From the isosurface map of electron density in Fig. 1a, the symmetrical distribution of electron in CH4@C60 was clearly shown. The electron density along the C-C bonds shared by one five-membered and one six-membered carbon ring ([5, 6] bond, labeled by 1 in Fig. 1) was smaller than C-C bonds shared by two six-membered carbon rings ([6, 6] bond, labeled by 2 in Fig. 1). This character was also clearly shown in isosurface map of gradient norm (Fig. 1b) and laplacian (Fig. 1c) of electron density.
The value of the laplacian function was defined as the trace of the electron density Hessian matrix at a point, which was the result of the Laplace operator applied to the electron density. The positive value of laplacian function means that the electron density was mainly divergent, otherwise the negative value mean that aggregation of electron density was dominant. In the isosurface map of laplacian of electron density of CH4@C60 (Fig. 1c), the positive value was indicated by green color and the purple color was used to indicate negative value. It can be clearly seen that the positive laplacian value at the center of carbon and the negative laplacian value along the C-C bonds and the center of hydrogen atom which indicated the flow direction of the electron in this molecule. The symmetrical distribution of electron density and the difference between the [5, 6] and [6, 6] bonds were also clearly shown in isosurface map of laplacian of electron density (Fig. 1c).
To illustrate the interaction between CH4 and the C60 in detail, the δg function of interaction area in the molecule were obtained through the multi wavefunction analysis software Multiwfn 3.7. It can be clearly shown that only the weak Van der Waals interaction of C-H…π exist between the CH4 and C60 which was same as the results of experiments before.
The distribution of molecular orbital of CH4@C60 was illustrated by Density-of-states (DOS) map in Fig. 3. The DOS curve reflected the number of molecular orbitals in unit energy interval at corresponding energy. The total DOS (TDOS) of CH4@C60 and the partial DOS (PDOS) of maps contributed by C60 and CH4 respectively were simultaneously shown in Figure. 3. Meanwhile isosurface maps of three molecular orbitals (orb92, HOMO, LUMO) with energy at about − 22.42eV, -8.10eV and − 2.12eV respectively at the current wB97XD3/def2-TZVP level were also drawn in the Figureure for comparation. It can be seen from Fig. 3 that the HOMO and LUMO of the CH4@C60 molecule were almost solely contributed by the C60 structure yet without the admixture of CH4 part. Within the occupied orbitals there were only several molecular orbitals with energy about − 23eV, -16eV and − 13eV in which the CH4 component made the contribution. This conclusion can also be deduced by the isosurface maps of molecular orbitals in Fig. 3.
The charge decomposition analysis (CDA) method was a valuable tool to analyze quantitatively the interactions between two molecular fragments in terms of the linear combination of the donor and acceptor fragment orbitals’ donation and polarization using quantum mechanical calculations. Within the CH4@C60 molecule, the C60 and CH4 were defined as two fragments to be analyzed using CDA method. According to the results of CDA analysis, the orbital interaction between two fragments of C60 and CH4 were illustrated in Fig. 4. The solid and dashed horizontal bar in the Figureure indicated the occupied and unoccupied molecular orbital respectively. The CH4@C60 molecular orbital was connected by red line with the fragments’ orbital which contributed more than 10% in that molecular orbital. It can be also seen in Fig. 4 that mostly the CH4@C60 molecular orbitals including HOMO and LUMO were contributed by the C60 structure. The CH4 part just played a role in few occupied and unoccupied CH4@C60 molecular orbitals which agreed with the conclusion from Fig. 3 mentioned before.
Fukui function and related dual descriptors are very popular methods defined under the framework of conceptual density functional theory for predicting reaction sites. Fukui functions and related dual descriptors are fine for most systems. However, for some systems which have higher order point group symmetry (tends to degenerate the frontline molecular orbitals) the Fukui function and the dual descriptor may give meaningless results, such as the distribution of the function does not satisfy the symmetry of the system, and therefore obviously violates the basic chemical intuition. Ricardo et al put forward the concept of orbital weighted Fukui function and dual descriptor of orbital weighted.[33, 34] Compared with the general form of Fukui function and double descriptors, the orbital weighted form has the advantage that it can be reasonably applied to the system of line orbital (quasi) degeneratation, and for the system with symmetry, the results completely satisfy the molecular symmetrical structure.
Here the orbital weighted double descriptors of CH4@C60 obtained from Multiwfn were illustrated in Fig. 5 to present the reaction sites of the molecule.
The negative value of orbital weighted double descriptors on [6, 6] bond (purple) indicated it was most easily electrophilic attacked. Otherwise, the positive value of orbital weighted double descriptors on [5, 6] bond (green) indicated it was most easily nucleophilic attacked. This result was similar as the former study on C60 and reflected the weak interaction between CH4 and C60. To further understand the vibrational properties of CH4@C60, the IR spectrum was calculated and displayed in Fig. 6. The several obvious peaks came from the vibration modes of C60 part which indicated the negligible interaction between C60 and CH4 at this energy level.