Doping the Buckminsterfullerene by Substitution : Density Functional Theory Studies of C 59 X ( X = B , N , Al , Si , P , Ga , Ge , and As )

e heterofullerenes C59X (X = B, N, Al, Si, P, Ga, Ge, and As) were investigated by quantum chemistry calculations based on density functional theory. ese hybrid cages can be seen as doping the buckminsterfullerene by heteroatom substitution. e geometrical structures, relative stabilities, electronic properties, vibrational frequencies, dielectric constants, and aromaticities of the doped cages were studied systemically and compared with those of the pristine C60 cage. It is found that the doped cages with different heteroatoms exhibit various electronic, vibrational, and aromatic properties.ese results imply the possibility tomodulate the physical properties of these fullerene-based materials by tuning substitution elements.


Introduction
Fullerenes and related materials have aroused considerable attention since the discovery of buckminsterfullerene C 60 [1].During the last two decades, a great number of studies have been carried out to investigate the structures and physical properties of the carbon cages and the derivates [2][3][4][5][6][7].Among various nanostructures derived from fullerenes, the heterofullerenes, in which one or more carbon atoms of the cage are substituted by heteroatoms, have especially caught the eyes of the researchers.e heterofullerenes exhibit unique structural, electronic, and nonlinear optical properties due to the existence of the heteroatom, which are considerably different from those of the pure carbon cages [8][9][10][11][12][13]. erefore, heterofullerenes should be interesting new nanoscaled materials to be expected in the future.
In 1991, six years aer the experimental discovery of the buckminsterfullerene, Chai et al. [14] claimed the spectroscopic observation of gas-phase formation of heterofullerene ions, indicating that the synthesis of heterofullerenes was achieved.And then, the doped cages obtained by N, B, Si, P, Ge, As, and transition-metal (such as Pt, Fe, and Co) substitution have been reported by several research groups [15][16][17][18][19][20][21].Most recently, N-, P-, and Si-doped single-walled carbon nanotubes (SWCNTs) are also synthesized using chemical vapor deposition method [22].
As for the theoretical side, several literatures have bad attention to the heterofullerenes [9,[11][12][13][23][24][25][26][27][28].However, most of the studies mainly focus on the geometries and ordinary electronic structures of the doped fullerenes.Up until now a systematic study on the relationship of structure and property for C 60 -based heterofullerenes by a single approach has not been reported according to our best knowledge.Furthermore, the doped carbon cages are good candidates of materials for hydrogen storage, optical device, and molecular sensor [13,29,30], and they have become the state-of-theart research subjects in recent years.Additionally, the heterofullerenes are of prominent importance since they are the building blocks of various polymerized fullerenes structures [9,11,21].us, in order to achieve a further understanding

Models and Computational Methods
It is known that the synthesized C 60 have  ℎ symmetry and all the 60 carbon atoms are equivalent.e heterofullerene structure C 59 X, obtained by only one carbon atom of the C 60 cage substituted by other atoms, is studied in this paper.e atom of several main group elements, including III (B, Al, and Ga), IV (Si and Ge), and V (N, P, and As) subgroups, are considered as the heteroatom to replace the carbon atom of the buckminsterfullerene cage.e obtained C 59 X cage only reserves a mirror plane, and the symmetry is reduced to   .Different spin states for the doped cages are also considered in our calculations.e ground state is treated as the lowest energy structure.e DFT hybrid functional B3LYP method [31] is adopted to calculate C 59 X cages.Both the geometrical optimizations and the electronic property calculations through out this paper are all performed using the Kohn-Sham selfconsistent �eld method at B3LYP/6-31G * level with Gaussian 09 program [32].In the DFT calculations, symmetry constraint is always adopted, and default values of convergence criteria in Gaussian 09 program are used.According to the previous calculations, the B3LYP method has been successfully applied to the theoretical studies on fullerenebased nanostructres [2-5, 8, 13, 23, 27], and the methods used here could give rather good results compared with those obtained by various different functionals and basis sets [33].

Results and Discussion
3.1.Structures.e optimized structures of the C 59 X (X = B, N, Al, Si, P, Ga, Ge and As) and C 60 cages are shown in Figure 1, and the bond lengths and atomic coordinates are listed in Tables S1 and S2  Figure 1 we can see that all the doped cages undergo some distortions due to the heteroatoms, though they still preserve closed cage structures.Here to evaluate the sphericity of the doped cages, the sphericity parameter (SP) is calculated through the equation [34,35]: where , , and  are the rotational constants (in GHz) of the corresponding cages.e structure with larger SP value is distorted more away from perfect sphere.As shown in Table 1, SP of C 60 is zero, and it comes without surprise because C 60 with  ℎ symmetry has the perfect sphere shape.
As for the doped cages, the values of SP are in the range of 0.055-3.637GHz −1 , indicating that deformations of the cage are occurred when the heteroatom is introduced into the pristine cage.It can be seen that SP are about 0.1 GHz −1 for C 59 X with X = B and N, while about 1.2 GHz −1 for X = Al, Si, and P, and the values of SP are even larger than 3.1 GHz −1 for X = Ga, Ge, and As.us, it is clearly that the cage with larger heteroatom gives larger SP and more obvious distortion.en we pay attention to the bond lengths of the heterofullerenes.It is well-known that there are two kinds of C-C bond in C 60 cage, the [6,6] bond and the [5,6] bond.e bond lengths are 1.395 and 1.454 Å for [6,6] and [5,6] bonds, respectively, based on our DFT calculations, which agree well with 1.391 (or 1.39) and 1.455 (or 1.46) Å by neutron diffraction experiments [36,37].When the carbon cage is doped by the heteroatom, the C-X bonds are presented.From Table S1, it can be seen that the C-X bond lengths are in the range of 1.404-1.950Å. e bond lengths increased obviously for X = B, Al, Si, P, Ga, Ge, and As, ranking from 1.526 Å to 1.950 Å.However, the C-N bonds in C 59 N are 1.408 and 1.424 Å, and thus the original [5,6] bond is even decreased by 0.03 Å compared with that in the pristine cage.It is also found that the C-X bond lengths increase more signi�cantly for the larger heteroatoms.For instance, the C-B bonds are 1.526 and 1.549 Å, while the C-X bonds (X = Al, Si, P) are calculated to be within 1.796-1.904Å, but the values are even 1.876-1.950Å for X = Ga, Ge, As. ese results also agree with our SP analyses as well as the previous studies [25][26][27][28].Now we focus on the C-C bonds in the doped fullerenes.As shown in Table S1, the original [6,6] and [5,6] bonds exhibit slight changes, with the lengths in the range of 1.385-1.427Å and 1.433-1.517Å, respectively.Moreover, it also can be seen that the C-C bond lengths change more signi�cantly near the region of the heteroatom, but almost inert in the region away from the heteroatom.

Energies and Relative
Stabilities.e doped cages with different spin-multiplicity states are calculated with openshell DFT B3LYP/6-31G * method to determine the ground state, and their energies are listed in Table S3 of supplementary material.It can be seen that the high-spin state structures always exhibit higher energies than those of the low-spin states according to the obtained energies (both corrected and uncorrected with zero-point vibrational energies).us, the spin multiplicity of the ground state of C 59 X is 1 for IV group elements, and 2 for III and V elements, respectively.
In order to study the thermodynamic stabilities of the doped cages, the cohesive energy ( coh ) per atom are calculated with the energies corrected with zero-point energy (ZPE), and the obtained results are listed in Table 1 and shown in Figure 2(a).Here the system with larger  coh is more stable.We can see that  coh of C 60 is calculated to be 6.813 eV/atom, and agrees with the previous results [9,38]. coh of the heterofullerene cages are in the range of 6.692-6.807eV/atom, and slightly smaller than those of the pristine cage.us the introduced heteroatoms would decrease the thermodynamic stability of the cages from viewpoint of cohesive energy.From Figure 2(a), we can see that  coh of C 59 X with X = B and N are larger than those of C 59 X with X = Al, Si, P, and even a bit more larger than those of C 59 X with X = Ga, Ge, and As.erefore, the doped cage with smaller heteroatom is more stable.e formation of the C 59 X cage can be seen in reaction (2).
en the energy difference of the above process, Δ, can be calculated by: where E(C 59 X), E(C 60 ), E(corannulene) and E(X-corannulene), are the energies of the species with the minimum structure, respectively.Furthermore, from theoretical point of view, the formation reaction of C 59 X cage from C 60 cage can be considered as two processes.In the �rst step, one carbon atom of the carbon cage is directly replaced by the heteroatom from the doped corannulene to form a hybrid structure, for which the skeleton of the hybrid cage is still the same as that of the pristine C 60 cage.In the next step the hybrid cage is then relaxed to reach its minimum structure.erefore, the replacing energy ( replace ) and the relaxing energy ( relax ) for the two processes are de�ned as follows: where E(C 59 X * ) is the energy of a C 59 X cage with the skeleton the same as that of the pristine C 60 cage.Based on (3)-( 4), it is easy to get Δ   replace +  relax .e calculated Δ,  replace , and  relax are shown in Table 1 and Figures We can see that Δ of the doped cages are all negative except for that of C 59 N, indicating the formations of the most doped cages are exothermic.Even for C 59 N, the obtained Δ is only 0.020 eV.us it is energetically favorable to form the C 59 X cages from viewpoint of total energy change.e obtained Δ are ranged from −1.926 to 0.020 eV for the heterofullerenes studied in this paper.From Figure 2(b), we can see that the curves of ΔE have the same trend as those of the  coh .us, contrary to the cohesive energy results, it seems that the formation of the doped cage with larger heteroatom is energetically more favorable from viewpoint of the total energy change of the reaction.
As for  replace of the heterofullerenes, C 59 B gives −0.170 eV, but others all exhibit positive values in the range of 0.144-13.273eV.is fact means that the directly replacement of a carbon atom by a heteroatom is an endoergic process for most of the doped cages.From Figure 2(c) for  replace of C 59 X where X belongs to III, IV and V Groups, it can be seen that  replace decrease monotonically for each group.Furthermore,  replace of C 59 B and C 59 N are the smallest among the eight doped cages studied in this paper.is is because the C-B and C-N bond lengths are more close to that of the C-C bond compared with those of other C-X bonds.us C 59 N has the least  replace , and C 59 B even exhibits exothermic replace process.Now we turn to the relaxing energy,  relax .From the obtained  relax shown in Table 1 and Figure 2(d), it can be seen that all obtained  relax of the doped cages are negative, indicating that the relaxing effect is exothermic.e calculated  relax are ranged from −0.124 to −15.027 eV for the heterofullerenes.
Since the doped cage becomes distorted structure from a perfect ball in the relaxing process, the asphericity parameter, ASP, is calculated to evaluate the geometrical distortion for the doped cages.ASP is introduced by Fowler et al. and can be calculated by [39]: where   is the radial distance of atom  from the cage central of mass, and  0 is the average radius.Here the structures with smaller ASP values are more close to a perfect sphere shape.e obtained ASP is listed in Table  an important role in chemical reaction for the reactant molecule, thus the frontier orbital analysis of the doped cages is necessary.In Table 2, we summarize the HOMO and LUMO energy levels of the heterofullerenes.It can be seen that HOMO levels of the doped cages are all increased compared with that of C 60 .However, the HOMO levels all vary not very large, except for that of C 59 N, which is increased by 1.4 eV.As for LUMO levels, they are decreased by about 0.5 eV when doping with Si and Ge atoms, but nearly unchanged for doping with other atoms compared with that of the pristine cage.Figure S2 in Supplementary Material shows the distributions of HOMO and LUMO for the cages studied in this paper.It can be seen that the frontier orbitals of C 60 cage are rather delocalized and spread over the whole surface of the cage.However, the frontier orbitals of several doped cages become localized obviously due the present of the heteroatoms.For example, the contributions from the Si, Ga, and Ge atoms are as large as 27.4%, 16.4%, and 35.3% to the HOMO, while 24.3%, 42.6% and 22.4% to LUMO, respectively according to our quantitative evaluations.
It is known that both the thermodynamic stability and kinetic stability have crucial in�uence on the relative abundances of different fullerene structures.It has been pointed out that higher kinetic stability is usually related with a larger HOMO-LUMO energy gap (  ) [40], because exciting electrons from a low HOMO to a high LUMO is energetically unfavorable, which would be necessary to activate a reaction.e calculated   of the doped cages are listed in Table 2.It can be found that all the doped cages present smaller   than that of C 60 cage.us kinetic stability of the cage is decreased by substitution from viewpoint of HOMO-LUMO gap.
Since the charge transport is one of the central issues for the performance of organic electronic devices, here the exciton binding energy (  ) is calculated to understand more about the transport properties of the doped fullerenes.Physically, the exciton binding energy can be seen as the energy required to decompose an exciton into a free electron and hole in the solid, and is de�ned as follows: where   is the transport gap and  opt is the optical gap.  can be treated as the orbital energy difference between the LUMO and the HOMO [41].As for  opt , it is calculated to be as the allowed lowest singlet optical transition energy with nonzero oscillator strength obtained by the time-dependent DFT (TD-DFT) calculations at B3LYP/6 [42] and 2.31 eV calculated by the UNO-CIS method [43,44].en 2.765 minus 2.099 eV gives   of C 60 cage with 0.666 eV.
From Table 2, we can see that   of the C 59 X cage with X to be IV and V Groups are in the range of 0.523-0.688eV, which are very close to that of the C 60 .However, C 59 X with X = B, Al and Ga exhibit obvious larger   (1.629, 1.239 and 1.238 eV, resp.).us compared with C 60 and other heterofullerenes, more energy is required to decompose the exciton for the cages doped with III Group elements.erefore, it seems that the charge transport behaviors would be quite different for the hole-type doped buckminsterfullerenes considering the exciton binding energy.

Vibrational Frequencies and Infrared Spectrum.
Based on the DFT calculations, the vibrational frequency analysis is performed with B3LYP/6-31G * method to verify whether these doped cages are local minima on the potential energy surface.e calculated vibrational frequencies for all the cages show no imaginary vibrational frequency, indicating that these doped structures all correspond to the true minima on the potential energy surface.e obtained infrared (IR) spectra were simulated as plotted in Figure 3, which have been scaled by a factor of 0.98, according to the DFT study on the infrared spectra of fullerene structures [45].From Figure 3 we can see that there are four IR-active absorptions at 530, 575, 1188, and 1431 cm −1 , respectively, for the C 60 cage.It is because only four  1U vibrational models are IR allowed by symmetry, though the perfect C 60 with  ℎ symmetry has 174 models totally.Our results agree quite well with the experimental (528, 577, 1183, and 1429 cm −1 ) [46] and other DFT results [47,48].As for the doped cages, their IR spectra are more complicated.It is found that the original four peaks are split and more absorptions are present.ese absorptions all exhibit somewhat red or blue shi due to the substitution of the heteroatoms compared with those of the pristine cage.Furthermore, it can be see that there are also several new peaks in the region of 600-1100 cm −1 .is is because the symmetry of the doped cages is decreased from  ℎ to   , and thus some of the original vibrational models forbidden by symmetry become IR-active.Additionally, it can be seen that the shapes of IR absorption spectra are different for the different doped cages.ese characteristic features in the IR spectra could be helpful to identify these heterofullerens from the experimental spectra.
3.5.Dielectric Constant.e dielectric constant is one of the important parameters for the materials of organic solid.In this paper a simple model based on the Clausius-Mossotti equation [49] is adopted.is model has been used to successfully evaluate the dielectric constant of C 60 and conjugated organic molecules [49,50].Under the framework of the Clausius-Mossotti model, the dielectric constant, , can be expressed as: where  is the �rst order polari�ability with   1    , in which   are the diagonal matrix elements of the tensor. is the volume occupied by a single molecule (tight option was taken for better accuracy). of C 60 is calculated to be 3.63 in this paper, which is comparable with previous experimental [51] and theoretical results [49].From Table 2, we can see that  of the doped cages are in the range of 3.72-3.95.us the substituted doping could increase  of the cages.It is also found that for C 59 X cage is increase by 2.8-9.2% by substituted doping compared with that of C 60 .However, doping with N atom even decreased  by 1.0%, and for other heteroatoms  is only increased by 1.5-4.0%.
Recall that larger  is related with larger  and smaller  values according to (7).As a result, all the doped cages exhibit larger dielectric constant than that of C 60 .

Aromaticity and Nuclear Independent Chemical Shi.
Aromaticity can be explained by the ring current theory and is a signi�cant concept in chemistry.In this paper the aromaticity of the cages is evaluated by using the nuclear independent chemical shi (NICS), which has proven to be a simple and efficient aromaticity probe [52][53][54].e NICS is de�ned as the negative of the isotropic magnetic shielding constant of a ghost atom located at the central of the cage.Negative NICS value means the aromaticity of the cage.In  this study, the NICS values listed in Table 2 is computed with the gauge-including atomic orbital (GIAO) method at B3LYP/6-31G * theory level.NICS of C 60 we obtained is −2.72, which indicates the weak aromaticity of C 60 and also agrees well with −2.8 by previous DFT calculation [55].
From Table 2, it can be seen that all the doped cages in this paper give negative NICS.us the eight doped cages studied are all aromatic.Among them, C 59 N cage has the NICS of −2.41, indicating it is slightly less aromatic than C 60 .As for other doped cages, the obtained NICS are more negative and thus they are more aromatic than the pristine cage, though the difference is small.Additionally, the C 59 B cage gives the most negative NICS, though it is only a bit distorted from the perfect sphere shape according to the SP and ASP analysis.erefore, it seems that there is no uniform correlation between the aromaticity and the sphere shape for the doped fullerenes.
Since it has been pointed that NICS at the cage centers have essentially the same values as the endohedral helium chemical shis [55,56], these obtained values are also helpful for the possible characterization of these doped fullerene cages.

Conclusion
eoretical studies of the C 59 X (X = B, N, Al, Si, P, Ga, Ge, and As) have been performed systemically based on the DFT calculations.e results of the geometrical structures, relative stabilities, electronic properties, vibrational frequencies, dielectric constants, and the aromaticities of the doped cages were discussed to achieve a further understanding of structure-property relationship of the doped cages.It is found that the hybrid cages undergo some distortions due to the substitution of the heteroatoms.According to the calculated cohesive energies, the C 59 X cage with smaller heteroatom is more stable.HOMOs of the heterofullerenes are all increased, but the HOMO-LUMO gaps are decreased compared with those of the C 60 .As for the exciton binding energy, the cages doped with III Group elements are obviously larger than other cages.e calculations also indicate that doping C 60 by substitution would give larger dielectric constant due to the increased polarizability.e obtained NICS show that most of the doped fullerenes are lightly more aromatic than the pristine cage.However, no correlation between the aromaticity and the sphere shape is found for the doped cages.
1.It can be seen that the calculated ASP are all nonzero for the heterofullerenes, with the values in the range of 0.001-0.074.is result also con�rms the distortions of the cage away from the perfect sphere.Furthermore, it seems that ASP of the cages has something to do with  relax .Here the doped cage with larger ASP values gives more negative  relax .For instance, C 59 N and C 59 B have the smallest ASP (less than 0.005) and the  relax are only −0.124 and −0.762 eV, respectively.C 59 Si and C 59 P have larger ASP (about 0.05) and the  relax are also more negative (about −8 and −9 eV).For C 59 Al, C 59 Ga, C 59 Ge, and C 59 As, they have the largest ASP (larger than 0.065) and also the most negative  relax (less than −13 eV).3.3.Electronic Properties.It is well-known that the frontier orbitals, the highest occupied molecular orbital (HOMO), and the lowest unoccupied molecular orbital (LUMO) play -31G * theory level in this paper.Here the obtained  opt of C 60 is 2.099 eV, which agrees with 1.95 eV by experiment of optical absorption spectrum T 2: e obtained HOMO, LUMO,   ,  opt ,   , , ,  and NICS of C 60 and the doped cages (HOMO, LUMO,   ,  opt and   in eV, and  and  in Å 3 ).