Atoms in Highly Symmetric Environments: H in Rhodium and Cobalt Cages, H in an Octahedral Hole in MgO, and Metal Atoms Ca-Zn in C 20 Fullerenes

: An atom trapped in a crystal vacancy, a metal cage, or a fullerene might have many immediate neighbors. Then, the familiar concept of valency or even coordination number seems inadequate to describe the environment of that atom. This difﬁculty in terminology is illustrated here by four systems: H atoms in tetragonal-pyramidal rhodium cages, H atom in an octahedral cobalt cage, H atom in a MgO octahedral hole, and metal atoms in C 20 fullerenes. Density functional theory deﬁnes structure and energetics for the systems. Interactions of the atom with its container are characterized by the quantum theory of atoms in molecules (QTAIM) and the theory of non-covalent interactions (NCI). We establish that H atoms in H 2 Rh 13 (CO) 243 − trianion cannot be considered pentavalent, H atom in HCo 6 (CO) 151 − anion cannot be considered hexavalent, and H atom in MgO cannot be considered hexavalent. Instead, one should consider the H atom to be set in an environmental ﬁeld deﬁned by its 5, 6, and 6 neighbors; with interactions described by QTAIM. This point is further illustrated by the electronic structures and QTAIM parameters of M@C 20 , M=Ca to Zn. The analysis describes the systematic deformation and restoration of the symmetric fullerene in that series.


Background
Chemical species can be systematically described with the assumption that individual atoms can be bound to specific numbers of neighbors. Local geometry and the strength of bonds are easily rationalized by, for example, the tetravalency of carbon, the divalency of oxygen, and similar assignments to other common atoms. Departures from such simple counts are exceptional (though not rare) and require special description. Molecules with larger than normal numbers of bound partners, i.e., "hypervalent compounds", were identified by Musher [1]. Many of these species may be written as X n GY m where G is a member of groups 14, 15, and 16. This includes for example ClF 3 , F 3 PO, PF 5 , XeF 6 , and IF 7 ; these species have unusual tri-, penta-or hepta-valency, with the number of valence pairs in the neighborhood of a central atom exceeding the familiar value of four. If all bonds are covalent and we adopt the Lewis model of pair bonding, then, from the elementary valence-bond view, d atomic orbitals (AOs) are needed to mix with s and p AOs to form local hybrids on the central atom. An alternative representation assigns ionic structures to the ligands, so that trigonal-bipyramidal PF 5 for example is assigned F 3 P 2+ (F 1− ) 2 with two negatively charged fluorides at the poles, and three PF bonds and a lone pair in the equatorial plane. These conceptual alternatives have been discussed mainly in pedagogical settings [2,3]. The question of the nature of bonding in such species is evaded by the noncommittal term "hypercoordinate" species apparently first coined by Schleyer [4].
remarked that a reasonable model of their results would represent C60 as a spherical shell of charge, polarizable and responsive to the central atom or ion.
Jalife et al. reviewed noble gas inclusion complexes in fullerenes [18]. In the intervening three decades, X-ray structures for C 60 inclusion complexes had become available for He [19], Xe [20] and Kr [21]. The QTAIM analysis reported by Cerpa et al. [22] revealed a multitude of cage C-to-noble gas paths and located associated line critical points with very small densities and small positive Laplacians. These authors remarked that the number of bond paths and BCPs was a consequence of the symmetry of the cage and had no necessary correspondence to chemical constructs. The apparent paradoxes in application of QTAIM to endohedral complexes was further explored by Popov and Dunsch [23]. Estrada-Salas and Valladares [24] studied endohedral and exohedral complexes of Mn, Fe, Co, Cu, and Zn with C 60 , optimizing structures with BPW91 and a DNP basis.
Modeling of neutral alkali metal and transition metal inclusion complexes in C 20 has been pursued for more than a decade. An and co-authors [25] studied Li, Na, K, Rb, and Cs atoms in icosahedral C 20 with DFT (PW91/DNP). Garg and co-workers [26] evaluated structures, energies, and magnetic moments with an alternative DFT model PBE with a pseudo potential representing core electrons and a DZP basis for valence electrons. Baei et al. [27] treated C 20 inclusion complexes of transition metals Sc-Zn with DFT (PBE/6-31G(d) for C, LANL2DZ for metals), finding that inclusion complexes M@C 20 of the later transition metals (Co-Zn) were not stable compared to the isolated metal atom M and the C 20 fullerene. Vibrational analysis showed that the centrosymmetric structures for Sc, Cr, Fe, Ni had saddle-point character and would spontaneously break symmetry. Gonzalez, Lujan, and Beran [28] modeled TM@C 20 for group 11 and 12 transition metals, Zhao, Li, and Wang [29] considered a broad sample of transition metals with configurations in 3d (Sc-Zn), 4d (Y-Cd) and 5d (Lu-Hg) shells, with the PBE functional and a numerical basis of double-zeta quality but apparently no relativistic corrections. They found maximum spin polarization for Mn, and most effective binding of Ti and V. Sarkar, Paul, and Misra [30] treated transition metals Ti, V, Cr, Mn, Fe, Co, and Ni in all spin states with UB3LYP expressed in a mixed basis, 6-31++G(d,p) for C and LANL2DZ for metals. They found that the structures all retained centrosymmetry and that maximum multiplicity state was most stable in every case except for Cr and Fe, which prefer the triplet. While details of binding energy and structure are not disclosed, the authors provide the range of M-C distances; these show that the endohedral complexes are nearly not perfectly centrosymmetric.
In this report we address first the description of coordination of H atom captured in metal cages and in a salt crystal [6][7][8][9][10]. QTAIM provides a unified description of the environment of the captured H atom. Despite claims to the contrary, the QTAIM results show that it is misleading to use the chemical terminology of pair bonds in these cases. Such time-honored representations of molecules are even less well suited to the description of the environment of atoms isolated in a small fullerene. The structures of C 20 systems enfolding metal atoms are not systematically established, nor is the thermodynamics of capture well established.

Systems Studied
The choice of systems was guided by theoretical concerns-the characterization of H atom confined in a structure with many immediate neighbors-and the need to refine the description of structure and bonding in the smallest metal atom endohedral fullerenes. Such analysis must precede the study of electrical, magnetic, and spectroscopic properties, which for these systems may be particularly interesting and useful.

H Atom Capture in Inorganic Vacancies
The high-symmetry environments we have chosen include a rhodium trianion with H atoms occupying nearly square-pyramidal cage sites [H 2 Re 13 (CO) 24 ] 3− , [7] and a cobalt nearly-octahedral cage of cobalt atoms enclosing a single H atom [HCo 6 (CO) 15 )] 1− , [8,9] a fragment of MgO crystal with a H atom occupying a hole left by removal of an O atom, [10] and fullerenes C20 with inclusion complexes of single atoms including uncharged transition metals and noble gases. We have experimental data for some of these species [7,9] and have constructed some computational models for others to advance the discussion. In the following discussion we provide brief initial discussions of these systems.

H Atoms in the Rhodium Square-Pyramidal Cages
In [H 2 Rh 13 (CO) 24 ] 3− which has a rhodium cage structure [7] there would seem to be six equivalent square-pyramidal open sites (three above the equator, three below). Idealized forms are shown in Figure 1. Each of the open sites is enclosed by four Rh atoms forming a square face of the polyhedron and the central metal atom. The electronic structure of the experimental system is represented by QTAIM interaction lines connecting nuclear centers in Figure 2. QTAIM critical points of types (3, −3)-that is, nuclear attractors, or atomic centersare shown in Figure 2, along with interaction lines (which have maximum density in two dimensions, a "ridge" in the density) connecting the nuclear attractors. The object is oriented to suggest a left-right symmetry, which is imperfect. The symmetry of the experimentally studied structure is reduced in part by the bridging carbon dioxide fragments which reduce the symmetry to C s . Symmetry is further reduced by the counterions in the crystal. Then the "square" pyramidal holes are deformed into three nonequivalent sets of two equivalent quadrilateral pyramidal holes. In the experimental system [7], the trapped H atoms occupy two non-equivalent holes.
More detailed description of the experimental system is in the Supplementary Material. Discussion of a symmetrized model is deferred to the section on results. The experimental X-ray structure [7] of the complex [H 2 Rh 13 (CO) 24 ] 3− as represented in the AIM topology. Oxygen atoms are color-coded red, rhodium atoms are dark green, carbon atoms dark gray, and hydrogen atoms are light gray. Of the 24 CO ligands, 12 are attached each to a single Rh and 12 bridge two rhodium atoms. The bridging reduces the global symmetry to C s . Even that plane of symmetry is broken in the crystal since the H atoms occupy nonequivalent sites.

H Atom in a Cobalt Octahedral Cage
Two views of the C 2v -symmetric framework in [HCo 6 (CO) 15 ] 1− anion [8,9] appear in Figure 3. This symmetric structure was obtained by geometry optimization with density functional theory, using the model ωB97XD/cc-pVTZ (see Methods and Software section for details). According to the QTAIM analysis, there are six interaction lines connecting Co atoms with the trapped H atom Figure 3. Two views of HCo 6 (CO) 15 1− with color code red = O atom, gray = C atom, pink = Co atom, light gray = H atom.
System optimized in C 2v symmetry with DFT model ωB97XD/def2-SVP. Connections are QTAIM features, ridges in the electron density linking nuclear centers. (a) A view along the C 2 axis (b) An offset view revealing the hexacoordinated H atom and displaying the lines connecting the H atomic center with Co atoms according to QTAIM.

H Atom in a Magnesium
Oxide Crystal Vacancy Figure 4 shows a fragment of a MgO crystal, with a neutral H atom replacing an oxygen atom [10]. In the perfect crystal each magnesium atom is coordinated with six oxygen atoms, and each oxygen atom by is coordinated with six magnesium atoms. The distortion of the crystal arising from the substitution of neutral H for the O dianion is evident. Some Mg atoms depart from the mean (MgO) n plane since they are more weakly attracted by H than by O −2 .

Endohedral M in C20 Fullerene
Illustrating use of QTAIM analysis [14,15] for M@C 20 , Figure 5 describes Ne@C20. The structure is optimized in ωB97XD/cc-pVDZ, and is shown in the QTAIM topology. Solid black lines connect nuclear attractors' (3, −3) critical points (atomic centers). The lines represent ridges in the electron density function and suggest that the central Ne atom interacts strongly with all 20 of the C atoms of the fullerene container. Green dots on these connectors mark a minimum density along the ridge; these points are termed line critical points (LCPs). These are termed (3, −1) extreme points. At those points, the gradient of the density is zero and the matrix of second derivatives has one positive root corresponding the increase in density along the interaction line (call this z), and the decrease in density in the other two dimensions (x and y).

Methods and Software
We used density functional theory to compute the ground state energies and harmonic vibrational frequencies, and electron distribution. For the fullerenes our choice of functional was ωB97XD [31]. This functional is gradient-corrected, its exchange is scaled for distance, and the functional includes an empirical term describing dispersion interactions, Grimme's D2 formulation [32]. The cc-pVDZ basis [33,34] was used for fullerenes and inclusion complexes. For the rhodium system, we used the Ahlrichs-Weigend basis SVP [35,36] and the Stuttgart basis SDD, [37] each of which incorporates a relativistic pseudopotential. These basis sets are all available for download [38]. Gaussian 09 [39] and Gaussian 16 [40] produced optimized structures and the wfx files required by QTAIM analysis. QTAIM treats the molecular charge density ρ and its gradients ∇ρ, and identifies interaction lines between atomic centers. Atomic centers have three negative roots of the matrix of second derivatives of the density with respect to Cartesian displacement coordinates [∂ 2 ρ/∂x i ∂x j ] and are denoted as (3, −3) critical points (CPs). The interaction lines are loci of points with maximum density (negative curvature) in two dimensions, a "ridge" in the density. The minimum density point along the ridge (a saddle point with two negative and one positive curvature) may be called a line critical point (LCP) of type (3, −1).
QTAIM reports the Laplacian of the density at LCPs, and energy densities G (kinetic) and potential (V) as well. The mathematical formulation of these quantities has been clearly described [41]. This report also defines the delocalization index (DI) which counts the number of shared electron pairs. These quantities have been used widely to characterize the strength and degree of covalency of bonding.
Non-covalent interactions (NCIs) appear in regions of low density and small values of the reduced density gradient (RDG) [42]. These may be attractive (as van der Waals forces or H-bonding) or repulsive (most closed shell-closed shell interactions, such as steric effects. A useful description of the RDG, its graphical characterization, and its interpretation is given by Contreras-Garcia et al. [43,44].
For the model structure for the rhodium cage, we began with idealized geometric forms (a dodecahedron of 12 Rh atoms with one more at the center of the figure) for the Rh 13 cluster, and introduced terminal and then bridging CO groups to mimic the connectivity of the X-ray structure. Two H atoms were introduced along the approximately fourfold axis passing through the central Rh and the tetragonal faces of Rh atoms. The energies of these starting forms were optimized in ωB97XD/SDD and ωB97XD/def2-tzvp models. For H@MgO, structures were taken from the X-ray analysis.
Initial M@C 20 structures were obtained by placing the M atom at the center of the D 3d form of C 20 . [46] The optimization did not enforce symmetry, and several systems were found to depart substantially from this initial symmetry.

Results
In this section we present details of the QTAIM and NCI analyses for the three cageconfined H atom systems (Section 4.1, parts Sections 4.1.1-4.1.3) and metal endohedral C20 fullerenes (Section 4.2, parts Sections 4.2.1-4.2.15). Summary remarks on the QTAIM analysis for members of the M@C20 series are to be found in Section 4.2.14, and an overview of the NCI results is to be found in Section 4.2.15.

Captured H Atoms
In the following sections we describe geometries of systems in which foreign atoms are captured. Some structures are simply borrowed from reported X-ray crystallography studies (as in the H-doped MgO solid sample and the experimentally known unsymmetric rhodium cage compound), while other structures are found by geometry optimization in ωB97XD/cc-pVDZ. These include all the M@C 20 fullerenes and the modeled highsymmetry rhodium and cobalt cage compounds.

H Atoms in Rhodium Cages
The unsymmetric experimental rhodium cage [7] and the symmetrized analog capture H atom in a square-pyramidal environment, as shown above in Figure 2. The QTAIM analysis is unwieldy owing to the low symmetry of the species; it can be found in supplementary information. An idealized D 4h structure of a simplified structure H 2 Re 13 (CO) 12 ] 3− , is shown in Figure 6a. Then H atoms occupy square-pyramidal holes and have five Rh atoms as neighbors in a square pyramid. Each Rh is bound to a CO carbon and four Rh neighbors, as in the experimental structure. A further idealization of the rhodium system with bridging carbonyls and formula H 2 Re 13 (CO) 24 ] 3− is shown in Figure 6b.  Figure 7 shows the linear H-Rh-H arrangement, and (less evident) the square base in which the H atoms are to be found. The connectivity is presented in schematic fashion in Figure 7b, which defines the labeling scheme used for Table 1.   24 ] 3− trianion. Interatomic distances are in Angstroms. Density ρ and its Laplacian ∇ 2 ρ are in atomic units, (absolute electrons per Bohr-cubed for the density, for example), as are the kinetic and potential energy densities G and V. Density was computed in ωB97XD/SDD.

Coordination
Length  24 3− trianion. The structure is obtained by optimization with ωB97XD/SDD. Symmetrization is far from perfect but it is clear that the distance from H to the central Rh (2.628 ± 0.04 Å) is larger than the distance from H to the Rh atoms making up the square bases (1.786 ± 0.026 Å).
The Laplacian values are all positive, indicating a depletion of charge at the H-Rh LCPs, characteristic of closed shell interactions. However, the delocalization index (DI), which reflects the number of shared electron pairs [48], suggests that covalent interactions are significant. If the ratio |V|/G < 1, we take the bonding to be mainly ionic, while if the ratio exceeds 2, the bonding is covalent. In this case, the ratio lies between 1 and 2, and bonding is considered "mixed" [48]. The bond strength measure −V/2 [49,50] assigns the interaction energy (about 0.05 to 0.06 Hartree) to the H-Rh interaction energy. For a detailed guide to the mathematical formulation of these QTAIM constructs see [48].
The delocalization index DI is an estimate of electron pairs shared between atoms. Negative values for H = G + V, or values of |V|/G > 1 (here they average about 1.4) suggest that covalent interactions are significant.
The DI index for H atom and the five surrounding Rh atoms is 1.304 ± 0.026, which does exceed the ideal limit of 1.00 for perfectly covalent dihydrogen. However, if a pentavalent species is to share five electron pairs, we see that it is seriously misleading to describe hydrogen atom in the Rh square pyramid as being pentavalent. It is of course pentacoordinate since it interacts with five immediate neighbors.
A view of the NCI surfaces for H 2 Rh 13 (CO) 24 3− trianion is given in Figure 8. NCI zones appear around the outer surface of the Rh dodecahedron, but there are no such zones near H atoms. This suggests that dispersion and steric interactions between H atom and Rh atoms making up its cage are not significant. The bonding is weak according to QTAIM but results predominantly from covalent/orbital mixing interactions. Figure 9a shows the cobalt cage compound HCo 6 (CO) 15 1− anion, by the density ridge lines connecting nuclear centers (in QTAIM terminology, (3, −3) critical points, meaning that density decreases in all three directions from the critical point. The density ridge or line critical points (LCPs) are minima along the line, and are called (3, −1) points. The trapped H atom interacts with all six Co atoms. Figure 9b schematizes those interactions and provides a labeling scheme, which is used in Table 2, summarizing QTAIM parameters. . Two views of the HCo 6 (CO) 15 1− anion (a) C s -symmetric HCo 6 (CO) 15 1− anion as represented in the AIM topology. Solid lines connect nuclear attractors (large spheres; H is light gray, Co is pink, Ne is light blue, carbon is dark gray) Green dots on these connectors represent line critical points and reinforce the suggestion that the central H atom interacts strongly with six Co atoms of the Co cage. (b) Labeling scheme for interaction lines. Values of QTAIM collected in Table 2 attest to weak Co-H bonding with a degree of covalency. The delocalization index DI is an estimate of electron pairs shared between atoms. Negative values for H, or values of |V|/G > 1 suggest that covalent interactions are significant. Here |V|/G averages about 1.3, so bonding is mixed. This is appropriate for these weakly charged systems and consistent with the modest values of the DI. The positive values of the Laplacian are characteristic of closed shell-closed shell interaction, so we must infer a delicate balance between weak covalent interaction and near-canceling steric repulsion and attractive dispersion.

The H Atom Captured in a Six-Coordinate Cobalt Cage
The total DI of 1.4033 electron pairs shared among the six immediate Co neighbors of H atom, is comparable to the DI of H in the rhodium cage. In analogy to the Rh case, if hexavalency implies that six electron pairs participate in bonding with H, it cannot be said that H is hexavalent. It is surely hexacoordinate; there is a small degree of covalency associated with those six interactions.
The NCI graphical analysis for the HCo 6 (CO) 15 1− anion ( Figure 10) shows noncovalent interactions between Co atoms but no evident NCI from Co to H atom. Evidently the dispersion and steric NCIs tend to cancel.

Hydrogen Atom in MgO
The H atom in the system H@MgO takes the place of an O atom in the crystal's rock salt structure. The interaction lines derived from QTAIM analysis are shown in Figure 11a.  The non-covalent interaction analysis produces surfaces enclosing regions of low density and low reduced density gradient. Figure 12 provides a view through the top layer of MgO. Behind the central Mg atom is a surface enclosing the H atom and blocking it from view. The blue-green color coding suggests that the Mg-H interaction is weak. Behind every edge-Mg lies an O atom, with a blue NCI zone between them. This is a strongly attractive ion-ion NCI. A similar but darker blue ring can be seen between edge O atoms and the Mg atoms behind them. The NCI visualization is entirely consistent with the semi-quantitative QTAIM description The following sections deal with a series of inclusion complexes M@C 20 . First (Section 4.2.1) we report the thermodynamic energy change associated with capture of neutral atoms in the C 20 cage. The reference point for bonding analysis (Section 4.2.2) is Ne@C 20 , which retains high (D 3d ) symmetry for the C 20 shell. Ne interacts almost identically with each of the shell carbons, in keeping with that high symmetry. NCI analysis shows regions of repulsion between Ne and each C.
Sections 4.2.3-4.2.13 are each devoted to a single endohedral complex M@C20 for M in the series Ca to Zn. In each case the interaction lines define a structure, and the QTAIM properties of line critical points (LCPs) are compiled. Key variables include the LCP electron density, its Laplacian, the delocalization index, and he kinetic and potential energy densities. Both the LCP electron density and the LCP potential energy density have been considered useful measures of interaction strength. The ratio of kinetic to potential energy density K/V, as well as their sum H has been used to judge the relative degree of ionic and covalent interaction, while the delocalization index is considered a measure of the number of shared electron pairs.
General description of behavior of the QTAIM quantities of the M@C 20 is found in Section 4.2.14, and an overall view of the NCI analysis is in Section 4.2.15.

Energetics for M + C 20 → M@C 20 for the Series M = Ca-Zn
In this section, we describe the set of M@C 20 fullerene inclusion complexes. The set of M lies in the first transition row and range from Ca to Zn. The atoms are all neutral, of configuration d n (n = 0 to 10). We chose maximum multiplicity since the set of d AOs span a single irreducible representation in the reference I h -symmetrical structure of C 20 . While all M@C 20 structures occupy a relative minimum in the potential surface (all vibrational frequencies are real) some species are thermodynamically unstable according to our unrestricted ωB97XD/cc-pVDZ calculations. Our estimates of the thermodynamic stability of the endohedral metal species M@C20 relative to separate M and C 20 (Table 4) disagree sharply with those of Garg et al. [26] and Gonzalez and co-workers [28]. Since these investigators used rather different computational models and the structures they used are not defined in detail the discrepancy is difficult to resolve. Misra and co-workers reported [30] that the triplet Fe@C 20 was lower in energy than the maximum-multiplicity quintet; we confirmed that finding, with our ωB97/XD/cc-pVDZ placing the triplet 0.09 eV lower than the quintet. The structure of the triplet is quite different from that of the quintet. Similarly, for Cr, we find the triplet more stable by 0.57 eV, and of distinct structure. Neither of these revisions alter the thermodynamic instability of the endohedral species.

The Ne@C 20 Reference System
As a reference for further discussion of the series M@C 20 with M being members of the first transition series of the periodic table, we report the QTAIM analysis for Ne@C20. Figure 13 displays the connectivity for this system. Solid black "interaction lines" follow ridges in the electron density and connect nuclear centers, i.e., (3, −3) critical points. Green dots mark line critical points (LCPs) of type (3, −1). (b) an adaptation of a Schlegel representation of the C 20 cage, with the Ne atom shown as a dot. In D 3d symmetry, there are three distinct Ne-C connections: along the D 3 axis; to a C which is a neighbor to a C on the axis; and to a C, which has no such neighbor on the axis The numerical results of a QTAIM analysis for Ne@C 20 are shown in Table 5. In D 3d symmetry there are four distinct types of CC interactions in the fullerene shell, with density at the LCPs ranging from 0.22 to 0.28 atomic units. This is typical of carboncarbon bonds. There are 20 Ne-C interaction lines, in three symmetry equivalent sets of two, six, and twelve. In contrast to the CC LCP densities, the Ne-C LCP densities are much smaller. The Laplacian values are negative for CC interactions, suggesting charge accumulation characteristic of covalent bonding. Laplacian values for Ne-C lines are smaller and positive, suggesting charge depletion at the LCPs, characteristic of closed shell-closed shell interactions. Energy density values at LCPs further distinguish the interactions. The Ne-C interaction, for which H is near zero and |V|/G is near 1, is not dominated by either extreme of ionic or covalent interaction. DI, an index of electron pair sharing, is very small for the Ne-C interactions, in contrast to the CC bonds which are highly covalent and have multiple bond character.
We use the properties of Ne-C bonds, as revealed in QTAIM analysis, as a basis for comparison for the series of M@C 20 systems. Explicitly these are LCP densities in the range 0.05 to 0.06; Laplacian values near 0.33; DI near 0.08; and balanced covalent/ionic interactions.
We take the averages of QTAIM parameters for Ne@C 20 as a reference for discussion of the M@C 20 species described below. That is, we will consider departures from 1: the D 3d symmetry; 2. The mean Ne-C distances, 2.02 (short) and 2.10 (long); The LCP density 0.055; the mean Laplacian +0.32; the mean DI 0.075; the kinetic energy density 0.082l; the potential energy density V; and the near-perfect balance between G and V (H = G + V near zero).
Though there are 20 interactions between Ne and C in Ne@C 20 identified by QTAIM, one would never claim twenty-fold valency of the noble gas. It is already remarkable that QTAIM would consider ca. 1.50 electron pairs to be shared between Ne and the C 20 shell.
The NCI analysis shown in Figure 14 displays small NCI zones at ring critical points and considerably larger zones between the central Ne and the fullerene shell. The color coding has blue as attractive and red as repulsive, so we infer that for the Ne-C connection the steric (Pauli pressure) repulsion is larger than the dispersive (London force) attraction.  20 . Color coding of NCI zones has red designating repulsive, blue attractive, and green weak interactions. Here we see small NCI repulsions at ring centers and larger zones between Ne and C.
In the following sections we present results for M@C 20 Figure 15a displays the C and Ca nuclei, that is, (3, −3) critical points, as spheres. Interaction lines between atoms and their (3, −1) critical points define the structure of the electron density distribution. Figure 15b provides a schematic view of the ten interactions between Ca and fullerene C atoms, and defines the labeling used in Table 6.   Table 6 shows that the fullerene inclusion complex Ca@C 20 retains high symmetry (all Ca-C distances are identical). A small density is found at the LCPs, and according to the Laplacian, charge is depleted. This is characteristic of closed shell-closed shell interactions, but the delocalization index (DI) indicates a degree of electron pair sharing. The small and negative value of H indicates a near balance of ionic and covalent bonding.  Table 7 indicate approximate symmetry, weak interactions, modest pair sharing, and a balance between covalent and ionic interaction between Sc and C.  There are 20 C-Sc interaction lines, with greater LCP density and DI than is found for Ca@C 20 . Sc is near the center of the fullerene. Counterintuitively, the basin charge for the Sc atom is +1.47 |e| with charge on C atoms-or more properly (3, Table 8 collects the Ti-C interatomic distances and QTAIM parameters for the Ti-C interaction LCPs. These have increased relative to Ne@C 20 , Ca@C 20 , and Sc@C 20 , indicating a strengthening interaction between the fullerene cage and the endohedral metal atom.  Figure 18a,b displays the interactions in V@C 20 , and the V-C interaction lines respectively. Figure 18b defines the labeling system for Table 9.   Table 9 shows that the symmetry for V@C 20 is disrupted, and some interactions have become strong. The negative value of H and its larger magnitude relative to values for Ca-Ti indicates that covalent bonding has become more significant.

Cr@C 20 (d 4 Spin Quintet)
QTAIM interaction lines characterize the electron density in Cr@C 20 (Figure 19a) and are the basis for the schematic representation in Figure 19b, which defines the labeling used in Table 10.  QTAIM entries for CrC 20 in Table 10 indicate a continuation of a trend toward greater distortion of the fullerene and stronger and more covalent interactions its carbon atoms with the endohedral atom. Figure 20a displays the QTAIM interactions lines characterizing the fullerene framework and connections between certain C atoms and the endohedral Mn atom. In Figure 20b we see the labeling scheme for entries in Table 11. Note the sudden decrease in the number of lines connecting Mn and C atoms.   Table 11 shows the shortening of Mn-C bonds and the enhanced, density, DI, and covalency in this system.

Fe@C 20 (d 6 Spin Quintet)
Interaction lines obtained from QTAIM analysis for Fe@C 20 are shown in Figure 21a. The notation used in Table 12 is defined in Figure 21b.  QTAIM parameters collected in Table 12 show that the Fe complex has more interaction lines than Mn, and that the covalency is higher for Fe than for Mn. The departure from high symmetry of the fullerene of Fe@C 20 is even greater than that seen in Mn@C 20 . Figure 22a shows QTAIM interaction lines for Co@C 20 ; note the lack of connection between C atoms in the base. Figure 22b defines the labeling scheme used in Table 13.    Figure 23a shows QTAIM interaction lines deduced from the density for Ni@C 20 ; note the lack of connection between C atoms in the base. The system of notation used in Table 14 is defined in Figure 23b.   Table 14 shows that the Ni@C 20 endohedral fullerene is still seriously compressed; the strength of Ni-C bonding is reflected in a relatively large LCP density, DI, and |V|/2.  Table 15.   Table 15 shows that while the fullerene in Cu@C 20 is compressed relative to C20 itself, the size reduction and the departure from symmetry is minor. Figure 25 shows the network of interaction lines as defined by a QTAIM analysis, and a schematic representation of interactions with the labeling scheme used in Table 16.  In Table 16 we find a very uniform set of QTAIM parameters, characteristic of a near-perfect symmetry in the Zn@C 20 system.

Summary Remarks on QTAIM Description of M@C 20
Calculations with a ωB97XD/cc-pVDZ density functional model have produced structures and energetics for the M@C 20 inclusion complexes of fullerenes, with metals M = Ca, Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn. A reference is provided by the Ne@C 20 and Ar@C 20 complexes, which are centrosymmetric. In some cases (vanadium, manganese, and iron especially-see Figure 26) the inclusion complexes depart seriously from the high symmetry assumed in initial model-building. All reported structures occupy relative minima in the potential surfaces (they are stable with respect to small distortions). However, the inclusion complexes are not always thermodynamically stable relative to symmetric C 20 and free gas phase metal. We take the averages of QTAIM parameters for Ne@C 20 as a reference for discussion of the M@C 20 species described below (see Table 17). That is, we will consider departures from (a) the D 3d symmetry; (b) the mean Ne-C distances, which range from 2.02 (short) to 2.10 (long); (c) the mean LCP density 0.055; (d) the mean Laplacian +0.32; (e) the mean DI 0.075; (f) the mean kinetic energy density 0.082; (g) the potential energy density V, and (h) the near-perfect balance between G and V (H = G + V near zero). A detailed numerical comparison with the Ne reference is compiled in Table 17. Entries show maximum and minimum values for the several parameters, which for distorted systems can span a considerable range.
A few examples will illustrate broad trends. Figure 26 shows M-C distances for the series Ca-Zn. The vertical lines through dots representing mean values are not error bars but rather represent the range of values of the bond distance. The range of values reflects the departure from a perfect sphere, which is minimal for Ca, Sc, and Ti at the beginning of the row, and Cu and Zn at the end. There is recognizable shortening beginning with V, and serious shortening for Co and Ni.
These features are imposed on the trend in van der Waals radii of the endohedral atoms [51]. The van der Waals radii decrease monotonically across the row, with argon then increasing sharply.  According to Figure 28, covalent bonding as measured by mean DI passes through a broad maximum extending from Mn to Ni. The vertical lines through dots representing mean values represent the range of values of the DI The range of values increases beginning with V, for which we see the first major distortion of the fullerene cage. Mean DI and its variance decrease sharply for Cu and Ni. The LCP density reaches a maximum about 0.14 at Fe and Co, more than twice the value for Ne@C 20 . The bonding strength measure −V/2 is largest for Mn through Ni, up to four times the value for the Ne inclusion complex. The mean delocalization index is large for Cr through Ni, reaching a fivefold increase over the value for Ni@C 20 . These features are reflected in the mean radius of the fullerene, which declines from the Ne@C 20 value (ca. 2.02 A) to 1.83 for Fe and Co.
The behavior of all measures of bonding suggests tighter M-C binding and greater distortion of the fullerene as we pass from weakly bound and very symmetric Ca@C 20 to the late transition metals. The C 20 cage distorts as the metals interact more strongly with particular areas of the surface. However, binding weakens and symmetry is re-established at the end of the passage with Cu and Zn. Neon provides a limit of minimal interaction between fullerene and its captured atom, while argon is large enough to have an impact on the cage structure.  We find only the repulsive NCI zones at ring critical points (RCPs) of the C 20 fivecarbon rings for calcium, scandium, chromium, manganese, copper and zinc. In contrast, the Neon endohedral system displays pronounced repulsive NCI zones between the captured atom and the C 20 fullerene. These compounds are approximately centrosymmetric. More interesting are the non-centrosymmetric cases chromium, manganese, and iron. We note the severe increase of a CC distance for cobalt and nickel complexes. Although these endohedral complexes occupy relative minima in potential energy, attested by the absence of imaginary vibrational frequencies, this suggests that the Co and Ni complexes can easily lose their captured atom.

Conclusions
The valency theory is challenged by chemical systems which enclose single atoms in a container. We have addressed H atoms in H 2 Rh 13 (CO) 24 3− trianion, for which each H is enclosed in a square-pyramidal rhodium cage; H atom in HCo 6 (CO) 15 1− anion, for which the H atom in trapped an octahedral cobalt cage, H@MgO in which the H atom occupies a hole from which O atom is absent. Ionic bonding dominates for MgO, but covalency as reflected in QTAIM theory is prominent in the H to metal interactions of the Rh and Co molecules. In these cases, the H atom shares about 1.4 electron pairs with its metallic neighbors, justifying a characterization of the bonding as hypervalent, but far short of penta-or hexa-valency.
We modeled endohedral complexes of metal atoms Ca-Zn in C 20 fullerenes using density functional theory, which provided energetics and structures for M@C 20 species. Our values for energetics of M + C 20 → M@C 20 are at variance with values from prior studies. Lack of detail of structures impedes resolution of this issue.
We found patterns in the density-based parameters of the QTAIM theory for M@C 20 . There seems to be a link between strength of the strength of individual M-C interaction and the number of unpaired electrons in these systems. Interactions between the trapped metal atom and the C 20 wrapping and the accompanying distortion of C 20 reach a peak late in the transition series (Mn-Ni), but decline for the more symmetric Cu and Ni species. This is reflected in the size of the C 20 shell, the extent of its distortion upon trapping of the metal atom, the magnitude of the density at line critical points, the number of electron pairs shared in bonding, and the extent of covalent bonding.
The confusion in terminology for the interaction of atoms with many neighbors is, we believe, resolved by the examples given for H atom captured in cages. Rather than speaking of valency or even coordination number in these cases, it is recommended to consider that a field is defined by the cage, with spatial properties well described by the constructs of QTAIM. This is reminiscent of the crystal field theory.
The properties of transition metals confined within a C 20 fullerene cannot be reliably described without first establishing stable structures and then reasonable energetics. This work will continue; once these properties are well characterized electrical and magnetic properties of these species can be estimated reliably.
Supplementary Materials: The following are available online at https://www.mdpi.com/article/ 10.3390/sym13071281/s1, Coordinates of all species as .mol2 files, and a sumviz file readable by AIMALL providing QTAIM parameters for the l H 2 Rh 13 (CO) 24 3− trianion in its X-ray structure.