Analysis of Nonclassical Fullerene C₂₄ Regioisomers Encapsulating H₂O using Hybrid Density Functional Methods B3LYP and M06-2X

Abstract

The atomic structures and electronic properties of six classical and nonclassical $H_2O$@$C_{24}$ fullerene regioisomers are systematically studied using the hybrid density functional B3LYP method and M06-2X method with empirical dispersion in conjunction with the 6-31G(d,p) basis sets. The charge transfer, frontier orbitals, dipole moment, energy gap between the HOMO and LUMO, and volume change of the $C_{24}$ cage are analyzed upon encapsulation of a $H_2O$ molecule in each $C_{24}$ regioisomer. All encapsulation processes are endothermic and the relative stabilities of six $C_{24}$ fullerene regioisomers change upon encapsulation of $H_2O$.

Introduction

Fullerenes are interesting carbon-based molecules because of their atomic structure and electronic features.1 Classical fullerenes are trivalent polyhedral carbon cages composed of pentagonal and hexagonal rings, which are generally more stable than nonclassical fullerenes, which include three-, four-, and seven-membered rings as well as larger analogs.2 It has been suggested that the inclusion of one or more fourmembered rings into fullerene cage can lead to energetically competitive isomers. Qian et al. used density functional theory and X-ray crystallography to identify an isomer of C62 that contains a four-membered ring.3

Among the various carbon clusters, C20 is the smallest possible classical fullerene and there is experimental evidence for the existence of three different isomers: cage, bowl, and ring.4 Density functional theory and ab initio calculations have been used to propose the structures and stabilities of ring, bowl, and fullerene C24 isomers.5-8 Previous calculations have indicated that the D6-symmetrical structure of the C24 classical fullerene is the most stable of the regioisomers. Recently, the relative stabilities of classical and nonclassical C24 fullerene regioisomers were reported.9

Endohedral fullerenes have attracted significant attention because of their interesting physical and chemical properties, which include negative thermal expansion,10 superconductivity, 11 ferroelectricity,12 and nonlinear optical properties.13 Experimental and theoretical studies have been performed on the structures and properties of endohedral fullerenes.14 The structures and electronic properties of TM＠C24 (TM = Mn, Fe, Co, Ni, Cu, and Zn),15 (TM = Cr, Mo, and W),16 and (TM= Sc, Y, and La)17 determined using the hybrid B3PW91 functional have been reported; however, these studies are restricted to C24 with D6d symmetry. To the best of our knowledge, there are no computational calculations regarding H2O＠C24 with six C24 regioisomers.

This study investigates H2O＠C24 fullerene isomers on the basis of six regioisomers of C24 using the hybrid density functional method B3LYP and the hybrid functional M06- 2X method with empirical dispersion in conjunction with the 6-31G(d,p) basis sets. A H2O molecule is encapsulated in the six regioisomers of the C24 fullerene. Hence, to elucidate the interactions between H2O and each C24 cage isomer, we assess how the atomic structure and electronic structure of each C24 fullerene regioisomer is affected by encapsulation of H2O. We also analyze the relative stabilities of the six neutral regioisomers of H2O＠C24.

 

Calculations

Hybrid density-functional theory (DFT) with Becke’s three-parameter hybrid method, Lee-Yang-Parr exchangecorrelation functional theory (B3LYP),18,19 and the hybrid meta exchange-correlational M06-2X functional20 was used to optimize the geometries of the C24 and H2O＠C24 regioisomers, as shown in Figure 1. The electron basis set of 6- 31G(d,p) was used in this study.21 The atomic geometries of all the C24 and H2O＠C24 regioisomers were fully optimized using the Gaussian 2003 B.04 and 2009 A.01 package suites for the B3LYP calculations and M06-2X calculations, respectively.22

All the stationary point geometries were analyzed by evaluating the harmonic vibrational frequencies at the same theoretical level. The cut-offs on the forces and step sizes were reduced using the pruned (99,590) grid (keywords: Opt = Tight, Grid = ultrafine) to obtain accurate geometries for all isomers of C24 with and without H2O encapsulation except for the default M06-2X calculations for isomer 1 (Opt, Scf = Tight, Grid = 75,302). The relative energies and HOMO and LUMO orbitals of the regioisomers were also analyzed.23

Figure 1.Atomic structures of fullerene C24 and H2O＠C24 isomers. Isomer 1 is a classical fullerene (C24_t0p12h2_D6); 2 is a nonclassical fullerene containing two 4-membered rings (4-MRs) (C24_t2p8h4_C2); 3 is a nonclassical fullerene containing one 4-MR (C24_t1p9h4_CS); 4 is a nonclassical fullerene containing two 4- MRs (C24_t2p8h4_CS); 5 is a nonclassical fullerene containing six 4-MRs (C24_t6p0h8_Oh); and 6 is a nonclassical fullerene containing two 4-MRs (C24_t2p8h4_D2h). Here, t, p, and h denote tetragonal, pentagonal, and hexagonal polygons and the numbers following the letters indicate the number of that type of polygon; e represents the encapsulated fullerene.

 

Results and Discussion

By performing calculations to fully optimize the atomic structures of the C24 and H2O@C24 regioisomers at the B3LYP/6-31G(d,p) and M06-2X/6-31G(d,p) levels without any constraints, we determined the relative energies (eV) of six regioisomers, including a classical fullerene, 1 (C24_t0p12h2_D6), and five nonclassical fullerenes, 2 (C24_ t2p8h4_C2), 3 (C24_t1p9h4_CS), 4 (C24_t2p8h4_CS), 5 (C24_ t6p0h8_Oh), and 6(C24_t2p8h4_D2h), as shown in Figure 1. Here, t, p, and h denote tetragonal (𝑖.𝑒., 4-MR), pentagonal (𝑖.𝑒., 5-MR), and hexagonal (𝑖.𝑒., 6-MR) polygons; the numbers following the letters indicate the number of that type of polygon.

Optimization of the geometries of the six regioisomers encapsulating H2O revealed that the relative energies of the C24 regioisomers are affected by encapsulation of H2O. Also, we analyzed the effect of encapsulation of H2O on the volume of the cage of neutral C24 and calculated the encapsulation energies corrected with the zero-point energies plus the basis-set superposition errors of the regioisomers of H2O＠C24, as shown in Table 1 and 2. The order of the relative energies of the C24 isomers, as shown in Table 1, obtained via both the B3LYP/6-31G(d,p) and M06-2X/6- 31G(d,p) calculations is the same as that obtained for PBE1PBE/cc-pVTZ.9 Thus, dispersion interactions do not affect the relative stability of these six C24 isomers. From the perspective of the dispersion interaction effect, isomers 6 and 3 have the highest and second highest relative energies, respectively, of the six C24 isomers

As shown in Table 2, all encapsulation processes, except for that of the 6e(O) isomer, are endothermic by ~10 eV; this is likely due to swelling of the C24 cage upon encapsulation of H2O. The encapsulation of H2O by 6e(O) is exothermic by 1.2 eV, which is likely due to a bond breaking in the C24 cage. Here, all nine regioisomers are at the local minima because they have all real frequencies. Upon encapsulation of H2O by C24, the order of the relative energies of the nine H2O＠C24 regioisomers differs from that of the C24 isomers: Isomer 4e, which has two tetragons, has the lowest energy and isomer 5e, which has six tetragons, has the second lowest energy. From the perspective of the dispersion interaction effect, isomer 6e(1) has the highest and isomer 1e has the second highest relative energy of the six H2O＠C24 isomers.

Also, the relative energies for the nine H2O@C24 isomers obtained using the B3LYP/6-31G(d,p) calculations, are in the same increasing order as the results obtained using the M06-2X/6-31G(d,p) calculations; this implies that the dispersion interactions affect the absolute values, but do not change the order of the relative stability of these H2O＠C24 isomers

In Table 1, the two columns of the volumes of the C24 cages show that the cage volumes of the C24 isomers obtained from the M06-2X calculations are smaller than the those obtained from the B3LYP calculations, which implies that the shrinkage effect arises from the empirical dispersion interaction of M06-2X on the C24 cages. Also, isomer 5, which has six 4-MRs, has the largest cage volume of the C24 regioisomers. Isomer 1, which is a classical fullerene, has the next largest cage volume. Thus, the cage volumes of the C24 isomers are not dependent on the number of tetragons.

Table 1.asee reference 9. Reference energy: −913.7037 a.u. (B3LYP) and −913.3996 a.u. (M06-2X).

Table 2.Here, EE = energy of H2O＠C24 − (energy of C24 + energy of H2O). Reference energy: −989.7248 a.u. (B3LYP) and −989.4100 a.u. (M06-2X)

As shown in Table 3, the shrinkage caused by the empirical dispersion contribution of the M06-2X is comparable to the cage volume obtained from the B3LYP calculations in the H2O＠C24 isomers. Isomer 5e, which has six 4-MRs, has the largest cage volume of the H2O＠C24 regioisomers in both calculations. However, the change in the cage volume of 5e is very dependent on the dispersion interaction; the volume change upon encapsulation of H2O is the smallest using B3LYP calculations and the largest using M06-2X calculations with the empirical dispersion interaction. Even though isomers 4 (C24_t2p8h4_CS) and 6 (C24_t2p8h4_D2h), which are nonclassical fullerenes containing two 4-MRs, eight 5-MRs, and four 6-MRs and have the same number of polygons, the volumes of the cages of 4e and 6e both with and without encapsulated H2O are different

Table 3.Here, ΔV% = 100 × (V2╶V1)/V1.

Our calculated results for the cage volumes of the regioisomers both with and without dispersion interactions show that the volume change upon encapsulation of H2O in C24 is less than 10%. The increase of the cage volume for all isomers upon encapsulation of H2O obtained from the M06-2X/ 6-31G(d,p) calculations, which includes empirical dispersion interactions, is smaller than that obtained from the B3LYP/6-31G(d,p) calculations, except for 5e.

Figure 2.HOMOs and LUMOs of six C24 fullerene regioisomers at the level of B3LYP and M06-2X theory.

Figure 3.HOMOs and LUMOs of six H2O＠C24 fullerene regioisomers.

Figure 4.The energy levels (eV) of the HOMO+1, HOMO, LUMO, and LUMO1 of six H2O＠C24 fullerene regioisomers obtained via B3LYP (represented by black lines) and M06-2X (represented by red lines) calculations.

The HOMO of 1 isomer is different than the HOMO of 1e, although their LUMOs are very similar. In contrast, the HOMO of 2 is similar to that of 2e, while the LUMOs of 2 and 2e are different. Similar patterns are evident in 1 and 1eand 5 and 5e, in which the frontier orbitals of C24 are oriented inward in H2O@C24. Similarly, the patterns of 2 and 2e and 4 and 4e are comparable.

As shown in Figure 4, analyses of the frontier orbital energy gaps indicate that isomers 2e and 4e are kinetically more stable than isomers 2 and 4, while 6 is more stable than 6e. Isomers 1e, 2e, and 5e have almost the same kinetic stability as their respective C24 isomers before encapsulation of H2O from the perspective that the energy gaps between the HOMO and LUMO of 1, 2, and 5 are almost the same as those encapsulating H2O. The M06-2X calculations result in greater energy gaps between the HOMOs and LUMOs than the B3LYP calculations

Figure 5 shows the interatomic distances between the carbon atoms of the cage and H2O and the atomic structures of the cages with H2O. Isomers 2e, 3e, and 4e show the covalent bond distances between the H sites and O site with respect to the sum of the covalent radii. We also analyzed the structures and interatomic distances using natural orbital analysis; the results are summarized in Table 4 and Tables S1-S8 in the supplementary material. As evident from Table 4, isomers 1e and 5e feature charge transfer from the encapsulated species to the C24 cage, while the opposite occurs for the other isomers. There are no covalent bonds between H and O in 2e(1), 2e(2), 3e, 4e, and 6e(1), one covalent bond between H and O in 1e, 6e(2), and 6e(O), and two covalent bonds between H and O for 5e. As shown in Tables S1―S8, the number of fragments without covalent interactions are as follows: Three (𝑖.𝑒., C24O, H, H) for 2e(1), 2e(2), 3e, 4e, and 6e(1) with both B3LYP and M06-2X calculations, two (i.e., HC24O, H) for 6e(2) with B3LYP calculations and for 6e(O) with M06-2X calculations, three (𝑖.𝑒., C24, OH, H) for 1e, and two (𝑖.𝑒., C24, H2O) for 5e.

These results suggest that the interatomic distances bet-ween the H and O sites of H2O are squeezed in the C24 cages. Isomers 1e and 5e have interatomic distances between the C and O sites of ∼2.2 Å, which are longer than the covalent bond distance between C and O atoms based on the covalent radii; this implies that there is no covalent bonding character. The nonbonding character in isomers 1e and 5e can be understood from the HOMO shown in Figure 3. Isomer 6e shows that the distances between the C and O sites are within the range of covalent bond distances based on the covalent radii; also, the natural orbital analysis suggests covalent bonding character.

Table 4.Natural charge populations and natural electron configurations of the lowest H2O＠C24 isomers computed using hybrid B3LYP and M06-2X calculations

 

Conclusions

In this paper, we systematically described the relative stability and atomic and electronic structures of six H2O＠C24 regioisomers using hybrid density functional theory with B3LYP and M06-2X methods. Our calculated results show that the volume change of C24 upon encapsulation of H2O is less than 10%. The order of the relative stability of the C24 isomers differs from that of H2O＠C24. All encapsulation processes are endothermic except that of 6(O). Analyses of the frontier orbital energy gaps indicate that isomers 3e and 6e are kinetically more stable than isomers3 and 6, but the relative stabilities of the other isomers are reversed. In addition, natural population analyses revealed that there are no covalent bonds between the C and O sites in the six H2O＠C24 isomers except isomers 1e and 5e, which have one OH bond and two OH bonds, respectively

Figure 5.The shortest atomic distances between carbon atoms of the C24 cage and the H and O atoms of the encapsulated H2O molecule (red lines represent the distances between carbon and oxygen atoms and blue lines represent the distances between carbon and hydrogen atoms).