Standard Enthalpies of Formation of Fullerenes and Their Dependence on Structural Motifs

Structural Chemistry of Pentagon-Fused C82 Fullerene Derivatives #39173C82(CF3)14,16,18 and #39173C82Cl28

High-temperature chlorination of the most stable Isolated-Pentagon-Rule (IPR) isomer of fullerene C₈₂, C₂-C₈₂(3), invariably produces non-IPR ^#39173C₈₂Cl₂₈, containing one pentagon-pentagon fusion in the carbon cage. High-temperature trifluoromethylation of ^#39173C₈₂Cl₂₈ followed by HPLC separation resulted in the isolation and structure elucidation of eight ^#39173C₈₂(CF₃)_n (n = 14, 16, 18) compounds. Structural chemistry of ^#39173C₈₂(CF₃)_14,16,18 and ^#39173C₈₂Cl₂₈ is characterized by the variation of the addition patterns in the region of a pentagon-pentagon fusion. The regiochemistry of CF₃ addition in the remaining cage region is similar to that of the known IPR C₈₂(3)(CF₃)_n compounds. Theoretical calculations revealed that ^#39173C₈₂(CF₃)_n possess lower thermodynamic stability than isomeric IPR derivatives.

Fullerenes Pose a Strain on Hybrid Density Functional Theory

The computational modeling of fullerenes plays a fundamental role in designing low-dimension carbon nanostructures. Nevertheless, the relative energies of fullerenes larger than C20 and C24 have not been comprehensively examined by means of highly accurate ab initio methods, for example, the CCSD(T) method. Here we report such an investigation for a diverse set of 29 C40 isomers. We calculate the energies of the C40 fullerenes using the G4(MP2) composite ab initio method, which approximates the CCSD(T) energy in conjunction with a triple-ζ-quality basis set (CCSD(T)/TZ). The CCSD(T)/TZ isomerization energies span 43.1-763.3 kJ mol-1. We find a linear correlation (R2 = 0.96) between the CCSD(T)/TZ isomerization energies and the fullerene pentagon signatures (P1 index), which reflect the strain associated with fused pentagon-pentagon rings. Using the reference CCSD(T)/TZ isomerization energies, we examine the relationship between the percentage of exact Hartree-Fock (HF) exchange in hybrid density functional theory (DFT) methods and the pentagon-pentagon strain energies. We find that the performance of hybrid DFT methods deteriorates with the pentagon-pentagon strain energy. This deterioration in performance becomes more pronounced with the inclusion of high amounts of HF exchange. For example, for B3LYP (20% HF exchange), the root-mean-square deviation (RMSD) relative to G4(MP2) increases from 8.9 kJ mol-1 for the low-strain isomers (P1 = 11) to 18.0 kJ mol-1 for the high-strain isomers (P1 > 13). However, for BH&HLYP (50% HF exchange) the RMSD increases from 23.0 (P1 = 11) to 113.2 (P1 > 13) kJ mol-1. A similar trend is observed for the M06/M06-2X pair of functionals. Namely, for M06 (27% HF exchange) the RMSD increases from 0.8 (P1 = 11) to 21.0 (P1 > 13) kJ mol-1, whereas for M06-2X (54% HF exchange) the RMSD increases from 16.7 (P1 = 11) to 77.7 (P1 > 13) kJ mol-1. Overall, we find that the strain associated with pentagon adjacency is an inherently challenging problem for hybrid DFT methods involving high amounts of HF exchange and that there is an inverse relationship between the optimal percentage of HF exchange and the pentagon-pentagon strain energy. For example, for BLYP the optimal percentages of HF exchange are 13% (P1 = 11), 10% (P1 = 12), 7.5% (P1 = 13), and 6% (P1 > 13).

Effects of orbital angles on the modeling of conjugated systems with curvature

Models with angle corrections give well predictions of both neutral and charged fullerenes. The integrals of nonparallel orbitals explain why angle features of designed and deep-learning models are necessary to describe conjugated systems.

Capturing the Missing Carbon Cage Isomer of C84 via Mutual Stabilization of a Triangular Monometallic Cyanide Cluster

Monometallic cyanide clusterfullerenes (CYCFs) represent a unique branch of endohedral clusterfullerenes with merely one metal atom encapsulated, offering a model system for elucidating structure-property correlation, while up to now only C₈₂ and C₇₆ cages have been isolated for the pristine CYCFs. C₈₄ is one of the most abundant fullerenes and has 24 isomers obeying the isolated pentagon rule (IPR), among which 14 isomers have been already isolated, whereas the C_2v(17)-C₈₄ isomer has lower relative energy than several isolated isomers but never been found for empty and endohedral fullerenes. Herein, four novel C₈₄-based pristine CYCFs with variable encapsulated metals and isomeric cages, including MCN@C₂(13)-C₈₄ (M = Y, Dy, Tb) and DyCN@C_2v(17)-C₈₄, have been synthesized and isolated, fulfilling the first identification of the missing C_2v(17)-C₈₄ isomer, which can be interconverted from the C₂(13)-C₈₄ isomer through two steps of Stone-Wales transformation. The molecular structures of these four C₈₄-based CYCFs are determined unambiguously by single-crystal X-ray diffraction. Surprisingly, although the ionic radii of Y³⁺, Dy³⁺, and Tb³⁺ differ slightly by only 0.01 Å, such a subtle difference leads to an obvious change in the metal-cage interactions, as inferred from the distance between the metal atom and the nearest hexagon center of the C₂(13)-C₈₄ cage. On the other hand, upon altering the isomeric cage from DyCN@C₂(13)-C₈₄ to DyCN@C_2v(17)-C₈₄, the Dy-cage distance changes as well, indicating the interplay between the encapsulated DyCN cluster and the outer cage. Therefore, we demonstrate that the metal-cage interactions within CYCFs can be steered via both internal and external routes.

Predicting the new carbon nanocages, fullerynes: a DFT study

In this study, based on density functional theory, we propose a new branch of pseudo-fullerenes which contain triple bonds with sp hybridization. We call these new nanostructures fullerynes, according to IUPAC. We present four samples with the chemical formula of C4nHn, and the structures derived from fulleranes. We compare the structural and electronic properties of these structures with those of two common fullerenes and fulleranes systems. The calculated electron affinities of the sampled fullerynes are negative, and much smaller than those of fullerenes, so they should be chemically more stable than fullerenes. Although fulleranes also exhibit higher chemical stability than fullerynes, but pentagon or hexagon of the fullerane structures cannot pass ions and molecules. Applications of fullerynes can be included in the storage of ions and gases at the nanoscale. On the other hand, they can also be used as cathode/anode electrodes in lithium-ion batteries.

Fullerene Thermochemical Stability: Accurate Heats of Formation for Small Fullerenes, the Importance of Structural Deformation on Reactivity, and the Special Stability of C60

We have used quantum chemistry computations, in conjunction with isodesmic-type reactions, to obtain accurate heats of formation (HoFs) for the small fullerenes C₂₀ (2358.2 ± 8.0 kJ mol^-1), C₂₄ (2566.2 ± 7.6), and the lowest-energy isomers of C₃₂ (2461.1 ± 15.4), C₄₂ (2629.0 ± 20.5), and C₅₄ (2686.2 ± 25.3). As part of this endeavor, we have also obtained accurate HoFs for several medium-sized molecules, namely 216.6 ± 1.4 for fulvene, 375.5 ± 1.5 for pentalene, 670.8 ± 2.9 for acepentalene, and 262.7 ± 2.5 for acenaphthylene. We combine the energies of the small fullerenes and previously obtained energies for larger fullerenes (from C₆₀ to C₆₀₀₀) into a full picture of fullerene thermochemical stability. In general, the per-carbon energies can be reasonably approximated by the "R+D" model that we have previously developed [Chan et al. J. Chem. Theory Comput. 2019, 15, 1255-1264], which takes into account Resonance and structural Deformation factors. In a case study on C₅₄, we find that most of the high-deformation-energy atoms correspond to the sites of the C-Cl bond in the experimentally captured C₅₄Cl₈. In another case study, we find that C₆₀ has the lowest value for the maximum local-deformation energy when compared with similar-sized fullerenes, which is consistent with its "special stability". These results are indicative of structural deformation playing an important role in the reactivity of fullerenes.

A Simple Model for Relative Energies of All Fullerenes Reveals the Interplay between Intrinsic Resonance and Structural Deformation Effects in Medium-Sized Fullerenes

Fullerenes are sheets of sp² carbon atoms wrapped around to form spheres. With this simple consideration, we have in the present study devised and (with over 3600 DFT data points) successfully validated a simple model, termed R+D, for estimating the relative energies of fullerenes. This model contains a resonance component to account for the intrinsic differences between the π-energies of different fullerenes, and a deformation component for treating the distortions from planarity. Notably, we find that both terms (and they alone) are required to obtain good relative energies, which lends support to the formulation of the R+D model. An interesting finding is that for some medium-sized IPR fullerenes, their isomers show similar variations in the two components. We deduce that these fullerenes may represent a good opportunity for tuning molecular properties for practical applications. We hope that the promising results of the present study will encourage further investigations into fullerenes from a fundamental perspective.

Gas-Phase Heat of Formation Values for Buckminsterfullerene (C60), C70 Fullerene (C70), Corannulene, Coronene, Sumanene, and Other Polycyclic Aromatic Hydrocarbons Calculated Using Density Functional Theory (M06 2X) Coupled with a Versatile Inexpensive Group-Equivalent Approach

A straightforward procedure using density functional theory (M06 2X) coupled with a group-equivalent approach is described that was used to calculate gas-phase heat of formation (Δ_f H°_g,298) values for buckminsterfullerene (C₆₀), C70 fullerene (C₇₀), corannulene, coronene, and sumanene. This procedure was also used to calculate exceptionally accurate Δ_f H°_g,298 values for a variety of single-ring aromatic and 2-7 ring polycyclic aromatic hydrocarbons (PAHs) as well as a large selection of other hydrocarbons and phenols. The approach described herein is internally consistent, and results for C₆₀, C₇₀, corannulene, coronene, and sumanene are in very close agreement with results reported by others who used higher-level computational theory. Statistical analysis of a test set containing benzene and 18 two to seven ring PAHs demonstrated that by using this approach a mean absolute deviation (MAD) and a root-mean-square deviation (RMSD) of 0.8 and 1.3 kJ/mol, respectively, were achieved for reference/experimental Δ_f H°_g,298 values versus calculated/predicted Δ_f H°_g,298 values. For statistical analysis of a larger test set containing 235 aromatic and aliphatic hydrocarbons and phenols, a MAD and a RMSD of 1.2 and 1.9 kJ/mol, respectively, were achieved for reference/experimental Δ_f H°_g,298 values versus calculated/predicted Δ_f H°_g,298 values.

Coarse-Grained Simulations Using a Multipolar Force Field Model

This paper presents a coarse-grained molecular simulation for fullerenes based on a multipolar expansion method developed previously. The method is enabled by the construction of transferable united atoms potentials that approximate the full atomistic intermolecular interactions, as obtained from ab initio electronic structure calculations supplemented by empirical force fields and experimental data, or any combination of the above. The resultant series contains controllable moment tensors that allow to estimate the errors, and approaches the all-atom intermolecular potential as the expansion order increases. We can compute the united atoms potentials very efficiently with a few interaction moment tensors, in order to implement a parallel algorithm on molecular interactions. Our simulations describe the mechanism for the condensation of fullerenes, and they produce excellent agreement with benchmark fully atomistic molecular dynamics simulations.

Comprehensive theoretical study of all 1812 C60 isomers

All 1812 C₆₀ isomers are investigated with high-level quantum chemical methods to benchmark semiempirical approaches and find appropriate stability criteria.

Generalized structural motif model for studying the thermodynamic stability of fullerenes: from C60to graphene passing through giant fullerenes

A generalized motif model to describe the stability of neutral fullerenes, covering the full range of cage sizes, starting from C₆₀, going through giant fullerenes, and ultimately leading to graphene.

The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96-145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website.

The first structural confirmation of a C102 fullerene as C102Cl20 containing a non-IPR carbon cage

The chlorination of a pristine C102 fullerene separated by HPLC from fullerene soot afforded crystals of C102Cl20 with a non-IPR (IPR = isolated pentagon rule) cage containing two pairs of fused pentagons; structural reconstruction of a two-step Stone-Wales rearrangement revealed the starting IPR isomer (no. 19) of C102.

Computational termochemistry study of the C₈₀ isomers and their endo lanthanum complexes by applying homodesmotic and isodesmic reactions

C₈₀ is a fullerene species which appears in different isomeric configurations. A general homodesmotic reaction previously designed to study the energy of fullerenes was implemented, in order to analyze the energy of this family of isomers. These results concur with some of the experimental data, but energy differences referring to all the configurations yield novel propositions about their particular behavior. The corresponding lanthanum complexes are also analyzed here and a new isodesmic reaction was designed for this particular case.

B3LYP-SVM METHOD FOR THE ESTIMATION OF MOLECULAR ENTHALPIES OF FORMATION

Support vector machine (SVM) is used to predict the enthalpies of formation at 298 K [Formula: see text] for 261 molecules based on B3LYP/6-311g (3df,2p) results. With data randomly separated into two parts: 195 for training set and 66 for test set, the resulting mean absolute deviation (MAD) and maximum deviation (MD) for training set are 1.51 kcal/mol and 9.23 kcal/mol (correlation coefficient R = 0.9995), and for test set they become to 1.78 kcal/mol and 7.31 kcal/mol (R = 0.9990). The result is improved according to G2 method.

Electronic structure and stability of fullerene C82 isolated-pentagon-rule isomers

All nine isolated-pentagon-rule isomers of fullerene C(82) were investigated by the DFT method with the B3LYP functional at the 6-31G, 6-31G*, and 6-31+G* levels. The distribution of single, double, and delocalized π-bonds in the molecules of these isomers is shown for the first time. The obtained results are fully supported by DFT quantum-chemical calculations of electronic and geometrical structures of these isomers. The molecules of isomers 7 (C(3v)), 8 (C(3v)), and 9 (C(2v)) contain some radical substructures (such as the phenalenyl-radical substructure), which indicates that they are unstable and cannot be obtained as empty molecules. Thus, there is a possibility of obtaining them only as endohedral metallofullerenes or exohedral derivatives. Isomers 1 (C(2)), 2 (C(s)), 4 (C(s)), 5 (C(2)), and 6 (C(s)) with closed electronic shell are supposed to be stable, resembling isomer 3 (C(2)), which has just been extracted experimentally as an empty fullerene. We assume they can be produced as empty molecules.

Identifying Tm@C82 isomers with density functional theory calculations

Density functional theory calculations have been performed to study the geometrical and electronic properties of endohedral metallofullerene Tm@C(82) isomers. Three energetically favorable isomers (with C(s), C(2) and C(2v) symmetry, respectively) are identified which are consistent with the nuclear magnetic resonance (NMR) observations. The simulated ultraviolet photoelectron spectra (UPS) based on the three structures agree well with the measurements. Particularly, the parent cage of the experimentally observed Tm@C(82) isomer with C(s) symmetry is newly assigned, which matches the experiments better than early assignments. In addition, strong interaction between an endohedral Tm atom and the C(82) cage is discussed and is thought to be responsible for the dramatic change in the relative stability of C(82) isomers when Tm is encapsulated.

Functionalized graphene sheet colloids for enhanced fuel/propellant combustion

We have compared the combustion of the monopropellant nitromethane with that of nitromethane containing colloidal particles of functionalized graphene sheets or metal hydroxides. The linear steady-state burning rates of the monopropellant and colloidal suspensions were determined at room temperature, under a range of pressures (3.35-14.4 MPa) using argon as a pressurizing fluid. The ignition temperatures were lowered and burning rates increased for the colloidal suspensions compared to those of the liquid monopropellant alone, with the graphene sheet suspension having significantly greater burning rates (i.e., greater than 175%). The relative change in burning rate from neat nitromethane increased with increasing concentrations of fuel additives and decreased with increasing pressure until at high pressures no enhancement was found.

Isolation and structural X-ray investigation of perfluoroalkyl derivatives of six cage isomers of C84

Perfluoroalkylation of a higher fullerene mixture with CF(3)I or C(2)F(5)I, followed by HPLC separation of CF(3) and C(2)F(5) derivatives, resulted in the isolation of several C(84)(R(F))(n) (n=12, 16) compounds. Single-crystal X-ray crystallography with the use of synchrotron radiation allowed structure elucidation of eight C(84)(R(F))(n) compounds containing six different C(84) cages (the number of the C(84) isomer is given in parentheses): C(84) (23)(C(2)F(5))(12) (I), C(84) (22)(CF(3))(16) (II), C(84) (22)(C(2)F(5))(12) (III), C(84) (11)(C(2)F(5))(12) (IV), C(84) (16)(C(2)F(5))(12) (V), C(84) (4)(CF(3))(12) (VI with toluene and VII with hexane as solvate molecules), and C(84) (18)(C(2)F(5))(12) (VIII). Whereas some connectivity patterns of C(84) isomers (22, 23, 11) had previously been unambiguously confirmed by different methods, derivatives of C(84) isomers numbers 4, 16, and 18 have been investigated crystallographically for the first time, thus providing direct proof of the connectivity patterns of rare C(84) isomers. General aspects of the addition of R(F) groups to C(84) cages are discussed in terms of the preferred positions in the pentagons under the formation of chains, pairs, and isolated R(F) groups.

Synthesis of a higher fullerene precursor-an "unrolled" C84 fullerene

We report the synthesis of the C(84)(20)-fullerene related hydrocarbon C(84)H(42), which possesses more than 83% of the C(84)(20)-fullerene connectivity, and its selective transformation to C(84) by flash vacuum pyrolysis (FVP).

Crystallographic characterization and identification of a minor isomer of C(84) fullerene

We report the synthesis and single crystal X-ray analysis of C(84) ().AgTPP (Ag tetraphenylporphyrin) cocrystal-the first ordered crystal structure containing a pristine higher fullerene.

Search for lowest-energy fullerenes: C98 to C110

By combining the semiempirical density-functional based tight-binding optimization with density-functional theory single-point energy calculation at the PBE1PBE/6-311G level, we propose an efficient computational approach to determine lowest-energy structures of large-sized carbon fullerenes. Our studies show that C(92) (D(3): 28) and C(94) (C(2): 43) are the new leading candidates for the lowest-energy structures of C(92) and C(94). Moreover, for the first time, the lowest-energy structures of C(98)-C(110) are identified on the basis of the density-functional theory calculation. The lowest-energy isomers C(102) (C(1): 603) and C(108) (D(2): 1771) are readily isolated experimentally because they are much lower in energy than their other low-lying IPR isomers.

Prediction of low-energy isomers of large fullerenes from C132 to C160

To predict energetically favored isomers, we used a topological scheme as a prescreening tool to select candidate isomers for each fullerene from C(106) to C(160). Comparison with the PM3 and tight-binding (TB) potential calculated results and few published data for the low-energy isomers of C(106) to C(130) indicates that the prescreening approach is feasible. For each fullerene from C(132) up to C(160), the selected 1000 candidate isomers were further optimized by PM3 and TB potential. The analysis of the semiempirical PM3 and TB results of C(106) to C(160) provides some qualitative features of the large fullerenes. Furthermore, calculations at the B3LYP/6-31G*//B3LYP/3-21G level of theory were carried out on the top ten PM3 and TB low-energy isomers of C(132) to C(160) to accurately predict the stable isomers, and the HOMO-LUMO gap, the ionization energy, and electron affinity of the lowest-energy isomers were also investigated at the same level.

Increasing Stability of the Fullerenes with the Number of Carbon Atoms: The Experimental Evidence

The values of the molar standard enthalpies of formation, Delta(f)H(o)(m)(C(76), cr) = (2705.6 +/- 37.7) kJ x mol(-1), Delta(f)H(o)(m)(C(78), cr) = (2766.5 +/- 36.7) kJ x mol(-1), and Delta(f)H(o)(m)(C(84), cr) = (2826.6 +/- 42.6) kJ x mol(-1), were determined from the energies of combustion, measured by microcombustion calorimetry on a high-purity sample of the D(2) isomer of fullerene C(76), as well as on a mixture of the two most abundant constitutional isomers of C(78) (C(2nu)-C(78) and D(3)-C(78)) and C(84) (D(2)-C(84), and D(2d)-C(84). These values, combined with the published data on the enthalpies of sublimation of each cluster, lead to the gas-phase enthalpies of formation, Delta(f)H(o)(m)(C(76), g) = (2911.6 +/- 37.9) kJ x mol(-1); Delta(f)H(o)(m)(C(78), g) = (2979.3 +/- 37.2) kJ x mol(-1), and Delta(f)H(o)(m)(C(84), (g)) = (3051.6 +/- 43.0) kJ x mol(-1), results that were found to compare well with those reported from density functional theory calculations. Values of enthalpies of atomization, strain energies, and the average C-C bond energy were also derived for each fullerene. A decreasing trend in the gas-phase enthalpy of formation and strain energy per carbon atom as the size of the cluster increases is found. This is the first experimental evidence that these fullerenes become more stable as they become larger. The derived experimental average C-C bond energy E(C-C) = 461.04 kJ x mol(-1) for fullerenes is close to the average bond energy E(C-C) = 462.8 kJ x mol(-1) for polycyclic aromatic hydrocarbons (PAHs).

Calculations of the C2 fragmentation energies of higher fullerenes C80 and C82

The C2 fragmentation energies of the most stable isolated-pentagon-rule (IPR) isomers of the C80 and C82 fullerenes were evaluated with second-order Møller-Plesset (MP2) theory, density-functional theory (DFT) and the semiempirical self-consistent charge density-functional tight-binding (SCC-DFTB) method. Zero-point energy, ionization energy and empirical C2 corrections were included in the calculation of fragmentation energies for comparison with experimental C2 fragmentation energies of the fullerene cations. In the case of the most probable Stone-Wales pathway of C2 fragmentation of C80, the calculated [Formula: see text] agree well with experimental data, whereas in the case of C(82) fragmentation, the calculated [Formula: see text] exceed by up to 1.2 eV the experimental ones, which suggests that other IPR isomers may be present in sufficient amounts in experimental samples. Computer-intensive MP2 calculations and DFT calculations with larger basis sets do not yield much improved C2 fragmentation energies, compared to those reported earlier with B3LYP/3-21G. On the other hand, semiempirical approaches such as SCC-DFTB, which are orders of magnitude less intensive, yield satisfactory fragmentation energies for higher fullerenes and may become a method of choice for routine calculations of fullerenes and carbon nanotubes.

High-resolution kinetic energy release distributions and dissociation energies for fullerene ions Cn+, 42 < or = n < or = 90

We have measured the kinetic energy released in the unimolecular dissociation of fullerene ions, Cn+ --> C(n-2)+ + C2, for sizes 42 < or = n < or = 90. A three-sector-field mass spectrometer equipped with two electric sectors has been used in order to ensure that contributions from isotopomers of different masses do not distort the experimental kinetic energy release distributions. We apply the concept of microcanonical temperature to derive from these data the dissociation energies of fullerene cations. They are converted to dissociation energies of neutral fullerenes with help of published adiabatic ionization energies. The results are compared with literature values.

Structure and stability of neutral polyoxometalate cages: (Mo2O6) m (m=1–13)

The structure and stability of neutral polyoxometalate cages (Mo2O6)m (m=1-13) have been computed systematically. These neutral cages can be viewed topologically as polyhedra containing triangles (f3) and squares (f4). The relative stability of these polyhedra is associated with the location and separation of the f3. The initial stable isomers were preselected by the number of shared triangle edges (N33), and the predicted stability was validated further at the GGA-PW91/DND level of density function theory with the fine quality of mesh size. For large clusters, the square neighbor signature (P4444), which is similar to the hexagon neighbor rule for fullerene, becomes more applicable. The calculated disproportionation energies indicate that Mo6O18 (O(h), Lindqvist), Mo12O36 (O(h), alpha Keggin), Mo18O54 (D(3h), Wells-Dawson) and Mo24O72 (O(h)) cages have enhanced stability. [structure in text]. Mo6O18 (O(h)), Mo12O36 (O(h)), Mo18O54 (D(3h)) and Mo(24)O72 (O(h)) are the most stable neutral polyoxometalate cages on the basis of the structural and energetic criteria. They can therefore be considered as the inorganic fullerenes.