Introduction

Every year more than 100 million tons of ethylene are produced from ethane [1]. The rate-limiting step of the conversion from ethane to ethene involves the C–H bond breaking in ethane. The energy barrier of this rate-limiting step can be reduced by using catalysts that effectively weaken or break a C–H bond in ethane. Considering the magnitude at which this reaction occurs, it is significant to investigate new catalysts to further bring the energy barrier down. Lowering the energy barrier for breaking a C-H bond in ethane, even by only a few kJ/mol, could potentially be valuable. Therefore, numerous efforts have been made to find novel catalysts to break the C–H bond more easily [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]. We have also been studying the atomic clusters of various transition metals (TM) as potential catalysts for the production of ethene via the oxidative dehydrogenation (ODH) of ethane (the catalytic cycle is depicted in Fig. 1) [15, 22, 23].

Fig. 1
figure 1

The catalytic cycle of the ODH of ethane. The step labeled “slow” is the presumably rate-limiting step of the catalytic cycle. Molecular formulas in red font are either vibrationally energetic in the TM complex or translationally energetic in the gas phase

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In our previous density functional theory (DFT) studies on the abilities of the Pt, Ir, and Rh atomic clusters to break a C–H bond in ethane (and propane) [15, 22, 23], several common features were observed. In all three cases, the C–H bond is weakened and eventually broken by the transition metal cluster, not because the d electrons of the TM cluster flow into the σ* bonding orbital of ethane, but because the σ bonding electrons in ethane flow into an empty d orbital of the transition metal cluster. The direction of the electron flow is evidenced directly by the Mulliken charge analysis; this is also evidenced indirectly by the correlation between the catalytic ability of a TM cluster and the diffusiveness of its LUMO that can accept electrons from ethane. In addition, we compared the TM cluster-mediated bond breaking in ethane and propane: the additional electron-pushing methyl group in propane lowers the energy barrier of the C–H bond cleavage. This substituent effect supports our conjecture that the C–H bond is weakened or broken by the Pt/Ir/Rh clusters because these TM clusters have empty d orbitals that readily accept the σ bonding electrons of ethane. In addition, we found that the natural bond orbital (NBO) charges [37, 38] and Wiberg bond indexes (WBIs) [39,40,41] may serve as reliable tools for prescreening of the various catalytic sites on the Ir and Rh clusters, and potentially other similar transition metal (e.g., Pt) clusters that activate the C–H bonds of ethane and other light alkane [23].

After the Pt/Ir/Rh clusters were studied, Pd, the remaining element in the 2 × 2 square of 45Rh, 46Pd, 77Ir, and 78Pt in the periodic table is studied with a focus on its atomic clusters’ thermal stability and activity toward ethane. The ground state electron configurations of Rh, Pd, Ir, and Pt are [Kr]4d85s1, [Kr]4d10, [Xe]4f145d76s2, and [Xe]4f145d96s1, respectively. Pd is special because, among these four elements, the Pd atom is the only one with completely filled d orbitals and thus may lack the ability to accommodate the σ bonding electrons of ethane. Since every subshell of the Pd atom is either completely filled or completely empty, it would be interesting to see how Pd atoms form metallic bonds between each other and how the Pd clusters interact with ethane. While numerous computational studies of the Pd atomic clusters or Pd-containing bimetal alloys and their interactions with hydrocarbons have been performed and published, most of these studies, if not all, have their accuracy levels limited by the computational methods, typically DFT methods that may not describe dispersion accurately, in their studies [42,43,44,45,46,47,48,49,50]. Therefore in this study, we used the B3LYP method coupled with Grimme’s empirical dispersion to obtain the optimized structures of the Pd clusters and the transition states (TS) for the Pd cluster-mediated C–H bond-breaking reactions. For an enhanced level of accuracy, we further used the “gold standard” of computational chemistry, the CCSD(T) method, to study the energetics of the Pd clusters and their reactions with ethane. The ideal sizes of the Pd atomic clusters were then proposed based on a balance between their energetic stability and activity toward ethane.

Computational methods

The B3LYP hybrid GGA method [51,52,53] was used in this study to carry out the optimization and frequency calculations of the Pd atomic clusters and the transition states (TS) of the Pdn + C2H6 → H−Pdn−C2H5 reactions. The D3 empirical dispersion formulated by Grimme with Becke–Johnson damping [54] (D3BJ) was included in the B3LYP calculations. Hereafter, this computing scheme is denoted by B3LYP-D3BJ. The vibrational frequency calculations ensure that the Pdn local minima have zero imaginary frequencies, whereas every TS structure has exactly one imaginary frequency that involves the interaction between the Pd cluster and a C–H bond of ethane. Some highly symmetric Pdn structures such as the linear Pd3 and the square planar Pd4 were found to have one or even more imaginary frequencies. The intrinsic reaction coordinate (IRC) calculations, a.k.a. the minimum energy path (MEP) calculations, were carried out to ensure that each identified transition state connects the reactants (Pdn + C2H6) and the desired product (H−Pdn−C2H5).

In the B3LYP-D3BJ optimization and frequency calculations, polarized valence double-ζ basis sets were used. The LANL2DZ basis set and the LANL2 effective core potential (ECP) [55], augmented with f-type polarization functions [56], were employed on the Pd atoms. The polarized valence double-ζ 6-31G(d) basis sets were employed on the C, H, and O atoms [57, 58]. Hereafter, this combination of the valence double-ζ basis sets is denoted by DZ. Overall this B3LYP-D3BJ/DZ computing scheme used in the optimization, frequency, and IRC calculations is the same as or very similar to those used in our previous studies on the C–H activation by the Pt, Ir, and Rh atomic clusters [15, 22, 23]; it is also similar to Liu et al.’s calculations on the functionalization of C-H bonds [59]. The natural bond orbital (NBO) [37, 38] analysis was carried out using version 3.1 of the NBO program [60].

The CCSD(T) single-point energy calculations were carried out on the B3LYP-D3BJ/DZ structures to achieve a higher level of accuracy. In the CCSD(T) calculations, The valence triple-ζ LANL2TZ basis set and the LANL2 effective core potential (ECP) [55], augmented with f-type polarization functions [56], were employed on the Pd atoms. The polarized valence triple-ζ 6-311G(d) basis sets were employed on the C, H, and O atoms [57, 58]. Hereafter, this well-balanced combination of valence triple-ζ basis sets is denoted by TZ. The CCSD(T) computing scheme is denoted by CCSD(T)/TZ//B3LYP-D3BJ/DZ or simply CCSD(T)//B3LYP in short for visual clarity. The 298 K internal energy, 298 K enthalpy (H), and 298 K Gibbs free energy (G) presented in this study are the sums of the CCSD(T)/TZ single-point electronic energy and the corresponding thermal corrections calculated at the B3LYP-D3BJ/DZ level of theory.

For the smallest Pd2 and Pd3 clusters, we were able to carry out the geometry optimization calculations at the CCSD(T)/TZ level to compare the CCSD(T)/TZ and B3LYP-D3BJ/DZ optimized structures. The CCSD(T) optimization calculations were completed numerically due to the lack of analytical gradients at the CCSD(T) level of theory. We also used the double-differentiation numerical method to obtain the CCSD(T) vibrational frequencies of the Pd2 clusters in the singlet and triplet electronic states.

The B3LYP and other DFT calculations were carried out with the Gaussian 09 program (Windows version) [61]. The CCSD(T) calculations were carried out with the Gaussian 16 program (Linux version) [62].

Results and discussion

CCSD(T)/TZ vs B3LYP-D3BJ/DZ

In this study, we compared the performance of the hybrid and pure DFT methods against the CCSD(T) benchmark and also against the experimental data (0.81 eV) [63] of the singlet–triplet (S-T) gap of the Pd atom. The S-T gap is simply the energy of the triplet electronic state of the Pd atom minus its energy in the singlet electronic state. Table 1 shows the comparison of the S-T gap of the Pd atom using the hybrid B3LYP and the pure BLYP DFT methods, along with the CCSD(T) method. Two different basis sets were used in the calculations: a smaller double-ζ LANL2DZ(f) basis set and a larger triple-ζ LANL2TZ(f) basis set. The table shows that as the basis set size increases, the calculated singlet–triplet gap decreases. The difference between the BLYP and B3LYP results is negligible (0.83 vs. 0.83 eV) when the double-ζ basis set is used but becomes more appreciable (0.72 eV vs. 0.69 eV) when the larger triple-ζ basis set is used. Among all these computing schemes, the B3LYP and BLYP calculations with the smaller DZ basis set result in the smallest discrepancies (only 0.02 eV) from the experimental value. Overall, the B3LYP values are even slightly more accurate than the CCSD(T) calculations with either the smaller DZ or the larger TZ basis set. In theory, the higher-level CCSD(T) method should be more accurate than the DFT methods when an infinitely large basis set is used. Therefore, the better performance of the B3LYP/DZ method is likely due to a fortuitous cancelation of errors for the studied system. Nonetheless, based on its agreement with experimental values of the S-T gap of the Pd atom as well as its success in the previous studies of the Ir and Rh systems [15, 23], we are sufficiently confident in using the B3LYP-D3BJ/DZ method for the geometry optimization and vibrational frequency calculations of the Pd clusters in this study.

Table 1 CCSD(T) and DFT calculated energies of the singlet and triplet Pd atom and the singlet–triplet (S-T) gap
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Fig. 2 shows the 23 atomic structures of Pdn (n = 3‒8) we optimized at the B3LYP-D3BJ/DZ level of theory. The structures of the Pd atom and the Pd2 dimer were not included in the figure because of their simplicity. For Pd3, the triangular and linear structures were locally optimized. For sizes larger than 3, linear structures are no longer studied due to their much higher energies [64]. For Pd4, the rhombus, square planar, and tetrahedral structures were optimized. Because the planar structures of Pd4 are much less stable than the nonplanar structures [64], planar structures were no longer considered beyond size 4. For Pd5, the square pyramidal and trigonal bipyramidal structures were optimized. For Pd6, the fused bi-tetrahedron, capped square pyramidal, trigonal prism, elongated/compressed octahedral, and octahedral structures were optimized. For Pd7, the capped trigonal prism, capped octahedral structures, and pentagon bipyramidal structures were optimized. For Pd8, the bicapped octahedral structure, the cubic structure, and various distorted cubic structures were optimized. In general, the choices of the initial structures of the Pd clusters were based on several previous studies [48, 64, 65].

Fig. 2
figure 2

Optimized structures of Pdn clusters (n = 3‒8). The CCSD(T) global minimum structures are underscored. The spin quantum number S is 1 for sizes 2, 4, 5, and 6; S is 0 for all other sizes

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For each atomic cluster structure, four different electronic structures with S = 0, 1, 2, 3 were studied, where S is the electron spin quantum number. The S = 3 spin states with six unpaired electrons were found to be highly unstable and therefore higher than S = 3 spin states were not examined in this study. The ground electronic state of the Pd atom is a singlet at the B3LYP and CCSD(T) levels. In partial agreement with some previous DFT studies [46, 48, 50, 64, 65], we found the CCSD(T) global minima of the Pd clusters are in a singlet electronic state at sizes 3, 7, and 8, whereas the global minima of Pd2, Pd4, Pd5, and Pd6 are in the triplet electronic state. It was experimentally verified that the global minimum of Pd2 is indeed in the triplet state [66, 67]. Our CCSD(T) predicted spin states of the Pdn global minima agree with the BLYP study [48] but differ from the PW91 DFT calculations by Zhang et. al. [64], which suggest that the Pdn global minima are all in the triplet electronic states except that Pd8 is in the singlet state [64]. Interestingly, another PW91 study conducted by Luo et. al. [50] suggests that Pd3 and Pd6 global minima are in the singlet electronic state whereas sizes 2, 4, 5, 7, and 8 are in the triplet electronic state. There are two reasons why these two PW91 studies made different predictions regarding the spin states of the Pdn global minima. One reason is that they used different basis sets: Zhang et al. used a plane wave basis set [64], whereas Luo et al. used a double numerical basis set [50]. Another reason is that the energy spacings between the Pdn global and local minima are typically so small that slightly different computing schemes (e.g., different basis sets) may result in different energy rankings. Nonetheless, the atomic structures of the Pdn global minima at the CCSD(T) level are the same as those proposed by Zhang et al. [64] and very close to those proposed by Luo et al. [50]. From size 3 to size 8, the global minimum structures at both the CCSD(T) and PW91 [64] levels of theory are triangular, tetrahedral, trigonal bipyramidal, nearly octahedral, pentagonal bipyramidal, and twisted cubic (in which the bottom square of the cube is rotated 45 degrees about the vertical axis). The PW91 study by Luo et. al. proposed the same global minimum structures for all sizes except for size 8: they suggest that the Pd8 lowest-energy structure is a bicapped octahedron instead of a twisted cube. The CCSD(T) predicted Pdn global minimum structures also agree with the BLYP [48] and BP86 [46] calculations in general. In these DFT studies, the global minimum structures of Pdn (n = 3–7) are also triangular, tetrahedral, trigonal bipyramidal, nearly octahedral, and pentagonal bipyramidal. Size 8 was not examined in the BLYP study [48] whereas the BP86 [46] calculations suggest that the Pd8 global minimum is a bicapped octahedron. Overall, the PW91, BP86, and BLYP DFT methods predicted the same lowest-energy structures as our CCSD(T)//B3LYP-D3BJ method with very few exceptions [46, 48, 50, 64]. While the prior DFT calculations and our CCSD(T) calculations do not completely agree on the spin states of the Pdn global minima, we are more confident with our CCSD(T) predictions for two reasons: one is that we included Grimme’s empirical dispersion in the DFT calculations to obtain more accurate optimized structures; the other is that the CCSD(T) method we used in this study is widely accepted as the gold standard of computational chemistry and is typically more accurate and reliable than the DFT methods both in theory and in practice.

We then studied these Pd atomic clusters’ abilities to activate a C–H bond in ethane. The 19 studied TS atomic structures of the Pdn + C2H6 → H−Pdn−C2H5 reactions (n = 1–8) are shown in Fig. 3. Fewer TS structures than the Pdn structures are presented because the TS search starting from two different Pdn structures may result in the same TS structure. For example, when a planar Pd4 atomic cluster inserts into a C-H bond of ethane, the four Pd atoms always become non-planar (nearly tetrahedral) in the TS structure. For each TS structure, four spin states (S = 0–3) were studied. The lowest-energy TS structures for nearly all sizes adopt the singlet electronic state (S = 0) except for size 7 (S = 1). This is likely because the introduction of ethane into the system breaks the high symmetry of the molecular orbitals (MOs) of the Pd atomic clusters and eliminates the (near) degeneracy of the MOs of the Pd clusters; the energy spacings between adjacent MOs become so large that the electrons occupy the lowest possible MOs, two by two, dictated by the Aufbau principle. As expected, the S = 3 high spin state of every TS structure has much higher energy than all other spin states in all cases. This is because too many electrons are forced to occupy high-energy MOs in the S = 3 electronic state. This energy increase cannot be sufficiently compensated by the energy decrease due to the extra negative exchange energy and the smaller e-e repulsion in the high-spin state. For this reaction, no higher spin state than S = 3 is of interest or further examined in this study.

Fig. 3
figure 3

The optimized TS structures of the Pdn + C2H6 → H−Pdn−C2H5 reactions (n = 1–8). The CCSD(T) global minimum structures are underscored. The spin quantum number S is 1 for size 7 and 0 for all other sizes

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Fig. 4 compares the CCSD(T)/TZ and B3LYP-D3BJ/DZ electronic energies for the Pd clusters (4a) and the TS structures (4b). The correlation is nearly perfect with R2 = 1 in both cases. The CCSD(T) energy is approximately 99.53% of the B3LYP-D3BJ energy for Pd atomic clusters and 99.55% for the TS structures. The very small 0.02% difference in percentage is mostly due to the introduction of ethane in the TS structure. The CCSD(T)/TZ singlet point energy of a free ethane molecule is 99.73% of its B3LYP-D3BJ/DZ energy.

Fig. 4
figure 4

CCSD(T)/TZ vs B3LYP/DZ electronic energies for a the Pdn atomic clusters and b the TS structures of the Pdn + C2H6 → H − Pdn − C2H5 reactions

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The correlations between the CCSD(T) and B3LYP-D3BJ energies are virtually perfect partly because of the vast difference in energy between the Pd clusters of different sizes. Therefore, we also compared the energy per Pd atom for all cluster sizes in Fig. 5. For the TS structures, we first subtract the energy of free ethane from the total energy, and then divide the remaining energy by the number of Pd atoms.

Fig. 5
figure 5

CCSD(T)/TZ vs B3LYP/DZ electronic energies per Pd atom for a the Pdn atomic clusters and b the TS structures (the ethane energy is subtracted)

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Fig. 5 demonstrates satisfactory linear correlations between the CCSD(T) and B3LYP-D3BJ energies per Pd atom. The CCSD(T) energy per Pd atom is 99.53% of the B3LYP/D3BJ energy in both Pd atomic clusters and in TS structures (the energy of free ethane is subtracted from the TS energy). The correlation between the CCSD(T) and B3LYP-D3BJ results for the TS structures (R2 = 0.9926) is notably stronger than that for the Pd clusters (R2 = 0.9363). This is because the near degeneracy among multiple electronic states and the resulting spin contamination prevalent in the Pd atomic clusters are suppressed in the TS structures due to the introduction of the ethane. As expected, less spin contamination in the TS structures results in a better correlation between the density functional theory and the coupled cluster theory. The strong correlation between the CCSD(T)/TZ and B3LYP-D3BJ/DZ energies, in total or per Pd atom, serves as indirect evidence of the satisfactory accuracy of the B3LYP-D3BJ optimized geometries. We further compared the CCSD(T)/TZ and B3LYP-D3BJ/DZ optimized structures for Pd2 in the singlet and triplet electronic states. For the singlet electronic state, the CCSD(T)/TZ and B3LYP-D3BJ/DZ bond lengths are 2.820 and 2.764 Å, respectively. Compared to the benchmark CCSD(T)/TZ result, the B3LYP-D3BJ/DZ method underestimates the bond length by 0.056 Å. Note that the CCSD(T)/TZ bond distance of Pd2 in the singlet electronic state (2.820 Å) is longer than the Pd–Pd distance in the face-centered cubic (FCC) crystalline structure of bulk Pd (2.748 Å) [68], which suggests an unusually weak bond in the Pd dimer in the singlet electronic state. For the triplet electronic state, the CCSD(T)/TZ and B3LYP-D3BJ/DZ bond lengths are 2.506 and 2.554 Å, respectively. Compared with the experimental bond length (2.48 Å) [69], the CCSD(T)/TZ and B3LYP-D3BJ/DZ methods overestimate the bond length by 0.026 and 0.074 Å, respectively. Others’ PW91, B3LYP, and BLYP calculations also overestimated the Pd–Pd bond distance by 0.01 [64], 0.06 [48], 0.05 [70], and 0.08 Å [71]. The PW91 [64] and CCSD(T) bond distances are more accurate than some other DFT calculations [48, 70, 71] and our own B3LYP-D3BJ calculations for the Pd dimer in the triplet electronic state. The discrepancies between the CCSD(T) and B3LYP-D3BJ bond distances (2.506 vs. 2.554 Å) of the Pd dimer do not significantly impact the accuracy of the CCSD(T) electronic energy: the CCSD(T)/TZ optimized energy is merely 0.4 kJ/mol or 0.8 kJ/mol lower than the CCSD(T)/TZ//B3LYP-D3BJ/DZ energy for the singlet or triplet electronic state, respectively. Both CCSD(T)/TZ and B3LYP-D3BJ/DZ calculations suggest that the Pd–Pd bond distance in the triplet state is significantly shorter than that of the singlet electronic state. This is because the theoretical Pd–Pd bond order is one in the triplet electronic state but zero in the singlet electronic state (see reasoning in “The Pdn atomic clusters”).

We also compared the CCSD(T)/TZ and B3LYP-D3BJ/DZ optimized structures for Pd3 in the singlet and triplet electronic states. For the singlet electronic state, the CCSD(T)/TZ and B3LYP-D3BJ/DZ optimized structures are both equilateral triangles (D3h point group) with a bond length of 2.531 and 2.509 Å, respectively. The B3LYP-D3BJ/DZ method slightly underestimates the bond length by 0.022 Å. For the triplet electronic state, the CCSD(T)/TZ and B3LYP-D3BJ/DZ structures are not equilateral triangles but isosceles triangles (C2v point group) due to the Jahn–Teller effect. The CCSD(T)/TZ bond lengths are 2.596, 2.596, and 2.496 Å, whereas the B3LYP/DZ bond lengths are 2.573, 2.573, and 2.566 Å. The discrepancies between the CCSD(T) and the B3LYP/DZ bond lengths of Pd3 (− 0.023, − 0.023, and 0.070 Å) do not significantly affect the accuracy of the CCSD(T) single-point energy either: the CCSD(T)/TZ optimized energy is only 0.5 kJ/mol or 1.9 kJ/mol lower than the CCSD(T)/TZ//B3LYP-D3BJ/DZ energy for the singlet or triplet electronic state, respectively.

The Pd n atomic clusters

To understand the size effect on the energetics of these Pd clusters, we computed the atomization energy of these clusters with S = 0, 1, 2, 3 at the CCSD(T)/TZ level of theory. The Pd atom in the ground electronic state ([Kr]4d105s0) is used as the zero reference in the calculation of atomization energies. Although multiple atomic structures, each with multiple electronic spin states, were calculated, only the maximum atomization energies per atom of the Pd clusters for each cluster size, and the corresponding second derivatives with respect to the cluster size, are shown in Fig. 6 for visual clarity. The singlet electronic state (S = 0) has the maximum atomization energy for sizes 1, 3, 7, and 8. Sizes 2, 4, 5, and 6 prefer the triplet state (S = 1). The lowest-energy structures of Pd3-8 adopt triangular, tetrahedral, square pyramidal, octahedral, pentagonal bipyramidal, and twisted cubic shapes, respectively. The expression of the energy second derivative for size n is [E(n + 1) + E(n–1) − 2E(n)]. A positive energy second derivative indicates energetic stability relative to the nearest neighboring sizes because the disproportionation reaction 2Pdn → Pdn−1 + Pdn+1 is endothermic and thus less likely to occur at a finite temperature, whereas a negative second derivative indicates energetic instability relative to the nearest neighboring sizes.

Fig. 6
figure 6

a Atomization energies per atom, abbreviated as Atom E., of the Pd atomic clusters, b the atomization energy second derivatives with respect to the atomic cluster size

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Fig. 6a shows that the atomization energy per atom increases monotonically with the cluster size as expected because the average number of bonds increases as the average number of surrounding Pd atoms increases as the cluster size increases. We were not able to calculate the energy second derivative for size 1 or 8 because they each have only one nearest neighbor in cluster size. In agreement with a global optimization study at the BLYP level [49], sizes 2, 3, 5, and 7 have negative second derivatives and thus are energetically unstable relative to their nearest neighbors. Pd2 is particularly unstable with the smallest atomization energy (29 kJ/mol) and the most negative energy second derivative (− 106 kJ/mol) at the CCSD(T) level.

Pd4 and Pd6 have a positive second derivative (56 and 26 kJ/mol, respectively) and are relatively stable. Specifically, size 4 might be considered the magic number of the small Pd clusters as Pd4 has the largest energy second derivative. Although we cannot calculate the energy second derivative for Pd8, the aforementioned global search at the BLYP level suggests that this size is also relatively stable [49].

Our CCSD(T) calculations suggest that 1Pd3 has much larger atomization energy per atom than 3Pd2 (74 vs 29 kJ/mol), which agrees with the multi-reference singles plus doubles configuration interaction (MRSDCI) calculations for the Pd dimer and trimer [72]. The Pd dimer in the singlet electronic state (1Pd2) has even smaller atomization energy per atom (17 kJ/mol) than the triplet electronic state (29 kJ/mol). The very weak bond strength of 1Pd2 can be well understood by visualizing its ten highest occupied MOs in Fig. 7.

Fig. 7
figure 7

The ten highest occupied MOs of 1Pd2. The dashed line separates the five d-d antibonding MOs from the five d-d bonding MOs

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Among the 10 highest occupied MOs of 1Pd2 in Fig. 7, from the bottom up, there is one σd-d orbital, two degenerate πd-d orbitals, two degenerate δd-d orbitals, two degenerate δ*d-d orbitals, two degenerate π*d-d orbitals, and one σ*d-d orbital. The bond order (B. O.) is zero in 1Pd2 because the ten d orbitals from the two Pd atoms form five bonding orbitals and five corresponding antibonding orbitals, all of which are doubly occupied. Note that Pd2 in the triplet electronic state (3Pd2) has a bond order of 1. This is because when an excited 3Pd atom with two unpaired electrons ([Kr]4d95s1) interacts with the 1Pd atom ([Kr]4d10) in the ground electronic state to form 3Pd2, the σ4d-4d and σ5s-5 s bond orders are both one half and therefore the overall bond order of 3Pd2 is one. The higher bond order of 3Pd2 than that of 1Pd2 is also evidenced by its higher CCSD(T) vibrational frequency (203 cm−1) than that of 1Pd2 (107 cm−1). Note that a higher vibrational frequency indicates a larger force constant (k) and a deeper potential energy well (De). Note that the Tayler expansion of the Morse expression, De(1 ‒ e^‒α(r-re))2, suggests that the depth of the potential energy well of a diatomic molecule De is approximately proportional to its force constant k. While the Pd–Pd bond strength is anticipated to be much greater in the triplet electronic state (B. O. = 1) than in the singlet electronic state (B. O. = 0) in the Pd dimer, 3Pd2 sits only 25 kJ/mol below 1Pd2 in the potential energy surface (PES). This is because the bond energy of the 3Pd2 is relative to a singlet Pd atom and a much higher-energy triplet Pd atom (3Pd lies 68 kJ/mol higher than 1Pd). Given the above data, we estimate the average strength of the σ4d-4d and σ5s-5 s bonds between two Pd atoms is approximately 25 + 68 = 93 kJ/mol.

But if the Pd atoms in the ground singlet electronic state have only completely filled subshells, why can 1Pd3 have a much larger atomization energy per atom (74 kJ/mol) than 1Pd2 (17 kJ/mol) and even 3Pd2 (29 kJ/mol)? The 15 highest occupied MOs of 1Pd3 in Fig. 8 shed light: while 14 out of the 15 highest occupied MOs of 1Pd3 are linear combinations of the d orbitals as expected, the HOMO, however, is a linear combination of three 5 s atomic orbitals, one from each Pd atom. The total bond order of 1Pd3 is thus two. Both of these two bonds are three-center two-electron σ bonds, one being the overlap of three 5 s atomic orbitals and the other being the overlap of three 4d atomic orbitals. The latter results from the removal of the two electrons from the antibonding σ*4d orbital illustrated in Fig. 9. The empty σ*4d orbital (which is the LUMO of 1Pd3) allows for the net σ bonding among the 4d orbitals (B. O. = 1) in 1Pd3. In addition, there is also net σ bonding among the 5 s orbitals (B. O. = 1) in 1Pd3. Given the above data (the atomization energy per atom is 74 kJ/mol and the total bond order is 2), we estimate the average strength of the σ4d and σ5s bonds among the three Pd atoms in 1Pd3 is approximately 74∙3/2 = 111 kJ/mol, which is notably larger than that of 3Pd2 (93 kJ/mol). This is because in a Pd dimer, each σ4d or σ5s bond results from only one region of constructive interaction between the two Pd atoms, while in a Pd trimer each σ4d or σ5s bond results from three regions of constructive interaction among the three Pd atoms in a triangle. In general, the atomization energy per atom in a Pd cluster increases as the cluster size increases mainly because the number of neighboring Pd atoms increases on average. For comparison, each Pd atom is surrounded by 12 nearest Pd atoms in the face-centered cubic structure (FCC) structure of bulk Pd, which has significantly larger atomization per atom (377 kJ/mol) than the small clusters studied in this work [68].

Fig. 8
figure 8

The top 15 highest occupied molecular orbitals of 1Pd3, among which 14 are interactions among the 4d orbitals, whereas the HOMO is the interaction among the 5 s orbitals

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Fig. 9
figure 9

The LUMO of 1Pd3, an empty σ*4d orbital that allows for the net σ bonding among the 4d orbitals (B. O. = 1) in.1Pd3

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The CCSD(T) global minimum of Pd4 adopts a nearly tetrahedral structure (with slight distortion due to the Jahn–Teller effect) in the triplet electronic state with an atomization energy per atom of 110 kJ/mol. Its singlet counterpart (1Pd4) has a higher energy mainly because all d-d antibonding orbitals are fully occupied in 1Pd4. The atomization energy per atom slowly increases as the cluster size increases from sizes 5 to 8: 121 kJ/mol, 131 kJ/mol, 135 kJ/mol, and 142 kJ/mol, respectively. Since these four values are still significantly smaller than the atomization energy per atom of bulk Pd (377 kJ/mol) [68], we suspect that the atomization energy per atom will continue to slowly increase well beyond size 8 and even well beyond size 55 as suggested by a PW91 study [64].

The TS of Pd n + C 2 H 6  → H−Pd n −C 2 H 5

The internal energy (0 K), enthalpy (298 K), and Gibbs energy (298 K) of activation of the Pdn + C2H6 → H−Pdn−C2H5 reactions are presented in Fig. 10. Their values along with the electronic energy of activation are also provided in Table 2.

Fig. 10
figure 10

The internal energy of activation at 0 K, enthalpy of activation at 298 K, and Gibbs energy of activation at 298 K at the CCSD(T)/TZ level

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Table 2 The CCSD(T) electronic energy of activation, the internal energy of activation at 0 K, the enthalpy of activation at room temperature, and the Gibbs free energy of activation at room temperature. All values are in kJ/mol
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The 0 K internal energy of activation (including the zero-point vibrational energy) is calculated using the following equation:

$$\Delta U^{\ddag } \, = \,U\,\left( {TS} \right)\, - \,U\,\left( {Pd_{n} } \right)\, - \,U\,\left( {{\text{ethane}}} \right)$$
(1)

In this equation, U(TS) is the lowest possible internal energy of all TS structures optimized for the Pdn + C2H6 → H−Pdn−C2H5 reaction for each cluster size n; U(Pdn) is the lowest possible energy of all Pdn local minimum (LM) structures. Although other higher-energy TS structures and other higher-energy Pdn structures also make contributions to the C–H activation by Pd clusters, Eq. 1 provides a simple yet representative means of quantifying the abilities of the Pdn clusters to activate a C-H bond in ethane. The electronic energy of activation (ΔEelec‡) is equivalent to ΔU‡ excluding the zero-point vibrational energy (ZPVE). ΔU‡ is consistently smaller than ΔEelec‡ by ~ 14 kJ/mol mainly because of one less C-H stretching vibrational mode in the TS. ΔH‡ and ΔG‡ at 298 K were calculated using the following equations:

$$\Delta H^{\ddag } \, = \,H\,\left( {TS} \right)\, - \,H\,\left( {Pd_{n} } \right)\, - \,H\,\left( {{\text{ethane}}} \right)$$
(2)
$$\Delta G^{\ddag } \, = \,G\,\left( {TS} \right)\, - \,G\,\left( {Pd_{n} } \right)\, - \,G\,\left( {{\text{ethane}}} \right)$$
(3)

In the above two equations, H(TS) or G(TS) is the lowest possible enthalpy or Gibbs energy of all TS structures for each cluster size n; H(Pdn) or G(Pdn) is the lowest possible enthalpy or Gibbs energy of all Pdn local minimum (LM) structures. Although the internal energy or Helmholtz energy of activation at room temperature is not presented in Fig. 10 or Table 2, they can be easily derived using the following equations:

$$\Delta U^{\ddag } \, = \, \, \Delta H^{\ddag } \, - \, \, \Delta \left( {PV} \right)^{\ddag } \, = \, \, \Delta H^{\ddag } \, - \,RT\Delta n^{\ddag } \, = \, \, \Delta H^{\ddag } \, + \,RT$$
(4)
$$\Delta A^{\ddag } \, = \, \, \Delta G^{\ddag } \, - \, \, \Delta \left( {PV} \right)^{\ddag } \, = \, \, \Delta G^{\ddag } \, - \,RT\Delta n^{\ddag } \, = \, \, \Delta G^{\ddag } \, + \,RT$$
(5)

where Δn‡ is the change in the number of the moles of gas species (Δn‡ = − 1 in a bimolecular reaction) and R is the gas constant. At 298 K, ΔU‡ or ΔA‡ is ~ 2.5 kJ/mol larger than ΔH‡ or ΔG‡, respectively, assuming the ideal behavior of all gas particles.

Pd4 has the largest 0 K energy of activation (55 kJ/mol) and 298 K enthalpy of activation (57 kJ/mol) among all sizes likely due to its ultrastability. Pd2/6/8 have the lowest energy and enthalpy of activation likely due to the large energy decrease when the Pdn–C2H6 reactant complex is formed before overcoming the energy barrier. In particular, sizes 2 and 6 both have negative energy and enthalpy of activation. A negative energy or enthalpy of activation is possible mainly because the energy released in the formation of the Pdn–C2H6 reactant complex can be greater than the energy barrier this complex overcomes to form the TS structure. Fig. 11 shows that the singlet TS of Pd2 + C2H6 → H−Pd2−C2H5 lies 37 kJ/mol below the separate reactants in the singlet state and 12 kJ/mol below the separate reactants in the triplet state. Either way, the energy barrier is negative. The minimum energy path of this reaction starts from the triplet Pd2 and ethane. In the formation of the Pd2–C2H6 reactant complex, a spin flip can take place (due to the spin–orbit coupling or the spin exchange with nearby species) and all electrons become paired. The singlet Pd2–C2H6 then overcomes a small 4 kJ/mol energy barrier for the hydrogen shift from ethane to Pd2 and forms the singlet H−Pd2−C2H5.

Fig. 11
figure 11

The CCSD(T) potential energy surface of Pd2 + C2H6 → H−Pd2−C2H5 in the singlet (solid line) and triplet states (dotted line). ZPVE is excluded from the potential energy. All numbers are in kJ/mol

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Negative energy barriers are not a unique phenomenon for Pdn + C2H6 → H−Pdn−C2H5. They are also observed in the interaction between ethane and the Pt, Rh, or Ir atomic clusters when the TMn–C2H6 complex (TM = Pt/Rh/Ir) has a lower energy than the TS of the TMn + C2H6 → H − TMn − C2H5 reaction for selected TM cluster sizes [15, 22, 23].

While Pd2 has the lowest energy, enthalpy, and Gibbs energy of activation and thus is considered the most active toward ethane, its small atomization energy per atom (29 kJ/mol) and negative second derivative of the atomization energy (− 106 kJ/mol) indicate its very poor thermal stability and a strong tendency to agglomerate or to undergo a disproportionation reaction. Among all studied cluster sizes, Pd6 and Pd8 well balance thermodynamic stability and catalytic activity. Their room-temperature Gibbs energies of activation were determined to be 30 and 47 kJ/mol (12RT and 19RT), respectively, when we considered only the global minima of the Pd cluster and TS for each of these two sizes.

The Gibbs energy of activation may be defined more thoroughly and thereby more accurately via a statistical means assuming the NPT condition in which the number of particles (N), the pressure (P), and temperature (T) remain constant. For example, the Gibbs energy of activation of Pdn under the isobaric condition may be calculated using the following equation:

$$\Delta {G}^{\ddagger }={G}^{\mathrm{apparent}}(\mathrm{TS})-{G}^{\mathrm{apparent}}({\mathrm{Pd}}_{n})-G\left(\mathrm{ethane}\right)$$
(6)

where \({G}^{\mathrm{apparent}}\) is defined statistically as follows:

$${e}^{-\frac{{G}^{\mathrm{apparent}}}{RT}}=\sum_{i}^{\mathrm{all structures}}{e}^{-\frac{{G}_{i}}{RT}}$$

Note that \(\sum_{i}^{\mathrm{all structures}}{e}^{-\frac{{G}_{i}}{RT}}\) provides a reasonable estimate of the thermodynamic contribution of all structures in an ensemble under the constant PT condition. Using Eq. 6, \(\Delta {G}^{\ddagger }\) for Pd1−8 were determined statistically over the very large number of Pdn and TS structures studied in this work. The resulting statistical \(\Delta {G}^{\ddagger }\) values are virtually the same as those obtained with Eq. 3 within a negligible discrepancy (0.2 kJ/mol for size 3 and 0.0 kJ/mol for all other sizes). Equations 3 and 5 may differ significantly when there exist multiple Pdn structures with similar Gibbs energies or when there exist multiple TS structures with similar Gibbs energies. Equation 3 underestimates \(\Delta {G}^{\ddagger }\) in the former case and overestimates \(\Delta {G}^{\ddagger }\) in the latter case.

Two electron-flow mechanisms of Pd n + C 2 H 6  → H−Pd n −C 2 H 5

How is the C-H bond in ethane weakened by the Pd clusters? Does the Pd cluster take away C–H bonding electrons or donate electrons to the C–H antibonding orbitals? The calculations of the natural bonding orbital (NBO) charges of the lowest-energy TS structure of the Pdn + C2H6 → H−Pdn−C2H5 (n = 1−8) reaction were carried out at the B3LYP-D3BJ level of theory to reveal the direction of the electron flow between the ethane molecule and the Pd cluster. Table 3 presents the total NBO charge on the Pd cluster in the TS structure. The total charges on the Pd clusters are less than 0.1 in magnitude for all eight sizes. This indicates that the net electron flow between the Pd cluster and the ethane molecule is minimal in their interactions.

Table 3 The total charge on Pdn in the lowest-energy TS structures of the Pdn + C2H6 → H−Pdn−C2H5 (n = 1–8) reaction
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We further examined the NBO charges on the individual Pd atoms in the Pd6 and Pd8 TS structures because they have a large atomization energy per atom (131 and 142 kJ/mol, respectively) and a small enthalpy of activation toward ethane (− 21 and 8 kJ/mol, respectively). The Pd atom partially bonded to the C and H atoms in the TS has a very small NBO charge (− 0.097 and − 0.050 for sizes 6 and 8 respectively). This minimal electron flow in the Pdn + C2H6 → H−Pdn−C2H5 reaction contrasts the much larger electron flow between ethane and Ir8 (a − 0.314 NBO charge on the Ir atom partially bonded to C and H) [15]. For a similar Rh8 + C2H6 reaction, the Rh atom partially bonded to C and H also has a large − 0.287 NBO charge [23]. It is clear that the net electron flow is from C2H6 to the empty d orbitals of the Ir and Rh clusters but this cannot be the entire story for the Pd clusters. Since the net electron flow between Pdn and C2H6 is very small compared to Ir and Rh systems, we conjecture that the Pd clusters weaken the C–H bond of ethane via two mechanisms concurrently. The first involves electron flow from the 4d orbitals of the Pd cluster to the σ* antibonding orbitals in ethane, and the second involves electron flow from the occupied σ bonding orbital of ethane to the empty 4d or 5 s orbitals of the Pd atom(s). The charge transfer between Pdn and C2H6 is small because the Pdn cluster accepts and donates electrons simultaneously. Despite the minimal net charge transfer between the Pd cluster and ethane, a very important change transpired: some C–H σ bonding electrons become σ* antibonding electrons in ethane and thereby a C–H bond is activated. Fig. 12 provides a rudimentary illustration of these two concurrent electron-flow mechanisms: A Pdn cluster can donate its 4d electrons to the antibonding orbitals of ethane and meanwhile the same Pdn cluster can also accept the C–H bonding electrons concurrently. Although the 4d orbitals of the free Pd atom in the ground electronic state are completely occupied, there are partially empty 4d orbitals in a Pdn cluster that can accommodate electrons from ethane. The NBO charge analysis for other cluster sizes suggests that the same two mechanisms account for the activation of the C–H bond in ethane.

Fig. 12
figure 12

Two mechanisms account for the Pdn-activated C–H bond in ethane

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Further NBO charge analysis of the separated ethane and Pd6 indicates that the shifted H atom, among all atoms, undergoes the most dramatic change in the atomic charge. The NBO charge of H in free ethane is calculated to be 0.22. In the reaction between ethane and Pd6, the shifted H accepts 0.06 e− from the Pd cluster and 0.14 e− from the C2H5 group during the formation of the TS structure. This shifted H atom has a very small positive NBO charge (0.02) in the TS structure partly because its Pauling electronegativity is the same as that of Pd (2.2). For Pd8, the shifted H atom donates 0.01 e− to the Pd cluster and accepts 0.15 e− from the C2H5 group in the formation of the TS structure. The shifted H atom also has a very small positive NBO charge (0.08) in the TS structure.

Conclusions

We used the CCSD(T)/TZ//B3LYP-D3BJ/DZ dual-level hybrid method to study the various local minima of the Pd atomic clusters (Pdn) and their abilities to activate a C–H bond in ethane. The global minimum structures of Pdn (n = 3 – 8) are triangular, tetrahedral, triangular bipyramidal, octahedral, pentagonal bipyramidal, and twisted cubic, respectively. The global minima of Pdn (n = 1, 3, 7, 8) are in the singlet electronic state, whereas Pdn (n = 2, 4, 5, 6) are in the triplet electronic state. The shapes of the Pdn in the lowest-energy TS of Pdn + C2H6 → H−Pdn−C2H5 largely resemble the corresponding free Pdn clusters (with noticeable distortion in the TS structure for sizes 7 and 8). Due to the interaction between ethane and Pdn that eliminates the (near) degeneracy of molecular orbitals, all lowest-energy TS structures are in the singlet electronic state except for size 7 (which is a triplet). Among the eight cluster sizes studied in this work, size 2 has the lowest energy, enthalpy, and Gibbs energy of activation but it also has poor thermal stability indicated by its small atomization energy and negative second derivative of atomization energy. Size 4 exhibits satisfactory thermal stability, but its ability to activate a C-H bond is significantly poorer than other sizes. Pd6 and Pd8 well balance the thermal stability and catalytic ability and thus are likely the ideal sizes for the effective activation of the C–H bonds in ethane. Pd6 and Pd8 are considered stable as they both have a large atomization energy per atom: 131 and 142 kJ/mol, respectively. They are considered active toward ethane as their room-temperature Gibbs energies of activation of Pdn + C2H6 → H−Pdn−C2H5 are only 30 and 47 kJ/mol (12RT and 19RT), respectively. In this work, we also quantified the C–H activation abilities of each Pd cluster size in two ways under constant PT conditions. One way is to take the difference between the lowest Gibbs energy of the TS structures and the lowest Gibbs energy of the reactants. The other is to take the contribution of all reactant structures and all TS structures into account in a statistical manner under the constant TP condition. These two different ways yield virtually the same result because of the lack of (near) degeneracy among the lowest-energy reactants and the lack of (near) degeneracy among the lowest-energy TS structures.