Density Functional Theory Study of the Hydrogenation of Carbon Monoxide over the Co (001) Surface: Implications for the Fischer–Tropsch Process

: The increasing demand for renewable fuels and sustainable products has encouraged growing interest in the development of active and selective catalysts for the conversion of carbon monoxide into desirable products. The Fischer–Tropsch process consists of the reaction of a synthesis gas mixture containing carbon monoxide and hydrogen (syngas), which are polymerized into liquid hydrocarbon chains, often using a cobalt catalyst. Here, ﬁrst-principles calculations based on the density functional theory (DFT) are used to investigate the reaction mechanism of the Fischer–Tropsch synthesis over the Co (001) surface. The most energetically favorable adsorption conﬁgurations of the species involved in the carbon monoxide hydrogenation process are identiﬁed, and the possible elementary steps of hydrogenation and their related transition states are explored using the Vienna Ab initio simulation package (VASP). The results provide the mechanisms for the formation of CH 4 , CH 3 OH and C 2 H 2 compounds, where the calculations suggest that CH 4 is the dominant product. Findings from the reaction energies reveal that the preferred mechanism for the hydrogenation of carbon monoxide is through HCO and cis-HCOH, and the largest exothermic reaction energy in the CH 4 formation pathway is released during the hydrogenation of cis-HCOH ( − 0.773 eV). An analysis of the kinetics of the hydrogenation reactions indicates that the CH production from cis-HCOH has the lowest energy barrier of just 0.066 eV, and the hydrogenation of CO to COH, with the largest energy barrier of 1.804 eV, is the least favored reaction kinetically.


Introduction
Carbon monoxide (CO) is a colorless, odorless and non-irritable gas [1,2].With a specific gravity of 0.97, CO is slightly lighter than air and is mainly produced by the incomplete combustion of organic compounds [3][4][5][6].Fischer-Tropsch synthesis [7][8][9] (FTS) is a process that has been used for many decades as it gives access to industrially important chemicals from CO [10][11][12][13].As the products of FTS are a complex mixture of a wide range of organic compounds, selectivity toward desired products is the most important issue in this reaction [14].In recent years, there has been an increasing motivation to deploy FTS at commercial scales, which has fueled the search for high-performance catalysts [15].
Several catalysts have been examined for their potential to catalyze CO hydrogenation [16][17][18][19].The rate of formation and the selectivity towards certain hydrocarbons are the key challenges in FTS and they depend on the catalyst used [20].Transition metal catalysis has long been recognized as a reliable and modular means of constructing complex molecules from simple, readily accessible starting materials [21].
Many studies based on density functional theory (DFT) calculations of FTS synthesis over metallic surfaces have been reported in the literature [22][23][24][25][26][27][28][29][30].For example, 24}, low-coverage pathways with activation energies that lie below the energy of gas-phase CO were identified.The existence of these low-energy pathways on the stepped surfaces allows a CO molecule from the gas phase to dissociate spontaneously [30].The elementary step from C 2 to C 6 and the α-olefin selectivity through the hydrogenation and dehydrogenation of n-alkyl groups on Co (0001) have been investigated in an early work by Cheng et al. [22] In another study, Cheng et al. studied CO hydrogenation on fcc Co (111), where they sought to study the formation of C 2 hydrocarbons on the surfaces of fcc Co, with significant results for the adsorption energies and activation energies [48].Petersen et al. investigated CO dissociation at step and kink sites on fcc Co (221) and Co (321) surfaces.In both cases, the direct CO dissociation path yields the lowest overall activation energy for CO dissociation, with H-assisted routes via HCO or COH intermediates being higher in energy [49].Helden et al. reported DFT results from a comparative study of the direct and hydrogen-assisted CO dissociation pathways on the surface of fcc Co (l00) [50,51], where they clearly showed that the hydrogen-assisted CO dissociation mechanism is an important contributor to the CO activation mechanism during the first step of FTS [51].
To the best of our knowledge, there is no comprehensive theoretical investigation of CO hydrogenation via FTS synthesis on the fcc Co (001) surface.As such, this paper presents the results of a DFT study of the CO hydrogenation mechanism and the reaction and activation energies towards different products [52,53], confirming that CH 4 is the main product, both thermodynamically and kinetically [54].
In this section, the adsorption of a range of intermediates on the Co (001) surface is examined.As illustrated in Figure 1, there are three different positions for the adsorption of molecules on the surface, i.e., the bridge, hollow and top sites.
The preferred adsorption positions for all intermediates on the Co (001) surface are presented in Figure 2, with geometric information and adsorption energies calculated using Equation (3), provided in Table 1.More information on the structural details of the adsorption geometries are found in Table S1 and Figure S1 of the Supplementary Information.The preferred adsorption positions for all intermediates on the Co (001) surface are presented in Figure 2, with geometric information and adsorption energies calculated using Equation 3, provided in Table 1.More information on the structural details of the adsorption geometries are found in Table S1 and Figure S1 of the Supplementary Information.
All possible adsorption configurations were studied, and the lowest-energy adsorption geometry for each intermediate was selected as the final configuration.The results show that the preferred site for adsorption is the hollow site, although some molecules adsorb on top, and just one adsorbs on a bridge site.In addition, the results show that all molecules prefer to interact with the surface through their carbon atom.
CO binds to the Co (001) surface through its C atom, which is located exactly in the hollow site between surface Co atoms, with a Co-C bond length of 2.03 Å and an adsorption energy of −2.268 eV.The negative value of the adsorption energy indicates that the adsorption is an exothermic process.This value is in good agreement with the literature values for the adsorption of CO [63][64][65][66].
COH adsorbs vertically and binds to the Co (001) surface through its C atom, with a Co-C bond length of 1.960 Å and an adsorption energy of −5.696 eV.This mode of adsorption is similar to that found by Psofogiannakis et al. on the Pt (111) surface [63], where COH is also adsorbed in a hollow site with a calculated adsorption energy of −5.64 eV.
When the HCO molecule is adsorbed, two oxygen and carbon atoms bind strongly to the surface atoms, with Co-O and Co-C bond lengths of 1.928 and 1.917 Å, respectively, while its hydrogen atom does not interact with the surface; the adsorption energy for this molecule is −4.03 eV.
Cis-and trans-HCOH are adsorbed with similar energies of −3.919 and −3.583 eV, respectively.Cis-HCOH adsorbs at a hollow position, while trans-HCOH is located exactly on a bridge position between two cobalt atoms.Another difference between the adsorption geometries of these two species is their binding to the surface, as cis-HCOH bonds to the cobalt surface atom through both its carbon and oxygen atoms, with Co-C and Co-O bond lengths of 1.970 and 2.365 Å, respectively, while trans-HCOH binds to the surface only through its carbon atom, with a Co-C bond length of 1.930 Å.
CH, CH2, CH3 and CH4 adsorb at the hollow position on the Co surface, with the former three species binding to the Co (001) surface through their carbon atoms with bond lengths of 1.942, 2.091, and 1.967 Å, respectively.The CH4 molecule approaches the surface via its C and H atoms, at average distances of 3.645 and 2.841 Å, respectively.The adsorption energies for CH, CH2, CH3 and CH4 were calculated at  All possible adsorption configurations were studied, and the lowest-energy adsorption geometry for each intermediate was selected as the final configuration.The results show that the preferred site for adsorption is the hollow site, although some molecules adsorb on top, and just one adsorbs on a bridge site.In addition, the results show that all molecules prefer to interact with the surface through their carbon atom.
CO binds to the Co (001) surface through its C atom, which is located exactly in the hollow site between surface Co atoms, with a Co-C bond length of 2.03 Å and an adsorption energy of −2.268 eV.The negative value of the adsorption energy indicates that the adsorption is an exothermic process.This value is in good agreement with the literature values for the adsorption of CO [63][64][65][66].
COH adsorbs vertically and binds to the Co (001) surface through its C atom, with a Co-C bond length of 1.960 Å and an adsorption energy of −5.696 eV.This mode of adsorption is similar to that found by Psofogiannakis et al. on the Pt (111) surface [63], where COH is also adsorbed in a hollow site with a calculated adsorption energy of −5.64 eV.
When the HCO molecule is adsorbed, two oxygen and carbon atoms bind strongly to the surface atoms, with Co-O and Co-C bond lengths of 1.928 and 1.917 Å, respectively, while its hydrogen atom does not interact with the surface; the adsorption energy for this molecule is −4.03 eV.
Cis-and trans-HCOH are adsorbed with similar energies of −3.919 and −3.583 eV, respectively.Cis-HCOH adsorbs at a hollow position, while trans-HCOH is located exactly on a bridge position between two cobalt atoms.Another difference between the adsorption geometries of these two species is their binding to the surface, as cis-HCOH bonds to the cobalt surface atom through both its carbon and oxygen atoms, with Co-C and Co-O bond lengths of 1.970 and 2.365 Å, respectively, while trans-HCOH binds to the surface only through its carbon atom, with a Co-C bond length of 1.930 Å.
CH, CH 2 , CH 3 and CH 4 adsorb at the hollow position on the Co surface, with the former three species binding to the Co (001) surface through their carbon atoms with bond lengths of 1.942, 2.091, and 1.967 Å, respectively.The CH 4 molecule approaches the surface via its C and H atoms, at average distances of 3.645 and 2.841 Å, respectively.The adsorption energies for CH, CH 2 , CH 3 and CH 4 were calculated at −7.964, −5.611, −2.972 and −0.204 eV, respectively.For comparison, the adsorption energies of CH and CH 2 on the Pt (111) surface were calculated by Psofogiannakis et al. at −7.55 and −4.56 eV, respectively, whereas the adsorption of CH 3 at the surface was calculated at −2.40 eV [63], and the adsorption energy for CH 4 on the Ru (0001) surface was calculated at −0.17 eV by Zhang et al. [31].
H 2 O adsorbs on top of a surface Co atom on the Co (001) surface via its oxygen atom, with an adsorption energy of −0.746 eV, forming a Co-O bond length of 2.141 Å. CH 2 OH prefers to be sited in a hollow position between four surface Co atoms, where it binds by its oxygen and carbon atoms with Co-O and Co-C bond lengths of 2.110 and 1.941Å and an adsorption energy of −2.74 eV.While Ashwell et al. [64] reported an energy of −1.68 eV for CH 2 OH adsorption on the Ni (110) surface, Psofogiannakis et al. calculated adsorption energies that were more similar to our result, obtaining −2.79 eV [63] as the adsorption energy of CH 2 OH on the Pt (111) surface.
The oxygen atom of CH 3 OH adsorbs above a surface Co atom, with an adsorption energy of just −0.718 eV, and forms a Co-O bond with a bond length of 2.117 Å.The adsorption geometry of C 2 H 2 shows that it adsorbs in a hollow site parallel to the surface by bonding to cobalt surface atoms, with a Co-C bond length of 1.343 Å and an adsorption energy of −3.241 eV; this is similar to the adsorption energy of C 2 H 2 on the Ni (111) surface, calculated by Medlin and Allendorf to be approximately −2.957 eV [67].
The adsorption energies of the studied intermediates on the Co (001) surface decrease in the order:

Hydrogenation
The hydrogenation of CO is the main goal of this study, and it is also an inseparable part of FTS.After calculating the adsorption of all relevant molecules at the Co (001 surface, we next introduced hydrogen on the surface near the adsorbed molecules to investigate the hydrogenation reactions.Due to the number of possible relative positions for the hydrogen atoms and molecules, several calculations were carried out to identify the lowest-energy positions for hydrogen in each adsorption configuration.The most stable co-adsorbed geometries of each intermediate and H are shown in Figure 3.
The hydrogenation of CO to obtain CH 3 OH, CH 4 , and C 2 H 2 goes first through COH and HCO, followed by the further hydrogenation of these intermediates to form cis-HCOH and trans-HCOH.In the next step, the hydrogenation of cis-HCOH and trans-HCOH can produce either CH 2 OH or CH + H 2 O, followed by the production of methanol CH 3 OH from CH 2 OH + H. Finally, CH 4 is produced through three intermediates: CH + H, CH 2 + H, and CH 3 + H.

Reactions
In this section, the reactions underpinning the mechanism of CO hydrogenation on the Co (001) surface are discussed.The reaction energies calculated via Equation ( 4) are presented in Table 2.
The hydrogenation of CO to obtain CH3OH, CH4, and C2H2 goes first through COH and HCO, followed by the further hydrogenation of these intermediates to form cis-HCOH and trans-HCOH.In the next step, the hydrogenation of cis-HCOH and trans-HCOH can produce either CH2OH or CH + H2O, followed by the production of methanol CH3OH from CH2OH + H. Finally, CH4 is produced through three intermediates: CH + H, CH2 + H, and CH3 + H.

Reactions
In this section, the reactions underpinning the mechanism of CO hydrogenation on the Co (001) surface are discussed.The reaction energies calculated via Equation 4 are presented in Table 2.
The transformation from CO to form HCO has a reaction energy of about 0.3 eV less than the reaction energy needed to form COH, with the reaction energy for CO + H → COH calculated at 0.853 eV.This is in perfect agreement with the work by Zhang et al. [68], who obtained a reaction energy of 0.85 eV to produce COH from the hydrogenation After the steps above, there are three ways to reach the desired products; first, the hydrogenation of CH 2 OH to CH 3 OH, second, a reaction between two CH species to produce acetylene (C 2 H 2 ), and finally, the production of CH 4 through a three-step hydrogenation reaction of CH→CH 2 →CH 3 →CH 4 .These three steps consist of the reactions: CH + H → CH 2 , CH 2 + H → CH 3 and CH 3 + H → CH 4 .The hydrogenation of CH 2 OH + H→CH 3 OH has a calculated reaction energy of 0.089 eV compared to Ashwell et al. [64], who calculated the reaction energy for this reaction to be 0.49 eV on the Cu (111) surface.Acetylene is produced by the reaction CH + CH, with an energy of 1.511 eV.The three-step process to produce CH 4 requires reaction energies for CH + H → CH 2 , CH + H → CH 3 and CH + H → CH 4 of 0.019, 0.679, and 0.491 eV, respectively.
In the above-described network of reactions, all reactions except three are endothermic.The transformation of cis-HCOH + H to CH + H 2 O is exothermic, with a reaction energy of −0.773 eV, whereas the other two exothermic reactions are the result of trans-HCOH hydrogenation, with reaction energies of −0.120 eV and −1.283 eV to produce CH 2 OH and CH + H 2 O, respectively.The transition states of all the elementary reactions were identified and are shown in Figure 4.In order to gain further insight, the activation barriers for the elementary reactions in the CO hydrogenation process over the Co (001) surface were calculated and are listed in Table 3, where the energies were calculated via Equation (5).

Transition States
The transition states of all the elementary reactions were identified and are shown in Figure 4.In order to gain further insight, the activation barriers for the elementary reactions in the CO hydrogenation process over the Co (001) surface were calculated and are listed in Table 3, where the energies were calculated via Equation 5.The energy barrier in the hydrogenation of CO is 1.804 eV for the production of COH, whereas it is 1.082 eV for the production of HCO, indicating that HCO is the preferred product from the first hydrogenation step.This finding is in good agreement with the literature, in which Zhu et al. [66] reported that the barrier for CO hydrogenation to COH on the Ni (111) surface is 1.97 eV, and Ashwell et al. [64] found a similar activation energy The energy barrier in the hydrogenation of CO is 1.804 eV for the production of COH, whereas it is 1.082 eV for the production of HCO, indicating that HCO is the preferred product from the first hydrogenation step.This finding is in good agreement with the literature, in which Zhu et al. [66] reported that the barrier for CO hydrogenation to COH on the Ni (111) surface is 1.97 eV, and Ashwell et al. [64] found a similar activation energy of 1.08 eV for the production of HCO over the Ni (110) surface.It is worth mentioning that among the reactions studied in this work, the reaction of CO + H → COH is the rate-determining step (RDS) on the Co (001) surface.
The energy barriers for COH hydrogenation to cis-HCOH and trans-HCOH are the same at 1.231 eV, which is comparable to the same reactions calculated by Amaya-Roncancio et al. [65] on the Fe (100) surface at 1.38 eV.The hydrogenation reactions of HCO to reach the cis-and trans isomers of HCOH must overcome energy barriers of 1.746 and 1.727 eV, respectively.
Next, we consider the production of CH + H 2 O and CH 2 OH from the cis and trans conformers of HCOH.The barriers for the production of CH 2 OH from the hydrogenation of either isomer are below 1 eV, at 0.662 and 0.131 eV for the cis-HCOH and trans-HCOH isomers, respectively, which is in good agreement with Qi et al. [72] who reported a barrier of 0.71 eV for cis-HCOH + H → CH 2 OH on the Co (0001) surface.However, there is a large difference between the energy barriers to be overcome for the dissociation into CH + H 2 O products from cis-HCOH or trans-HCOH in the presence of hydrogen.cis-HCOH dissociation produces CH + H 2 O with a barrier of just 0.066 eV, but the production of CH + H 2 O from trans-HCOH requires an activation energy of 1.581 eV.These results show that CH + H 2 O and CH 2 OH are more likely to be produced by the hydrogenation of cis-HCOH and trans-HCOH conformers, respectively, see Table 3.
The activation energy barrier for the synthesis of C 2 H 2 from the reaction of CH + CH on the Co (001) surface is 1.556 eV, whereas the production of CH 3 OH from the reaction of CH 2 OH + H has a barrier of 0.725 eV.As discussed above, CH can be produced from either reaction of cis-HCOH + H → CH + H 2 O or trans-HCOH + H → CH + H 2 O, followed by further reaction with adsorbed hydrogens to produce CH 4 along three continuous reaction steps: CH + H, CH 2 + H, and CH 3 + H, with energy barriers of 0.065, 0.969, and 1.089 eV, respectively, see Table 3.These calculated energy barriers are in agreement with Zhu et al. [66], Cheng et al. [73], and Niu et al. [74], who reported activation energies of 0.69, 0.81, and 1.187 eV for this reaction on the Ni (111), Fe 5 C 2 (100), and Pt (111) surfaces, respectively.

Reaction Pathways
The energies of the reaction routes to methanol, methane and acetylene production on the surface are shown in Figures 5-7, respectively.For each product, there are four pathways (Tables 4-6) through each of the routes of CO → HCO → cis-HCOH, CO → HCO → trans-HCOH, CO → COH → cis-HCOH, and CO → COH → trans-HCOH.Each product is reached through a main pathway with a favored reaction mechanism.
product is reached through a main pathway with a favored reaction mechanism.
CH4 production can occur through CO hydrogenation via HCO and cis-HCOH intermediates.The reaction begins with the hydrogenation of CO.Then, according to the Figure 5 and Table 4, the pathways passing through HCO are the preferred route because the activation energy barrier for CO + H→COH is about 0.8 eV larger than for CO + H→HCO.In the next step, HCO is hydrogenated to HCOH isomers (HCO + H → cis-HCOH and HCO + H → trans-HCOH).Although these two reactions are kinetically the same, thermodynamically, cis-HCOH is the preferred intermediate, resulting in the reaction sequences CH + H → CH2, CH2 + H→ CH3 and CH3 + H→CH4 being the most favorable route for CH4 production (Figure 5).This pathway is completely exothermic, with an overall reaction energy of −1.53 eV, and the hydrogenation of cis-HCOH + H → CH + H2O is the most favorable reaction in this pathway, with an energy of −0.773 eV.In contrast, the least favourable reaction is HCO + H → cis-HCOH, which requires 1.393 eV of energy.Kinetically, the hydrogenation of CH + H → CH2 is the optimum reaction in this pathway, with an energy barrier of only 0.065 eV.The kinetic and thermodynamic outcomes for the byproducts indicate that the favored pathways resulting in CH3OH and C2H2 (presented in Tables 5 and 6) have the same intermediates in the first two steps, beginning with CO + H → HCO and then passing through HCO + H → cis-HCOH.In the production of CH3OH, the cis-HCOH + H → CH2OH reaction costs 0.39 eV, with a barrier of 0.662 eV, followed by the reaction of CH2OH + H → CH3OH, which requires 0.089 eV and must overcome a barrier of 0.725 eV to proceed.Cis-HCOH is hydrogenated to C2H2 by the reaction sequences of cis-HCOH + H → CH + H2O and CH + CH → C2H2, with reaction energies of −0.773 and 1.511 eV and energy barriers of 0.066 and 1.556 eV (Figures 6 and 7).The optimum pathways for the production of CH3OH and C2H2 are entirely endothermic, with overall reaction energies of 0.563, and 0.991 eV, respectively.The most favorable reaction in the production of C2H2 is cis-HCOH to CH + H2O, with an energy of −0.773 eV, and in the CH3OH formation, it is the hydrogenation of cis-HCOH to CH2OH, with an energy of 0.39 eV.CH 4 production can occur through CO hydrogenation via HCO and cis-HCOH intermediates.The reaction begins with the hydrogenation of CO.Then, according to the Figure 5 and Table 4, the pathways passing through HCO are the preferred route because the activation energy barrier for CO + H→COH is about 0.8 eV larger than for CO + H → HCO.In the next step, HCO is hydrogenated to HCOH isomers (HCO + H → cis-HCOH and HCO + H → trans-HCOH).Although these two reactions are kinetically the same, thermodynamically, cis-HCOH is the preferred intermediate, resulting in the reaction sequences CH + H → CH 2 , CH 2 + H→ CH 3 and CH 3 + H → CH 4 being the most favorable route for CH 4 production (Figure 5).This pathway is completely exothermic, with an overall reaction energy of −1.53 eV, and the hydrogenation of cis-HCOH + H → CH + H 2 O is the most favorable reaction in this pathway, with an energy of −0.773 eV.In contrast, the least favourable reaction is HCO + H → cis-HCOH, which requires 1.393 eV of energy.Kinetically, the hydrogenation of CH + H → CH 2 is the optimum reaction in this pathway, with an energy barrier of only 0.065 eV.Table 6.Four paths resulting in C2H2.Energies are related to the CO in the gas phase plus the energy of one hydrogen atom and one CH2 on the surface.The symbols in parentheses are the atoms added along the paths to balance the number of atoms during reaction profile.The kinetic and thermodynamic outcomes for the byproducts indicate that the favored pathways resulting in CH 3 OH and C 2 H 2 (presented in Tables 5 and 6) have the same intermediates in the first two steps, beginning with CO + H → HCO and then passing through HCO + H → cis-HCOH.In the production of CH 3 OH, the cis-HCOH + H → CH 2 OH reac-tion costs 0.39 eV, with a barrier of 0.662 eV, followed by the reaction of CH 2 OH + H → CH 3 OH, which requires 0.089 eV and must overcome a barrier of 0.725 eV to proceed.Cis-HCOH is hydrogenated to C 2 H 2 by the reaction sequences of cis-HCOH + H → CH + H 2 O and CH + CH → C 2 H 2 , with reaction energies of −0.773 and 1.511 eV and energy barriers of 0.066 and 1.556 eV (Figures 6 and 7).The optimum pathways for the production of CH 3 OH and C 2 H 2 are entirely endothermic, with overall reaction energies of 0.563, and 0.991 eV, respectively.The most favorable reaction in the production of C 2 H 2 is cis-HCOH to CH + H 2 O, with an energy of −0.773 eV, and in the CH 3 OH formation, it is the hydrogenation of cis-HCOH to CH 2 OH, with an energy of 0.39 eV.

Path
Table 5.Four paths resulting in CH 3 OH.Energies are related to the CO in the gas phase plus the energy of four hydrogen atoms on the surface.The symbols in parentheses are the atoms added along the paths to balance the number of atoms during reaction profile.Based on kinetic and thermodynamic considerations, path C appears the most likely pathway, shown in Figure 8, leading to CH 4 as the product.The selectivity of this pathway can also be attributed to its lower activation energy and higher thermodynamic stability, which favors the formation of the CH 4 .Based on kinetic and thermodynamic considerations, path C appears the most likely pathway, shown in Figure 8, leading to CH4 as the product.The selectivity of this pathway can also be attributed to its lower activation energy and higher thermodynamic stability, which favors the formation of the CH4.

Methods
Periodic plane-wave density functional theory (DFT) [76][77][78] calculations were carried out to study the CO adsorption and its reactivity with adsorbed hydrogen on the Co (001) surface.All parts of this study employed the Vienna Ab initio Simulation Package (VASP) [79][80][81].To determine the electronic ground state, VASP makes use of efficient iterative matrix diagonalization techniques, computing an approximate solution to the many-body Schrödinger equation.Ion-electron interactions were represented by the Projector-Augmented-Wave (PAW) method [82,83].The total energy calculations were performed using the Perdew−Burke−Ernzerhof (PBE) [84] form of the Generalized Gradient Approximation (GGA).The inclusion of the long-range Van der Waals (vdW) forces improved the energy description of each system, and we therefore employed the DFT-D3 method of Grimme, as implemented in VASP [85].The widths of the smearing and the global break condition for the electronic SC-loop during structure relaxations were set to 0.2 eV/Å and 10 −4 eV, respectively.The electron wave functions were expanded using plane waves with a cutoff energy of 450 eV for the cobalt bulk and surface structure.The energies of the transition states (TSs) were calculated using the nudged elastic band (NEB) [52,86] and dimer methods [53,54], implemented in VASP to increase the potential energy surface from minimum to saddle points [53,54,[87][88][89].The KPOINTS file specified the Bloch vectors (k-points) used to sample the Brillouin zone.Converging this sampling is one of the essential tasks in many calculations concerning the electronic minimization [90].Finally, 6 × 6 × 6 and 7 × 7 × 1 Monkhorst pack grids of k-points were used to sample the Brillouin zone in the bulk and Co (001) surface, respectively.

Model
Metallic cobalt can crystallize in two different crystal structures: a hexagonal closedpacked (hcp) structure and a face-centered cubic (fcc) structure [91][92][93][94].These two phases possess similar energetic stabilities; hence, small temperature or pressure variations give rise to changes in the crystal phase.This similar stability also renders theoretical predictions difficult for either the bulk or nanoparticles [91].The fcc bulk crystal structure of cobalt was selected in this study, and we investigate the Fischer-Tropsch synthesis mechanism on its (001) plane.Figure 9 shows the primitive cell in the fcc crystal system with a lattice parameter of 3.42 Å, which was downloaded from MaterialsProject [95] with the name "Co_mp-102".According to the literature, the Fischer-Tropsch synthesis of carbon monoxide hydrogenation over hcp cobalt goes through the direct dissociation of the C-O bond, while H-assisted dissociation of the C-O is the preferred mechanism over fcc cobalt [96].In this study of the fcc Co (001) plane, we investigated H-assisted CO dissociation, followed by further hydrogenation reactions.
Metallic cobalt can crystallize in two different crystal structu packed (hcp) structure and a face-centered cubic (fcc) structure [9 possess similar energetic stabilities; hence, small temperature or rise to changes in the crystal phase.This similar stability also re tions difficult for either the bulk or nanoparticles [91].The fcc cobalt was selected in this study, and we investigate the Fischeranism on its (001) plane.Figure 9 shows the primitive cell in the lattice parameter of 3.42 Å, which was downloaded from Mater name "Co_mp-102".According to the literature, the Fischer-Tro monoxide hydrogenation over hcp cobalt goes through the direc bond, while H-assisted dissociation of the C-O is the preferred m [96].In this study of the fcc Co (001) plane, we investigated H-a followed by further hydrogenation reactions.The optimized bulk structure was cleaved to obtain the (001) surface using Materials studio [59].The supercell was expanded to 3 × 3 × 1 with dimensions of 6.841 × 6.841 × 21.841 Å to ensure we had enough space for the adsorption of the molecules on the surface.Slabs with different thicknesses of three to eight layers were created, and their surface energies, E sur f , were calculated using Equation ( 1): where A is the cross-sectional area of the surface slab, E tot (bulk) refers to the energy of a unit cell of the bulk metal per atom, E tot (slab) is the total energy of the slab, and n denotes the number of atoms in the slab.The surface energy was calculated for all slabs with different thicknesses.According to Table 7, the five-layer slab of cobalt atoms, shown in Figure 10, converged sufficiently and offered the optimum balance between a sufficient number of layers to enable surface relaxation and speed of calculation.The calculated surface energy for the Co (001) surface (Table 7) agrees well with values in the literature (3.40 J/m 2 ) [97].The adsorption energy of the adsorbates (Eads) can be calculated using

𝐸 = 𝐸 − (𝐸 + 𝐸 )
where  is the energy of the relaxed molecule on the relaxed surfa lowest energy of the optimized molecule in a vacuum, and  is the tota relaxed surface.

Conclusions
Calculations based on the density functional theory were employed conversion of a mixture of hydrogen and carbon monoxide into hydrocarbo (001) surface, which has provided valuable insights into the mechanism of C tion over the Co (001) surface in Fischer-Tropsch synthesis.We also identifi To determine the optimum number of relaxed layers in the slab, we examined slabs with different numbers of layers, from one to four, that were allowed to relax unrestrainedly, while the rest of the layers were fixed in their bulk positions.The surface energy was then calculated as: where E slab,relaxed is the energy of the slab with a number of relaxed and fixed layers, whereas E slab,unrelaxed is the energy of the fixed-layer slab.Table 8 shows the relaxed surface energies with respect to the number of fixed layers for different slabs.According to the results, the surface energy converged for the slab with two fixed and three relaxed layers, i.e., apart from the constrained bottom two layers of the slab, all atoms were allowed to relax explicitly upon optimization.The vacuum space was introduced on top of the slab to avoid interactions between the slab images in the Z direction of the cell.The adsorption energy of the adsorbates (E ads ) can be calculated using Equation ( 3): where E slab+mol is the energy of the relaxed molecule on the relaxed surface, E mol is the lowest energy of the optimized molecule in a vacuum, and E slab is the total energy of the relaxed surface.The reaction (E reaction ) and activation (E activation ) energies of each reaction can be calculated using Equations ( 4) and ( 5), respectively: E activation = E transitionstate − E reactant (5)

Conclusions
Calculations based on the density functional theory were employed to unravel the conversion of a mixture of hydrogen and carbon monoxide into hydrocarbons over the Co (001) surface, which has provided valuable insights into the mechanism of CO hydrogenation over the Co (001) surface in Fischer-Tropsch synthesis.We also identified several key intermediates and transition states involved in the elementary reactions, which can be used to guide the design of more efficient and selective Co-based catalysts for industrial applications.
The calculated adsorption energies of different intermediates on the Co (001) surface show that the CH 4 and CH 3 OH products adsorbed with energies of −0.204 and −0.718 eV at hollow and top positions, respectively, whereas C 2 H 2 adsorbed with an energy of −3.241 eV in a hollow site, confirming that CH 4 and CH 3 OH are more easily desorbed from the Co (001) surface than C 2 H 2 .
The formation of methane, methanol, and acetylene was found to proceed via the hydrogenation of the carbon end of CO to HCO, followed by hydrogenation to cis-HCOH, and via the CH 2 OH intermediate to methanol, which can then be further hydrogenated to methane and acetylene through CH intermediates.The preferred mechanism resulting in CH 4 as the favored product begins with the reactions CO + H→HCO and HCO + H→cis-HCOH, followed by cis-HCOH + H→CH + H 2 O. Next, CH is hydrogenated to CH 4 along the reactions CH + H→CH 2 , CH 2 + H→CH 3 and CH 3 + H→CH 4 .For the other products, the preferred mechanisms are the same until cis-HCOH formation, whence CH 3 OH is produced through the CH 2 OH intermediate, and C 2 H 2 results from the reaction between two CHs that were produced through the hydrogenation of cis-HCOH.The optimum pathways for CH 4 , CH 3 OH, and C 2 H 2 production proceed with overall energies of −1.53, 0.563, and 0.991 eV, respectively.The reaction HCO + H→cis-HCOH, with an activation energy of 1.746 eV, has the highest energy barrier in the selected pathways.
We consider that this study has provided important understanding of the catalytic processes involved in the hydrogenation of carbon monoxide over the Co (001) surface with implications for Fischer-Tropsch synthesis.

Figure 2 .
Figure 2. The adsorption geometries of the studied molecules during CO hydrogenation on the Co (001) surface.

Figure 3 .
Figure 3. Side views of the lowest-energy co-adsorption configurations of each intermediate and H on Co (001) surface.The gray, white, red, and blue balls represent carbon, hydrogen, oxygen, and cobalt atoms, respectively.

Figure 3 .
Figure 3. Side views of the lowest-energy co-adsorption configurations of each intermediate and H on Co (001) surface.The gray, white, red, and blue balls represent carbon, hydrogen, oxygen, and cobalt atoms, respectively.The transformation from CO to form HCO has a reaction energy of about 0.3 eV less than the reaction energy needed to form COH, with the reaction energy for CO + H → COH calculated at 0.853 eV.This is in perfect agreement with the work by Zhang et al. [68], who obtained a reaction energy of 0.85 eV to produce COH from the hydrogenation of CO over the InZr 3 surface.The other intermediate produced from the reaction between CO and H is HCO, with the reaction of CO + H → HCO requiring an energy of 0.574 eV to proceed.Hirunsit [69] reported a reaction energy of about 0.53 eV for the same reaction, which is very close to our result.Following this initial CO hydrogenation, both COH and HCO can react with hydrogen, which results in two different isomers of HCOH.The reaction energies of COH + H → cis-HCOH and COH + H → trans-HCOH are 0.858 and 1.004 eV, respectively, whereas the reaction energies for HCO + H → cis-HCOH and HCO + H→ trans-HCOH differ by 0.2 eV, i.e., to produce cis-HCOH from HCO by HCO + H → cis-HCOH, the reaction energy is 1.393 eV, while it is 1.539 eV for the reaction HCO + H → trans-HCOH.The migration of H to the nearby HCOH isomers can lead to either HCOH hydrogenation to form CH 2 OH or HCOH dissociation to form CH and H 2 O.The former reactions, from either cis-HCCOH + H → CH 2 OH or trans-HCCOH + H → CH 2 OH, have reaction energies of 0.39 eV and −0.12 eV, respectively, while the dissociation reactions of cis-HCCOH + H → CH + H 2 O and trans-HCCOH + H → CH + H 2 O are exothermic, with reaction energies of −0.773 and −1.283 eV, respectively.

Figure 4 .
Figure 4.The reactions with their corresponding transition states on the Co (001) surface.

Figure 4 .
Figure 4.The reactions with their corresponding transition states on the Co (001) surface.

Figure 6 .
Figure 6.Four reaction pathways illustrating how the energies change in each path resulting in CH3OH production.Figure 6.Four reaction pathways illustrating how the energies change in each path resulting in CH 3 OH production.

Figure 6 .
Figure 6.Four reaction pathways illustrating how the energies change in each path resulting in CH3OH production.Figure 6.Four reaction pathways illustrating how the energies change in each path resulting in CH 3 OH production.

Figure 7 .
Figure 7. Four reaction pathways illustrating how the energies change in each path resulting in C2H2 production.

Figure 7 .
Figure 7. Four reaction pathways illustrating how the energies change in each path resulting in C 2 H 2 production.

Table 4 .
Four paths resulting in CH 4 .Energies are related to the CO in the gas phase plus the energy of four hydrogen atoms on the surface.The symbols in parentheses are the atoms added along the paths to balance the number of atoms during reaction profile.

Figure 8 .
Figure 8.The most favorable reaction pathway to reach CH4.

Figure 8 .
Figure 8.The most favorable reaction pathway to reach CH 4 .

:
Side views of the lowest-energy adsorption configurations of each intermediate on Co (001) surface.The bond lengths are shown in Å.The gray, white, red, and blue balls represent carbon, hydrogen, oxygen, and cobalt atoms, respectively.;Table S1: The structural details of the preferred adsorption geometries for all intermediates on the Co (001) surface.Author Contributions: Conceptualization, S.S.T. and N.H.d.L.; methodology, S.S.T.; software, S.S.T. and M.T.; validation, S.S.T. and N.H.d.L.; formal analysis, S.S.T. and N.H.d.L.; investigation, M.T.; resources, N.H.d.L.; data curation, M.T.; writing-original draft preparation, M.T.; writing-review and editing, S.S.T. and N.H.d.L.; visualization, M.T.; supervision, S.S.T. and N.H.d.L.; project administration, S.S.T.; funding acquisition, S.S.T. and N.H.d.L.All authors have read and agreed to the published version of the manuscript.

Table 1 .
The preferred adsorption geometries and energies for all intermediates on the Co (001) surface.

Table 2 .
Calculated reaction energies for all hydrogenation elementary reactions on the Co (001) surface.

Table 6 .
Four paths resulting in C 2 H 2 .Energies are related to the CO in the gas phase plus the energy of one hydrogen atom and one CH 2 on the surface.The symbols in parentheses are the atoms added along the paths to balance the number of atoms during reaction profile.

Table 8 .
Calculated relaxed surface energies in (J⁄m 2 ) for Co (001) slabs with the different number of fixed layers.