Pentacoordinate Carbon Atoms in a Ferrocene Dication Derivative—[Fe(Si

: Pentacoordinate carbon atoms are theoretically predicted here in a ferrocene dication derivative in the eclipsed-( 1 ; C 2 v ), gauche-( 2 ; C 2 ) and staggered-[Fe(Si 2 - η 5 -C 5 H 2 ) 2 ] 2 + ( 3 ; C 2 h ) forms for the ﬁrst time. Energetically, the relative energy gaps for 2 and 3 range from − 3.06 to 16.74 and − 2.78 to 40.34 kJ mol − 1 , respectively, when compared to the singlet electronic state of 1 at different levels. The planar tetracoordinate carbon (ptC) atom in the ligand Si 2 C 5 H 2 becomes a pentacoordinate carbon upon complexation. The ligand with a ptC atom was predicted to be both a thermodynamically and kinetically stable molecule by some of us in our earlier theoretical works. Natural bond orbital and adaptive natural density partitioning analyses conﬁrm the pentacoordinate nature of carbon in these three complexes ( 1 – 3 ). Although they are hypothetical at the moment, they support the idea of “hypercoordinate metallocenes” within organometallic chemistry. Moreover, ab initio molecular dynamics simulations carried out at 298 K temperature for 2000 fs suggest that these molecules are kinetically stable.


Results and Discussion
The zero-point vibrational energy (ZPVE) corrected-relative energies and Gibbs free energies obtained for 1-3 at different levels for the singlet and quintet electronic states are shown in Table 1. It is noted here that calculations were carried out for the triplet electronic states of these complexes. However, they turned out to be lying above singlets and quartets and thus for brevity they are given in the supporting information (see Table S3 for details). Bonding pattern obtained for eclipsed conformer (1) through AdNDP analysis is shown in Figure 2.

Energetics
As far as relative stability is concerned among the three, different DFT functionals gave us different results (see Table 1). The quintet electronic state of the staggered (3) form is the most stable at the ωB97X-D/def2-TZVP level. However, with the same def2-TZVP basis set, functionals such as M06-L, TPSSh, and TPSSh-D3BJ predict that the singlet electronic state of the eclipsed form (1) is the most stable. With the popular functional B3LYP, though we obtain the quintet of 3 as the most stable, adding empirical dispersion corrections on top of B3LYP (i.e, B3LYP-D3BJ) changes the result yet again because at the B3LYP-D3BJ/def2-TZVP level, the quintet of gauche-form (2) is the most stable. Like ferrocene, the relative energy gap (∼4 kJ mol −1 ) between staggered and eclipsed forms is quite small [8]. In the parent molecule, as per gas-phase calculations, the staggered form is a saddle-point (transition state) and the eclipsed form is a minimum [74]. In the derivatives studied here, all the three different forms in their singlet ground electronic state are minima at all levels. The quintet electronic states of staggered (3) and gauche (2) conformers are also minima whereas the eclipsed (1) conformer of quintet either turns out to be a transition-state or a second-order saddle-point at different levels. Our main motivation in this study is to analyze the bonding scenario and thus we leave this discussion with a caveat that various DFT functionals including the popular B3LYP underestimate the barrier-heights [75,76].

Bonding
Let us analyze the bonding scenario in these cations (1-3) as each isomer contains two pentacoordinate carbon atoms. The C-C bond length in 1 (C 2v ) range from 1.42 to 1.47 Å (see Figure 1a), whereas in 2 (C 2 ) and 3 (C 2h ) it varies from 1.41 to 1.46 Å (see Figure 1c,e). Compared to ferrocene [77], where the mean C-C bond length is equal to 1.431 Å, these bond lengths are slightly varied, which is reasonable due to the ionic character (dication) in these complexes apart from the presence of silicon atoms. Likewise, the Fe-C bond length in 1 range from 2.07 to 2.10 Å, whereas in 2 and 3 it varies from 2.07 to 2.11 Å and 2.07 to 2.12 Å, respectively. In ferrocene, the mean Fe-C bond length is equal to 2.059 Å and here they are slightly longer. The Si-C bond length connected to the pentacoordinate carbon is 1.98 Å in 1, 1.97 and 1.99 Å in 2, and 1.99 Å in 3, which reflects its single bond characteristics whereas the Si-C bond on the sides are shorter with a bond length of 1.79 Å in all the three forms. In principle, the isolated Si 2 C 5 H 2 ligand almost behaves like a cyclopentadienyl anion (C 5 H − 5 ), with a slight exception that the former contains a 3center-2electron (3c-2e) σ bond (see Figure 2) around the Si-C-Si region [61,62]. This is evidently seen even when it makes complexation with Fe(II).

Wiberg Bond Indices
The presence of pentacoordinate carbon atoms could be further justified with the WBIs calculated for 1, 2, and 3, in Figure 1b, 1d, and 1f, respectively. In all cases, the WBI values for Fe-C are in the range of 0.39 to 0.42. This indicates that they are indeed single bonds. The hypercoordinate C-Si WBI values in 1 and 3 are 0.40 and 0.38, respectively, whereas for 2 we obtained two values due to reduced symmetry and they are 0.37 and 0.41, respectively. All these values also reflect the single bond characteristics of these bonds. The Si-C WBI values in 1 to 3 are in the range of 0.71 to 0.70 showing single bond characteristics. WBI values for all C-C bond lengths are greater than one, which indicates resonance stabilization plus double bond characteristics. On the basis of these values, one could certainly conclude that the central carbon atoms are hypercoordinate (penta) in all the three forms. It is emphasized here that each hypercoordinate carbon obeys the octet-rule as the total WBI for each pentacoordinate carbon is 3.62 for 1 and 2, whereas it is 3.61 for 3. However, some of the bonds (Si-C and C-Fe) are electron-deficient bonds with fewer than two electrons as mentioned elsewhere in the example of C(CH 3 ) + 5 [45].

AdNDP Analysis
To clarify the chemical bonding pattern further in these complexes, we carried out adaptive natural density partitioning (AdNDP) analysis [78,79] using the Multiwfn [80] program. For brevity, only the bonding patterns of the eclipsed conformer alone are shown here. In total, 66 valence electrons 1 are taken into account. Seven lone-pairs (LPs), four 2c-2e C-H bonds, ten 2c-2e C-C bonds, four 2c-2e C-Si bonds, two 3c-2e Si-C-Si σ bonds, and six 19c-2e dπ-pπ are shown in Figure 2. Among them, the latter are more important as they support the pentacoordinate nature of the central carbon atom . Though 1-3 are hypothetical examples at the moment, theoretically, these complexes support the idea of "hypercoordinate metallocenes" within organometallic chemistry.

MD Simulations
To verify the kinetic stability of the ferrocene derivatives (1-3), we have carried out ab initio molecular dynamics simulations using ADMP method [72]. These simulations are done at 298 K temperature and 1 atm pressure for 2000 fs time. For brevity, the time evolution of total energy for the eclipsed isomer 1 computed at the ωB97XD/def2-TZVP level for 2000 fs time scale is shown in Figure 3. Similar plots for isomers 2 and 3 are shown in the supporting information. To clearly depict the alteration of the structure over the 2000 fs of time, snapshots at 400 fs interval have been added. Considering the low-energy gap among the three different forms, it is not surprising to see that the structure rotates quite freely and at the time of 1200 fs itself, one could notice a structure that is very close to the gauche-form (2). However, none of the structures broke altogether, which indicates that all these three different forms of [Fe(Si 2 -η 5 -C 5 H 2 ) 2 ] 2+ are kinetically stable.

Conclusions
In conclusion, three ferrocene derivatives (1-3) are theoretically identified here using DFT at different levels. All of them contain two pentacoordinate carbon atoms. NBO and AdNDP analyses confirm that they are indeed pentacoordinate carbons. Ab initio MD simulations carried out at 298 K temperature for 2000 fs assert that they are kinetically stable molecules. For each pentacoordinate carbon, one 3c-2e Si-C-Si σ-bond (total two for each molecule) and three 19c-2e dπ-pπ-bonds (total six for each molecule) exist and they follow the existing pattern of (2σ + 6π)-dual aromaticity, which is a well-established pattern in ptC and hypercoordinate carbon molecules. Since the ligand here (Si 2 C 5 H 2 ) contains a ptC atom before complexation, these results are surprising. Theoretically, this new class of molecules could be termed "hypercoordinate metallocenes".
Supplementary Materials: The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/chemistry4040074/s1, Supplementary file S1: Supporting information for this paper.  Acknowledgments: Additional computational support provided to SS and KT at VIT, Vellore, India, is also gratefully acknowledged. VST thanks Andrew L. Cooksy (SDSU, San Diego) for providing additional computing time.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: