Theoretical Investigation of Single-Atom Catalysts for Hydrogen Evolution Reaction Based on Two-Dimensional Tetragonal V₂C₂ and V₃C₃

Abstract

Developing stable and effective catalysts for the hydrogen evolution reaction (HER) has been a long-standing pursuit. In this work, we propose a series of single-atom catalysts (SACs) by importing transition-metal atoms into the carbon and vanadium vacancies of tetragonal V₂C₂ and V₃C₃ slabs, where the transition-metal atoms refer to Ti, V, Cr, Mn, Fe, Co, Ni, and Cu. By means of first-principles computations, the possibility of applying these SACs in HER catalysis was investigated. All the SACs are conductive, which is favorable to charge transfer during HER. The Gibbs free energy change (ΔG_H*) during hydrogen adsorption was adopted to assess their catalytic ability. For the V₂C₂-based SACs with V, Cr, Mn, Fe, Ni, and Cu located at the carbon vacancy, excellent HER catalytic performance was achieved, with a |ΔG_H*| smaller than 0.2 eV. Among the V₃C₃-based SACs, apart from the SAC with Mn located at the carbon vacancy, all the SACs can act as outstanding HER catalysts. According to the ΔG_H*, these excellent V₂C₂- and V₃C₃-based SACs are comparable to the best-known Pt-based HER catalysts. However, it should be noted that the V₂C₂ and V₃C₃ slabs have not been successfully synthesized in the laboratory, leading to a pure investigation without practical application in this work.

1. Introduction

Energy is an important issue related to the economically and environmentally sustainable development of the world [1,2]. The conventional sources of energy, such as coal, petroleum, and natural gas, are available in limited quantity and not renewable and environmentally friendly [3,4]. To alleviate the looming energy crises, there is a global drive to explore and develop renewable and clean alternatives to fossil fuels [5]. This pressing need has sparked significant scientific efforts aimed at developing sustainable energy technologies to satisfy the growing energy demands of society while reducing carbon emissions as well as environmental degradation. As a clean and sustainable energy carrier, hydrogen (H₂) possesses high gravimetric energy density and emits only water as a byproduct when utilized in a fuel cell [6]. Hydrogen production driven by renewable energy sources has become a promising strategy for generating carbon-free fuel from sustainable energy at low cost.

Electrochemical water splitting on the cathode is an efficient approach for hydrogen production, where the hydrogen evolution reaction (HER) plays a key role in producing H₂ [7]. During the HER process, highly effective catalysts are required to reduce the energy barrier and boost H₂ production. Among the various types of electrocatalysts, Pt-based catalysts have demonstrated exceptional electrocatalytic performance and have been acknowledged as the most effective HER catalysts [8,9]. However, the limitation in availability and the high cost of Pt obstruct large-scale implementation, which leads to the exploration of alternatives as substitutes for Pt-based catalysts [10,11,12]. Up to now, a variety of efficient catalysts based on abundant elements have been developed with the advancement of nanotechnology. Zhao et al. successfully synthesized a series of cobalt-doped Ni_0.85Se chalcogenides (Co_xNi_0.85-xSe, x = 0.05, 0.1, 0.2, 0.3, and 0.4) as electrocatalysts for HER [13]. The results indicate that Co_0.1Ni_0.75Se displays the highest HER performance. When adopting graphene oxide (rGO) in this catalyst as the support, the supported Co_0.1Ni_0.75Se exhibits even better performance than unsupported Co_0.1Ni_0.75Se, which is determined by the decrease in the HER overpotential of Co_0.1Ni_0.75Se/rGO (103 mV) compared to Co_0.1Ni_0.75Se (153 mV) at a current density of 10 mA cm⁻², and the lower Tafel slope (43 mV dec⁻¹) and kinetic resistance (21.34 Ω) compared to Co_0.1Ni_0.75Se (47 mV dec⁻¹, 30.23 Ω). Ito and coworkers reported that N, S co-doped nanoporous graphene possesses high HER catalytic ability at low operating potential, comparable to two-dimensional (2D) MoS₂ [14]. Wang et al. reported a double-deck carbon-coated V₈C₇ network as an efficient HER electrocatalyst. The HER catalytic performances of the V₈C₇ network are comparable to those of Pt in all pH conditions, with an overpotential of 38, 77, and 47 mV at a current density of −10 mA cm⁻² in 0.5 M H₂SO₄, 0.1 M phosphate buffer, and 1 M KOH, respectively [15]. The significant progress of non-Pt-based catalysts encourages researchers to explore more noble-metal-free catalytic materials for HER.

Since 2D materials possess special physical and chemical properties, such as a large specific surface area [16], excellent optical transparency [17], and good mechanical properties [18], their potential application in HER catalysis has been intensively studied for decades [19]. Among the various 2D materials, 2D transition-metal carbides (TMCs) have exhibited great potential as electrocatalysts for HER due to their outstanding characteristics, including their excellent structural stability, high electrical conductivity, and large active surface. A class of 2D TMCs known as MXenes show great potential to act as HER catalysts. Bai et al. introduced many applications of MXene-based HER electrocatalysts from theoretical and experimental perspectives [20]. According to their review, many MXene-based materials, such as O-terminated Ti₃C₂, nanoribbons of Ti₃C₂, transition-metal-promoted V₂CO₂, and Co-substituted Mo₂CT_x, are promising candidates for HER catalysts. In addition to MXenes, many other 2D TMCs have also been reported to possess HER catalytic ability. By performing first-principles calculations, Yu et al. found that NbC₂, TaC₂, and MoC₂ possess excellent HER catalytic ability under the reaction controlled by the Volmer–Heyrovsky mechanism [21]. These studies show that 2D TMCs possess significant potential in catalyzing HER.

With the development of catalyst preparation methods, the rational design of potential catalysts for HER has become fascinating. Among the different types of catalysts, single-atom catalysts (SACs), with only isolated atoms dispersed on the support surface, have attracted considerable interest due to their maximum atom utilization efficiency and high selectivity [22,23,24]. The high dispersion of isolated atoms can greatly reduce the usage of metals, which can greatly lower the application cost of SACs. Importantly, SACs possess unique spatial and electronic structures, and the electronic structure can be adjusted via heteroatom doping, thus achieving a specific coordination environment of the isolated atoms [25]. The active sites in SACs typically consist of individual metal atoms and the coordinating atoms of the support material [26]. Up to now, many species of metal atoms (precious and transition metals) and different support materials have been adopted in SACs with great HER catalytic ability. For example, Qiu et al. reported that isolated nickel atoms anchored to nanoporous graphene exhibit excellent HER catalysis with a low overpotential of about 50 mV and a Tafel slope of 45 mV dec⁻¹ in 0.5 M H₂SO₄, together with high cycling stability [27]. Zhang and coworkers developed an SAC by importing single Pt atoms into the Mo vacancies of Mo₂TiC₂T_x MXene, and this SAC shows excellent HER catalytic performance with low overpotentials of 30 and 77 mV at current densities of 10 and 100 mA cm⁻², respectively [28]. Furthermore, this Mo₂TiC₂T_x-based SAC possesses a mass activity about 40 times greater than that of a Pt-on-carbon catalyst. Deng et al. demonstrated that the HER catalytic ability of the in-plane S atoms of 2D MoS₂ can be enhanced by Pt atom doping [29]. In order to explore more suitable SACs for HER, it is essential to investigate novel combinations of different support materials and single atoms.

Using experimental methods, researchers can prepare SACs based on 2D materials with isolated transition-metal atoms dispersed on the surface of substrates [30,31]. On the other hand, there are many works that theoretically investigate SACs based on 2D materials [32,33,34]. Inspired by these theoretical studies, in this work, we rationally designed a series of SACs theoretically by embedding various transition-metal atoms in the C and V vacancies of the stable 2D tetragonal vanadium carbides V₂C₂ and V₃C₃ that were identified by previous structural prediction on V-C phases [35,36], where the transition-metal atoms refer to Ti, V, Cr, Mn, Fe, Co, Ni, and Cu. The thermally, dynamically, and mechanically stable V₂C₂ and V₃C₃ possess good electrical conductivity, which is favorable to electrocatalysis. In addition, different from most 2D materials, each sub-layer of V₂C₂ and V₃C₃ has the same coordination for the C/V atom in the xy plane, which may result in unique HER catalytic properties. After the construction of V₂C₂- and V₃C₃-based SACs, the HER catalytic performances of the proposed SACs at different adsorption sites were investigated theoretically.

2. Computational Details

All the calculations based on spin-polarized density functional theory (DFT) were carried out with the Vienna ab initio simulation package (VASP 5.4.4) [37,38]. The exchange correlation effect was treated using generalized gradient approximation (GGA) [39] in the Perdew–Burke–Ernzerhof (PBE) form [40]. The kinetic energy cut-off was set as 500 eV, and the convergence criteria for electron self-consistent loop and geometry optimization were set as 10⁻⁵ eV and 0.02 eV/Å, respectively. A vacuum space larger than 15 Å in the z direction was chosen to eliminate spurious interactions between adjacent images. The DFT-D3 scheme [41] was employed to capture the long-range van der Waals interaction between the slabs. The 4 × 4 supercells of the V₂C₂ and V₃C₃ slabs were used to design SACs, and a 4 × 4 × 1 Gamma-centered k-point mesh was utilized to sample the Brillouin zone. In addition, to inspect the thermal stability of the studied SACs, ab initio molecular dynamics (AIMD) simulations were conducted at 600 K with a time step of 2 fs for 10 ps. The AIMD calculations were performed using the Andersen thermostat [42] and NVT ensemble.

To evaluate the possibility of forming a single vacancy on the V₂C₂/V₃C₃ slab, the defect formation energy was calculated via the following equation:

$$E_{\mathrm{form}}({\mathrm{V}}_{\mathrm{i}})=E_{\mathrm{perfect}-{\mathrm{V}}_{\mathrm{i}}}-E_{\mathrm{perfect}}+n{\mu }_{\mathrm{i}}$$

(1)

where $E_{\mathrm{perfect}-{\mathrm{V}}_{\mathrm{i}}}$ and $E_{\mathrm{perfect}}$ are the energies of the V₂C₂/V₃C₃ slab with and without isolated vacancies, respectively, μ_i is the chemical potential of type i atom, and n is the number of vacancies, which is 1 in this study.

The binding energy of transition-metal atoms (TM = Ti, V, Cr, Mn, Fe, Co, Ni, and Cu) embedded in the vacancies of the V₂C₂/V₃C₃ slab was determined as follows:

$$E_{\mathrm{bind}}=E_{\mathrm{TM}+\mathrm{S}}-E_{\mathrm{S}}-E_{\mathrm{TM}}$$

(2)

where E_TM+S, E_S, and E_TM represent the energies of the designed SACs, defective V₂C₂/V₃C₃ slabs, and a single transition-metal atom in its bulk phase, respectively. According to the defined criteria, more negative values of E_bind correspond to more stable structures.

It is widely accepted that HER occurs via a multi-step electrochemical process. Specifically, in acidic solutions, the mechanism of the HER process involves hydrogen adsorption (Volmer reaction) followed by hydrogen desorption (Heyrovsky reaction and/or Tafel reaction) [43]:

H⁺ + e⁻ +* → H* (Volmer)

(3)

H* + H⁺ + e⁻ → H₂ + * (Heyrovsky)

(4)

H* + H* → H₂ + * (Tafel)

(5)

where the asterisk refers to the adsorption sites on the surfaces of the catalysts. Both Volmer and Heyrovsky/Tafel reactions influence the HER activity, and the Volmer reaction is a dominant step for the overall HER process, which can be applied to assess the HER catalytic ability of the catalyst [44]. The change in Gibbs free energy for hydrogen adsorption (∆G_H∗) during the Volmer step is an effective descriptor for measuring HER catalytic activity and can be obtained by the following expression:

$$\Delta G_{\mathrm{H}*}=\Delta E_{\mathrm{H}}+\Delta E_{\mathrm{ZPE}}-T\Delta S_{\mathrm{H}}$$

(6)

In this expression, ΔE_H is the hydrogen adsorption energy and is calculated using the following equation:

$$\Delta E_{\mathrm{H}}=E_{(\mathrm{catalyst}+\mathrm{H})}-E_{\mathrm{catalyst}}-{\displaystyle \frac{1}{2}}{E}_{{\mathrm{H}}_2}$$

(7)

where E_catalyst+H and E_catalyst denote the energies of the SACs with and without an adsorbed hydrogen atom, respectively, and $E_{{\mathrm{H}}_2}$ is the energy of the hydrogen molecule. ΔE_ZPE is the zero-point energy (E_ZPE) difference, and the contributions from the substrate to E_ZPE are negligible. Hence, ΔE_ZPE is computed by

$$\Delta E_{\mathrm{ZPE}}=E_{\mathrm{ZPE}}^{\mathrm{H}}-{\displaystyle \frac{1}{2}}{E}_{\mathrm{ZPE}}^{{\mathrm{H}}_2}$$

(8)

where $E_{\mathrm{ZPE}}^{\mathrm{H}}$ and $E_{\mathrm{ZPE}}^{{\mathrm{H}}_2}$ are the E_ZPE of the adsorbed hydrogen atom and H₂ in the gas phase, respectively. T is the temperature (T = 298.15 K), and TΔS_H is the vibrational entropy difference during hydrogen adsorption. It should be noted that the contributions from the substrate to entropy are insignificant. Thus, TΔS_H can be approximated as the entropy difference between the adsorbed hydrogen atom and hydrogen in a gaseous state. At 298.15 K, the entropy of gaseous H₂ is approximately 0.40 eV [45].

3. Results and Discussion

3.1. Structures, Stability, and Active Sites

Pristine 2D V₂C₂ and V₃C₃ are tetragonal structures, and they possess two and three sub-layers, respectively [36]. For each sub-layer of these two vanadium carbides, each C/V atom is four-fold coordinated with its neighboring V/C atoms in the xy plane. For (4 × 4)-V₂C₂, the vacancy formation energies (E_form) of C and V vacancies are 8.52 and 10.54 eV, respectively. Given that the surfaces of V₃C₃ are more likely to produce vacancy defects compared to the middle sub-layer, the E_form of the C_surf (surface C) and V_surf (surface V) vacancies of (4 × 4)-V₃C₃ were calculated, and the corresponding E_form are 7.25 and 10.82 eV, respectively. It is evident that both V₂C₂ and V₃C₃ show greater potential to form carbon defects on the surface with relatively low E_form compared to vanadium defects. The binding energies (E_bind) of transition-metal atoms embedded in the vacancies are listed in

Table 1. The binding energies E_bind (eV) of transition-metal-atom-substituted C or V monovacancies of (4 × 4)-V₂C₂.

	Ti	V	Cr	Mn	Fe	Co	Ni	Cu
C monovacancy	1.62	2.22	2.27	2.12	1.51	1.58	1.11	0.92
V monovacancy	−5.02		−3.81	−3.34	−2.84	−2.49	−2.14	−1.14

and

Table 2. The binding energies E_bind (eV) of transition-metal-atom-substituted C_surf or V_surf monovacancies of (4 × 4)-V₃C₃.

	Ti	V	Cr	Mn	Fe	Co	Ni	Cu
Csurf monovacancy	1.72	2.62	2.90	2.78	2.41	1.86	1.04	0.77
Vsurf monovacancy	−5.61		−3.90	−3.38	−2.80	−2.31	−2.03	−1.17

for (4 × 4)-V₂C₂ and (4 × 4)-V₃C₃, respectively. The E_bind of transition-metal atoms at the C vacancy of (4 × 4)-V₂C₂ range from 0.92 to 2.27 eV, while those at the V vacancy range from −5.02 to −1.14 eV. The E_bind at the vacancies of (4 × 4)-V₃C₃ exhibit a similar range to those of defective (4 × 4)-V₂C₂. The Cr atom imported into the carbon defects shows the biggest binding energies (2.27 and 2.90 eV for V₂C₂ and V₃C₃ supercells, respectively), while the Ti atom in the vanadium defects exhibits the smallest binding energies (−5.02 and −5.61 eV for V₂C₂ and V₃C₃ supercells, respectively). Although the vanadium defects have higher formation energies compared with the carbon defects, they exhibit negative binding energies for all the imported transition-metal atoms. Taking into account both formation energies and binding energies, both carbon and vanadium defects are considered to be replaced by transition-metal atoms in this study.

The lattice constants of (4 × 4)-V₂C₂ with C and V vacancies are 11.400 and 11.389 Å, respectively, slightly smaller than those of (4 × 4)-V₂C₂ (11.415 Å). Figure 1a,b present the configurations of the V₂C₂-based SACs, along with the considered adsorption sites for HER, and Figures S1 and S2 (Supplementary Materials) show the relaxed structures of the SACs. It can be seen that there are significant height differences between the substrate and transition-metal atoms located at the C vacancy (Figure S1). This is because the radii of transition-metal atoms are bigger than the radius of a carbon atom, and a C vacancy does not have enough space to accommodate a transition-metal atom. When transition-metal atoms are stabilized at the V vacancy, the encased transition-metal atoms and surface of defective (4 × 4)-V₂C₂ are almost at the same height (Figure S2). The SACs are labeled TM@(4 × 4)-V₂C₂-V_C and TM@(4 × 4)-V₂C₂-V_V, as shown in Figures S1 and S2, respectively. For TM@(4 × 4)-V₂C₂-V_C and TM@(4 × 4)-V₂C₂-V_V, the anchored transition-metal atoms are surrounded by five V atoms and five C atoms, respectively. In addition, the lattice constants of optimized TM@(4 × 4)-V₂C₂-V_C and TM@(4 × 4)-V₂C₂-V_V are listed in Tables S1 and S2 (Supplementary Materials), respectively. The lattice constants of most TM@(4 × 4)-V₂C₂-V_C are bigger than those of pristine (4 × 4)-V₂C₂, while those of most TM@(4 × 4)-V₂C₂-V_V are smaller than those of (4 × 4)-V₂C₂. The lattice constant changes are related to the sizes of the vacancies, the radii of imported transition-metal atoms, and the heights of imported atoms from the substrate. Larger vacancies, smaller radii of imported atoms, and higher anchoring heights are conducive to smaller lattice constants.

The lattice constants of (4 × 4)-V₃C₃ with C_surf and V_surf vacancies are 11.508 and 11.493 Å, respectively, slightly smaller than those of pristine (4 × 4)-V₃C₃ (11.510 Å) and larger than those of defective (4 × 4)-V₂C₂. Figure 1c,d display the configurations of the V₃C₃-based SACs, along with the considered adsorption sites for HER, and Figures S3 and S4 (Supplementary Materials) show the relaxed structures of the SACs. The relative position between the imported transition-metal atoms and substrates is similar to the situation in the V₂C₂-based SACs. The V₃C₃-based SACs are labeled TM@(4 × 4)-V₃C₃-V_surf-C and TM@(4 × 4)-V₃C₃-V_surf-V, as shown in Figures S3 and S4, respectively. The lattice constants of all the TM@(4 × 4)-V₃C₃-V_surf-C are bigger than those of pristine (4 × 4)-V₃C₃ (Table S3 of the Supplementary Materials), while those of most TM@(4 × 4)-V₃C₃-V_surf-V are smaller than those of (4 × 4)-V₃C₃ (Table S4 of the Supplementary Materials). Overall, the differences in lattice constants are less than 1%, indicating a good stiffness of the substrates.

The thermal stability of the studied SACs was evaluated by AIMD simulations at 600K with a time step of 2 fs. The AIMD results for the V₂C₂- and V₃C₃-based SACs are shown in Figures S5–S8 (Supplementary Materials). During the simulations, the energy profiles for each catalyst did not fluctuate violently. Furthermore, the studied systems did not undergo any degradation after the 10 ps AIMD simulations. These results demonstrate that all the designed SACs are thermally stable at high temperature.

The considered adsorption sites for HER are marked in Figure 1. For both the V₂C₂- and V₃C₃-based SACs, when transition-metal atoms (Ti, V, Cr, Mn, Fe, Co, Ni, and Cu) are anchored at the C/C_surf vacancy, the tops of the transition-metal atoms (site TM¹), V atoms (sites V¹, V², and V³), and C atoms (sites C¹, C², C³, C⁴, and C⁵) are chosen as the adsorption sites for HER. For the SACs with transition-metal atoms (Ti, Cr, Mn, Fe, Co, Ni, and Cu) embedded in the V/V_surf vacancy, the nine adsorption sites are the tops of the transition-metal atoms (site TM¹), C atoms (sites C¹, C², and C³), and V atoms (sites V¹, V², V³, V⁴, and V⁵).

3.2. Electronic Conductivity

Electronic conductivity is closely related to the efficiency of charge transfer during the catalytic process [46]. The densities of states (DOSs) of the defective V₂C₂/V₃C₃ supercells and constructed SACs are displayed in Figures S9–S12 (Supplementary Materials). As shown in Figures S9a and S10a, the V₂C₂ slab with a C/V vacancy exhibits good electronic conductivity, with the Fermi level falling into a continuum of energy states. Furthermore, all the TM@(4 × 4)-V₂C₂-V_C and TM@(4 × 4)-V₂C₂-V_V also possess excellent conductivity (Figures S9b–i and S10b–h). According to the projected density of states (PDOS), the electronic conductivity of the V₂C₂-based SACs is dominated by V-d orbitals near the Fermi level. In addition, the DOSs of the defective (4 × 4)-V₃C₃ and V₃C₃-based SACs have similar characteristics to the DOSs of the V₂C₂-based structures (Figures S11 and S12). Overall, similar to other excellent HER catalysts, such as Mo₂B₂-, MoS₂-, and MXene-based catalysts [32,47,48], all the designed SACs exhibit high electronic conductivity, which can ensure efficient charge transfer in HER.

3.3. Hydrogen Evolution Reaction Activity of V₂C₂-Based Catalysts

Generally, the HER process can be divided into three states, including the initial state H⁺ + e⁻, the intermediate adsorbed H*, and the final H₂ state. The Gibbs free energy difference between H* and H₂ (ΔG_H*) is considered a key descriptor of the HER activity. In this work, the HER catalytic ability of 2D V₂C₂- and V₃C₃-based SACs at different adsorption sites is determined by ΔG_H*. A smaller absolute value of ΔG_H* means a better HER performance, and the optimal value of ΔG_H* is zero. In general, catalysts with a | ΔG_H*| smaller than 0.2 eV are regarded as excellent HER catalysts.

For TM@(4 × 4)-V₂C₂-V_C and TM@(4 × 4)-V₂C₂-V_V, the calculation details for the free energy differences are listed in Tables S5–S19 (Supplementary Materials), including the energies (E), E_ZPE, vibrational entropy (TS_H), Gibbs free energies (G), and ΔG_H* of every V₂C₂-based SAC at different active sites. For each TM@(4 × 4)-V₂C₂-V_C, the E_ZPE value of the catalyst with the hydrogen atom adsorbed on the top of the imported transition-metal atom is smaller than that of its counterparts adsorbing H on the top of the C or V atoms. In addition, almost all the TS_H values are less than 0.1 eV, much smaller than the E_ZPE values. Hence, compared with TS_H, E_ZPE contributes more to the free energy.

The ΔG_H* values of TM@(4 × 4)-V₂C₂-V_C at different adsorption sites are listed in

Table 3. Calculated Gibbs free energies at different adsorption sites for TM@(4 × 4)-V₂C₂-V_C. The units are in electron volts.

	C1	C2	C3	C4	C5	V1	V2	V3	TM1
Ti@(4 × 4)-V2C2-VC	0.38	0.47	0.53	0.45	0.42	−0.33	0.52	0.53	0.30
V@(4 × 4)-V2C2-VC	0.43	0.45	0.51	0.47	0.43	−0.23	0.52	0.53	0.13
Cr@(4 × 4)-V2C2-VC	0.45	0.48	0.45	0.48	0.47	−0.22	0.53	0.53	−0.06
Mn@(4 × 4)-V2C2-VC	−0.13	0.00	0.02	−0.04	−0.05	−0.78	0.07	0.06	−0.04
Fe@(4 × 4)-V2C2-VC	0.40	0.49	0.49	0.45	0.43	−0.14	0.54	0.53	0.04
Co@(4 × 4)-V2C2-VC	0.35	0.43	0.43	0.41	0.40	−0.23	0.50	0.48	−0.23
Ni@(4 × 4)-V2C2-VC	0.39	0.47	0.48	0.46	0.41	−0.18	0.56	0.52	−0.19
Cu@(4 × 4)-V2C2-VC	0.35	0.51	0.49	0.42	0.42	−0.04	0.53	0.51	0.19

. For each TM@(4 × 4)-V₂C₂-V_C, the ΔG_H* at adsorption site C¹ is not higher than that at sites on the top of other C atoms, and the ΔG_H* at adsorption site V¹ is smaller than that at sites on the top of other V atoms. Compared with other adsorption sites, C¹ and V¹ are closer to the imported transition-metal atoms, and the great impact of the transition-metal atoms on hydrogen adsorption leads to the relatively small ΔG_H* values. Apart from (Ti and Co)@(4 × 4)-V₂C₂-V_C, other TM@(4 × 4)-V₂C₂-V_C exhibit high HER catalytic ability, with a |ΔG_H*| smaller than 0.2 eV at some active sites. Specifically, there are up to eight adsorption sites that display great HER catalytic activity on the surface of Mn@(4 × 4)-V₂C₂-V_C, including C¹, C², C³, C⁴, C⁵, V², V³, and TM¹. In particular, the ΔG_H* of Mn@(4 × 4)-V₂C₂-V_C at site C² is zero, suggesting the best catalytic performance. (Fe, Ni and Cu)@(4 × 4)-V₂C₂-V_C have two active sites (V¹ and TM¹) with a |ΔG_H*| smaller than 0.2 eV, and (V and Cr)@(4 × 4)-V₂C₂-V_C have only one active site (TM¹) possessing excellent catalytic capacity. Among all the V₂C₂-based SACs, (Cr, Mn, Fe, and Cu)@(4 × 4)-V₂C₂-V_C possess active sites with a smaller |ΔG_H*| compared to Pt (0.09 eV) [49]. In addition, for each TM@(4 × 4)-V₂C₂-V_C, the |ΔG_H*| value of some sites is lower than that of Mo₂B₂ (0.37 eV) [32], and the |ΔG_H*| value of all the sites is lower than that of MoS₂ (2.0 eV) [50]. It should be noted that pristine tetragonal V₂C₂ is not suitable for HER catalysis [36], and the construction of SACs based on tetragonal V₂C₂ has enhanced its HER catalytic ability. Besides V₂C₂, the HER catalytic performance of Mo₂B₂, graphene, and MoS₂ has also been greatly improved after importing transition-metal atoms [32,34,50], indicating the huge potential of SACs in HER catalysis.

Table 4. Calculated Gibbs free energies at different adsorption sites for TM@(4 × 4)-V₂C₂-V_V. The units are in electron volts.

	C1	C2	C3	V1	V2	V3	V4	V5	TM1
Ti@(4 × 4)-V2C2-VV	0.43	0.46	0.47	0.52	0.54	0.52	0.52	0.51	0.78
Cr@(4 × 4)-V2C2-VV	0.46	0.46	0.47	0.51	0.50	0.52	0.50	0.51	0.31
Mn@(4 × 4)-V2C2-VV	0.42	0.45	0.47	0.57	0.50	0.51	0.51	0.53	0.31
Fe@(4 × 4)-V2C2-VV	0.38	0.46	0.51	0.59	0.50	0.53	0.50	0.54	0.35
Co@(4 × 4)-V2C2-VV	0.33	0.51	0.55	0.67	0.54	0.58	0.56	0.60	0.47
Ni@(4 × 4)-V2C2-VV	0.34	0.46	0.53	0.67	0.51	0.57	0.52	0.57	0.91
Cu@(4 × 4)-V2C2-VV	0.44	0.47	0.51	0.63	0.50	0.62	0.52	0.56	1.48

lists the ΔG_H* values of TM@(4 × 4)-V₂C₂-V_V at different adsorption sites. Similar to TM@(4 × 4)-V₂C₂-V_C, adsorption site C¹ exhibits a relatively low ΔG_H* among the sites on the top of the C atoms. Overall, the ΔG_H* values of sites on the top of the C atoms are lower than those of sites on the top of the V atoms. For each TM@(4 × 4)-V₂C₂-V_V, the free energy differences of all the considered adsorption sites are larger than 0.3 eV, suggesting inefficient HER catalytic performance.

3.4. Hydrogen Evolution Reaction Activity of V₃C₃-Based Catalysts

For TM@(4 × 4)-V₃C₃-V_surf-C and TM@(4 × 4)-V₃C₃-V_surf-V, the calculation details for the free energy differences are listed in Tables S20–S34 (Supplementary Materials). It can be seen that the E_ZPE values of the SACs with hydrogen atoms adsorbed on the top of the C atoms (around 0.24 eV) are larger than those of the SACs adsorbing H on the top of the V atoms (0.17–0.19 eV) or imported transition-metal atoms (no more than 0.20 eV). In addition, the TS_H values are smaller than 0.1 eV, indicating less influence on the free energy compared with E_ZPE.

The ΔG_H* values of TM@(4 × 4)-V₃C₃-V_surf-C at different adsorption sites are listed in

Table 5. Calculated Gibbs free energies at different adsorption sites for TM@(4 × 4)-V₃C₃-V_surf-C. The units are in electron volts.

	C1	C2	C3	C4	C5	V1	V2	V3	TM1
Ti@(4 × 4)-V3C3-Vsurf-C	−0.16	0.19	0.17	0.09	0.02	−0.21	0.66	0.66	0.18
V@(4 × 4)-V3C3-Vsurf-C	−0.36	0.13	0.17	0.09	0.03	−0.21	0.66	0.66	0.05
Cr@(4 × 4)-V3C3-Vsurf-C	−0.91	−0.70	−0.77	−0.87	−0.88	−1.14	−0.30	−0.32	−0.09
Mn@(4 × 4)-V3C3-Vsurf-C	−1.23	−1.04	−1.10	−1.18	−1.20	−1.46	−0.62	−0.63	−1.07
Fe@(4 × 4)-V3C3-Vsurf-C	−0.71	−0.63	−0.64	−0.73	−0.75	−0.89	−0.14	−0.16	−0.01
Co@(4 × 4)-V3C3-Vsurf-C	−0.14	0.02	0.02	−0.06	0.06	−0.23	0.51	0.49	−0.08
Ni@(4 × 4)-V3C3-Vsurf-C	0.15	0.21	0.19	0.10	0.09	−0.03	0.68	0.66	0.10
Cu@(4 × 4)-V3C3-Vsurf-C	0.23	0.28	0.21	0.12	0.11	0.04	0.68	0.66	0.41

. For each TM@(4 × 4)-V₃C₃-V_surf-C, the ΔG_H* values of the active sites on the top of the C atoms are lower than those of the V² and V³ sites. In addition, adsorption sites V² and V³ exhibit similar values of Gibbs free energy. Apart from Mn@(4 × 4)-V₃C₃-V_surf-C, other TM@(4 × 4)-V₃C₃-V_surf-C possess active sites with a |ΔG_H*| smaller than 0.2 eV, suggesting their great HER catalytic ability. Specifically, (Ti and Co)@(4 × 4)-V₃C₃-V_surf-C display excellent catalytic performance at sites C¹, C², C³, C⁴, C⁵, and TM¹. Ni@(4 × 4)-V₃C₃-V_surf-C also has six sites (C¹, C³, C⁴, C⁵, V¹, and TM¹) exhibiting excellent catalytic performance. (V, Fe and Cu)@(4 × 4)-V₃C₃-V_surf-C have more than two active sites with a |ΔG_H*| smaller than 0.2 eV. Only Cr@(4 × 4)-V₃C₃-V_surf-C has one adsorption site with great catalytic performance. These results indicate that most of the designed TM@(4 × 4)-V₃C₃-V_surf-C have potential as promising HER catalysts.

The ΔG_H* values of TM@(4 × 4)-V₂C₂-V_surf-V suggest that they are all outstanding HER catalysts (

Table 6. Calculated Gibbs free energies at different adsorption sites for TM@(4 × 4)-V₃C₃-V_surf-V. The units are in electron volts.

	C1	C2	C3	V1	V2	V3	V4	V5	TM1
Ti@(4 × 4)-V3C3-Vsurf-V	0.15	0.12	0.07	0.63	0.68	0.63	0.63	0.63	0.94
Cr@(4 × 4)-V3C3-Vsurf-V	−0.04	0.04	0.08	0.65	0.61	0.64	0.63	0.63	0.45
Mn@(4 × 4)-V3C3-Vsurf-V	−0.05	0.08	0.10	0.65	0.65	0.65	0.64	0.63	0.57
Fe@(4 × 4)-V3C3-Vsurf-V	−0.12	0.08	0.12	0.66	0.65	0.65	0.64	0.63	0.59
Co@(4 × 4)-V3C3-Vsurf-V	−0.25	0.03	0.08	0.64	0.61	0.66	0.61	0.63	0.60
Ni@(4 × 4)-V3C3-Vsurf-V	−0.18	0.08	0.13	0.66	0.65	0.67	0.64	0.62	0.98
Cu@(4 × 4)-V3C3-Vsurf-V	−0.04	0.10	0.10	0.64	0.67	0.67	0.65	0.63	1.37

). All the adsorption sites on the top of the C atoms exhibit a |ΔG_H*| smaller than 0.2 eV for all the TM@(4 × 4)-V₂C₂-V_surf-V except Co@(4 × 4)-V₂C₂-V_surf-V. Co@(4 × 4)-V₂C₂-V_surf-V shows excellent HER catalytic performance at sites C² and C³. Different from the sites on the top of the C atoms, the sites on the top of the V atoms and anchored transition-metal atoms have values not less than 0.45 eV and display inefficient catalytic activity. On the whole, we can conclude that all the TM@(4 × 4)-V₂C₂-V_surf-V can act as great HER catalysts.

Overall, V₃C₃-based SACs have better HER catalytic capacity than their V₂C₂-based counterparts. Based on the |ΔG_H*| values, apart from Mn@(4 × 4)-V₃C₃-V_surf-C, the HER catalytic abilities of other V₃C₃-based SACs are comparable to those of Pt- and Ru-based catalysts [51,52], suggesting the potential to replace precious-metal-based catalysts for HER. The |ΔG_H*| value of every adsorption site is lower than that of MoS₂ (2.0 eV) [50]. In addition, the minimum |ΔG_H*| values of most V₃C₃-based SACs are comparable to the ΔG_H* of Cr₂TiC₂O₂ and Cr₂VC₂O₂ MXenes [53]. Cr₂TiC₂O₂ possesses excellent HER catalytic performance when hydrogen coverage is 1/4 (ΔG_H* = −0.17 eV), 3/8 (ΔG_H* = 0.16 eV), and 1/2 ML (ΔG_H* = 0.20 eV), while Cr₂VC₂O₂ exhibits great HER catalytic ability for 3/8 hydrogen coverage (ΔG_H* = −0.03 eV). This indicates that transition-metal carbides have potential application value in HER catalysis.

4. Summary

In summary, we have constructed a battery of HER SACs by importing transition-metal atoms into the carbon and vanadium vacancies of 2D tetragonal V₂C₂ and V₃C₃. By means of first-principles computations, the feasibility of applying these SACs in HER catalysis was examined. The results indicated that all the SACs are thermally stable and conductive, which is favorable to catalysis. The Gibbs free energy change during hydrogen adsorption was used to evaluate their catalytic ability. Among the V₂C₂-based SACs, Mn@(4 × 4)-V₂C₂-V_C possesses eight adsorption sites with excellent HER catalytic ability, and (V, Cr, Fe, Ni and Cu)@(4 × 4)-V₂C₂-V_C also exhibit great HER catalytic performance at some active sites. These excellent V₂C₂-based SACs have a smaller |ΔG_H*| at some sites than pristine V₂C₂, Mo₂B₂, and MoS₂, indicating the great advantage of SACs. Among the V₃C₃-based SACs, apart from Mn@(4 × 4)-V₃C₃-V_surf-C, all the SACs can act as outstanding HER catalysts. The minimum |ΔG_H*| values of these outstanding V₃C₃-based SACs are comparable to the ΔG_H* of Cr₂TiC₂O₂ and Cr₂VC₂O₂ MXenes. Overall, these great V₂C₂- and V₃C₃-based SACs can be compared with the best-known Pt-based HER catalysts. It should be noted that this is a pure theoretical study as the V₂C₂ and V₃C₃ slabs have not been successfully synthesized in the laboratory. This work offers researchers an incentive to synthesize these two materials. After successfully synthesizing these two materials, the stability of them in real electrochemical HER environments should be ensured before practical application.