (PAW) method.^66,67 The valence electrons were expanded on a set of plane waves with a kinetic cutoff of 400 eV. The dispersion interactions were included using Grimme's D3 correction.⁶⁸ The reciprocal space was sampled in order to provide converged results. A 5 × 5 × 1 Monkhorst–Pack grid was adopted.⁶⁹ The convergence criteria for the electronic and ionic loops were set at 10⁻⁶ eV and 10⁻³ eV Å⁻¹ respectively. In the case of PBE+U, the U values for TM were taken from the literature. These values have been discussed for SACs, tested and benchmarked elsewhere (see Table S1, ESI†).^70,71

We fully optimized a 4 × 4 supercell of graphene, with a = b = 9.87 Å and γ = 120°.^72,73 This was used to build the 4N-Gr support, where a 15 Å-thick vacuum layer was included along the non-periodic direction to avoid interaction between the periodic replicas of the system. A C-divacancy was created in the cell, and four C atoms were replaced with N atoms to build the 4N-Gr (pyridine) support. The TM atom was embedded into the coordination site resulting in the TM@4N-Gr structure.^30,74,75Fig. 1 shows the prototype structure of the SACs and the investigated TM atoms.

Fig. 1 (a) Top view and (b) side view of the TM-embedded N-doped graphene structure. The inset shows the TM considered in this study.

In each case, the atomic coordinates have been fully relaxed. The approximation of keeping the lattice vectors frozen at those of graphene implies a negligible error, less than 0.1 eV,⁷⁰ and therefore it can be considered acceptable for the purpose of the study. The adhesion energies and magnetization of the TMs are reported in Table S1 (ESI†). When adsorbed in the cavity, the metal atom can either remain in the same plane as the C atoms or it can protrude from the surface layer. Fig. S1 (ESI†) shows the structure of two representative cases, namely, Mo@4N-Gr and Ni@4N-Gr.

The adsorption energy (ΔE_ads) of an adsorbed species on the TM@4N-Gr SAC is calculated as:

ΔE_ads = E_SAC+mol − E_SAC − E_mol (1)

where E_SAC+mol, E_SAC and E_mol are the energies of the catalyst with the molecule adsorbed, of the bare catalyst and of the molecule, respectively. In this work we neglect any reaction barrier different from those arising from thermochemistry, following the approach proposed by Norskov and co-workers.⁷⁶ This implies assuming that some correlation exists between the thermodynamic energies of the reaction steps and the corresponding activation energy. Since the specific purpose of this study is not to provide absolute predictions, but to show the effect of the adopted functional on the activation of CO₂ on SACs, we can consider this approximation acceptable. The reaction Gibbs free energy of each reaction step was obtained by using Norskov's Computational Hydrogen Electrode (CHE) approach,^77–79

ΔG = ΔE_ads − TΔS + ΔE_zpe (2)

where ΔE_zpe and ΔS are the changes in zero-point energy and entropy, respectively. The values of ZPE and entropic contributions for CO₂ and H₂O gas phase molecules and the reaction intermediates were taken from the literature⁸⁰ and NIST database, respectively. The entropies of solid-state species were neglected.⁷⁸ A way to overcome this approximation is to estimate the vibrational entropy of solids by means of the partition function formalism working in a harmonic fashion. If one neglects this contribution, the expected error is about 0.1–0.2 eV, which can be considered acceptable given the specific purpose of the study. It should be mentioned, however, that if one aims at reproducing the experimental complexity, this effect should be accounted for, together with many other effects such as solvation, applied voltage and pH.^81–85 Further details and working equations are reported in the ESI.†

3. Results

As mentioned before, we investigated the performance of three popular DFT exchange–correlation functionals. The first is the PBE functional,⁵³ one of the most popular for the study of SACs. It is known that this functional may lead to some inaccuracies when compared to high level benchmark calculations,^36,75 mainly because it tends to over delocalize electrons in a system. However, given its efficiency, it is widely used for screening and predictive studies.^{33,44,45,86–89} One approach to overcome the issues with PBE is to employ hybrid functionals, i.e., including a given fraction of exact Fock exchange into the functional.^90–92 A popular choice is the PBE0 parametrization.^55,56 A previous work by Patel et al. demonstrated that PBE0 shows negligible differences from high level CCSD(T) calculations of a SAC used as a test case.³⁶ Therefore, PBE0 can be considered a solid benchmark for other functionals. Hybrid functionals are however rather expensive when associated with plane wave approaches. A compromise between accuracy and computational cost is represented by the DFT+U approach.⁵⁴ In this framework one adds an ad hoc correction term mimicking the effect of the exact Fock exchange in hybrid functionals. The extra computational cost is negligible with respect to PBE, but the choice of the U correction term is delicate.

3.1. Benchmarking PBE and PBE+U against PBE0

We start by benchmarking the reliability of PBE and PBE+U against the PBE0 hybrid functional in the activation of CO₂, eqn (3a).

CO₂ + * → *CO₂ (3a)

As reported elsewhere,³¹ at the PBE level only few SACs are able to effectively chemisorb CO₂, resulting in bending of the O–C–O angle. In particular, Sc, Ti, Mo, Ru, W, and Os@4N-Gr bind and activate CO₂. In the other cases (among those showed in Fig. 1) CO₂ remains physisorbed to the SAC. To assess the performance of PBE and PBE+U against PBE0, we focus on the SACs that activate CO₂. For comparison, we also included a representative SAC that is not able to activate the molecule, Ni@4N-Gr. Then, we simulated the formation of the reaction intermediates resulting from the addition of the first H⁺ and e⁻ couple on the SACs. CO₂ reduction can occur via formation of *COOH (eqn (3b)) or *OCHO (eqn (3c)) intermediates.^93–96

CO₂ + H⁺ + e⁻ * → *COOH (3b)

CO₂ + H⁺ + e⁻ * → *OCHO (3c)

Table 1 presents the reaction energies, reaction free energies and structural geometries related to *CO₂, *COOH, and *OCHO adsorbed species on TM@4N-Gr obtained using PBE, PBE+U, and PBE0 approaches (* indicates the site where an adsorbed molecular fragment is bound). In the case of *COOH adsorption on Ni@4N-Gr with PBE0 we encountered unsurmountable convergence issues, and therefore we are not reporting any value in Table 1. Reaction free energies reported in Table 1 are calculated according to eqn (3a)–(3c) and refer to the adsorption of reaction intermediates to the SACs.

Table 1 Reaction energies, E_ads, and reaction free energies, ΔG, of adsorbed CO₂ and reaction intermediates formed on TM@4N-Gr, obtained with PBE0, PBE+U and PBE approaches. Values are in eV

	PBE0						PBE+U						PBE
	*CO2		*COOH		*OCHO		*CO2		*COOH		*OCHO		*CO2		*COOH		*OCHO
TM	E ads	ΔG	E ads	ΔG	E ads	ΔG	E ads	ΔG	E ads	ΔG	E ads	ΔG	E ads	ΔG	E ads	ΔG	E ads	ΔG
Sc	−1.01	−0.33	−1.72	−0.69	−3.39	−2.38	−0.92	−0.24	−1.47	−0.44	−3.14	−2.13	−0.84	−0.16	−1.38	−0.35	−3.01	−2
Ti	−1.39	−0.71	−1.66	−0.63	−2.84	−1.83	−1.31	−0.63	−1.41	−0.38	−2.78	−1.77	−1.58	−0.9	−1.57	−0.54	−2.61	−1.61
Ni	0.02	0.70	—	—	1.12	2.13	−0.13	0.56	1.03	2.06	0.86	1.87	−0.13	0.55	1.04	2.07	0.78	1.79
Mo	−0.84	−0.16	−0.37	0.66	−1.75	−0.74	−0.73	−0.05	−0.59	0.44	−1.84	−0.83	−1.41	−0.73	−1.45	−0.41	−2.21	−1.2
Ru	0.38	1.06	−0.77	0.26	−0.09	0.92	0.16	0.84	−0.64	0.39	0.07	1.08	−0.11	0.57	−0.99	0.04	−0.16	0.85
W	−1.56	−0.88	−1.15	−0.12	−2.5	−1.49	−1.65	−0.97	−1.27	−0.24	−2.47	−1.46	−1.92	−1.24	−1.96	−0.92	−2.6	−1.59
Os	−0.47	0.21	−1.17	−0.14	−0.69	0.33	−0.45	0.23	−1.01	0.02	−0.65	0.36	−0.74	−0.06	−1.12	−0.09	−0.71	0.3

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PBE offers large deviations from PBE0, with a mean absolute error (MAE) of 0.30 eV, 0.43 eV, and 0.23 eV for *CO₂, *COOH, and *OCHO, respectively. The maximum deviation is very high, being 0.57 eV, 1.08 eV, and 0.46 eV for *CO₂, *COOH, and *OCHO respectively. This result already shows the problems connected with the use of the PBE functional: in the adsorption of CO₂ the differences can be up to 0.6 eV and for the formation of the intermediates, they can be even larger, exceeding 1 eV in some cases, Table 1 and Fig. 2. PBE+U provides estimates closer to PBE0, as the MAE is 0.11 eV, 0.19 eV, and 0.13 eV for *CO₂, *COOH, and *OCHO, respectively. Similarly, the maximum deviation is nearly the same in all the cases, 0.22 eV (*CO₂), 0.25 eV (*COOH), and 0.26 eV (*OCHO), Fig. 2. Thus, the energy computed at the PBE+U level is very similar to that obtained at the PBE0 level, with a MAE always lower than 0.2 eV and a maximum deviation always smaller than 0.3 eV. This shows that PBE+U is sufficiently reliable for providing quantitative estimates of the reaction energetics at an acceptable computational cost. These results are in line with previous findings showing that PBE+U estimates of H atom adsorption on TM@4N-Gr SACs closely match with the PBE0 ones (MAE less than 0.2 eV).⁷⁵

Fig. 2 Deviation of Gibbs free energies comparing different levels of theory for *CO₂, *COOH, and *OCHO adsorption on TM@4N-Gr.

3.2. CO₂ activation

The results show that PBE+U provides results that closely resemble those obtained with the benchmark PBE0 functional. Therefore, in the following, discussion is restricted to the comparison of PBE and PBE+U results.

We concentrate first on the activation of CO₂ on the full set of 19 SACs, Fig. 1. Fig. 3 shows representative cases of CO₂ adsorption. The CO₂ molecule can be attached to the SAC by means of TM–O and TM–C interactions (Mo@4N-Gr), or through a TM–C bond (Ru@4N-Gr). In the first case, the complex shows bidentate bonding, while in the latter case the interaction occurs only via the C atom of the CO₂ molecule. In the case of physisorption, the CO₂ molecule is distant from the SAC (∼3.0 Å). Fig. S2 (ESI†) shows the structure of *CO₂ adsorbed on various SACs. Table 2 presents the calculated adsorption energy, Gibbs free energies of *CO₂ and structural parameters.

Fig. 3 CO₂ adsorption on TM@4N-Gr: chemisorption versus physisorption. Left: PBE; right: PBE+U.

Table 2 Reaction free energies and structural parameters for CO₂ adsorption on TM@4N-Gr with PBE and PBE+U approaches. In bold cases where CO₂ is chemically adsorbed to the SAC^a

TM	PBE				PBE+U
	E ads (eV)	ΔG (eV)	<O–C–O	TM–C (Å)	E ads (eV)	ΔG (eV)	<O–C–O	TM–C (Å)
a MAE of PBE with respect to PBE+U: MAE chemisorption 0.41 eV; MAE physisorption 0.04 eV.
Sc	−0.84	−0.16	139	2.42	−0.92	−0.24	139.48	2.44
Ti	−1.58	−0.90	133	2.05	−1.31	−0.63	133.73	2.06
V	−0.08	0.60	177	3.19	−0.07	0.61	178.53	3.23
Cr	−0.11	0.57	179	3.40	−0.10	0.58	179.7	3.40
Mn	−0.10	0.58	180	3.44	−0.09	0.59	179.7	3.45
Fe	−0.10	0.58	180	3.32	−0.09	0.59	179.94	3.32
Co	0.93	1.61	180	3.24	−0.11	0.57	179.06	3.29
Ni	−0.13	0.55	179	3.24	−0.13	0.56	179.09	3.26
Cu	0.04	0.72	180	3.31	−0.10	0.58	179.86	3.32
Mo	−1.41	−0.73	131	2.03	−0.73	−0.05	132.70	2.04
Ru	−0.11	0.57	147	2.16	0.16	0.84	148.60	2.20
Rh	−0.18	0.50	176	3.22	−0.15	0.53	177.85	2.30
Pd	−0.13	0.55	179	3.32	−0.13	0.55	178.95	3.32
Ag	−0.17	0.51	179	4.20	−0.37	0.31	178.73	4.22
W	−1.92	−1.24	129	2.02	−1.65	−0.97	130.75	2.06
Os	−0.74	−0.06	139	2.02	−0.45	0.23	140.14	2.02
Ir	−0.19	0.49	176	3.28	−0.16	0.52	177.60	3.34
Pt	−0.14	0.54	178	3.35	−0.14	0.54	178.45	3.35
Au	−0.12	0.56	180	3.44	−0.12	0.56	179.83	3.44

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The number of SACs capable of activating CO₂ remains unchanged when adopting the PBE+U approach, indicating that PBE is sufficiently reliable for obtaining qualitative predictions. The picture changes when looking at quantitative estimates. In fact, on average, the PBE values deviate by 0.41 eV from the PBE+U ones, with cases where the difference reaches 1 eV (Co@4N-Gr), see Fig. 4. Not surprisingly, PBE+U impacts the results only when CO₂ is chemisorbed. If one restricts to the physisorption cases, the MAE is 0.04 eV, Fig. 4.

Fig. 4 Deviation of Gibbs free energies computed at PBE and PBE+U levels for CO₂ adsorption on TM@4N-Gr.

3.3. Reaction intermediates

After discussing the activation of CO₂, we now move to the reaction intermediates arising from the addition of the first H⁺ and e⁻ couple to CO₂, the first step in CO₂ electroreduction. Fig. S3 (ESI†) shows the structure of *COOH and *OCHO molecular fragments adsorbed on the SACs considered.

When comparing the stability of the *COOH intermediate at PBE and PBE+U levels of theory, we see that the MAE is still high, 0.32 eV, comparable to that of CO₂ activation (0.41 eV), Table 3. Even the maximum deviation is large, 0.86 eV (Mo@4N-Gr), close to 1 eV, as found before. Besides Mo@4N-Gr, two other systems are critical, namely, Ru@4N-Gr and W@4N-Gr, with a deviation equal to 0.35 eV and 0.69 eV, respectively. This implies that the stability of reaction intermediates is very sensitive to the adopted functional and the usage of PBE can lead to significant errors.

Table 3 Reaction energies, E_ads, and reaction free energies, ΔG, for reaction intermediates on TM@4N-Gr obtained with PBE and PBE+U approaches. Values are in eV

	PBE				PBE+U
	*COOH		*OCHO		*COOH		*OCHO
TM	E ads	ΔG	E ads	ΔG	E ads	ΔG	E ads	ΔG
Sc	−1.38	−0.35	−3.01	−2.00	−1.47	−0.44	−3.14	−2.13
Ti	−1.57	−0.54	−2.61	−1.61	−1.41	−0.38	−2.78	−1.77
Ni	1.04	2.07	0.78	1.79	1.03	2.06	0.86	1.87
Mo	−1.45	−0.41	−2.21	−1.20	−0.59	0.44	−1.84	−0.83
Ru	−0.99	0.04	−0.16	0.85	−0.64	0.39	0.07	1.08
W	−1.96	−0.92	−2.60	−1.59	−1.27	−0.24	−2.47	−1.46
Os	−1.12	−0.09	−0.71	0.30	−1.01	0.02	−0.65	0.36

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The stability of OCHO* seems to be less sensitive to the functional adopted, since the MAE is 0.17 eV and the maximum deviation is 0.37 eV only, Table 3. Regarding CO₂ adsorption, PBE is still reliable in providing qualitative predictions, since the relative stability of the two intermediates remains unchanged. In particular, with both PBE and PBE+U, OCHO* is more stable than COOH* on Sc, Ti, Ni, Mo, and W@4N-Gr, while COOH* is more stable on Ru and Os@4N-Gr.

3.4. Competing action of water

In this section, we analyze the impact of the exchange–correlation functional on another relevant point for the study of the electrochemical reduction of CO₂ with SACs: the role of water. In electrochemical reactions, a fundamental role is played by the solvent, water. An appropriate way to account for it requires a dynamic approach that explicitly considers the catalyst/water interface. Interestingly, we have recently reported that on SACs, water does not only act as a solvent in the reaction but can also act as a ligand competing with CO₂ and the other reaction intermediates in binding to the TM center.³¹ In this respect, this finding is another demonstration of the similarity between SACs and organometallic compounds. Here we discuss the effect of the adopted functional on the prediction of the “coordination” effect played by water. A dedicated study is planned to address the complex problem of solvation.

We first adsorbed a water molecule on the SACs considered in this work, as shown in Table 4, where the calculated Gibbs free energies at PBE and PBE+U levels are reported along with the relevant structural information. If one considers the structure of the complex, negligible changes are found between the two functionals, as shown in Fig. 5 where the representative case of Mo@4N-Gr is shown.

Table 4 Reaction energies, reaction free energies and structural parameters for H₂O adsorption on TM@4N-Gr, with PBE and PBE+U approaches

TM	PBE				PBE+U
	E ads (eV)	ΔG (eV)	<H–O–H	d TM–O (Å)	E ads (eV)	ΔG (eV)	<H–O–H	d TM–O (Å)
Sc	−1.00	−0.42	107	2.30	−1.05	−0.47	107	2.30
Ti	−0.97	−0.39	106	2.20	−0.91	−0.33	106	2.20
Ni	−0.14	0.44	103	3.31	−0.14	0.44	103	3.26
Mo	−0.63	−0.05	105	2.30	−0.49	0.09	105	2.30
Ru	−0.25	0.33	105	2.70	−0.29	0.29	105	2.70
W	−0.73	−0.15	106	2.30	−0.74	−0.16	106	2.30
Os	−0.06	0.52	105	2.80	−0.25	0.33	105	2.80

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Fig. 5 H₂O adsorption on Mo@4N-Gr with PBE and PBE+U methods.

The analysis of the energetics suggests that in this case PBE provides acceptable results, since the MAE is less than 0.1 eV (0.07 eV) and the maximum deviation is 0.19 eV only, Fig. 6. Comparing the Gibbs free energy of *CO₂ and *H₂O to that of SACs, one can evince that the two ligands have comparable interactions with the TM center and in fact compete for bonding to it. A comparison of the Gibbs free energies of adsorption is reported in Tables 2 and 4. On Sc, Ni, and Ru@4N-Gr, water adsorption is more favorable than CO₂ adsorption. In the latter case, *H₂O is more stable by 0.55 eV than *CO₂, Tables 2 and 4. Of course, this is a simplified model since here the nature of liquid water is not considered. Other approaches should be adopted in order to properly describe the catalyst/water interface including bulk water and explicit solvation spheres.^82,97–101

Fig. 6 Deviation of Gibbs free energies computed at PBE and PBE+U levels for H₂O adsorption (green) and CO₂ co-adsorption (blue) in the presence of *H₂O on TM@4N-Gr.

Finally, we considered the adsorption of CO₂ in the presence of *H₂O. Here we assume that the adsorption occurs on the opposite side. Further work in the future will be dedicated to the exploration of the several possible co-adsorption configurations. The adsorption energy is calculated as follows.

Table 5 presents the calculated Gibbs free energies. Fig. S5 (ESI†) shows the structure of *CO₂ in the presence of *H₂O on the various SACs. Even in this case PBE and PBE+U provide similar results, with a MAE of 0.19 eV, but with relevant exceptions. In fact, in the case of Ti@4N-Gr, PBE and PBE+U differ by 0.63 eV, Fig. 6. This value largely contributes to the MAE, as MAE decreases to 0.11 eV by excluding Ti@4N-Gr. It this respect, the adoption of PBE+U in place of PBE looks more solid.

Table 5 Reaction energies, E_ads, and reaction free energies, ΔG, for CO₂ and H₂O co-adsorption on TM@4N-Gr, using PBE and PBE+U functionals. Values in eV

TM	PBE		PBE+U
	E ads	ΔG	E ads	ΔG
Sc	−0.08	0.60	−0.12	0.56
Ti	−0.79	−0.11	−0.16	0.52
Ni	−0.14	0.54	−0.14	0.54
Mo	0.39	1.07	0.25	0.93
Ru	−0.15	0.53	−0.14	0.54
W	−1.37	−0.69	−1.07	−0.39
Os	−0.36	−0.32	−0.14	0.54

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4. Conclusions

In this work we performed a computational study of the initial phases of CO₂ electroreduction on single atom catalysts consisting of TM atoms stabilized in nitrogen-doped graphene. We simulated the bonding and activation of CO₂ as well as the first hydrogenation step leading to the formation of *COOH and *OCHO intermediates, as they usually represent the most complex steps to overcome. The focus is on the role of the exchange–correlation functional used to simulate CO₂ activation and reduction on SACs. Previous studies have demonstrated that the PBE functional, one of the most commonly used functionals for this kind of calculation, is sufficiently reliable when looking at trends and general rules, while it may fail in providing quantitative estimates. The picture can be improved by invoking the PBE+U approach, or even better hybrid functionals, such as the PBE0 functional that we used as a benchmark. The results of this study show that PBE+U reproduces the PBE0 results with satisfactory accuracy, at a much lower computational cost. Both PBE+U and PBE0 provide more reliable results than the standard PBE functional. In particular, for CO₂ adsorption, PBE estimates deviate on average by 0.3 eV from PBE0 ones, while the maximum deviation can reach up to 0.5 eV. The effect can be tamed by invoking PBE+U as the MAE decreases to 0.1 eV. The picture is similar when looking at *COOH and *OHCH intermediates, as the deviation between PBE and PBE0 is 0.43 eV and 0.23 eV, respectively, with a maximum deviation of up to one eV. The difference between PBE0 and PBE+U is much lower, 0.19 eV and 0.13 eV. Therefore, PBE+U is an acceptable compromise between accuracy and computational cost. Despite the important quantitative effect on the reaction energies, PBE looks sufficiently reliable when looking at qualitative trends, such as the energetic ordering of stability of *COOH and *OCHO intermediates, which remains unchanged when the self-interaction correction is introduced.

Finally, we investigated the competing effects of water adsorption and CO₂ binding to the active site. We first considered simple water adsorption, finding that the discrepancy between PBE and PBE+U is small, as shown by a MAE of 0.07 eV, and a maximum deviation of 0.19 eV. Next, we considered co-adsorption on the same site of water and CO₂. Here the MAE between PBE and PBE+U becomes more relevant, 0.19 eV.

In general, the work serves as another example highlighting the importance of moving beyond the PBE functional when trying to provide quantitative predictions of the catalytic activity of SACs. In fact, while PBE looks sufficiently reliable for a preliminary screening of the catalytic activity, significant differences arise when the system is treated with the more rigorous PBE+U or even better PBE0 functionals. This level of theory becomes essential when one is interested in determining the reaction profile and the expected activity of a potential new catalyst.

Further work will be dedicated to searching for correlations between thermodynamic and activation energies, aiming to provide insight to verify if the universal relations discovered for extended catalysts are valid also in the context of SACs.

Author contributions

D. M. performed the calculations and analysis. G. D. L. and G. P. supervised the project. G. D. L. wrote the first draft. All authors contributed to designing the study, analyzing the results, and preparing the final draft.

Data availability

The data reported in this article can be found in the cited literature and can be requested from the authors upon reasonable inquiry.

Conflicts of interest

■■■■There are no conflicts to declare.

Acknowledgements

D. M. is grateful to the Science and Engineering Research Board (SERB), Govt. of India, for the SIRE fellowship (award no. SIR/2022/000690) which supported her three-month visit to University of Milano-Bicocca in 2022, where most of the calculations have been carried out. We acknowledge the financial support from the Italian Ministry for University and Research MUR, through the projects UNDERSAC and SACtoH2. Access to CINECA supercomputing resources was granted via ISCRAB and ISCRAC. We also thank the COST Action 18234 supported by COST (European Cooperation in Science and Technology).

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Footnote

† Electronic supplementary information (ESI) available: additional DFT data, supporting figures and tables. See DOI: https://doi.org/10.1039/d4cp00175c

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