How Important is Metal-Carbon Back-Bonding for the Stability of Fullerene-Transition Metal Complexes? Role of Cage Sizes, Encapsulated Ions and Metal Ligands How Important is Metal-Carbon Back-Bonding for the Stability of Fullerene-Transition Metal Complexes? Role of Cage Sizes, Encapsulated Ions and Metal Ligands

A density functional study of {η 2 -(X@C n )}ML 2 complexes with various cage sizes (C 60 , C 70 , C 76 , C 84 , C 90 , C 96 ), encapsulated ions (X = F − , 0, Li + ) and metal fragments (M = Pt, Pd) is per- formed, using M06/LANL2DZ levels of theory. The importance of π back-bonding to the thermodynamic stability of fullerene-transition metal complexes ({η 2 -(X@C n )}ML 2 ) and the effect of encapsulated ions, metal fragments and cage sizes on the π back-bonding are determined in this study. The theoretical computations suggest that π back-bonding plays an essential role in the formation of fullerene-transition metal complexes. The theoretical evidence also suggests that there is no linear correlation between cage sizes and π back-bonding, but the encapsulated Li + ion enhances π back-bonding and F − ion results in its deterioration. These computations also show that a platinum center produces stronger π back-bonding than a palladium center. It is hoped that the conclusions that are pro vided by this study can be used in the


Introduction
The first fullerene-transition metal complex, (η 2 -C 60 )Pt(Ph 3 ) 2 , was prepared and structurally characterized by Fagan et al. in 1991 [1]. It was the starting point for a new class of study for fullerene chemistry. Since then, various fullerene-transition metal complexes have been synthesized and these have potential applications in solar cells, spintronics, catalysis and drug delivery [2]. Balch et al. then studied the reactions of C 60 with electron-rich fragments, IrCl(CO)(PPh 3 ) 2 and produced the fullerene-iridium complex (η 2 -C 60 )IrCl(CO)(PPh 3 ) 2 [3]. The formation of fullerene-iridium complex is a reversible process and the reversible binding of IrCl(CO)(PPh 3 ) 2 to fullerenes can be used as a structural probe because the adducts can build ordered single crystals that are suitable for X-ray diffraction [4,5]. Fullerene-iridium complexes that contain an enantiomeric phosphine ligand are used as solar photoelements [6]. One of the significant characteristics of fullerenes is that they are capable of encaging atoms, ions and small molecules to form endohedral complexes. Endohedral metallofullerenes (EMFs) are those that encapsulate metal atoms within a hollow carbon cage. Proft et al. theoretically studied the interactions between encapsulated monoatomic ions (Li + to Rb + and F − to I − ) and C 60 and its Si and Ge analogues [7] and found that, for these families, the interactions between Li + (Na + ) and F − (Cl − ) ions and C 60 are strongest and exothermic, which confirms the possibility of the existence of these species. Recently, Li + @C 60 was successfully synthesized and isolated by Watanabe et al. [8].
Understanding the strength and nature of metal-ligand bonding is crucial for the design of new fullerene-transition metal complexes because the structure and stability of various intermediates are important to the formation of organometallics [9]. In an earlier work by the authors [10], {η 2 -(X@C 60 )}ML 2 complexes (M = Pt, Pd; X = 0, Li + , L = PPh 3 ) were studied and it was found that there is a relationship between thermodynamic stability and π backbonding; that is, the greater the π back-bonding, the greater is thermodynamic stability. This shows that thermodynamic stability can be modified by tuning the π back-bonding. As far as the authors are aware, π back-bonding could be affected by several factors, including the encapsulated ions, the metal fragments and the cage sizes. This study determines the importance of π back-bonding to the thermodynamic stability of {η 2 -(X@C n )}ML 2 complexes by using M = Pt, Pd; X = F − , 0, Li + and n = 60, 70, 76, 84, 90 and 96 to ascertain the role of these factors in π back-bonding. Since the system is very large, methyl-substituted N-heterocyclic carbenes (NHC) are used as a ligand (L), instead of PPh 3 . NHC is one of the frequently used and most powerful tools in organic chemistry [11]. In this work, the following reactions are studied:

Computational details
The following fullerenes that comply with the isolated pentagon rule are used to develop a correlation: Ih-C 60 , D 5h -C 70 , D 2 -C 76 , D 2d (23)-C 84 , D 5h (1)-C 90 and D 3d (3)-C 96 . These are experimentally isolated and identified [12][13][14]. The symmetry and numbering scheme for fullerene isomers are in accordance with an approved classification [15]. Hückel molecular orbital calculations show that the 6:6 ring junctions at the poles of the molecules usually have highest π bond orders (B) and are expected to be the most reactive, so these are the sites of attack (see Scheme 1) [12,16].
The geometry optimizations are performed without any symmetry restrictions by using the M06 [17]/LANL2DZ [18,19] level of theory. The vibrational frequency calculations at 298.15 K and 1 atm use the same level of theory. The stationary points are confirmed by the absence of imaginary frequencies. The natural charges are obtained using NBO 5.9, as implemented in the Gaussian 09 program [20].
The interatomic interactions are determined using energy decomposition analysis (EDA). Two types of EDA are used in this work. The first is the basic EDA that was developed individually by Yang et al. [21] and by Ziegler and Rauk [22]. For this basic EDA, the bonding energy (∆E) is partitioned into two terms, ∆E = ∆E(DEF) + ∆E(INT). In this work, basic EDA is used for the optimized ML 2 X@C n complexes, which are categorized into transition metal complexes (A), carbon cages (B) and metal ions (C) as shown in Scheme 2. The deformation energy (∆E(DEF)) Scheme 1. The sites of attack for addition to the fullerenes Ih-C 60 , D 5h -C 70 , D 2 -C 76 , D 2d (23)-C 84 , D 5h (1)-C 90 and D 5h (1)-C 96 . The Hückel π bond orders (B) were calculated using the freeware program, HuLiS [16].
is the sum of the deformation energy of A (∆E(DEF) A , which is defined as the energy of A in the product relative to the optimized isolated structure (A 0 ) and B (∆E(DEF) B )). The interaction energy term, ∆E(INT) A(BC) , is the interaction energy between A and (BC) for their respective optimized product structures.
Advanced EDA unites the natural orbitals for chemical valence (NOCV), so it is possible to separate the total orbital interactions into pairwise contributions [23]. The advanced EDA (i.e., EDA-NOCV) further divides the interaction energy (∆E(INT)) into three main components: ∆E(INT) = ∆E elstat + ∆E Pauli + ∆E orb . It is used for a quantitative study of π back-bonding to fullerene ligands that uses the M06/TZ2P level of theory with the ADF 2016 program package [24]. The relativistic effect is considered by applying a scalar zero-order regular approximation (ZORA) [25]. The interaction energy and its decomposition terms are obtained from a singlepoint calculation using the M06/TZ2P basis set from the Gaussian 09 optimized geometry.

Geometric changes
The structures of {η 2 -(X@C n )}ML 2 complexes for M = Pt, Pd; X = F − , 0, Li + and n = 60, 70, 76, 84, 90 and 96 were fully optimized at the M06/LANL2DZ level of theory. The geometries that are obtained are illustrated in Figure 1. The key structural parameters of the stationary points are listed in Table 1 (the structural parameters for n = 70, 76, 84, 90 and 96 are presented elsewhere). For the Pt-C 60 complex in the absence of encapsulated ions, the respective lengths of the metal-carbon bonds are 2.12 Å and 2.12 Å ( Table 1). When the Li + ion is encapsulated into the cage, the metal-carbon bonds remain unaltered and the respective distances between C 1 , C 2 and Li + are 2.29 Å and 2.29 Å. As the encapsulated ion is changed to F − , the metal-carbon bonds remain almost unchanged (2.13 and 2.13 Å), but the distance between C 1 , C 2 and encapsulated ions (F − ) increases (3.18 and 3.18 Å). The Li + ion is located at a site that is close to the transition metals because of electrostatic interaction. The metal-coordinated carbon atoms of C 60 are negatively charged because there is π back-donation from the metal center. For the Pt-C 60 complex without encapsulated ions, the natural population analysis (NPA) shows that the atomic charges on the C 1 (C 2 ) atoms are −0.27 (−0.27). When the cage is encapsulated by a Li + ion, the atomic charges on the C 1 (C 2 ) atoms are increased to −0.32 (−0.32) and the atomic charge on the Li atom is +0.86. Therefore, the encaged Li + ion is attracted toward these negatively charged C atoms. However, as the encapsulated ion is changed to F − , NPA shows that the atomic charges on C 1 (C 2 ) atoms are decreased to −0.23 (−0.23) and the atomic charge on the F atom is negative (−0.93), so the encaged F − ion is repelled by the negatively charged C atoms. In terms of Pd-C 60 complexes, it is worthy of note that the geometrical distances are generally similar to the corresponding distances for Pt-C 60 complexes, but the charge distributions are different. Specifically, the encaged Li atom has a charge (+0.86) but the charges on C 1 (C 2 ) atoms are reduced to −0.27 (−0.27). The negative charges on metal-coordinated carbon atoms are also less for X = 0 and F − . Similar geometric changes and charge distributions are seen for n = 70, 76, 84, 90 and 96 and are presented elsewhere.  is more stable. The relative thermodynamic stability increases in the order: ∆E(X = F − ) < ∆E (X = 0) < ∆E(X = Li + ), as shown in Table 2. In addition, |∆∆E(DEF)| is small and |∆∆E(INT) A(BC) | is large, so the latter must be responsible for an increase in thermodynamic stability. Similar results are obtained for Pd-C 60 complexes, but, the distortion in fragments A or B is smaller than that for the Pt-C 60 complex, as is the interaction energy, so the complex is less stable. Therefore, an increase in the cage size has no effect on the basic EDA results.

Advanced energy decomposition analysis (advanced EDA)
In an earlier study by the authors, structural parameters and spectral characteristics were used to estimate the strength of π back-bonding for {η 2 -(X@C 60 )}ML 2 (M = Pt, Pd; X = 0, Li + , L = PPh 3 ) complexes [10]. The changes in bond length (Δr/r 0 ), bond angle (Δθ av ), vibrational frequency (Δν) and the chemical shift (Δδ) were used to describe the character of the π-complex s. In this study, the strength of the π back-bonding strength is estimated from an energetic viewpoint using an advanced EDA method. This analysis shows the effect of encapsulated ions, metal fragments and cage sizes on π back-bonding.

The effect of encapsulated ions on π back-bonding
In an earlier discussion (Section 3.2), it was proven that thermodynamic stabilities increase in the order: ∆E(X = F − ) < ∆E(X = 0) < ∆E(X = Li + ), because the interaction energy (∆E(INT)) The difference is relative to corresponding quantity at X = 0. d The reaction energy without zero-point energy (ZPE) correction for the product, relative to the corresponding reactants. is increased. The interaction between the metal fragment and X@C n is now studied using advanced EDA, which further decomposes the interaction energy into electrostatic interaction (∆E elstat ), repulsive Pauli interaction (∆E Pauli ) and orbital interaction (∆E orb ) terms. The orbital interactions are the most important of these three and only the most important pairwise contributions to ΔE orb are considered. The advanced EDA method is used for {η 2 -(X@ C n )}ML 2 complexes, as shown in Tables 3 and 4 (the results for n = 70, 76, 84, 90 and 96 are presented elsewhere). A plot of the deformation density and a qualitative drawing of the orbital interactions between the metal fragment and X@C 60 are shown in Figure 2. In terms of the Pt-C 60 complexes, Table 3 shows that both the electrostatic interaction (∆E elstat ) and the orbital interaction (∆E orb ) stabilize the complexes because they are negative terms, but the percentage of ∆E orb increases in the order: ∆E orb (X = F − ) < ∆E orb (X = 0) < ∆E orb (X = Li + ). Therefore, the enhanced orbital interaction must be responsible for the increase in the thermodynamic stability. Table 3 also shows that ΔE 1 contributes significantly to ΔE orb : 69.5% for X = F − , 75.2% for X = 0 and 76.4% for X = Li + . The deformation densities show that these come from π backdonation from a filled d orbital of the metal to the π* orbitals of C 60 (charge flow is yellow to green at the top of Figure 2c). The large contributions of ΔE 1 to ΔE orb are in agreement with the results of previous studies. The metal-carbon bonds are principally formed by π backdonation [8]. It is also seen that the order of ΔE 1 is |ΔE 1 (X = F − )| = 94.4 < |ΔE 1 (X = 0)| = 118.6 < |ΔE 1 (X = Li + )| = 142.8 kcal/mol. Therefore, ΔE 1 is increased when there is the encapsulated Li + ion but decreased when there is a F − ion. The second contribution of ΔE 2 to ΔE orb is comparatively small: 18.0% for X = F − , 13.1% for X = 0 and 10.0% for X = Li + . This results from The values in parentheses give the percentage contribution to the total orbital interactions, ΔE orb . σ-donation from a filled π orbital of C 60 to the π* orbital of the metal (middle of Figure 2c). The computational results show that π back-bonding is crucial to the thermodynamic stability of Pt-C 60 complexes and that an encapsulated Li + ion increases π back-bonding but an encapsulated F − has the opposite effect.

The effect of metal fragments on π back-bonding
Pd-C 60 complexes appear to be similar to Pt-C 60 complexes, but a comparison of the results in Tables 3 and 4 shows that the value ΔE 1 for a Pd-C 60 complex is smaller than the corresponding value for a Pt-C 60 complex, which demonstrates that the π back-bonding for a palladium center is weaker than that for a platinum center. For example, |ΔE 1 (M = Pd, X = Li + )| = 121.0 < |ΔE 1 (M = Pt, X = Li + )| = 142.8 kcal/mol. This is consistent with the earlier results that were obtained using structural parameters and spectral characteristics [10]. Figure 3 shows a plot of the ΔE 1 values versus cage sizes that are calculated for {η 2 -(X@C n )} PtL 2 complexes (n = 60, 70, 76, 84, 90 and 96). It is seen that there is no linear relationship and there is one obvious peak for each X at n = 84 [26]. This demonstrates the effect of a difference in size of the carbon clusters on π back-bonding for a metal center, but the correlation is not simply monotonic. Therefore, a larger (smaller) cage size does not necessarily imply that there is stronger (weaker) π back-bonding, which results in greater (lower) thermodynamic stability. The values in parentheses give the percentage contribution to the total orbital interactions, ΔE orb .  (a) A qualitative drawing of the orbital interactions between the metal fragment and Li + @C 60 ; (b) the shape of the most important interacting occupied and vacant orbitals of the metal fragments and Li + @C 60 ; (c) a plot of the deformation densities, Δρ, for the pairwise orbital interactions between the two fragment in their closed-shell state, the associated interaction energies, ΔE orb (in kcal/mol), and the eigenvalues ν. The eigenvalues, ν, indicate the size of the charge flow. The direction of the charge flow is from yellow to the green.

Conclusion
This computational study uses density functional theory to determine the thermodynamic stability of {η 2 -(X@C n )}ML 2 complexes (M = Pt, Pd; X = F − , 0, Li + and n = 60, 70, 76, 84, 90 and 96). The calculations show the reaction is more stable when the Li + ion is encapsulated within C n but the complex becomes unstable if there is a F − ion. Basic EDA shows that there is an increase in the interaction between the metal fragment and C n if there is an encapsulated Li + ion but F − ion has the opposite effect.
The advanced EDA results show that π back-bonding is crucial to thermodynamic stability and that thermodynamic stability is increased by the presence of a Li + ion but the presence of a F − ion has the opposite effect. These computations also show that a platinum center results in stronger π back-bonding than a palladium center and that there is no linear relationship between cage size and π back-bonding.