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

Abstract

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 CH₄, CH₃OH and C₂H₂ compounds, where the calculations suggest that CH₄ 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 CH₄ 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.

1. 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, Zhang et al. [31] studied the hydrogenation mechanism of carbon dioxide and carbon monoxide over Ru(0001), where they found that during CO hydrogenation, CO may dissociate via either a COH or CHO intermediate, resulting in C and CH species, respectively [31]. A broad array of palladium catalytic systems, mainly based on Pd salts and complexes in the presence of a base, are currently employed as efficient, chemoselective and productive homogeneous or heterogeneous catalysts to promote C–C cross-coupling reactions [12,32,33,34,35,36,37,38,39,40,41,42]. The suppression of methane production through optimization of the physical properties of Fe has allowed Hirsa and co-workers to further understand and develop the performance of iron-based catalysts [43]. Iron has a high water–gas shift activity and is therefore suitable for syngas feedstocks of a low H₂/CO ratio, such as those derived from coal gasification [44].

Cobalt is generally preferred over Fe and Ru for FTS as it possesses high activity and selectivity in the production of long-chain hydrocarbons from syngas [45,46,47,48]. Ge et al. [30] reported a density functional theory study which was used to analyze the first steps in the mechanism of Fischer–Tropsch synthesis, i.e., CO adsorption and activation over the close-packed {0001}, corrugated {$11\stackrel{-}{2}0$} and stepped {$10\stackrel{-}{1}2$} and {$11\stackrel{-}{2}4$} Co surfaces. The adsorption energy of CO tends to increase as the CO coverage is reduced. If chemisorbed CO is used as the reference state, the reaction on Co {0001} and {$11\stackrel{-}{2}0$} becomes endothermic whereas it remains exothermic on Co {10$\stackrel{-}{1}$2} and {$11\stackrel{-}{2}4$}. On Co {$10\stackrel{-}{1}2$} and {$11\stackrel{-}{2}4$}, 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 C₂ to C₆ 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 C₂ 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 CH₄ is the main product, both thermodynamically and kinetically [54].

2. Results and Discussion

2.1. Thermodynamic Analysis

2.1.1. Adsorption of Molecules

All molecule structures were downloaded from PubChem [55,56,57] at the National Center for Biotechnology Information [58]. They were then edited and designed through Materials Studio [59], VESTA [60], and P4vasp [61,62].

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.

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 −7.964, −5.611, −2.972 and −0.204 eV, respectively. For comparison, the adsorption energies of CH and CH₂ on the Pt (111) surface were calculated by Psofogiannakis et al. at −7.55 and −4.56 eV, respectively, whereas the adsorption of CH₃ at the surface was calculated at −2.40 eV [63], and the adsorption energy for CH₄ on the Ru (0001) surface was calculated at −0.17 eV by Zhang et al. [31].

H₂O 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. CH₂OH 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 CH₂OH 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 CH₂OH on the Pt (111) surface.

The oxygen atom of CH₃OH 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 C₂H₂ 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 C₂H₂ 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: CH ˃ COH ˃ CH₂ ˃ HCO ˃ cis-HCOH ˃ trans-HCOH ˃ C₂H₂ ˃ CH₃ ˃ CH₂OH ˃ CO ˃ H₂O ˃ CH₃OH ˃ CH₄.

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

Species	Site, Atom, Bond Length (Å)	Eads (eV)	Eads in Literature (eV)
CO	hollow, carbon, 2.03	−2.268	−2.34(Pt(111)) [63], −1.91 (Ni(110)) [64], −2.00 (Fe(100)) [65], −1.92 (Ni(111)) [66]
COH	hollow, carbon, 1.960	−5.696	−5.64(Pt(111)) [63], −4.01 (Ni(110)) [64], −6.21(Fe(100)) [65],
HCO	hollow, carbon,1.917	−4.03	−2.60 (Ni(110)) [64], −6.49 (Fe(100)) [65]
Cis-HCOH	hollow, carbon,1.970	−3.919	−3.51 (Ni(110)) [64], −4.04 (Fe(100)) [65]
Trans-HCOH	bridge, carbon, 1.930	−3.583	−3.25 Ni(110) [64], −4.04 (Fe(100)) [65]
CH	hollow, carbon, 1.942	−7.946	−6.43 (Ni(111)) [66], −7.55(Pt(111)) [63]
CH2	hollow, carbon, 2.091	−5.611	−4.01 (Ni(111)) [66], −4.56(Pt(111)) [63]
CH3	hollow, carbon, 1.967	−2.972	−2.40(Pt(111)) [63]
CH4	hollow, carbon, 3.645	−0.204	−0.17 (Ru(0001) [31]
H2O	top, oxygen, 2.141	−0.746	−0.29 (Ni(111)) [66]
CH2OH	hollow, oxygen, 2.110 carbon, 1.941	−2.74	−2.79(Pt(111)) [63], −1.68 (Ni(110)) [64]
CH3OH	top, oxygen, 2.117	−0.718	−0.45 (Ni(110)) [64]
C2H2	hollow, carbon, 1.843	−3.241	−2.957 (Ni(111)) [67]

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

Species	Site, Atom, Bond Length (Å)	Eads (eV)	Eads in Literature (eV)
CO	hollow, carbon, 2.03	−2.268	−2.34(Pt(111)) [63], −1.91 (Ni(110)) [64], −2.00 (Fe(100)) [65], −1.92 (Ni(111)) [66]
COH	hollow, carbon, 1.960	−5.696	−5.64(Pt(111)) [63], −4.01 (Ni(110)) [64], −6.21(Fe(100)) [65],
HCO	hollow, carbon,1.917	−4.03	−2.60 (Ni(110)) [64], −6.49 (Fe(100)) [65]
Cis-HCOH	hollow, carbon,1.970	−3.919	−3.51 (Ni(110)) [64], −4.04 (Fe(100)) [65]
Trans-HCOH	bridge, carbon, 1.930	−3.583	−3.25 Ni(110) [64], −4.04 (Fe(100)) [65]
CH	hollow, carbon, 1.942	−7.946	−6.43 (Ni(111)) [66], −7.55(Pt(111)) [63]
CH2	hollow, carbon, 2.091	−5.611	−4.01 (Ni(111)) [66], −4.56(Pt(111)) [63]
CH3	hollow, carbon, 1.967	−2.972	−2.40(Pt(111)) [63]
CH4	hollow, carbon, 3.645	−0.204	−0.17 (Ru(0001) [31]
H2O	top, oxygen, 2.141	−0.746	−0.29 (Ni(111)) [66]
CH2OH	hollow, oxygen, 2.110 carbon, 1.941	−2.74	−2.79(Pt(111)) [63], −1.68 (Ni(110)) [64]
CH3OH	top, oxygen, 2.117	−0.718	−0.45 (Ni(110)) [64]
C2H2	hollow, carbon, 1.843	−3.241	−2.957 (Ni(111)) [67]

2.1.2. 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 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.

2.1.3. 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 of CO over the InZr₃ 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 CH₂OH or HCOH dissociation to form CH and H₂O. The former reactions, from either cis-HCCOH + H → CH₂OH or trans-HCCOH + H → CH₂OH, have reaction energies of 0.39 eV and −0.12 eV, respectively, while the dissociation reactions of cis-HCCOH + H → CH + H₂O and trans-HCCOH + H → CH + H₂O are exothermic, with reaction energies of −0.773 and −1.283 eV, respectively.

After the steps above, there are three ways to reach the desired products; first, the hydrogenation of CH₂OH to CH₃OH, second, a reaction between two CH species to produce acetylene (C₂H₂), and finally, the production of CH₄ through a three-step hydrogenation reaction of CH→CH₂→CH₃→CH₄. These three steps consist of the reactions: CH + H → CH₂, CH₂ + H → CH₃ and CH₃ + H → CH₄. The hydrogenation of CH₂OH + H→CH₃OH 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 CH₄ requires reaction energies for CH + H → CH₂, CH + H → CH₃ and CH + H → CH₄ 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 + H₂O 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 CH₂OH and CH + H₂O, respectively.

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

Reaction	Ereaction (eV)	Ereaction (eV) in Literature
$CO+H\to COH$	0.853	0.85(InZr3(110)) [68], 1.04(PdCu3(111)) [70]
$CO+H\to HCO$	0.574	0.80(Ni(110)) [64], 0.53(Cu3Ag(211)) [69], 0.75(Cu(211)) [69]
$COH+H\to cis-HCOH$	0.858	−0.37(PdCu3(111)) [70], 0.14(Cu(111)) [71]
$COH+H\to trans-HCOH$	1.004
$HCO+H\to cis-HCOH$	1.393
$HCO+H\to trans-HCOH$	1.539
$cis-HCOH+H\to CH+H_{2}O$	−0.773
$cis-HCOH+H\to C{H}_{2}OH$	0.39	0.01(PdCu3(111)) [70], 0.84(Cu(111)) [71]
$trans-HCOH+H\to CH+H_{2}O$	−1.283
$trans-HCOH+H\to C{H}_{2}OH$	−0.12	0.01(PdCu3(111)) [70], 0.77(Cu(111)) [71]
$CH+H\to C{H}_2$	0.019	0.35(InZr3(110)) [68]
$C{H}_2+H\to C{H}_3$	0.679	0.36(InZr3(110)) [68]
$C{H}_3+H\to C{H}_4$	0.491
$C{H}_{2}OH+H\to C{H}_{3}OH$	0.089	0.49(Ni(110)) [64], 0.90(Cu(111)) [71]
$CH+CH\to C_2{H}_2$	1.511

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

Reaction	Ereaction (eV)	Ereaction (eV) in Literature
$CO+H\to COH$	0.853	0.85(InZr3(110)) [68], 1.04(PdCu3(111)) [70]
$CO+H\to HCO$	0.574	0.80(Ni(110)) [64], 0.53(Cu3Ag(211)) [69], 0.75(Cu(211)) [69]
$COH+H\to cis-HCOH$	0.858	−0.37(PdCu3(111)) [70], 0.14(Cu(111)) [71]
$COH+H\to trans-HCOH$	1.004
$HCO+H\to cis-HCOH$	1.393
$HCO+H\to trans-HCOH$	1.539
$cis-HCOH+H\to CH+H_{2}O$	−0.773
$cis-HCOH+H\to C{H}_{2}OH$	0.39	0.01(PdCu3(111)) [70], 0.84(Cu(111)) [71]
$trans-HCOH+H\to CH+H_{2}O$	−1.283
$trans-HCOH+H\to C{H}_{2}OH$	−0.12	0.01(PdCu3(111)) [70], 0.77(Cu(111)) [71]
$CH+H\to C{H}_2$	0.019	0.35(InZr3(110)) [68]
$C{H}_2+H\to C{H}_3$	0.679	0.36(InZr3(110)) [68]
$C{H}_3+H\to C{H}_4$	0.491
$C{H}_{2}OH+H\to C{H}_{3}OH$	0.089	0.49(Ni(110)) [64], 0.90(Cu(111)) [71]
$CH+CH\to C_2{H}_2$	1.511

2.2. Analysis of the Kinetics

2.2.1. 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 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 + H₂O and CH₂OH from the cis and trans conformers of HCOH. The barriers for the production of CH₂OH 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 → CH₂OH on the Co (0001) surface. However, there is a large difference between the energy barriers to be overcome for the dissociation into CH + H₂O products from cis-HCOH or trans-HCOH in the presence of hydrogen. cis-HCOH dissociation produces CH + H₂O with a barrier of just 0.066 eV, but the production of CH + H₂O from trans-HCOH requires an activation energy of 1.581 eV. These results show that CH + H₂O and CH₂OH 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 C₂H₂ from the reaction of CH + CH on the Co (001) surface is 1.556 eV, whereas the production of CH₃OH from the reaction of CH₂OH + H has a barrier of 0.725 eV. As discussed above, CH can be produced from either reaction of cis-HCOH + H → CH + H₂O or trans-HCOH + H → CH + H₂O, followed by further reaction with adsorbed hydrogens to produce CH₄ along three continuous reaction steps: CH + H, CH₂ + H, and CH₃ + 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), Fe₅C₂ (100), and Pt (111) surfaces, respectively.

Table 3. Calculated activation energies (E_a) for all reactions on Co (001) surface.

Reaction	Ea (eV)	Ea (eV) In literature
$CO+H\to COH$	1.804	1.55(Co(0001)) [72], 1.07(Fe(100)) [65], 1.97(Ni(111)) [66]
$CO+H\to HCO$	1.082	1.08(Ni(110)) [64]
$COH+H\to cis-HCOH$	1.231	1.38(Fe(100)) [65], 1.522(Pt(111)) [74]
$COH+H\to trans-HCOH$	1.231	1.38(Fe(100)) [65], 1.522(Pt(111)) [74]
$HCO+H\to cis-HCOH$	1.746	1.59(Co(0001)) [75]
$HCO+H\to trans-HCOH$	1.727	1.59(Co(0001)) [75]
$cis-HCOH+H\to CH+H_{2}O$	0.066
$cis-HCOH+H\to C{H}_{2}OH$	0.662	0.71(Co(0001)) [72], 0.43(Co(0001)) [75]
$trans-HCOH+H\to CH+H_{2}O$	1.581
$trans-HCOH+H\to C{H}_{2}OH$	0.131	0.71(Co(0001)) [72], 0.43(Co(0001)) [75]
$CH+CH\to C_2{H}_2$	1.556	0.77(Ru(0001) [31]
$CH+H\to C{H}_2$	0.065	0.69(Ni(111)) [66]
$C{H}_2+H\to C{H}_3$	0.969	0.81(Fe5C2(100)) [73], 1.360(Pt(111)) [74]
$C{H}_3+H\to C{H}_4$	1.089	1.187(Pt(111)) [74], 0.96(Fe5C2(100)) [73], 0.90(Ni(111)) [66]
$C{H}_{2}OH+H\to C{H}_{3}OH$	0.725	1.04(Ni(110)) [64], 0.69(Ni(111)) [66], 0.82(Co(0001)) [75]

Table 3. Calculated activation energies (E_a) for all reactions on Co (001) surface.

Reaction	Ea (eV)	Ea (eV) In literature
$CO+H\to COH$	1.804	1.55(Co(0001)) [72], 1.07(Fe(100)) [65], 1.97(Ni(111)) [66]
$CO+H\to HCO$	1.082	1.08(Ni(110)) [64]
$COH+H\to cis-HCOH$	1.231	1.38(Fe(100)) [65], 1.522(Pt(111)) [74]
$COH+H\to trans-HCOH$	1.231	1.38(Fe(100)) [65], 1.522(Pt(111)) [74]
$HCO+H\to cis-HCOH$	1.746	1.59(Co(0001)) [75]
$HCO+H\to trans-HCOH$	1.727	1.59(Co(0001)) [75]
$cis-HCOH+H\to CH+H_{2}O$	0.066
$cis-HCOH+H\to C{H}_{2}OH$	0.662	0.71(Co(0001)) [72], 0.43(Co(0001)) [75]
$trans-HCOH+H\to CH+H_{2}O$	1.581
$trans-HCOH+H\to C{H}_{2}OH$	0.131	0.71(Co(0001)) [72], 0.43(Co(0001)) [75]
$CH+CH\to C_2{H}_2$	1.556	0.77(Ru(0001) [31]
$CH+H\to C{H}_2$	0.065	0.69(Ni(111)) [66]
$C{H}_2+H\to C{H}_3$	0.969	0.81(Fe5C2(100)) [73], 1.360(Pt(111)) [74]
$C{H}_3+H\to C{H}_4$	1.089	1.187(Pt(111)) [74], 0.96(Fe5C2(100)) [73], 0.90(Ni(111)) [66]
$C{H}_{2}OH+H\to C{H}_{3}OH$	0.725	1.04(Ni(110)) [64], 0.69(Ni(111)) [66], 0.82(Co(0001)) [75]

2.2.2. Reaction Pathways

The energies of the reaction routes to methanol, methane and acetylene production on the surface are shown in Figure 5, Figure 6 and Figure 7, respectively. For each product, there are four pathways (Table 4, Table 5 and Table 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.

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.

Table 4. Four paths resulting in CH₄. 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.

Path A		Path B		Path C		Path D
State	E (eV)	State	E (eV)	State	E (eV)	State	E (eV)
CO(g)+(4H)	0.00	CO(g)+(4H)	0.00	CO(g)+(4H)	0.00	CO(g)+(4H)	0.00
CO+H+(3H)	−2.538	CO+H+(3H)	−2.538	CO+H+(3H)	−2.538	CO+H+(3H)	−2.538
TS1+(3H)	−0.734	TS1+(3H)	−0.734	TS7+(3H)	−1.456	TS7+(3H)	−1.456
COH+(3H)	−1.685	COH+(3H)	−1.685	HCO+(3H)	−1.964	HCO+(3H)	−1.964
COH+H+(2H)	−1.636	COH+H+(2H)	−1.636	HCO+H+(2H)	−2.173	HCO+H+(2H)	−2.173
TS2+(2H)	−0.405	TS5+(2H)	−0.405	TS8+(2H)	−0.425	TS9+(2H)	−0.444
cis-HCOH+(2H)	−0.778	trans-HCOH+(2H)	−0.632	cis-HCOH+(2H)	−0.778	trans-HCOH+(2H)	−0.632
cis-HCOH+H+(H)	−0.773	trans-HCOH+H+(H)	−0.263	cis-HCOH+H+(H)	−0.773	trans-HCOH+H+(H)	−0.263
TS10+(H)	−0.707	TS14+(H)	1.318	TS10+(H)	−0.707	TS14+(H)	1.318
CH+H2O+(H)	−1.546	CH+H2O+(H)	−1.546	CH+H2O+(H)	−1.546	CH+H2O+(H)	−1.546
CH+H+(H2O)	−0.991	CH+H+(H2O)	−0.991	CH+H+(H2O)	−0.991	CH+H+(H2O)	−0.991
TS11+(H2O)	−0.93	TS11+(H2O)	−0.93	TS11+(H2O)	−0.93	TS11+(H2O)	−0.93
CH2+(H2O)	−0.976	CH2+(H2O)	−0.976	CH2+(H2O)	−0.976	CH2+(H2O)	−0.976
CH2+H+(OH)	−2.249	CH2+H+(OH)	−2.249	CH2+H+(OH)	−2.249	CH2+H+(OH)	−2.249
TS12+(OH)	−1.28	TS12+(OH)	−1.28	TS12+(OH)	−1.28	TS12+(OH)	−1.28
CH3+(OH)	−1.57	CH3+(OH)	−1.57	CH3+(OH)	−1.57	CH3+(OH)	−1.57
CH3+H+(O)	−2.225	CH3+H+(O)	−2.225	CH3+H+(O)	−2.225	CH3+H+(O)	−2.225
TS13+(O)	−1.136	TS13+(O)	−1.136	TS13+(O)	−1.136	TS13+(O)	−1.136
CH4+(O)	−1.734	CH4+(O)	−1.734	CH4+(O)	−1.734	CH4+(O)	−1.734
CH4(g)+(O)	−1.53	CH4(g)+(O)	−1.53	CH4(g)+(O)	−1.53	CH4(g)+(O)	−1.53

Table 4. Four paths resulting in CH₄. 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.

Path A		Path B		Path C		Path D
State	E (eV)	State	E (eV)	State	E (eV)	State	E (eV)
CO(g)+(4H)	0.00	CO(g)+(4H)	0.00	CO(g)+(4H)	0.00	CO(g)+(4H)	0.00
CO+H+(3H)	−2.538	CO+H+(3H)	−2.538	CO+H+(3H)	−2.538	CO+H+(3H)	−2.538
TS1+(3H)	−0.734	TS1+(3H)	−0.734	TS7+(3H)	−1.456	TS7+(3H)	−1.456
COH+(3H)	−1.685	COH+(3H)	−1.685	HCO+(3H)	−1.964	HCO+(3H)	−1.964
COH+H+(2H)	−1.636	COH+H+(2H)	−1.636	HCO+H+(2H)	−2.173	HCO+H+(2H)	−2.173
TS2+(2H)	−0.405	TS5+(2H)	−0.405	TS8+(2H)	−0.425	TS9+(2H)	−0.444
cis-HCOH+(2H)	−0.778	trans-HCOH+(2H)	−0.632	cis-HCOH+(2H)	−0.778	trans-HCOH+(2H)	−0.632
cis-HCOH+H+(H)	−0.773	trans-HCOH+H+(H)	−0.263	cis-HCOH+H+(H)	−0.773	trans-HCOH+H+(H)	−0.263
TS10+(H)	−0.707	TS14+(H)	1.318	TS10+(H)	−0.707	TS14+(H)	1.318
CH+H2O+(H)	−1.546	CH+H2O+(H)	−1.546	CH+H2O+(H)	−1.546	CH+H2O+(H)	−1.546
CH+H+(H2O)	−0.991	CH+H+(H2O)	−0.991	CH+H+(H2O)	−0.991	CH+H+(H2O)	−0.991
TS11+(H2O)	−0.93	TS11+(H2O)	−0.93	TS11+(H2O)	−0.93	TS11+(H2O)	−0.93
CH2+(H2O)	−0.976	CH2+(H2O)	−0.976	CH2+(H2O)	−0.976	CH2+(H2O)	−0.976
CH2+H+(OH)	−2.249	CH2+H+(OH)	−2.249	CH2+H+(OH)	−2.249	CH2+H+(OH)	−2.249
TS12+(OH)	−1.28	TS12+(OH)	−1.28	TS12+(OH)	−1.28	TS12+(OH)	−1.28
CH3+(OH)	−1.57	CH3+(OH)	−1.57	CH3+(OH)	−1.57	CH3+(OH)	−1.57
CH3+H+(O)	−2.225	CH3+H+(O)	−2.225	CH3+H+(O)	−2.225	CH3+H+(O)	−2.225
TS13+(O)	−1.136	TS13+(O)	−1.136	TS13+(O)	−1.136	TS13+(O)	−1.136
CH4+(O)	−1.734	CH4+(O)	−1.734	CH4+(O)	−1.734	CH4+(O)	−1.734
CH4(g)+(O)	−1.53	CH4(g)+(O)	−1.53	CH4(g)+(O)	−1.53	CH4(g)+(O)	−1.53

The kinetic and thermodynamic outcomes for the byproducts indicate that the favored pathways resulting in CH₃OH and C₂H₂ (presented in Table 5 and Table 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 (Figure 6 and Figure 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.

Table 5. Four paths resulting in CH₃OH. 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.

Path E		Path F		Path G		Path H
State	E (eV)	State	E (eV)	State	E (eV)	State	E (eV)
CO(g)+(4H)	0.00	CO(g)+(4H)	0.00	CO(g)+(4H)	0.00	CO(g)+(4H)	0.00
CO+H+(3H)	−2.538	CO+H+(3H)	−2.538	CO+H+(3H)	−2.538	CO+H+(3H)	−2.538
TS1+(3H)	−0.734	TS1+(3H)	−0.734	TS7+(3H)	−1.456	TS7+(3H)	−1.456
COH+(3H)	−1.685	COH+(3H)	−1.685	HCO+(3H)	−1.964	HCO+(3H)	−1.964
COH+H+(2H)	−1.636	COH+H+(2H)	−1.636	HCO+H+(2H)	−2.173	HCO+H+(2H)	−2.173
TS2+(2H)	−0.405	TS5+(2H)	−0.405	TS8+(2H)	−0.425	TS9+(2H)	−0.444
cis-HCOH+(2H)	−0.778	trans-HCOH+(2H)	−0.632	cis-HCOH+(2H)	−0.778	trans-HCOH+(2H)	−0.632
cis-HCOH+H+(H)	−0.773	trans-HCOH+H+(H)	−0.263	cis-HCOH+H+(H)	−0.773	trans-HCOH+H+(H)	−0.263
TS3+(H)	−0.111	TS6+(H)	−0.132	TS3+(H)	−0.111	TS6+(H)	−0.132
CH2OH+(H)	−0.383	CH2OH+(H)	−0.383	CH2OH+(H)	−0.383	CH2OH+(H)	−0.383
CH2OH+H	−0.244	CH2OH+H	−0.244	CH2OH+H	−0.244	CH2OH+H	−0.244
TS4	0.481	TS4	0.481	TS4	0.481	TS4	0.481
CH3OH	−0.155	CH3OH	−0.155	CH3OH	−0.155	CH3OH	−0.155
CH3OH (g)	0.563	CH3OH (g)	0.563	CH3OH (g)	0.563	CH3OH (g)	0.563

Table 5. Four paths resulting in CH₃OH. 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.

Path E		Path F		Path G		Path H
State	E (eV)	State	E (eV)	State	E (eV)	State	E (eV)
CO(g)+(4H)	0.00	CO(g)+(4H)	0.00	CO(g)+(4H)	0.00	CO(g)+(4H)	0.00
CO+H+(3H)	−2.538	CO+H+(3H)	−2.538	CO+H+(3H)	−2.538	CO+H+(3H)	−2.538
TS1+(3H)	−0.734	TS1+(3H)	−0.734	TS7+(3H)	−1.456	TS7+(3H)	−1.456
COH+(3H)	−1.685	COH+(3H)	−1.685	HCO+(3H)	−1.964	HCO+(3H)	−1.964
COH+H+(2H)	−1.636	COH+H+(2H)	−1.636	HCO+H+(2H)	−2.173	HCO+H+(2H)	−2.173
TS2+(2H)	−0.405	TS5+(2H)	−0.405	TS8+(2H)	−0.425	TS9+(2H)	−0.444
cis-HCOH+(2H)	−0.778	trans-HCOH+(2H)	−0.632	cis-HCOH+(2H)	−0.778	trans-HCOH+(2H)	−0.632
cis-HCOH+H+(H)	−0.773	trans-HCOH+H+(H)	−0.263	cis-HCOH+H+(H)	−0.773	trans-HCOH+H+(H)	−0.263
TS3+(H)	−0.111	TS6+(H)	−0.132	TS3+(H)	−0.111	TS6+(H)	−0.132
CH2OH+(H)	−0.383	CH2OH+(H)	−0.383	CH2OH+(H)	−0.383	CH2OH+(H)	−0.383
CH2OH+H	−0.244	CH2OH+H	−0.244	CH2OH+H	−0.244	CH2OH+H	−0.244
TS4	0.481	TS4	0.481	TS4	0.481	TS4	0.481
CH3OH	−0.155	CH3OH	−0.155	CH3OH	−0.155	CH3OH	−0.155
CH3OH (g)	0.563	CH3OH (g)	0.563	CH3OH (g)	0.563	CH3OH (g)	0.563

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.

Path I		Path J		Path K		Path L
state	E (eV)	state	E (eV)	State	E (eV)	state	E (eV)
CO(g)+(H)+(CH2)	0.00	CO(g)+(H)+(CH2)	0.00	CO(g)+(H)+(CH2)	0.00	CO(g)+(H)+(CH2)	0.00
CO+H+(CH2)	−2.538	CO+H+(CH2)	−2.538	CO+H+(CH2)	−2.538	CO+H+(CH2)	−2.538
TS1+(CH2)	−0.734	TS1+(CH2)	−0.734	TS7+(CH2)	−1.456	TS7+(CH2)	−1.456
COH+(CH2)	−1.685	COH+(CH2)	−1.685	HCO+(CH2)	−1.964	HCO+(CH2)	−1.964
COH+H+(CH)	−2.298	COH+H+(CH)	−2.298	HCO+H+(CH)	−2.833	HCO+H+(CH)	−2.833
TS2+(CH)	−1.067	TS5+(CH)	−1.077	TS8+(CH)	−1.087	TS9+(CH)	−1.106
cis-HCOH+(CH)	−1.44	trans-HCOH+(CH)	−1.294	cis-HCOH+(CH)	−1.44	trans-HCOH+(CH)	−1.294
cis-HCOH+H+(C)	−2.178	trans-HCOH+H+(C)	−1.668	cis-HCOH+H+(C)	−2.178	trans-HCOH+H+(C)	−1.668
TS10+(C)	−2.112	TS14+(C)	−0.087	TS10+(C)	−2.112	TS14+(C)	−0.087
CH+H2O+(C)	−2.951	CH+H2O+(C)	−2.951	CH+H2O+(C)	−2.951	CH+H2O+(C)	−2.951
CH+CH+(OH)	−3.761	CH+CH+(OH)	−3.761	CH+CH+(OH)	−3.761	CH+CH+(OH)	−3.761
TS15+(OH)	−2.205	TS15+(OH)	−2.205	TS15+(OH)	−2.205	TS15+(OH)	2.205
C2H2+(OH)	−2.25	C2H2+(OH)	−2.25	C2H2+(OH)	−2.25	C2H2+(OH)	−2.25
C2H2(g)+(OH)	0.991	C2H2(g)+(OH)	0.991	C2H2(g)+(OH)	0.991	C2H2(g)+(OH)	0.991

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.

Path I		Path J		Path K		Path L
state	E (eV)	state	E (eV)	State	E (eV)	state	E (eV)
CO(g)+(H)+(CH2)	0.00	CO(g)+(H)+(CH2)	0.00	CO(g)+(H)+(CH2)	0.00	CO(g)+(H)+(CH2)	0.00
CO+H+(CH2)	−2.538	CO+H+(CH2)	−2.538	CO+H+(CH2)	−2.538	CO+H+(CH2)	−2.538
TS1+(CH2)	−0.734	TS1+(CH2)	−0.734	TS7+(CH2)	−1.456	TS7+(CH2)	−1.456
COH+(CH2)	−1.685	COH+(CH2)	−1.685	HCO+(CH2)	−1.964	HCO+(CH2)	−1.964
COH+H+(CH)	−2.298	COH+H+(CH)	−2.298	HCO+H+(CH)	−2.833	HCO+H+(CH)	−2.833
TS2+(CH)	−1.067	TS5+(CH)	−1.077	TS8+(CH)	−1.087	TS9+(CH)	−1.106
cis-HCOH+(CH)	−1.44	trans-HCOH+(CH)	−1.294	cis-HCOH+(CH)	−1.44	trans-HCOH+(CH)	−1.294
cis-HCOH+H+(C)	−2.178	trans-HCOH+H+(C)	−1.668	cis-HCOH+H+(C)	−2.178	trans-HCOH+H+(C)	−1.668
TS10+(C)	−2.112	TS14+(C)	−0.087	TS10+(C)	−2.112	TS14+(C)	−0.087
CH+H2O+(C)	−2.951	CH+H2O+(C)	−2.951	CH+H2O+(C)	−2.951	CH+H2O+(C)	−2.951
CH+CH+(OH)	−3.761	CH+CH+(OH)	−3.761	CH+CH+(OH)	−3.761	CH+CH+(OH)	−3.761
TS15+(OH)	−2.205	TS15+(OH)	−2.205	TS15+(OH)	−2.205	TS15+(OH)	2.205
C2H2+(OH)	−2.25	C2H2+(OH)	−2.25	C2H2+(OH)	−2.25	C2H2+(OH)	−2.25
C2H2(g)+(OH)	0.991	C2H2(g)+(OH)	0.991	C2H2(g)+(OH)	0.991	C2H2(g)+(OH)	0.991

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.

3. Computational Detail

3.1. 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.

3.2. Model

Metallic cobalt can crystallize in two different crystal structures: a hexagonal closed-packed (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.

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, $E_{surf}$, were calculated using Equation (1):

$$E_{surf}=\frac{\left[E_{tot}\left(slab\right)-n{E}_{tot}\left(bulk\right)\right]}{2A}$$

(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. Calculated surface energies in (J⁄m²) for Co (001) slabs with different thicknesses.

Number of Layers	${\mathit{E}}_{\mathit{s}\mathit{u}\mathit{r}\mathit{f}}\left(\raisebox{1ex}{$\mathbf{J}$}\!\left/ \!\raisebox{-1ex}{${\mathbf{m}}^2$}\right.\right)$
3	3.590
4	3.647
5	3.639
6	3.626
7	3.639
8	3.632

, 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/m²) [97].

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:

$${\gamma }_r=\frac{E_{slab,relaxed}-n{E}_{bulk}}{A}-\frac{E_{slab,unrelaxed}-n{E}_{bulk}}{2A}$$

(2)

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. Calculated relaxed surface energies in (J⁄m²) for Co (001) slabs with the different number of fixed layers.

Number of the Fixed Layers	${\mathit{\gamma }}_{\mathit{r}}(\mathbf{J}/{\mathbf{m}}^2)$
1	4.492
2	4.495
3	4.4955
4	4.4955

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):

$$E_{ads}=E_{slab+mol}-(E_{mol}+E_{slab})$$

(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:

$$E_{reaction}=E_{product}-E_{reactant}$$

(4)

$$E_{activation}=E_{transitionstate}-E_{reactant}$$

(5)

4. 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 CH₄ and CH₃OH products adsorbed with energies of −0.204 and −0.718 eV at hollow and top positions, respectively, whereas C₂H₂ adsorbed with an energy of −3.241 eV in a hollow site, confirming that CH₄ and CH₃OH are more easily desorbed from the Co (001) surface than C₂H₂.

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 CH₂OH intermediate to methanol, which can then be further hydrogenated to methane and acetylene through CH intermediates. The preferred mechanism resulting in CH₄ as the favored product begins with the reactions CO + H→HCO and HCO + H→cis-HCOH, followed by cis-HCOH + H→CH + H₂O. Next, CH is hydrogenated to CH₄ along the reactions CH + H→CH₂, CH₂ + H→CH₃ and CH₃ + H→CH₄. For the other products, the preferred mechanisms are the same until cis-HCOH formation, whence CH₃OH is produced through the CH₂OH intermediate, and C₂H₂ results from the reaction between two CHs that were produced through the hydrogenation of cis-HCOH. The optimum pathways for CH₄, CH₃OH, and C₂H₂ 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.