Bonding Analysis of Compounds with Unusual Coordination of Carbon: Proposed Symmetric Systems with Six-Coordinate Carbon

The possibility of carbon tetravalence in geometries other than tetrahedral and of carbon hypervalence has been taken seriously since the 1970s. Computational modeling and subsequent experimental validation have established the existence of molecules with carbon atoms with planar tetravalence and as many as six objects in carbon’s coordination sphere. In this work, we develop insight into the nature of bonding to carbon in these unusual environs as provided by Bader’s Atoms in Molecules (AIM) analysis of the electron density, along with the electron localization function (ELF) and the non-covalent index (NCI). We review several well-established systems (spiropentadiene dication, hexamethyl benzene dication, dimethanospiro[2.2]octaplane dication, and 1,8-dimethoxy-9-dimethoxyanthracene cation) and propose new D2d–symmetric variants of a hexacoordinated species.

Radom's alkaplane systems [14] rely entirely on strain to force the central CC4 fragment toward planar tetracoordination of Carbon. The tpC with its immediate C neighbors in D2h dimethanospiro [2.2] octaplane 3 is shown in Figure 1. However, the stability of the neutral D2hsymmetric form is questionable. Radom and Rasmussen [13] report that a MP2/6-31G(d) calculation (by Mark S. Gordon, unpublished) produces all-real vibrational frequencies, suggesting the D2h structure occupies a relative minimum on the potential surface. B3LYP/6-31G(d) calculation and our ωB97XD/cc-pVTZ calculation both produce a substantial imaginary frequency (ca. 300i) suggesting the symmetric form is a transition state between two twisted forms of the CC4 fragment in the dimethanospiro [2.2]octaplane. Figure 2 displays the isosurface of the HOMO for neutral 3, which is dominated by the troublesome C lone pair. This form of the HOMO is in general agreement with results of calculations at lower levels of theory [14,15].
The planarity can be established by ionization; the monocation radical has a planar tpC [15]. We have confirmed that complete removal of two electrons from the HOMO more firmly establishes stability of the D2h symmetry of Radom's dimethanospiro [2.2]octaplane dication.
Substitution of two Carbon atoms by Boron atoms, forming systems isoelectronic with the hydrocarbon dications, accomplishes planarization. This formation of neutral analogs to hydrocarbon cations is called the charge compensation principle by its inventors [13] and is applied to this issue by Wang and Schleyer [16]. A tactic recommended by Hoffmann [7,8] is to surround the central carbon by species which are strong sigma donors and pi acceptors, which then may effect redistribute the troublesome planar-Carbon lone pair toward its surroundings. A series of molecules incorporating Copper and Nickel substituents on the tpC investigated by Schleyer and Boldyrev [17]  Radom's alkaplane systems [14] rely entirely on strain to force the central CC 4 fragment toward planar tetracoordination of Carbon. The tpC with its immediate C neighbors in D 2h dimethanospiro [2.2] octaplane 3 is shown in Figure 1. However, the stability of the neutral D 2h -symmetric form is questionable. Radom and Rasmussen [14] report that a MP2/6-31G(d) calculation (by Mark S. Gordon, unpublished) produces all-real vibrational frequencies, suggesting the D 2h structure occupies a relative minimum on the potential surface. B3LYP/6-31G(d) calculation and our ωB97XD/cc-pVTZ calculation both produce a substantial imaginary frequency (ca. 300i) suggesting the symmetric form is a transition state between two twisted forms of the CC 4 fragment in the dimethanospiro [2.2]octaplane. Figure 2 displays the isosurface of the HOMO for neutral 3, which is dominated by the troublesome C lone pair. This form of the HOMO is in general agreement with results of calculations at lower levels of theory [14,15].
The same stratagem has prompted computational studies of planar penta-and hexa-coordinated Carbon. For example, Exner and Schleyer [22] used B3LYP and CCSD(T)/6-311+G(d,p) to describe a planar CB6 2− system as well as isoelectronic and neutral C3B4 and other variants. In this report we confine our attention to metal-free systems and examine hypercoordinated Carbon atoms with O and S members in the coordination sphere. The coordination sets in our systems are often nonplanar. Our purpose is to characterize the interactions of the hypercoordinated carbon with its neighbors. To assist intuition we employ diagnostics and visualization aids from Bader's Atoms-in-Molecules theory [23], and related quantities such as the electron localization function [24]  The planarity can be established by ionization; the monocation radical has a planar tpC [15]. We have confirmed that complete removal of two electrons from the HOMO more firmly establishes stability of the D 2h symmetry of Radom's dimethanospiro [2.2]octaplane dication.
Substitution of two Carbon atoms by Boron atoms, forming systems isoelectronic with the hydrocarbon dications, accomplishes planarization. This formation of neutral analogs to hydrocarbon cations is called the charge compensation principle by its inventors [13] and is applied to this issue by Wang and Schleyer [16]. A tactic recommended by Hoffmann [7,8] is to surround the central carbon by species which are strong sigma donors and pi acceptors, which then may effect redistribute the troublesome planar-Carbon lone pair toward its surroundings. A series of molecules incorporating Copper and Nickel substituents on the tpC investigated by Schleyer and Boldyrev [17] reverse this motif, calling for strong pi-donor and sigma-acceptors on a periphery. A central example is CAl 4 anion [16]. Structures with many other metals have been evaluated computationally [18][19][20][21].
The same stratagem has prompted computational studies of planar penta-and hexa-coordinated Carbon. For example, Exner and Schleyer [22] used B3LYP and CCSD(T)/6-311+G(d,p) to describe a planar CB 6 2− system as well as isoelectronic and neutral C 3 B 4 and other variants.
In this report we confine our attention to metal-free systems and examine hypercoordinated Carbon atoms with O and S members in the coordination sphere. The coordination sets in our systems are often nonplanar. Our purpose is to characterize the interactions of the hypercoordinated carbon with its neighbors. To assist intuition we employ diagnostics and visualization aids from Bader's Atoms-in-Molecules theory [23], and related quantities such as the electron localization function [24] and the non-covalent index [25].

Software
We use Gaussian 09 [26] and 16 [27] for electronic structure calculations including geometry optimization and vibrational analysis. The density functional model is ωB97XD [28] and the basis is cc-pVTZ [29] except as otherwise noted. With one exception (5) in which the experimental (X-ray) structure is used, all reported structures are local minima according to ωB97XD/cc-pVTZ, with no imaginary vibrational frequencies. Bader AIM analysis is conducted by AIMALL [30] and Multiwfn software [31]. Construction of the electron localization function (ELF) and the non-covalent index (NCI) is accomplished with Multiwfn and NCIplot [32]. In the following sections we describe AIM, ELF, and NCI methods of characterizing the electron density. Those familiar with these analyses may wish to pass directly to Section 3, in which we describe applications.

AIM and Bader Terminology
The Atoms-in-Molecules (AIM) analysis characterizes bonding with reference to the electron density, an observable of a molecular system. The review by Kumar et al. is a useful introduction, with many examples [33]. At critical points where ∇ρ = 0 we can evaluate the charge density ρ, its Laplacian L = ∇ 2 ρ, and the eigenvalues {λ k } of the matrix {H ij } = {∂ 2 ρ/∂x i ∂x j } along with various energy densities. Diagonalization of H yields three eigenvalues. There can be three negative eigenvalues, defining NCP nuclear critical points (attractors) coded (3, −3); two negative and one positive value defining BCP bond critical points, coded (3, −1), one negative and two positive values defining RCP ring critical points coded (3, +1), or three positive values defining CCP cage critical points coded (3, +3). The number of critical points of all kinds should obey the Poincare-Hopf rule, which in this case requires that NCP − BCP + RCP − CCP = 1. This is the case for all systems described here.
AIM divides Cartesian space into basins, each containing an attractor-most often a nucleus. The boundaries of the basins are defined by a zero-flux (∇ρ = 0) surface. Integration of electronic charge within a basin including a nucleus produces an AIM atomic charge.
For familiar covalent chemical bonds, the electron density ρ at a BCP would be on the order of tenths of density units, electrons per cubic bohr. For weak mainly electrostatic interactions such as H bonds, the density at the BCP would be much smaller (at best, hundredths of atomic units) and the Laplacian would be positive. In all the examples to follow, we find the usual array of nuclear attractors with very large densities and large negative Laplacian values; BCPs for covalent sigma bonds; nonpolar aromatic CC bonds in the anthryl rings; and polar bonds with smaller densities. In AIM analysis the kinetic and potential energy densities G and V at the bond critical points offer insight into the nature of bonding. A useful diagnostic of covalent bonding is the ratio |2G(r bcp )/V(r bcp )|; if this ratio is less than unity then the interaction is covalent [34].
While AIM provides a well-defined methodology for processing molecular electron density (obtained either by theoretical methods or by processing experimental data) its link to intuitive chemical constructs-especially bonds and atomic charges-has been the subject of considerable debate. We refer to only a very small sample, which will convey the flavor [35][36][37][38]. We judge that reductio ad absurdum does not undermine the practical value of QTAIM.

ELF
The electronic localization function (ELF) was defined by Becke and Edgecombe [24]. The probability density that a second like-spin electron is found near a "localized" electron is presumably small. This density arising from Fermions in the molecular system is written as: The sum is over all the occupied molecular spinorbitals; ρ σ (r) is the electron density of electrons with σ spin which in the Kohn-Sham approximation is given by ρ σ (r) = Σ σ i ψ i 2 . The smaller the quantity D, the more highly localized is the reference electron. A dimensionless index χ is defined as the ratio of the density D to the corresponding value for the uniform electron gas: ELF is confined to the range [0, 1] by the transform ELF = 1/(1 + χ 2 ). The limit of maximum localization is χ = 1.

NCI
A useful description of the non-covalent index, its graphical characterization, and its interpretation is given by Contreras-Garcia, et al. [39]. The reduced (dimensionless) density gradient s (also called RDG) is defined as Two NCI graphic realizations are useful. A two-dimensional plot of s vs. the density ρ (given the sign of the second eigenvalue of H) will produce a broad sweep from one extreme at small density but large s to large density and small s. This is characteristic of an exponentially-decaying density. Superimposed on this sweep may be spikes in s extending to values near zero at modest values of density. In Figure 3, plots derived from ωB97XD/cc-pVTZ computations for water dimer and formic acid dimer are shown. The high-density (but low s) regions near nuclei are far to the right or left of the range of densities shown. The low-density region is located primarily in Cartesian space close to the density tails for which s is large. However, s can be small precisely where the density is disturbed by non-covalent interactions, such as van der Waals/dispersion and electrostatic forces. Figure 3 shows the plot for water dimer and formic acid dimer. For water dimer, we find one such case at signed density about −0.025, and for formic acid dimer we find two, at +0.01 and −0.05. Molecules 2020, 25, x FOR PEER REVIEW 5 of 24 Mapping those zones of small density and small s (RDG) into three-dimensional space ( Figure  4), we isolate regions of significant NCI in isosurfaces of value 0.05. For water dimer there is a single zone of non-covalent interaction, which we must associate with H-bonding. One spike from Figure 3 can refer to more than one zone. In the formic acid dimer (Figure 4), we find two attractive H-bonding NCI regions and also a destabilizing zone corresponding to a ring critical point.

Example 1: Spiropentadiene Dication
Our ωB97XD/cc-pVTZ calculations and earlier reports [9] verify that D2h-symmetric spiropentadiene dication 1 occupies a relative minimum energy point. (The unsubstituted monocation and neutral counterparts however are nonplanar.) The neutral system can be constrained to coplanarity of the three-membered rings by enclosing the spiropentadiene core within aromatic seven-membered rings [40]. In this case the troublesome electron pair is exported from the p-pi AO of tpC into the aromatic region.

AIM Analysis
Firme et al. [41,42] presented an AIM analysis of the planar dication of spiropentadiene. Critical points and bond paths from our AIM2000 analysis of the density obtained with ωB97XD/cc-pVTZ are Mapping those zones of small density and small s (RDG) into three-dimensional space (Figure 4), we isolate regions of significant NCI in isosurfaces of value 0.05. For water dimer there is a single zone of non-covalent interaction, which we must associate with H-bonding. One spike from Figure 3 can refer to more than one zone. In the formic acid dimer (Figure 4), we find two attractive H-bonding NCI regions and also a destabilizing zone corresponding to a ring critical point.  Mapping those zones of small density and small s (RDG) into three-dimensional space ( Figure  4), we isolate regions of significant NCI in isosurfaces of value 0.05. For water dimer there is a single zone of non-covalent interaction, which we must associate with H-bonding. One spike from Figure 3 can refer to more than one zone. In the formic acid dimer (Figure 4), we find two attractive H-bonding NCI regions and also a destabilizing zone corresponding to a ring critical point.

Example 1: Spiropentadiene Dication
Our ωB97XD/cc-pVTZ calculations and earlier reports [9] verify that D2h-symmetric spiropentadiene dication 1 occupies a relative minimum energy point. (The unsubstituted monocation and neutral counterparts however are nonplanar.) The neutral system can be constrained to coplanarity of the three-membered rings by enclosing the spiropentadiene core within aromatic seven-membered rings [40]. In this case the troublesome electron pair is exported from the p-pi AO of tpC into the aromatic region.

AIM Analysis
Firme et al. [41,42] presented an AIM analysis of the planar dication of spiropentadiene. Critical points and bond paths from our AIM2000 analysis of the density obtained with ωB97XD/cc-pVTZ are

Example 1: Spiropentadiene Dication
Our ωB97XD/cc-pVTZ calculations and earlier reports [10] verify that D 2h -symmetric spiropentadiene dication 1 occupies a relative minimum energy point. (The unsubstituted monocation and neutral counterparts however are nonplanar.) The neutral system can be constrained to coplanarity of the three-membered rings by enclosing the spiropentadiene core within aromatic seven-membered rings [40]. In this case the troublesome electron pair is exported from the p-pi AO of tpC into the aromatic region.

AIM Analysis
Firme et al. [41,42] presented an AIM analysis of the planar dication of spiropentadiene. Critical points and bond paths from our AIM2000 analysis of the density obtained with ωB97XD/cc-pVTZ are shown in Figure 5. Values of density, the Laplacian, G and V at selected BCPs are shown in Table 1. Entries in red are from Firme, et al. [41] and agree pretty nearly with our ωB97XD/cc-pVTZ values. We see that the BCPs along the paths from a peripheral Carbon to the tetracoordinate planar Carbon have much smaller values of the Laplacian of the density compared with the values at BCPs along the two bond paths connecting peripheral C-Cs. We infer that the tpC-Cx bond is weaker than the Cx-Cx bond. It is remarkable that the tpC is has a net negative AIM basin charge (−0.193 absolute electrons or |e|); the positive charge in this dication is dispersed to the surrounding C and H atoms (+0.169 and +0.379 |e| respectively.) Molecules 2020, 25, x FOR PEER REVIEW 6 of 24 shown in Figure 5. Values of density, the Laplacian, G and V at selected BCPs are shown in Table 1. Entries in red are from Firme, et al. [41] and agree pretty nearly with our ωB97XD/cc-pVTZ values. We see that the BCPs along the paths from a peripheral Carbon to the tetracoordinate planar Carbon have much smaller values of the Laplacian of the density compared with the values at BCPs along the two bond paths connecting peripheral C-Cs. We infer that the tpC-Cx bond is weaker than the Cx-Cx bond. It is remarkable that the tpC is has a negative AIM basin charge (−0.193 absolute electrons or |e|); the positive charge in this dication is dispersed to the surrounding C and H atoms (+0.169 and +0.379 |e| respectively.)

ELF Analysis
The electron localization function shown in Figure 6 displays the four bonding regions C to H and the two single CC bonds, top and bottom. The pi character of the C=C bonds is visible at the bottom. The ELF zones for the CH bonds are extensive, compared with CC bonds. From the AIM basin charges, ( Table 1) it appears that upon double ionization electron density from the HC=CH fragments has migrated so that tcC obtains a net negative charge.

ELF Analysis
The electron localization function shown in Figure 6 displays the four bonding regions C to H and the two single CC bonds, top and bottom. The pi character of the C=C bonds is visible at the bottom. The ELF zones for the CH bonds are extensive, compared with CC bonds. From the AIM basin charges, ( Table 1) it appears that upon double ionization electron density from the HC=CH fragments has migrated so that tcC obtains a net negative charge.

NCI Analysis
The plot of the reduced density gradient s(ρ) against the density as signed by the second eigenvalue of the Laplacian of the density shows no departures from monotone decline in s(ρ) at left and right (See Figure S1, Supplementary Materials). That is, there are no zones of non-covalent interaction in this dication.

Example 2: Hexamethylbenzene Dication
The structure of hexamethylbenzene dication 2 found by ωB97XD/cc-pVTZ is a pentagonal pyramid, in which a -CMe cap subtends a five-membered ring. The carbon of the cap could be considered hexacoordinate, and termed hcC. The crystal structure has been reported [43] along with proton and C-13 NMR. Bonding is characterized by NICS [44] the system displays three-dimensional aromaticity. Apical to basal CC bond orders are small (ca. 0.5), and the apical (hexa-coordinate) Carbon employs only four electrons in bonding. That is, the octet rule is not violated. The values of the Laplacian at AIM bond critical points (shown in Table 2) suggest that the C (ring or apex) to C (methyl) bonds are conventional single bonds (the Laplacian is about −0.5, density about 0.27) while the CC bonds within the ring are appreciably stronger, (the Laplacian is about -0.8, density about 0.28). The BCPs for the bonds from apical C to ring Cs are distinct; density is about half that of the single covalent bonds just mentioned and the Laplacian > 0, suggesting that charge is expelled from that region, covalency is weak, and there is some electrostatic component to the interaction. Klein, Havenith, and Knizia [45] consider the compound to be best characterized as a donor-acceptor complex of anionic cyclopentadiene and a "highly Lewis-acidic Carbon atom… capable of acting as an electron-pair donor to a formal CH3 + group." This is an appealing way to describe the bonding partly because it lends itself so well to interpretation of bonding by means of an interaction diagram. The result of charge migration, as expressed in the basin charges of AIM, is to produce the base ring of five C atoms with associated basin charges near 6.02 electrons (|e|) on

NCI Analysis
The plot of the reduced density gradient s(ρ) against the density as signed by the second eigenvalue of the Laplacian of the density shows no departures from monotone decline in s(ρ) at left and right (See Figure S1, Supplementary Materials). That is, there are no zones of non-covalent interaction in this dication.

Example 2: Hexamethylbenzene Dication
The structure of hexamethylbenzene dication 2 found by ωB97XD/cc-pVTZ is a pentagonal pyramid, in which a -CMe cap subtends a five-membered ring. The carbon of the cap could be considered hexacoordinate, and termed hcC. The crystal structure has been reported [43] along with proton and C-13 NMR. Bonding is characterized by NICS [44] the system displays three-dimensional aromaticity. Apical to basal CC bond orders are small (ca. 0.5), and the apical (hexa-coordinate) Carbon employs only four electrons in bonding. That is, the octet rule is not violated. average, with an outer ring of five methyl C atoms with basin charges near 6.037 on average. The basin charge corresponding to the apical six-coordinate C atom is 6.088, and its methyl C basin charge is 6.016 (|e|). That is, even in this dication the C frame is net negative in charge. The positive charge making up the net +2 is borne by H atoms.    The values of the Laplacian at AIM bond critical points (shown in Table 2) suggest that the C (ring or apex) to C (methyl) bonds are conventional single bonds (the Laplacian is about −0.5, density about 0.27) while the CC bonds within the ring are appreciably stronger, (the Laplacian is about −0.8, density about 0.28). The BCPs for the bonds from apical C to ring Cs are distinct; density is about half that of the single covalent bonds just mentioned and the Laplacian >0, suggesting that charge is expelled from that region, covalency is weak, and there is some electrostatic component to the interaction. Klein, Havenith, and Knizia [45] consider the compound to be best characterized as a donor-acceptor complex of anionic cyclopentadiene and a "highly Lewis-acidic Carbon atom . . . capable of acting as an electron-pair donor to a formal CH 3 + group." This is an appealing way to describe the bonding partly because it lends itself so well to interpretation of bonding by means of an interaction diagram. The result of charge migration, as expressed in the basin charges of AIM, is to produce the base ring of five C atoms with associated basin charges near 6.02 electrons (|e|) on average, with an outer ring of five methyl C atoms with basin charges near 6.037 on average. The basin charge corresponding to the apical six-coordinate C atom is 6.088, and its methyl C basin charge is 6.016 (|e|). That is, even in this dication the C frame is net negative in charge. The positive charge making up the net +2 is borne by H atoms.   Figure 8 (bottom) displays non-covalent interaction zones for hexamethyl benzene dication. These include repulsions between adjacent methyl groups in the (CH3C)5 base ring, and a feature at the center of that ring. The plot of the reduced density gradient (RDG) s(ρ) vs. the signed density (Supplemental information) suggests that the NCI zones identify two kinds of weak van der Waals interactions, which we can see are the ring methyl-methyl interactions and the NCI zone at the center of the five-carbon ring.  Figure 8 (bottom) displays non-covalent interaction zones for hexamethyl benzene dication. These include repulsions between adjacent methyl groups in the (CH 3 C) 5 base ring, and a feature at the center of that ring. The plot of the reduced density gradient (RDG) s(ρ) vs. the signed density (Supplementary Materials) suggests that the NCI zones identify two kinds of weak van der Waals interactions, which we can see are the ring methyl-methyl interactions and the NCI zone at the center of the five-carbon ring.

Example 3: Dimethanospiro[2.2]octaplane
To our knowledge there is no experimental realization of the C 5 H 4 dication 1 itself, but Radom's ways of stabilizing this structure by steric constraint and charge delocalization might point the way toward a synthesis. The dimethanospiro[2.2]octaplane dication 3 discussed above contains a tetracoordinate planar Carbon (tpC) similar in structure to the spiropentadiene dication. The central CC 4 fragment is forced toward planarity primarily by the surrounding rigid cage structure. The constraint is most effective for the dication, and the stable minimum has D 2h symmetry 3.3.1. AIM Analysis Figure 9 displays bond critical points, (white), ring critical points (red) and cage critical points (blue). The Bader parameters for cage CC and CH bond critical points ( Table 3) are characteristic of saturated hydrocarbons. Note that there are bond critical points and bond paths between two pairs of C atoms coordinated to the tpC, consistent with the rectangular arrangement and the overall D 2h symmetry, as in spiropentadiene dication 1. As in the case for 1, the tpC has a negative AIM basin charge (−0.165 |e|); the positive charge in this dication is dispersed not substantially to the surrounding C atoms in the C 5 plane (+0.028 |e| each) but rather mainly to the constraining cage atoms.  Table 3) are characteristic of saturated hydrocarbons. Note that there are bond critical points and bond paths between two pairs of C atoms coordinated to the tpC, consistent with the rectangular arrangement and the overall D2h symmetry, as in spiropentadiene dication 1. As in the case for 1, the tpC has a negative AIM basin charge (−0.165 |e|); the positive charge in this dication is dispersed not substantially to the surrounding C atoms in the C5 plane (+0.028 |e| each) but rather mainly to the constraining cage atoms.

ELF Analysis
The ELF isosurface diagram shown in Figure 10 (left) reveals the essentially covalent character of the bonds of tpC to its neighbors and the absence of localized charge perpendicular to the plane containing the tpC and its four bonded C atoms. The lack of localized pi charge at tpC is the consequence of double ionization to the dication.

ELF Analysis
The ELF isosurface diagram shown in Figure 10 (left) reveals the essentially covalent character of the bonds of tpC to its neighbors and the absence of localized charge perpendicular to the plane containing the tpC and its four bonded C atoms. The lack of localized pi charge at tpC is the consequence of double ionization to the dication.

NCI Analysis
The plot of the reduced density gradient (RDG) s(ρ) vs. the signed density (Supplementary Materials) shows van der Waals features and also strongly repulsive interaction. The NCI zones shown in Figure 10 (right) lie at cage critical points and between the >CH2 bridges of the cyclooctane fragments at top and bottom of the structure. Curious NCI zones appear above and below the CC4 central plane.

Example 4: 1,8-Dimethoxy-9-Dimethoxymethylanthracene Cation
Akiba and co-workers [46][47][48] have reported synthesis and isolation of a compound with roughly trigonal bipyramidal five-coordinated carbon (tbpC). The C2v-symmetric structure from ωB97XD/cc-pVTZ is shown in Figure 1 as species 4. The X-ray analysis shows that the distance between the Anthryl methoxy Oxygen and the cationic 5-coordinated tbpC (2.44 ± 0.01 A) is smaller than the distance between the ipso carbons of anthracene to which they are attached (2.50 ± 0.02 A). Computational characterization [40] of the system also reflects a shortening, which suggests an attractive MeO…tbpC electrostatic interaction. Figure 11 shows BCPs and RCPs for 1,8-dimethoxy-9-dimethoxymethylanthracene cation (species 4 in Figure 1). The broken lines representing bond paths connecting anthryl-methoxy oxygens and the tbpC indicate a weak interaction. The Bader parameters for BCPs (Table 4) bear out the expectation that the tbpC…OMe interaction is ionic. The Lapacian's positive sign shows that charge is exported from the BCP, and the |2G/V| ratio indicates that the interaction is not covalent.

NCI Analysis
The plot of the reduced density gradient (RDG) s(ρ) vs. the signed density (Supplementary Materials) shows van der Waals features and also strongly repulsive interaction. The NCI zones shown in Figure 10 (right) lie at cage critical points and between the >CH 2 bridges of the cyclooctane fragments at top and bottom of the structure. Curious NCI zones appear above and below the CC4 central plane.

Example 4: 1,8-Dimethoxy-9-Dimethoxymethylanthracene Cation
Akiba and co-workers [46][47][48] have reported synthesis and isolation of a compound with roughly trigonal bipyramidal five-coordinated carbon (tbpC). The C 2v -symmetric structure from ωB97XD/cc-pVTZ is shown in Figure 1 as species 4. The X-ray analysis shows that the distance between the Anthryl methoxy Oxygen and the cationic 5-coordinated tbpC (2.44 ± 0.01 A) is smaller than the distance between the ipso carbons of anthracene to which they are attached (2.50 ± 0.02 A). Computational characterization [40] of the system also reflects a shortening, which suggests an attractive MeO . . . tbpC electrostatic interaction. Figure 11 shows BCPs and RCPs for 1,8-dimethoxy-9-dimethoxymethylanthracene cation (species 4 in Figure 1). The broken lines representing bond paths connecting anthryl-methoxy oxygens and the tbpC indicate a weak interaction. The Bader parameters for BCPs (Table 4) bear out the expectation that the interaction between MeO and tbpC is ionic. The Lapacian's positive sign shows that charge is exported from the BCP, and the |2G/V| ratio indicates that the interaction is not covalent. We have already suspected from the short distance between methoxy O and tbpC that the tbC cationic center exerts an electrostatic attraction on the methoxy oxygens. The basin charges for the tbpC (+1.539) and methoxy oxygens (−0.140) are consistent with this inference.

ELF Analysis
The view of the ELF isosurfaces for Akiba's five-coordinated C species (Figure 12, left) shows the tbpC-OMe (the clearest view is at lower left) and the OMe connected to the anthryl ring; there is no ELF between that OMe's O atom and the tbpC. This argues for an interaction dominated by electrostatics.

ELF Analysis
The view of the ELF isosurfaces for Akiba's five-coordinated C species (Figure 12, left) shows the tbpC-OMe (the clearest view is at lower left) and the OMe connected to the anthryl ring; there is no ELF between that OMe's O atom and the tbpC. This argues for an interaction dominated by electrostatics.

NCI Analysis
The NCI regions displayed in Figure 12 (right) include the interiors of aromatic rings and the interaction zone between the anthryl methoxy fragment and a nearby CH bond. There is an NCI zone between the OMe attached to the tbpC and the anthryl Carbon to which the tbpC is attached. Most significant, we see an NCI between each anthryl methoxy O atom and the tbpC. There is important interaction, but suggests there is no covalent bonding between the anthryl methoxy O atoms and the tbpC.

Proposed Symmetric Variants on a Known Example of Approximately Octahedral Coordination of Carbon
Yamaguchi et al. reported the synthesis of a system with six entities in the coordination sphere of a carbon atom [49].
The design principle is shown in Figure 13. In Figure 14 showing the X-ray structure of the dication with -SMe(+) bridges, (5) the golden rectangle encloses the central six-coordinate carbon (6cC) and its two allenic C neighbors, while the green arrow points to one of the four methoxy oxygen atoms in the coordination sphere of 6cC. The system is far from symmetrical, probably a consequence of packing requirements of the counterions. A variety of related hexacoordinated-C compounds have been explored computationally; for example, in the coordination sphere -OPh can replace -OMe as R and the bridging >SMe (+) can be replaced by >SO2, >CH, and >S as B (Figure 13). Substituents can also be chosen which influence the strength of interactions in the coordination sphere [50][51][52].
Since the behavior of this structure is complex, we address only a few details of its topology. Figure 15 displays the results of topological analysis of Yamaguchi's system, in its crystal geometry. As seen in earlier examples, bond critical points appear on bond paths (heavy black lines) corresponding to links in the molecular drawing. Ring critical points (in red) are found within each of the aromatic rings of the anthryl fragments. Fine white lines connect RCPs and BCPs. Heavy broken lines represent weaker interactions. The four methoxy oxygens are linked by such lines to the central hexacoordinate C. We also observe heavy broken lines connecting the methoxy oxygen atoms with H atoms of the thiomethyl bridging groups, among other surprising connections.

NCI Analysis
The NCI regions displayed in Figure 12 (right) include the interiors of aromatic rings and the interaction zone between the anthryl methoxy fragment and a nearby CH bond. There is an NCI zone between the OMe attached to the tbpC and the anthryl Carbon to which the tbpC is attached. Most significant, we see an NCI between each anthryl methoxy O atom and the tbpC. There is important interaction, but suggests there is no covalent bonding between the anthryl methoxy O atoms and the tbpC.

Proposed Symmetric Variants on a Known Example of Approximately Octahedral Coordination of Carbon
Yamaguchi et al. reported the synthesis of a system with six entities in the coordination sphere of a carbon atom [49].
The design principle is shown in Figure 13. In Figure 14 showing the X-ray structure of the dication with -SMe(+) bridges, (5) the golden rectangle encloses the central six-coordinate carbon (6cC) and its two allenic C neighbors, while the green arrow points to one of the four methoxy oxygen atoms in the coordination sphere of 6cC. The system is far from symmetrical, probably a consequence of packing requirements of the counterions. A variety of related hexacoordinated-C compounds have been explored computationally; for example, in the coordination sphere -OPh can replace -OMe as R and the bridging >SMe (+) can be replaced by >SO 2 , >CH, and >S as B (Figure 13). Substituents can also be chosen which influence the strength of interactions in the coordination sphere [50][51][52].
Molecules 2020, 25, x FOR PEER REVIEW 14 of 24 Figure 13. Design for hexacoordinate Carbon (hcC). B is a bridging group, and R is an electron donor. Figure 13. Design for hexacoordinate Carbon (hcC). B is a bridging group, and R is an electron donor. Figure 13. Design for hexacoordinate Carbon (hcC). B is a bridging group, and R is an electron donor. We may expect that the Bader parameters for the X-ray structure, obtained by analysis of a charge distribution that does not correspond to an energy minimum within that computational method, might be called into question. We should judge the results by their consistency with more rigorous calculations on similar systems, as described below. The density and Laplacian values at the BCPs for the allene fragment are conventional for double bonds, and the interaction between methoxy O atoms and the hcC is similar to that found in Akiba's system for its methoxy O atoms and the pcC. (Tables 4 and 5). That is, the BCPs for the bond path between MeO and the hcC have small positive ∇ 2 ρ values and |2G/V| ratios > 1. This indicates charge depletion at the BCP and predominantly e electrostatic interactions. Since the AIM charges of these O atoms and the hcC are both negative, we can confirm that the electrostatic interaction is repulsive contrary to the suggestion that the Since the behavior of this structure is complex, we address only a few details of its topology. Figure 15 displays the results of topological analysis of Yamaguchi's system, in its crystal geometry. As seen in earlier examples, bond critical points appear on bond paths (heavy black lines) corresponding to links in the molecular drawing. Ring critical points (in red) are found within each of the aromatic rings of the anthryl fragments. Fine white lines connect RCPs and BCPs. Heavy broken lines represent weaker interactions. The four methoxy oxygens are linked by such lines to the central hexacoordinate C. We also observe heavy broken lines connecting the methoxy oxygen atoms with H atoms of the thiomethyl bridging groups, among other surprising connections.   We may expect that the Bader parameters for the X-ray structure, obtained by analysis of a charge distribution that does not correspond to an energy minimum within that computational method, might be called into question. We should judge the results by their consistency with more rigorous calculations on similar systems, as described below. The density and Laplacian values at the BCPs for the allene fragment are conventional for double bonds, and the interaction between methoxy O atoms and the hcC is similar to that found in Akiba's system for its methoxy O atoms and the pcC. (Tables 4  and 5). That is, the BCPs for the bond path between MeO and the hcC have small positive ∇ 2 ρ values and |2G/V| ratios > 1. This indicates charge depletion at the BCP and predominantly e electrostatic interactions. Since the AIM charges of these O atoms and the hcC are both negative, we can confirm that the electrostatic interaction is repulsive contrary to the suggestion that the interaction is stabilizing by mixing of the methoxy O lone pair with the allyl π* MO and the -OMe fragments are forced into the hcC coordination sphere solely by steric constraint. Owing to the difficulty of visualization in this asymmetric system, we do not discuss the ELF and NCI figures. Instead we turn to discussion of our proposed symmetric variants.

A Neutral Symmetric System with Six-Coordinate Carbon
In the example 6 shown in Figure 16 the bridge is >C=O, replacing the >SMe(+) of the first experimental compound 5. High symmetry allows a clearer view of the geometry of our variants. We wish to encourage synthesis of this neutral system and its -SCH 3 analog described below. A neutral charge allows for the possibility of high symmetry in the solid phase, thereby simplifying the X-ray analysis.

AIM Analysis
Symmetry equivalence also allows a compact presentation of AIM results, shown in Figure 17 for the >C=O system 6. Some key numerical values for density, the Laplacian, and energy densities along with the ratio |2G/V| are collected in Table 6. The parameters reflect the double bond character of the allenic core containing the hcC and the polar nature of the H 3 C-O and C=O bonds. The small values of density and the Laplacian L, and the positive sign of L as well, attest to the weak and predominantly electrostatic nature of the interaction.

AIM Analysis
Symmetry equivalence also allows a compact presentation of AIM results, shown in Figure 17 for the >C=O system 6. Some key numerical values for density, the Laplacian, and energy densities along with the ratio |2G/V| are collected in Table 6. The parameters reflect the double bond character of the allenic core containing the hcC and the polar nature of the H3C-O and C=O bonds. The small values of density and the Laplacian L, and the positive sign of L as well, attest to the weak and predominantly electrostatic nature of the interaction. Figure 17. AIM analysis of CO system 6. Figure 17. AIM analysis of CO system 6.

ELF Analysis
The electron localization function ( Figure 18) shows the central allene system, with the anthryl methoxy oxygens in the coordination sphere of the hexacoordinated C. There is no indication of ELF amplitude in the bonding region between CH 3 O and the hcC, but rather an accumulation of charge on the methoxy oxygen and depletion of charge near the hcC. This is entirely consistent with the description of the interaction between MeO and hcC as weak and electrostatic, inferred from the AIM analysis.

ELF Analysis
The electron localization function (Figure 18) shows the central allene system, with the anthryl methoxy oxygens in the coordination sphere of the hexacoordinated C. There is no indication of ELF amplitude in the bonding region between CH3O and the hcC, but rather an accumulation of charge on the methoxy oxygen and depletion of charge near the hcC. This is entirely consistent with the description of the interaction between MeO and hcC as weak and electrostatic, inferred from the AIM analysis.

NCI Analysis
The NCI zones displayed in Figure 19 include interactions between the bridging >C=O groups and ipso CH groups of the anthryl fragments, destabilized regions within six-membered rings, and interactions between -OCH3 methyl groups and nearby CH groups of the anthryl frame. Most significant for our purpose are noncovalent interactions between -OCH3 and the allenyl group's central hexacoordinated carbon.

NCI Analysis
The NCI zones displayed in Figure 19 include interactions between the bridging >C=O groups and ipso CH groups of the anthryl fragments, destabilized regions within six-membered rings, and interactions between -OCH 3 methyl groups and nearby CH groups of the anthryl frame. Most significant for our purpose are noncovalent interactions between -OCH 3 and the allenyl group's central hexacoordinated carbon.

A Neutral System with Attractive Interactions in the Coordination Sphere
To enhance possible soft-soft interaction between the six-coordinate Carbon and members of its coordination set we replace the methoxy O atoms by Sulfur atom, retaining D2d symmetry. The structure 7 is shown in Figure 20.

A Neutral System with Attractive Interactions in the Coordination Sphere
To enhance possible soft-soft interaction between the six-coordinate Carbon and members of its coordination set we replace the methoxy O atoms by Sulfur atom, retaining D 2d symmetry. The structure 7 is shown in Figure 20.

A Neutral System with Attractive Interactions in the Coordination Sphere
To enhance possible soft-soft interaction between the six-coordinate Carbon and members of its coordination set we replace the methoxy O atoms by Sulfur atom, retaining D2d symmetry. The structure 7 is shown in Figure 20.   According to the AIM data (Table 7), the strength of the CH3S…C (central) interaction is almost as strong as the CH3O…C (central) interaction. Compare the Laplacian value of 0.0619 for 7 with +0.0755 for 6. According to the AIM data (Table 7), the strength of the CH 3 S . . . C (central) interaction is almost as strong as the CH 3 O . . . C (central) interaction. Compare the Laplacian value of 0.0619 for 7 with +0.0755 for 6. As was the case for the methoxy system's O . . . hcC BCP, the S . . . hcC BCP has a positive Laplacian, indicating charge depletion at that point, and predominantly electrostatic interaction. But in this case the interaction is attractive; hcC is negatively charged in each case, but the CH 3 S sulfur atom has a positive AIM charge whereas the CH 3 O oxygen atom has a strongly negative AIM charge. (Table 8) This qualitative difference is unaffected by enhancing the basis from cc-pVTZ to aug-cc-pVTZ, though the charges in the thia system are appreciably altered.

ELF Analysis
In Figure 22, the ELF is again shown as a set of isosurfaces enclosing regions of high charge localization. Along the axis linking the carbonyl oxygens and passing through the allenyl carbons we see features corresponding to the carbonyl oxygen lone pairs (top and bottom) and also the orthogonal pi bonds between the central hcC and its C neighbors in the allene. Two of the four S atoms (from anthryl -SMe substituents) are in view, at right and left. Lone pairs on S atoms are prominent, but there is no evident localization between S atoms and the hcC. This is consistent with the low value of electron density at the BCP on the path fromS to hcC, and confirms the essentially electrostatic nature of the interaction.

NCI Analysis
In Figure 23, we find regions of non-covalent interaction (a) between anthryl carbonyl groups and nearby anthryl CH hydrogen atoms; (b) between -SCH 3 methyls and nearby anthryl CH hydrogens; and (c) within anthryl rings. Most prominent are the NCI regions between S atoms and the central hcC.

Conclusions
We examined some well-established examples of unusual coordination of carbon atom. Systems 1 and 3 containing tetracoordinate planar carbon (tpC) are stabilized by removal of two electrons; in Radom's octaplanes the tpCC 4 2+ fragment is further constrained by surrounding hydrocarbon rings.
To our knowledge no X-ray structures are known for these systems, though many analogs with metal substituents have been characterized. [17][18][19][20][21] We recommend that the synthesis and spectroscopic characterization of hydrocarbon octaplanes be re-visited. A signal experimental success is the structural study of hexamethyl benzene dication containing hexacoordinated carbon at the apex of a pentagonal pyramid. Bond paths confirm hexacoordination, but need not correspond to electron pair bonds. This electron-deficient species, like many carbocations with unusual carbon coordination, does not threaten the octet rule.
Building on a theme first illustrated in a cationic 1,8-dimethoxy-9-dimethoxymethylanthracene, a series of five-and six-coordinated species have been synthesized [46][47][48][49][50]. In the pentacoordinate system two of the O atoms in the coordination sphere are bound weakly. AIM, ELF, and NCI analysis all confirm that for an experimental system possessing six entities in a C atom's coordination sphere, and for symmetrized variations proposed here, there is evidence for six significant interactions between the allenyl central C and neighbors. The four interactions with O (6) or S (7) are electrostatic, not covalent, in view of the low level of charge sharing and density at the bond critical points. The C is hexacoordinate but not hexavalent. Again, no violation of the octet rule model is claimed.
Replacing the oxygen in the methoxy group with sulfur alters the electrostatic interaction from repulsive to attractive. We encourage synthesis of species 6 and 7, and a thorough structural characterization including X-ray, vibrational, and NMR spectra. This should distinguish the repulsive character in 6 from the attractive character of 7.