Mechanistic insight into the hydrogenation of acetylene on the Pd2/g-C3N4 catalyst: effect of Pd clustering on the barrier energy and selectivity

In this work, the hydrogenation of acetylene on the Pd2/g-C3N4 catalyst is investigated by the density functional theory (DFT) and quantum theory of atoms in molecules (QTAIM) calculations. The pre-reactant (R), transition states (TSs), and the intermediates (IMs), involved in the hydrogenation process, are characterized from the point of view of energy and structure. The calculated energy barriers for the hydrogen transfer to the acetylene and ethylene are 6.77 and 12.28 kcal/mol, respectively, which shows that the Pd2/g-C3N4 catalyst has good selectivity for the conversion of acetylene to ethylene rather than ethane. Comparing the values of these energy barriers with those of the hydrogenation of acetylene on the Pd/g-C3N4 catalyst (21.53 and 38.88 kcal/mol, respectively) shows that the increase in the number of the Pd atoms decreases the energy barriers of the hydrogenation reaction and increases the selectivity of the catalyst for the ethylene production.


Introduction
Ethylene is an important raw material with wide applications in petrochemical industries [1]. Generally, the production of ethylene is performed via the steam cracking of hydrocarbons. The main disadvantage of this method is the catalyst poisoning leading to the reduction of catalyst lifetime [2]. So, the catalytic hydrogenation of acetylene followed by the classical Horiuti-Polanyi mechanism has been considered as the best method for ethylene production on large scales [3]. An efficient catalyst for acetylene hydrogenation should have high activity and selectivity for ethylene production [4]. In this regard, the noble transition metals such as palladium (Pd)-based catalysts are widely used because of their good selectivity for ethylene production due to their superior hydrogen solubility and good ability for dissociation of H 2 molecules [5,6]. There are several experimental works aiming production of ethylene via acetylene hydrogenation in a selective manner in which the structure of the Pd-based catalyst plays a critical role in achieving a good selectivity [7][8][9][10][11][12].
Huang et al. prepared the dispersed Pd atoms on the g-C 3 N 4 using atomic layer deposition and performed the hydrogenation of acetylene on the Pd 1 /g-C 3 N 4 catalyst [7]. Boucher et al. performed the hydrogenation of styrene and acetylene on the catalyst composed of the Pd atoms binding to the Cu surface [8]. Panpranot et al. investigated the catalytic efficiency of Pd catalyst supported on nanocrystalline TiO 2 (Pd/TiO 2 ) for the selective hydrogenation of acetylene in the presence of excess ethylene [11]. The same group also examined the effect of the phase composition of the TiO 2 crystalline support on the physicochemical and catalytic properties of the Pd/TiO 2 catalyst on the selective hydrogenation of acetylene [12].
There are several theoretical studies of the clarification of the hydrogenation mechanism of acetylene on the Pd-based catalysts and the interaction between the different forms of Pd and the substrates [13][14][15][16]. Meng et al. studied the hydrogenation of acetylene on the TiO 2 -supported Pd a Ag b (a+b=4) catalyst using the density functional theory (DFT)+U method [13]. The hydrogenation of acetylene using the Pd 4 cluster on the anatase TiO 2 (101) support was studied by Yang et al. [14]. Ma et al. performed the DFT study of the selective hydrogenation of acetylene on the P-doped Cu (111) surface [15]. The effect of the size of Pd cluster (Pd n ) and its geometry on the hydrogenation of acetylene has been investigated [17][18][19][20]. Li et al. investigated the effect of the size of Pd clusters on acetylene hydrogenation in the absence of any support [17]. Pu et al. investigated the effect of Pd n clusters with different sizes (n=2-8) on acetylene hydrogenation by the DFT calculations. They proposed two pathways for acetylene hydrogenation including (a) ethene formation as the main product via vinyl intermediate and (b) the vinylidene, ethylidene, and ethylidene as the intermediates participating in the hydrogenation process. Also, they showed that the reaction pathway depends strongly on the size of Pd n clusters so that for n<4, the hydrogenation reaction proceeded through the vinyl intermediate, while the other pathway via the vinylidene intermediate was the main pathway for the larger Pd clusters [17].
Due to the high activity of Pd catalysts, the overhydrogenation of ethylene is always a deep concern. Therefore, designing a selective catalyst for ethylene production to eliminate extra purification steps in its synthesis and prevent ethane production is considered a challenging task in acetylene hydrogenation [21,22]. Various strategies have been so far reported for the promotion of the performance of Pd-based catalysts for the acetylene hydrogenation which one of the important of them is the doping of the Pd cluster with other noble metals such as Au [4], Pb [23,24], Cu [25], Ag [26][27][28], and Ga [29,30]. The other important strategy is using single-atom alloys (SAA) and single-atom catalysts (SAC), immobilized on solid supports such as ZnO [31], fiberglass [32], graphene [33], Al 2 O 3 [34], and graphitic carbon nitride (g-C 3 N 4 ) [6,7] which has shown interesting results in acetylene hydrogenation. Among the solid supports, g-C 3 N 4 is always considered a suitable candidate for the deposition of Pd-active sites [35][36][37]. It has some outstanding features such as low-cost fabrication, short time and easy synthesis, adjustable surface area, good dispersion potential for active sites, and high coordination ability for metal clusters and nanoparticles. Huang et.al showed that Pd single-atom immobilized on the surface of g-C 3 N 4 has a better selectivity in acetylene hydrogenation compared to nanoparticle ones [7]. Zhao et al.'s theoretical study aiming to demonstrate the acetylene hydrogenation mechanism over the Pd 1 /g-C 3 N 4 catalyst showed that the Pd atom tended to adsorb on the top of the sixth fold cavity of g-C 3 N 4 as the most accessible site. Moreover, the Pd 1 /g-C 3 N 4 catalyst showed better selectivity to ethylene production in comparison to ethane production as expected from the experimental studies [38]. Kang et al. investigated the catalytic activity of the Pd 1 /g-C 3 N 4 catalyst for acetylene hydrogenation by replacing the nitrogen and carbon atoms of g-C 3 N 4 with the sulfur atom (S) [39]. It was observed that the Pd 1 /g-C 3 N 4 containing sulfur-doped carbon atoms have higher activity for the dissociation of H 2 molecules, while better selectivity is achieved for sulfur-doped nitrogen ones.
To the best of our knowledge, there is no theoretical mechanistic study of acetylene hydrogenation over Pd n /g-C 3 N 4 structures with n>1 in the literature. Our recent theoretical study on the adsorption of small-size Pd n clusters (n=2, 4, 6, and 8), immobilized on the g-C 3 N 4 quantum dot, reveals that the Pd 2 cluster has the highest adsorption energy among the other selected Pd clusters [40]. In this work, the DFT calculations are performed to understand the exact mechanism of acetylene hydrogenation on the Pd 2 /g-C 3 N 4 catalyst and see the effect of the added Pd atom on the activity of catalyst compared to that of Pd 1 /g-C 3 N 4 catalyst reported in references 39 and 40. Our calculated results show less activation energy for acetylene hydrogenation using the Pd 2 /g-C 3 N 4 compared to the Pd 1 /g-C 3 N 4 catalyst and better selectivity for ethylene production which is in good accordance with the experimental data in the literature [38,[41][42][43].

Computational details
The structure of all reactants (Pd 2 /g-C 3 N 4 , C 2 H 2 , and H 2 ), prereactants (R), transition states (TSs), and intermediates (IMs) were optimized in the gas phase using the DFT method employing B3PW91 functional [44]. The 6-311+G(d,p) and Los Alamos effective core pseudo-potential (ECP) (LANL2DZ) basis set were used for the g-C 3 N 4 and Pd 2 , respectively. The frequency calculations were performed on all optimized structures to check their correct positions on their potential energy surfaces in the hydrogenation pathway and determine their electronic energies, enthalpies, Gibbs free energies, and zero-point energies. The suitability of the selected DFT functional and basis set for the considered systems has been investigated by Zhao et al [38]. The intrinsic reaction coordinate (IRC) calculations were performed to make sure that each TS structure was related to its corresponding reactants and products in the reaction mechanism. The highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) of the structures were calculated to obtain the bandgap energies and donor-acceptor properties of structures. The Quantum Theory of Atoms in Molecules (QTAIM) calculations were performed using Multiwfn 3.4 software [45] for better structural characterization of the TSs and confirming the proposed mechanism of the reaction. All DFT calculations were performed using the Gaussian 09 Quantum Chemistry Package [46].

Results and discussion
The optimized structure of the Pd 2 /g-C 3 N 4 catalyst and its charge distribution A quantum dot of g-C 3 N 4 is selected as a model of substrate for the deposition of the Pd 2 cluster. The g-C 3 N 4 has two main structural forms based on the s-triazine and tri-s-triazine building blocks so that the latter form is always considered as the most stable form of g-C 3 N 4 [38][39][40]. A layer of g-C 3 N 4 composed of three tri-s-triazine units was selected and optimized (see Fig. 1). The nitrogen atoms in this model are classified into two classes: (a) saturated nitrogen atoms which are bonded to three carbon atoms (N1) and (b) pyridine-like nitrogen atoms which are bonded to two carbon atoms and are located at the corners of the sixth fold cavity (N2). Since 2p orbitals of N2 atoms have the main contribution in the valance band of g-C 3 N 4 , their lone pair electrons have the main role in the chemical activity of g-C 3 N 4 compared to the N1 atoms [40]. On the other hand, 2p orbitals of the N1 and C atoms orbitals have a major contribution to the conduction band of g-C 3 N 4 [40].
The Pd 2 cluster is optimized separately and placed on the top of the sixth fold cavity of g-C 3 N 4 aiming to achieve the maximum interaction with the lone pairs of N2 atoms. The comparison of the structure of g-C 3 N 4 in the Pd 2 /g-C 3 N 4 complex with the optimized structure of isolated g-C 3 N 4 shows that the presence of the Pd 2 creates more wrinkles and waves in the structure of g-C 3 N 4 ( Fig. 1). The bond length of the Pd 2 cluster in the optimized structure of the Pd 2 /g-C 3 N 4 is 2.74 Å which shows the elongation of the Pd-Pd bond length compared to the isolated Pd 2 cluster due to the strong interaction between the Pd atom, directed to the support, and the carbon atom of one of the tri-s-triazine units. The calculated partial atomic charges of these Pd and C atoms in the Pd 2 /g-C 3 N 4 are 0.37e and −1.458e, respectively. Comparison of the charge of this C atom in the complex with that in the isolated g-C 3 N 4 (0.297e) shows a strong charge transfer from the Pd 2 to g-C 3 N 4 confirming a strong interaction between the Pd and carbon atom (see Fig. 2). The angle between the Pd 2 axis and the g-C 3 N 4 plane is 46.0°. Table 1 reports the energies of the HOMO and LUMO of C 2 H 2 , H 2 , g-C 3 N 4 , Pd 2 /g-C 3 N 4 , and C 2 H 2 .Pd 2 /g-C 3 N 4 structures along with their energy gaps. The shapes of the HOMO and LUMO of g-C 3 N 4 , Pd 2 /g-C 3 N 4 , and C 2 H 2 ·Pd 2 /g-C 3 N 4 structures have been shown in Fig. 3. According to our previous work [40], immobilization of the Pd 2 cluster on the surface of g-C 3 N 4 causes the decrease of the energy gap of the Pd 2 /g-C 3 N 4 complex compared to the isolated g-C 3 N 4 because of creating new electronic states between the HOMO and LUMO of g-C 3 N 4 [40]. These new electronic states cause the increase of the photocatalytic activity of the Pd 2 /g-C 3 N 4 complex compared to the bare g-C 3 N 4 . Comparison of the difference between the energy of the HOMO of Pd 2 /g-C 3 N 4 and that of LUMO of C 2 H 2 (−4.25 eV) with the energy difference between the HOMO of C 2 H 2 and LUMO of Pd 2 /g-C 3 N 4 (−5.24 eV) shows that the C 2 H 2 is an electron acceptor species. A similar comparison between the Pd 2 /g-C 3 N 4 and H 2 shows that the H 2 is also an electron acceptor. The smaller value of the energy difference between the HOMO of C 2 H 2 and LUMO of Pd 2 /g-C 3 N 4 compared to the corresponding Adsorption of H 2 and C 2 H 2 on Pd 2 /g-C 3 N 4

HOMO and LUMO of reactants
Essentially for a heterogeneous catalyst containing the Pd cluster, it is important to use the maximum active site of the cluster after immobilization. Therefore, it is necessary to find the most stable site for the adsorption of H 2 and C 2 H 2 molecules on the surface of the Pd 2 /g-C 3 N 4 catalyst. For this purpose, the H 2 molecule approaches the catalyst in the dissociated and bonded forms, separately (see Figures S1(a) and (b)). As seen in Figure S1(a), the H 2 molecule is located near one Pd atom which is farther from the surface of g-C 3 N 4 . In Figure S1(b), it is assumed that the H 2 molecule has been dissociated on the Pd 2 /g-C 3 N 4 catalyst and each of its H atoms is interacting with one of the Pd atoms. The two initial structures lead to the same structure after the optimization so that the two hydrogen atoms of the H 2 molecule are interacting with one Pd atom (H 2 ·Pd 2 /g-C 3 N 4 ; see Figure S1(c)). The reason for the interaction of the H 2 molecule with this Pd atom could be attributed to its negative Mulliken charge which provides a good potential for the participation in electron donation to reactants.
In the next step, the C 2 H 2 is approached to the H 2 ·Pd 2 /g-C 3 N 4 complex to form the pre-reactant (C 2 H 2 ·H 2 /g-C 3 N 4 ) for the start of the reaction. Figure S2 shows two proposed structures for the pre-reactant (R1 and R2). The best selected initial orientation of the C 2 H 2 relative to the Pd 2 in the C 2 H 2 ·H 2 Pd 2 / g-C 3 N 4 complex is such that its molecular axis is in a vertical position relative to the axis of the Pd 2 (R1 in Figure S2). In this case, the simultaneous interaction of each C atom of C 2 H 2 with both Pd atoms is possible. It will be shown that the interaction of the C atoms of C 2 H 2 with the Pd atom, located far away from the surface of g-C 3 N 4 , is stronger than the interaction with the other Pd atom. As previously shown in Fig. 2b, the Pd atom located further away from the g-C 3 N 4 has a more negative Mulliken charge compared to the other Pd atom. Therefore, this Pd atom has good potential for electron donation to the reactants (H 2 and C 2 H 2 ). Therefore, the C 2 H 2 and H 2 interact mainly with the upper Pd atom in the optimized structure of C 2 H 2 ·H 2 Pd 2 /g-C 3 N 4 complex. Due to the adsorption, the H-H bond length increases to 0.93 Å compared to that of an isolated H 2 molecule (0.74 Å). Moreover, the increase of the C≡C bond length is seen for the acetylene in the C 2 H 2 ·H 2 Pd 2 /g-C 3 N 4 complex (1.28 Å) compared to isolated acetylene molecule (1.19 Å). These observations confirm the activation of the H 2 and C 2 H 2 molecules by the catalyst and indicate that the initial complex (R1 in Figure S2) is suitable for starting the hydrogenation process. Considering the adsorption energies of reactants, it can be concluded that the adsorption of acetylene on the catalyst (−28.84 kcal/mol) is stronger than that of the H 2 molecule (−13.23 kcal/mol) which is in agreement with the frontier molecular orbital analysis performed in the previous section.
The energy difference between the HOMO of C 2 H 2 ·Pd 2 /g-C 3 N 4 and the LUMO of H 2 (−5.93 eV) is smaller than the energy difference between the HOMO of H 2 and the LUMO Fig. 2 Mulliken partial atomic charge distribution of (a) g-C 3 N 4 quantum dot and (b) Pd 2 /g-C 3 N 4 complex where E complex is the electronic energy of H 2 ·C 2 H 2 Pd 2 /g-C 3 N 4 complex. The E acetylene , E catalyst , and E H2 are the electronic energies of isolated optimized acetylene, Pd 2 /g-C 3 N 4 , and H 2 molecule, respectively Mechanism of acetylene hydrogenation on the Pd 2 /g-C 3 N 4 catalyst As previously mentioned, two initial structures, R1 and R2 ( Figure S2), were considered as the starting structures for the hydrogenation mechanism. Zhao et al. demonstrated that the hydrogenation pathway via the R1 complex in the presence of Pd 1 /g-C 3 N 4 single-atom catalyst requires less activation energy compared to the R2 ones [38]. Also, they reported that the Fig. 3 Frontier molecular orbitals of the g-C 3 N 4 , Pd 2 /g-C 3 N 4 , and C 2 H 2 ·Pd 2 /g-C 3 N 4 structures hydrogenation pathway via the R1 complex had more selectivity to the ethylene production rather than ethane production [38]. Therefore, the R1 structure is selected as the initial structure for the hydrogenation mechanism on the Pd 2 /g-C 3 N 4 catalyst.
Considering the R1 as a starting complex, two consecutive pathways are proposed for the hydrogenation mechanism. In the first pathway, the acetylene is converted to ethylene (IM3) via the three transition states (TS1, TS2, and TS3) (see Fig. 4). In the second pathway, ethylene (IM3) is hydrogenated more and converted to ethane (P) via two transition states (TS4 and TS5). To obtain the activation energy of each step, the E coadsorption of each complex involved in each step is calculated using Eq. (1). The energy difference between the E co-adsorption of the reactant and TS of each step is considered as the activation energy (Fig. 4). Figures 5, 6, 7, 8, and 9 show the optimized structures of the complexes (R1, IMs, and TSs) involved in the pathways shown in Fig. 4. Also, the important interaction paths for each complex, obtained from the QTAIM calculations, are shown in the figures. The position of the bond critical points (BCPs) (dots) and their calculated electron density (ρ) have been identified in the figures.

R1→ TS1→ IM1
The calculated value of the activation energy for the conversion of the R1 complex to IM1 via TS1 is 6.77 kcal/mol (see Fig. 5). The optimized structure of TS1 is similar to the R1 complex except that the H 2 bond length increases to 2.77 Å in the structure of TS1 indicating the complete dissociation of the H 2 molecule on the catalyst. The calculated vibrational mode of TS1 with the imaginary vibrational frequency (ν=−470i cm −1 ) is shown in Figure S3. The direction of the displacement vector of one of the H atoms of the dissociated H 2 molecule is toward the C atom of the acetylene indicating the H transfer from the Pd to C atom. Comparison of the arrangement of the BCPs in the R1 complex with that of TS1 shows that the BCP No.1 (BCP1; ρ=0.1732 a.u), related to the interaction path between two H atoms of H 2 molecule in the R1 complex, disappears. Also, a new BCP (BCP4) is formed between one of the H atoms of the H 2 molecule and the C atom of acetylene in the TS1 complex. As seen, this H atom has simultaneous interaction with the Pd and C atoms via the interaction paths related to the BCP2 (ρ=0.1189 a.u) and BCP4 (ρ=0.1444 a.u) in the structure of TS1. Due to the interaction of this H atom with the C atom of acetylene, the C-H bond length increases from 1.08 in the R1 complex to 1.43 Å in the IM1 complex. These observations confirm the transfer of the H atom from the Pd site to the C atom of acetylene for the formation of the vinyl compound. It is important to note that there is no BCP between two Pd atoms in the structure of the complexes shown in Fig. 5 and each Pd atom is interacting with the C atoms of acetylene independent of the other one. In the R1 complex, each of the Pd atoms is interacting with one C atom of acetylene, while one of the Pd atoms is in the direct interaction with two C atoms, simultaneously in the TS1 and IM1 complexes via the two BCPs (Nos. 5 and 6 in TS1 and Nos. 3 and 4 in IM1). The increase of the value of ρ for the BCP of Pd-H bond (BCP2, BCP1, and BCP1 in the R1, TS1, and IM1 complexes, respectively) shows that this bond becomes stronger due to the H transfer. Also, the strength of the Pd-N bond between the Pd atom and g-C 3 N 4 becomes weaker in the IM1 (BCP No. 5; ρ=0.7159 a.u.) complex compared to the R1 complex (BCP No. 5; ρ=0.7015 a.u.) due to the decrease in its ρ value.
IM1→ TS2→ IM2 Before the transfer of the second H atom in the structure of IM1 to the C atom of vinyl for forming IM3 (see Fig. 7), a step is necessary (IM1→TS2→IM2). The second H atom is interacting with the Pd atom via the BCP1 in the structure of IM1. Notably, this H transfer becomes easier when the orientation of the complex above the g-C 3 N 4 changes so that this Pd atom can directly interact with the N atoms of the substrate (see Fig. 6; IM2). Due to this interaction, the IM2 complex is more stable than the IM1 complex, and the orientation of the Pd-H bond is more suitable for the second H Fig. 4 Energy diagram for the acetylene hydrogenation over the Pd 2 /g-C 3 N 4 catalyst transfer to the C atom of vinyl compared to what is observed in the IM1. The bond angle of C-Pd-H in the structure of IM1 is 160.8°and reaches 83.9°in the structure of IM2 so that the distance between the second H atom and the C atom of vinyl decreases (see Fig. 6). As seen, the orientation of the complex on the g-C 3 N 4 changes so that the two Pd atoms are in direct interaction with the N atoms of g-C 3 N 4 . The strength of the H-Pd bond does not change due to this configuration change because of the small change of its ρ value. The calculated value of ρ of the H-Pd bond in the IM1 (BCP1) is 0.1484 a.u., and the corresponding value in the IM2 is 0.1502 a.u. (BCP5). Moreover, a decrease is seen in the bond length of the formed C-H bond in step 1 in the IM2 complex (2.36 Å) compared to that in the IM1 complex (2.60 Å) which shows that this bond is stronger. Therefore, the transfer of the second H atom from the Pd atom to the C atom in the IM2 needs less activation energy compared to that of this process in the IM1. A considerable decrease in the ρ values of the C-Pd interaction paths of the TS2 structure compared to those of the C-Pd interaction paths of the IM1 structure is seen. For example, the ρ of the BCPs Nos. 3 and 4 (0.9617 and 0.9885 a.u, respectively), related to the simultaneous interaction of one Pd atom with two C atoms in the structure of IM1, reduces to 0.3508 and 0.4902 a.u., respectively, in the structure of TS2 showing the decrease of the strength of these bonds (see Fig.  6). Also, the arrangement of the interaction paths in the IM2 shows that each Pd atom is mainly interacting with one C atom unlike what is observed for the IM1 and TS2 structures in which one of the Pd atoms is in the direct interaction with two C atoms, simultaneously. Notably, the orientation of the complex on the g-C 3 N 4 in the structure of TS 2 is so that the interaction of complex with the substrate takes place mainly via the H atoms (see BCPs No. 4, 7, and 8 in Fig. 6). The imaginary vibrational mode of the TS2 is shown in Figure S3. As seen, the H atom which is interacting with the g-C 3 N 4 via the BCPs Nos. 7 and 8 moves out of the substrate plane.
IM2→ TS3→ IM3 Figure 7 shows the transfer of the second H atom from the Pd atom to the C atom of vinyl via the TS3 structure and form of the IM3. The comparison of the structure of TS3 with IM2 shows that the bond angle of C-Pd-H related to BCPs No. 2 and 5 in IM2 decreases in the TS3 structure (see BCPs No. 2 and 4). Comparison of the value of ρ of the BCP5 of the structure of IM2 (0.1502 a.u.) with that of the corresponding BCP of the structure of TS3 (BCP4; ρ= 0.1341 a.u.) shows the decrease of the strength of the Pd-H bond due to the decrease of the C-Pd-H bond angle. The imaginary vibrational mode of the TS3 is shown in Figure S4. The displacement vector of the H atom shows its transfer from the Pd atom to the C atom. Notably, a BCP (ρ=0.5594 a.u.) is seen between two Pd atoms in the structure of IM3 showing the interaction of two Pd atoms with each other which is absent in the structure of IM2 and TS3.
IM4→ TS4→ IM5 Figure 8 shows the first step of the hydrogenation of ethylene to form ethane via TS4. A new H 2 molecule is added to the IM3 for forming the IM4. For this purpose, the H 2 molecule is located near the Pd atoms to feasible Fig. 5 The optimized structure of the R1, TS1, and IM1 of the first step of hydrogenation of acetylene on the Pd 2 /g-C 3 N 4 catalyst. The red points show the position of the bond critical points (BCPs) obtained from the QTAIM calculations. The value of electron density (ρ) for each BCP has been given Fig. 6 The optimized structure of the IM1, TS2, and IM2 of the second step of hydrogenation of acetylene on the Pd 2 /g-C 3 N 4 catalyst. The red points show the position of the bond critical points (BCPs) obtained from the QTAIM calculations. The value of electron density (ρ) for each BCP has been given the breaking of the H 2 bond. As seen in the figure, the dissociation of the H 2 molecule takes place in the TS4 structure so that one of the H atoms of H 2 is connected to one of the Pd atoms via the BCP2, and the other one is pulled toward the carbon atom of ethylene so that it is in the direct interaction with three atoms via the three BCPs (BCP4, BCP5, and BCP3), simultaneously. The imaginary vibrational frequency of the TS4 is shown in Figure S4. As seen, the displacement vector of one of the H atoms shows its tendency for forming the bond with the C atom of ethylene. The imaginary frequency of this vibrational mode and the barrier energy for this hydrogen transfer is 409i cm −1 and 12.28 kcal/mol, respectively. The comparison of the structure of IM4 with TS4 shows the decrease of the Pd-Pd distance so that a new BCP appears between two Pd atoms in the IM4 structure. The strength of the Pd-Pd interaction in IM5 is lower than that in IM4 because of the decrease of the ρ of BCP3 from IM4 to IM5. Comparison of the ρ value of C-H bond in TS4 (BCP4; 0.1557 a.u.) with the corresponding value in IM5 (BCP1; 0.2384 a.u.) shows the complete formation of the new C-H bond in IM5 compared to TS4.
IM5→ TS5→ P Figure 9 shows the last step of the hydrogenation mechanism related to the second H transfer from the Pd atom to the C atom of CH 3 CH 2 via the TS5 structure to form ethane. As seen, this H atom interacts with the Pd atom via the BCP5 (ρ= 0.1960 a.u.) in the structure of IM5. Also, the increase of the number of BCPs between the Pd atom and g-C 3 N 4 is seen in the TS5 structure (BCPs Nos. 3 and 4) compared to the IM5 structure (BCP No. 6) which helps to the hydrogen transfer by the weakening of the Pd-H bond as well as the presence of the BCP1 between the H and the C atom in this structure. Comparison of the calculated value of ρ of the BCP2 (0.1484 a.u.) in the structure of TS5 with that of BCP5 (0.1960 a.u.) in the structure of IM5 shows the decrease of the strength of the Pd-H interaction and confirms the H transfer from the Pd atom to C atom. The calculated value of vibrational frequency of the imaginary vibrational mode of TS5 (see Figure S4) related to the H transfer is 535i cm −1 which is in agreement with the literature [38]. After the transfer of the last H atom to the CH 3 CH 2 and the formation of ethane (structure P in Fig. 9), one of the Pd atoms becomes close to the substrate which creates a folding in the structure of g-C 3 N 4 .
A comparison has been made between the imaginary vibrational frequencies of the TSs of the hydrogenation of acetylene using the Pd 2 /g-C 3 N 4 catalyst with that reported in the literature [38] for Pd 1 /g-C 3 N 4 (see Table 2). As seen, the calculated vibrational frequencies for the R1→TS1→ IM1, IM4→TS4→IM5, and IM5→TS5→P steps of our work are smaller than the value reported by Zhao et al. [38]. The value of the imaginary vibrational frequency of the TS of IM1→TS2→IM2 is nearly equal to that reported by Zhao et al. for the Pd 1 /g-C 3 N 4 catalyst. The imaginary frequency of the transition state of IM2→TS3→IM3 is greater than that reported for the Pd 1 /g-C 3 N 4 catalyst. Fig. 7 The optimized structure of the IM2, TS3, and IM3 of the third step of hydrogenation of acetylene on the Pd 2 /g-C 3 N 4 catalyst. The red points show the position of the bond critical points (BCPs) obtained from the QTAIM calculations. The value of electron density (ρ) for each BCP has been given Fig. 8 The optimized structure of the IM4, TS4, and IM5 of the first step of hydrogenation of ethylene on the Pd 2 /g-C 3 N 4 catalyst. The red points show the position of the bond critical points (BCPs) obtained from the QTAIM calculations. The value of electron density (ρ) for each BCP has been given

Selectivity and reactivity
As previously mentioned, the effective performance of a heterogeneous catalyst in the hydrogenation of acetylene is the production of ethylene with a smaller value of activation energy compared to that of ethane production [38,39]. According to Fig. 4, the activation energy for the first H transfer reaction (R1-TS1-IM1) is equal to 6.77 kcal/mol, and it is smaller than the values reported in the literature for the other heterogeneous catalyst (see Table 3). Also, the comparison of the activation energy of the R1-TS1-IM1 step obtained in this work using the Pd 2 /g-C 3 N 4 catalyst (6.77 kcal/mol) with the corresponding value reported by Zhao et al. [38] using the Pd 1 /g-C 3 N 4 catalyst (21.53 kcal/mol) shows that the increase of the number of Pd atom has a considerable effect on the decrease of activation energy of this step.
The selectivity of a heterogeneous catalyst in the acetylene hydrogenation, for the production of ethylene, increases by the decrease of the activation energy of the ethylene production step compared to that of the ethane production step. The selectivity (ΔE) is considered as the difference between the hydrogenation barrier (E hydrogenation ) and desorption barrier (E desorption ) of ethylene [38,39]. The higher value of ΔE leads to the higher selectivity of the catalyst for ethylene production. The calculated value of ΔE obtained in this work is 21.90 kcal/ mol (see Table 3) which is more than that reported by Zhao et al. for the Pd 1 /g-C 3 N 4 catalyst. This shows that the increase in the number of Pd atoms of the catalyst is accompanied by the increase of the selectivity of the catalyst for ethylene production.

Conclusion
In this work, a theoretical mechanistic study of the hydrogenation of acetylene on the Pd 2 /g-C 3 N 4 heterogeneous catalyst was performed using the DFT and QTAIM calculations. The selectivity of this catalyst for the ethylene production was compared with those of the other catalysts reported in the literature (Pd(111) [41], Al 13 CoO 4 [42], AlPd [43], Pd1/g-C 3 N 4 [38]). It was observed that the Pd 2 /g-C 3 N 4 has more selectivity and activity for ethylene production compared to Fig. 9 The optimized structure of the IM5, TS5, and P of the second step of hydrogenation of ethylene on the Pd 2 /g-C 3 N 4 catalyst. The red points show the position of the bond critical points (BCPs) obtained from the QTAIM calculations. The value of electron density (ρ) for each BCP has been given Table 2 The imaginary frequencies of the transition states (TSs) involved in the different steps of the acetylene hydrogenation mechanism taken from Zhao's group work [38] for the Pd1/g-C 3 N 4 with those obtained in this work