In Silico Studies on Selected Neutral Molecules, CGa2Ge2, CAlGaGe2, and CSiGa2Ge Containing Planar Tetracoordinate Carbon

Density functional theory (DFT) was used to study the structure, stability, and bonding in some selected neutral pentaatomic systems, viz., CGa2Ge2, CAlGaGe2, and CSiGa2Ge containing planar tetracoordinate carbon. The systems are kinetically stable, as predicted from the ab initio molecular dynamics simulations. The natural bond orbital (NBO) analysis showed that strong electron donation occurs to the central planar carbon atom by the peripheral atoms in all the studied systems. From the nucleus independent chemical shift (NICS) analysis, it is shown that the systems possess both σ- and π- aromaticity. The presence of 18 valence electrons in these systems, in their neutral form, appears to be important for their stability with planar geometries rather than tetrahedral structures. The nature of bonding is understood through the adaptive natural density partitioning analysis (AdNDP), quantum theory of atoms in molecules (QTAIM) analysis, and also via Wiberg bond index (WBI) and electron localization function (ELF).


Introduction
Nowadays, planar tetracoordinate carbon (ptC) molecules showed great attention to experimentalists [1][2][3][4][5][6][7] as well as theoreticians [8][9][10][11][12][13][14][15] because it deviates from the usual ideas of tetrahedral tetracoordinate carbon. The tetrahedral carbon concept was given by van't Hoff and Le Bel in 1874 [16,17]. The idea of the ptC was introduced in 1968 by H. J. Monkhorst as a transition state of methane in a non-dissociative racemization process [18]. After two years, Hoffmann and co-workers showed two strategies for the stabilization of molecules containing ptC and those methods constitute a useful guide to design new ptC molecules [8,19]. The first one is the electronic strategy that involves appropriate substituents with σ-donors and π-acceptors simultaneously to stabilize the ptC species. The second one is the mechanical strategy that involves aromatic systems, cages, and transition metals to design and stabilize ptC systems. Using those two proposals, Schleyer and co-workers, in 1976, designed lithium substituted cyclopropane (C 3 H 3 Li 2 ), the first insilico studied ptC molecule [9]. After that, many theoretical studies from different research groups were reported on ptC systems [12,[20][21][22][23][24][25][26][27]. In a theoretical work in 1991, Schleyer and Boldyrev proposed cis and trans geometries of CSi 2 Al 2 molecules containing ptC [28]. In order to understand the stability of the ptC systems, the "18 valence electron rule" was composed and many systems were shown to obey this rule [2]. However, there are some examples in which more or less than 18 valence electrons are present and the ptC systems are still stable [29,30]. One of the most recent theoretical work shows that the CAl 4 Mg system in its neutral and mono-anionic state show global minima with ptC [30]. The neutral ptC system shows 18 valence electrons and the anionic one possesses 19 valence electrons.
Boldyrev and Simons [2] studied on three neutral pentaatomic molecules (CSi 2 Al 2 , CSi 2 Ga 2 , and CGe 2 Al 2 ) and showed the planar tetracoordinate central carbon (ptC) in these systems. They used molecular orbital analysis to rationalize the preference of the planar geometry over the tetrahedral one and they concluded that molecules containing one central C and two Si or Ge ligand atoms and two Al or Ga ligand atoms should have stable planar geometries. Inspired by this work, we designed three neutral pentaatomic systems, viz., CGa 2 Ge 2 , CAlGaGe 2, and CSiGa 2 Ge, containing 18 valence electrons. Both cis and trans isomers of the systems are considered in this present work. The electronic interaction among the middle carbon atom and the peripheral atoms is responsible for the planar arrangement of these molecules obeying the "electronic strategy method" for their stabilization. All these three systems follow the 18 valence electrons rule to gain stability in planar form. With the help of the DFT method, we studied the geometries, stability, and the nature of bonding in the molecules. Now, the nature of bonding includes molecular orbital analysis, AdNDP analysis, NBO analysis, and electron density analysis.

Computational Details
The geometry optimization and the subsequent frequency calculation of all the systems considered in this study were carried out using BP86 functional [40,41] coupled with D3 correction [42]. The BP86 functional is one of the commonly used DFT methods for electronic structure calculations for main group elements. The correlation consistent tripleζ quality basis set has been taken for this computation. For C, Al, and Si atoms cc-pVTZ basis set and for Ga and Ge atoms cc-pVTZ-PP basis set along with relativistic effective core potential (RECPs), ECP10MDF have been used [43][44][45][46]. All real harmonic vibrational frequencies correspond to the energy minima on the potential energy surfaces, as observed. Using the same computational method the aromaticity analysis of the studied ptC systems was performed, with the help of NICS. All of these computations were performed with the help of the Gaussian 16 program package [47].
The atom-centered density matrix propagation (ADMP) simulation [48][49][50] for the studied systems was performed to obtain information about their kinetic stability at 300 K and 500 K temperatures, and 1 atm pressure over 2000 fs of simulation time. The simulation was performed with the BP86-D3/def2-SVP method by taking the initially optimized geometries.
To know about the charge distribution in the systems and the Wiberg bond indices (WBI) [51] between any two atoms connected through a bond, natural bond orbital (NBO) [52] analysis was performed using NBO 3.1 [52,53] as implemented in Gaussian 16, using the same computational method that was used for the geometry optimization.
The atoms-in-molecules (AIM) [54] analysis was carried out at the same level that was used for the geometry optimization, using the Multiwfn program package [55]. We have generated various bond critical points (BCPs) and also analyzed the ELF basin [56].

Geometries
The optimized geometries of the cyclic rings (L) and the ptC systems, both in cis and trans orientations, are given in Figure 1. The L and ptC systems were optimized without any symmetry constraint. In this work, we considered only the singlet state of the ptC systems because the higher spin states correspond to a higher energy than the singlet state. Moreover, the mono-anionic and mono-cationic states of these ptC systems are not planar. So, we considered only neutral states of the studied systems. For all L systems, cis isomers of AlGaGe 2 and SiGa 2 Ge rings have C s symmetry and the cis Ga 2 Ge 2 ring has C 2v symmetry. For AlGaGe 2 and SiGa 2 Ge rings, the trans isomers correspond to the C 2v point group of symmetry while, for the Ga 2 Ge 2 ring, the trans isomer has D 2h symmetry. The carbon atom at the middle of the rings bonded with four peripheral atoms of each L and the ptC systems were generated. We took the tetrahedral structures of the systems as an initial guess but, after optimization, the structures are converted to more stable planar forms. The energy differences between cis and trans isomers are 2.45, 3.70, and 1.50 kcal/mol for CGa 2 Ge 2 , CAlGaGe 2 , and CSiGa 2 Ge systems, respectively. At the second-order Møller−Plesset perturbation (MP2) method [57] with the same basis set, the energy differences are 1.40, 2.27, and 0.72 kcal/mol for CGa 2 Ge 2 , CAlGaGe 2 , and CSiGa 2 Ge systems, respectively. So, we considered both isomers throughout the work. The trans isomer of the CGa 2 Ge 2 system has D 2h point group of symmetry while the trans isomers of CAlGaGe 2 and CSiGa 2 Ge systems have C 2v point group. However, the cis isomers of CAlGaGe 2 and CSiGa 2 Ge systems have C s symmetry, and the cis CGa 2 Ge 2 system has C 2v symmetry. The bond distances for all the bonds in the systems are presented in Figure 1. Boldyrev and Simons [2] explained that tetrahedral structures of their systems are Jahn-Teller unstable and subsequent distortion should lead to a planar structure. For this purpose, they compared the occupancy pattern of the valence MOs of tetrahedral CF 4 molecule with the tetrahedral structures of their systems. The CF 4 molecule is the 32 valence electronic system and the occupancy pattern of the occupied MOs is 1a 1 2 1t 2 6 2a 1 2 2t 2 6 1e 4 3t 2 6 1t 1 6 . They assumed that other tetrahedral molecules or nearly tetrahedral structures would follow this occupancy pattern (except for symmetry-imposed degeneracies) and that the 18 valence electronic tetrahedral structures showed a 1a 1 2 1t 2 6 2a 1 2 2t 2 6 1e 2 pattern of occupancy. Due to this partially filled e orbital, the tetrahedral structures of their systems show Jahn-Teller instability and obtain distorted to a planar structure. Our designed systems are very similar to the systems studied by Boldyrev and Simons. The systems have 18 valence electrons and the carbon atom is present at the central position. So, our systems in tetrahedral structures also show a similar occupancy pattern of the occupied MOs and that is 1a 1 2 1t 2 6 2a 1 2 2t 2 6 1e 2 . Due to this partially filled e orbital the tetrahedral structures of our designed systems show Jahn-Teller instability and distorted to a planar structure. So, considering Jahn-Teller distortion the ground state of our systems show planar geometries.
Ab initio molecular dynamics simulations were performed at 300 K and 500 K temperatures and 1 atm pressure over 2000 fs of time to check the kinetic stability for both isomers using ADMP approach. The time evolution of energy of all the systems is presented in Figure 2. The oscillation in the plots arises because of the increase in the nuclear kinetic energy during structural deformation throughout the whole simulation. Some snapshots at different time steps of the simulation for cis and trans isomers are given in Figures 3 and 4, respectively, to show the structural deformation of these systems at these temperatures. The steady fluctuations in energy and consistency in the geometry suggest the kinetic stability of these systems both in cis and trans isomers at these temperatures.

Molecular Orbitals
The molecular orbital analysis for the systems was carried out to obtain more information about the bonding. The molecular orbitals along with associated energies for all the systems in cis and trans isomers are presented in Figure 5. The lowest unoccupied molecular orbitals (LUMOs) are the π-type orbitals distributed over the four membered rings. The HOMO-3 molecular orbitals in the systems are π-type molecular orbitals and are delocalized over the whole molecule. In the HOMO-3 π-delocalized MO of the systems, significant contribution comes from the perpendicular 2p z orbital and 2p x orbital of central ptC in cis and trans isomers, respectively. This σand πdelocalization of electron density support the electronic stabilization in the designed ptC systems. We discussed the electronic delocalization within the systems based on multi-center-2e bonding in the AdNDP section in detail. The highest occupied molecular orbital (HOMO), HOMO-1 and HOMO-2 orbitals, are σ-type molecular orbitals. The energy differences (∆E H-L ) between the HOMO and the LUMO are 1.99, 1.80, 1.97 eV, and 1.26, 0.98, 1.28 eV, for cis and trans isomers for CGa 2 Ge 2 , CAlGaGe 2 , and CSiGa 2 Ge systems, respectively. The cis isomers showed higher ∆E H-L values than that of the trans isomers indicating greater stability [58][59][60].

Natural Bond Orbital (NBO) Analysis
The distribution of natural charge in the studied ptC molecules and the WBI values for a bond was analyzed using the natural bond orbital scheme. This analysis showed that a significant amount of charge transfer occurred from the peripheral atoms to the central carbon atom and the C atom is highly negative in each ptC system. The values of the natural charges on the atoms are shown in Table 1. The natural charges on central C atoms vary from −2.14 |e| to −2.23 |e| in the ptC systems and the peripheral atoms show positive charges. The CAlGaGe 2 system showed the highest negative charges and the CGa 2 Ge 2 system showed the lowest negative charges on the central C atom, respectively. The charges on C atoms in the ptC systems are the same or almost the same in both cis and trans isomers but the charges on the peripheral atoms are different for the two isomers. The valence electronic configurations of C atoms in the systems are presented in Table 1. It is well known that, in ptC systems, the central C atoms act as both a σ-acceptor and a π-donor. In our present study, the σ-acceptor is evident from the highly negative charge on C centers in all the systems (see Table 1). From the valence electronic configuration of C centers as shown in Table 1, in the case of cis isomers, the population of 2p z orbital is lower than that of the 2p x and 2p y orbitals, and, in trans isomers the population of 2p x orbital is lower than that of the 2p y and 2p z orbitals of all the ptC systems. These lower populations of 2p z and 2p x orbitals for cis and trans isomers, respectively, is a consequence of π-donation from the central carbon atom [34]. The cis isomers of the systems lie on the XY plane and the trans isomers of the systems lie on the YZ plane. The valence p orbitals along these perpendicular axes, i.e., the 2p z orbital and 2p x orbital of central carbon atom in the cis and trans isomers of the systems, respectively, take part in delocalized π-bonding. The WBI values for C-Al, C-Si, C-Ga, and C-Ge bonds were computed for both isomers of all the ptC systems, and the numerical values are presented in Table 2. The WBI values for the C-Si bonds are 1.09 and 1.18 for cis and trans isomers of CSiGa 2 Ge system, indicating pure covalent bonds. Similarly for C-Ge bonds, the WBI values are in the range of 0.99 to 1.04 for cis isomers and 1.09 to 1.13 for trans isomers of the systems. These WBI C-Ge values indicate the covalent nature of the bonds. The WBI values for C-Al and C-Ga bonds are comparatively lower than that of the C-Si and C-Ge bonds, indicating the partial covalent character of these bonds. The WBI values for the C-Ga bonds are comparable with that reported in the theoretical study by Zhou et al. [61]. The WBI values for the C-Si and C-Ge bonds in trans isomers are a little bit higher than that in the cis isomers. However, for C-Al and C-Ga bonds, the WBI values in cis isomers are a little bit higher than those in the trans-isomers.

Adaptive Natural Density Partitioning (AdNDP) Analysis
We carried out AdNDP analysis [62,63] in bonding context for both cis and trans isomers of the studied systems. We used this technique to investigate the presence of n-center-two-electron (nc-2e) bonds (n ranges from 1 (lone pair) to the maximum number of atoms (completely delocalized bonding)). Figures 6 and 7 represent the results from this analysis for cis and trans isomers, respectively. The results showed the appearance of four lone pairs (1-center-two-electron bonds) on the four peripheral atoms and the occupation numbers (ON) are in between 1.91-1.96 |e| and 1.90-1.96 |e| in cis and trans isomers, respectively. There are two 2-center-two-electron (2c-2e) σ bonds in both isomers of the systems. Additionally, one 3-center-two-electron (3c-2e) σ bond, one 4-center-twoelectron (4c-2e) σ bond, and one 5-center-two-electron (5c-2e) π-bond in cis isomers; and one 4-center-two-electron (4c-2e) σ bond, one 5-center-two-electron (5c-2e) σ bond, and one 5-center-two-electron (5c-2e) π-bond in trans isomers are present for all the systems. The occupation numbers of these nc-2e bonds in the systems under consideration are presented in Figures 6 and 7 for cis and trans isomers, respectively. This analysis shows that in the designed ptC systems the central carbon atom is present in the 2π/6σ framework. Thus, the 2π/6σ frameworks and peripheral 4c-2e σ bonds are crucial for the stability of these ptC species. Both the σand πdelocalization of electron densities supported the stability of the designed system in planar form.

Atoms in Molecule (AIM) Analysis
We generated various electron density descriptors at all the (3, −1) bond critical points (BCPs) of the studied systems, and the numerical values are shown in Table 3. The contour plots of ∇ 2 ρ(r) along the bond paths for the systems are presented in Figure 8. From the presented BCPs in Table 3, the existence of bonds between the middle carbon atom (C) and the four peripheral atoms is confirmed in all the systems. The low ρ(r c ) values and positive ∇ 2 ρ(r c ) values suggest that there is a closed-shell type of bonding. From the negative values of H(r c ) at the corresponding BCPs, it may be argued that the bonds have a partial covalent character. The values of −G(r c )/V(r c ) at different BCPs are between 0.5 and 1.0, indicating the absence of a noncovalent interaction. Table 3. Electron Density (ρ(r c )), Laplacian of Electron Density (∇ 2 ρ(r c )), Kinetic Energy Density (G(r c )), Potential Energy Density (V(r c )), Total Energy Density (H(r c )) for the ptC systems.   The ELF plots for both the isomers of the systems were generated, and the plots are presented in Figure 9. The ELF plots show the interaction among the middle C atom and the peripheral atoms and the electron densities are delocalized within the whole molecule. These ELF plots are in support of stability of the systems in planar geometries in terms of electron delocalization within the ptC systems.

Aromaticity Analysis
Aromatic molecules are cyclic, planar, conjugated, and have (4n + 2) π electrons. To predict the aromaticity of molecules, NICS is an efficient criterion proposed by Schleyer and co-workers [64,65]. NICS is calculated based on the use of magnetic shielding tensor for a dummy magnetic dipole considered at the center of an under-examination ring. It is called a nucleus-independent chemical shift, because there is no nucleus at the center of the molecular ring to experience the effective magnetic field. Molecules are considered aromatic based on the negative NICS values, and the positive NICS values indicate anti-aromatic systems. The non-aromatic systems have NICS values close to zero.
We computed both NICS(0) and NICS(1) at the middle of each triangle by placing a ghost atom at the plane and 1 Å above the plane. We also computed NICS(1) of the four membered ring by placing the ghost atom at 1 Å above the central C atom for all the systems for cis and trans isomers. The results of this analysis are presented in Figure 10. Both the NICS(0) and NICS(1) values for each triangle and the NICS(1) values at 1 Å above the central C atom of the ring are negative. These negative values of NICS(0) and NICS (1) are the indicators of σand πaromaticity, albeit approximately. In aromatic systems, the electron densities are delocalized throughout the molecule. In all of our designed systems, the electron density is delocalized, as shown from the molecular orbitals, ELF plots, and the multi-center-2e-bonds of all the systems. So, these computed negative NICS values support the stability of the designed ptC systems.

Summary and Conclusions
The three neutral ptC systems considered in this work possess 18 valence electrons and show stability in planar form. The stability of both cis and trans isomers of these systems is reported in this work. The NICS calculations showed that the systems possess both σand πaromaticities that support the stability of the designed systems. The ADMP simulation at 300 K and 500 K temperatures over 2000 fs of time revealed the kinetic stability of the molecules. The AIM analysis suggested the presence of bonds between the central C atom and the four peripheral atoms. We believe that this theoretical study will help in designing new ptC molecules.
Experimentally ptC systems were characterized in the gas phase. It is also our hope that the designed ptC systems could be interesting to the experimentalists. Considering all the supported results for the planar geometries of these studied systems, the experimental realization of these ptC systems could be possible in the gas phase or matrix isolation. In isolation, the systems may exist in planar form. When the systems are interacting with other species, the structures of the systems may or may not be planar; however, if the bound species create sufficient steric forces, the systems may exist in planar form.