Stable Structures Of Boron/Nitrogen Substituted Fullerenes

Heterofullerenes,the fullerenes cages doped by other atoms,have been attractive because of various exotic properties and application prospects.The most concerned heterofullerenes are borafullerenes and azafullerenes,in which the elements substituting for carbon are boron and nitrogen,respectively.At present,only mono-azafullerenes have been isolated and characterized,hence theoretical studies are the main means for heterofullerenes.The theoretical studies of the structural stability of heterofullerenes help us to gain deep insights into the structures of fullerens and provide theoretical guidance to obtain more exotic fullerenes structures,for example,to produce novel structures which are non IPR cages or have rings other than pentagon and hexagon.People usually adopt first-principles methods to calculate the electronic structure for the structural stability.But almost all concerned theoretical calculations performed are on some selected geometries,and lots of isomers are not considered.We try to solve this matter by a new structure recognition method based on the symmetric group of C₆0fullerene.Using the minimum structural index as the ID of an isomer,we manage to conduct the congruence check and enumerate all the inequivalent isomers.We calculate all the isomers of C₆0-nX_n?X=N,B;9?=1⁴)and found that boron atoms tends to be on the para positions of consecutive hexagons,which is not followed by azafullerenes except C₅8N₂,and boron tends to be more dispersed than boron.The obtained minimal energy of C₅7B₃ is about 0.73 eV lower than that of the reported counterpart;For multi-azafullerenes,all the ground state structures,except C₅8N₂,have lower energy than the their reported counterparts,respectively.With the increase of the number of heteroatoms,the total number of heterofullerenes isomers increased so dramatically that the calculations of all the structures by first-principles method are over expensive.Therefore,we set up a fitting method based on cluster expansion to predict the energy of high concentration heterofullerenes.In the method,the structure is expanded as different substructures.By linear regression method,the interactive energies of their substructures are obtained.For heterofullerenes of higher heteroatom concentration,the interactive energies of the subclusters that they contains are combined linearly to predict their energies.The substructures with the same number of heteroatoms share the same fitting coefficient i.e.the concentration dependent fitting.With more structures calculated,the fitting coefficients are adjusted,the errors approach normal distribution,and the number of low lying structures converges.We finally obtain the low lying structures under various heteroatom concentrations.Compared with all the reported ground states of C₆0-nB_n?n=5⁶?and C₆0-nN_n?n=5⁹?,the structures we obtain have lower energies.For C₆0-nN_n?n=10¹2?heterofullerenes,the total numbers of the structures are astronomical,the concentration dependent fitting becomes less efficient.According to the results of C₆0-nN_n?n=5⁹?,for the 2-body subcluster of the low energy structures,the proportion of?1,7?remains the largest,the structures of?1,15?,?1,31?and?1,49?are in the majority,but for?1,2?,?1,3?and?1,9?,the proportions are almost 0,and?1,6?also have a very small proportion.Excluding gradually the structures with subcluster of?1,2?,?1,3?,?1,9?and?1,6?,we obtain the reported two lowest energetic structures of C₄8N₁2,between which 24new low energetic structures are also obtained.The obtained lowest energetic structures of C₄9N₅1 and C₅0N₁0 both have lower energy than those reported in the literatures.We adopt a new model with more coefficients to train and test the calculated structures.The learning model is found to be well generalized,and the predicted values are in good agreement with the DFT results.We then obtain the new lower energetic structures of C₅3N₇,C₅2N₈ and C₅0N₁0.Compared with the concentration dependent fitting method,this method has higher efficiency for structure search.