1、Matlab 编程的 Hckel 分子轨道法计算研究掺杂纳米碳管的化学反应性（英文）杨龙飞 张业新 都时禹 上海大学材料科学与工程学院 中国科学院宁波材料技术与工程研究所 摘 要： 使用 Matlab 自编简单 Hckel 分子轨道法 (SHMO) 计算程序, 分析空位、Stone-Wales 缺陷位、N 和 B 原子掺杂的 CNT (5, 5) 碳纳米管, 计算 电子密度和前线分子轨道 (HOMO 和 LUMO) 为研究掺杂相对碳纳米管的化学反应性提供依据.具有不同电特性的掺杂相打破了碳纳米管的 电子、HOMO 和 LUMO 的均衡分布.掺杂相和/或邻近的碳原子为 HOMO 或 LUMO 贡
2、献了较其它原子更大的轨道系数, 在不同的化学反应中表现出良好的亲核性或亲电性.此外, HOMO-LUMO 能量差很好地反映了掺杂纳米碳管的导电性.计算结果与已报道的实验和理论结果吻合良好.关键词： 碳纳米管; 掺杂; Hckel 分子轨道理论; Matlab 编程; 作者简介：YANG Longfei (1991-) , male, master candidate, major in doped carbon materials and application in energy storage, E-mail:Y作者简介：ZHANG Yexin (1980-) , male, Ph D,
3、associate professor, major in carbon nanotubes and environmental catalysis, E-mail: 收稿日期：2016-10-09基金：Supported by the National Natural Science Foundation of China (No.51422212 and 21307142) Study of Chemical Reactivity of Doped Carbon Nanotubes by Simple Hckel Molecular Orbital Calculations with Ma
4、tlab ProgrammingYANG Longfei ZHANG Yexin DU Shiyu School of Materials Science and Engineering, Shanghai University; Institute of New Energy Technology, Ningbo Institute of Material Technology Abstract： A Matlab program was self-developed for simple Hckel molecular orbital calculations, with which ca
5、rbon nanotubes of CNT (5, 5) doped with vacancy, Stone-Wales defect, N and B atoms were analyzed.Calculated-electron density and frontier molecular orbitals (HOMO and LUMO) provide a basis to study chemical reactivity of CNTs.Homogenous distributions of-electron, HOMO and LUMO are broken by dopants
6、with different electronic character.Furthermore, dopants and/or nearby carbon atoms contribute more in HOMO or LUMO, behaving nucleophile or electrophile in different reactions, respectively.Calculated HOMO-LUMO gaps reflect electrical conductivity of doped CNTs well.Calculated results are in agreem
7、ent with experimental and theoretical results reported elsewhere.Keyword： carbon nanotubes; doping; Hckel molecular orbital theory; Matlab programming; Received： 2016-10-090 IntroductionCarbon nanotubes (CNTs) are always attracting tremendous attentions because of their unique nanostructure of hollo
8、w cylinder which can offer several advantages over many chemical applications such as energy, environment and catalysis.However, the direct use of CNTs encounters their inert chemical properties due to the seamless graphitic structure, in which the homogeneous distribution of-electrons leads to the
9、electroneutrality of CNTs.In order to break the electroneutrality, doping with defects or heteroatoms becomes necessary, which can rearrange the distribution of-electrons to tailor the chemical properties of CNTs for special applications1-3.For example, the defect doping improves the capacitance of
10、CNTs4and the conductance sensitivity to chemical vapors5while N doping improves the activities of CNTs for some catalytic reactions such as Knoevenagel condensation6.Both N-doped and B-doped CNTs exhibited better reactivity for electrocatalytic oxygen reduction reaction (ORR) 7-9.Several studies hav
11、e been devoted to interpret theoretically the improved reactivity of doped CNTs.Most of them were based on Ab initio method10or density functional theory (DFT) 11, in which some commercial quantum chemistry software, such as VASP, Gaussian and Materials Studio, were employed.However, these calculati
12、ons were time-consuming since a great number of atoms were involved for CNTs models.Actually, the-electrons on CNTs structure can be considered as separable from the-skeleton electrons, and both electrons can be assumed to have little influence upon each other.Accordingly, the distribution of-electr
13、ons on CNTs can be separately calculated based on simple Hckel molecular orbital (SHMO) method, a classical semi-empirical method for conjugatedelectron system.Sato et al.12-15studied the electronic structure of CNTs with SHMO method, in which the frontier molecular orbitals (FMO) , including the hi
14、ghest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) , were obtained.According to the theory of frontier molecular orbitals, the-electrons of HOMO and LUMO significantly contribute to a chemical reaction16.It should be expected that the electronic structure and
15、FMO of doped CNTs can be calculated by SHMO method.The calculation may providea clear and concise theoretical basis to study the chemical reactivity of doped CNTs without complication of full quantum chemical treatment.So far, the relevant reports are still missing.In this work, SHMO calculations fo
16、r doped CNTs with defects, N and B atoms were performed via programming in Matlab, a multi-paradigm numerical computing environment with a rich graphical interface.The powerful matrix manipulations of Matlab allow coping with complex equations and matrices of a great number of variables correspondin
17、g to the atoms in the CNTs model.The calculated results here, including-electron density and HOMO and LUMO, can be used to study the reactivity of doped CNTs with different dopants and defects, which are in good agreement with experimental and/or theoretical results reported elsewhere.1 Calculation1
18、.1 Calculation modelsThe calculation models based on CNT (5, 5) are as shown in Fig.1.From Fig.1 (a) to 1 (i) , the pristine (CNT) , vacancy-doped (v-CNT) , Stone-Wales defect-doped (SW-CNT) , graphitic N-doped (g N-CNT) , pyridinic N-doped (p N-CNT) , pyrrolic N-doped (py N-CNT) , B-doped (B-CNT) ,
19、 N and B separating co-doped (B_N-CNT) , and N and B bonding co-doped (BN-CNT) models are displayed in turn.All of the dangling bonds at the edge of models are assumed to be terminated by hydrogen atoms.Geometrical optimization is not required here since the electronic structures are fully described
20、 by the bonding feature between adjacent atoms in the model which can reproduce well the experimental observations.1.2 Calculation of molecular orbitalsAccording to Hckel theory, a molecular orbital () is expressed by linear combination of atomic orbitals ( r) where coefficient crrepresents contribu
21、tion of the rth atomic orbital to the molecular orbital.By optimizing the energy of molecular orbital (E) using variational method, crare given by the secular equationFig.1 Schematic models (a) pristine CNT (5, 5) ; (b) doped CNT (5, 5) including vacancy-doped; (c) Stone-Wales (SW) defect-doped; (d)
22、 graphitic N-doped; (e) pyridinic N-doped; (f) pyrrolic N-doped; (g) B-doped; (h) B and N separating co-doped; (i) B and N bonding co-doped 下载原图where ris Coulomb integral on atom site r while rscorresponds to the overlap integral between bonded atoms sites r and s.If atom sites r and s are not bonde
23、d, rsis assumed to be zero.Equation (2) can be written in the matrix formFor the non-trivial solutions to the equation, the determinant of coefficients (secular determinant) vanishes, i.e., For all carbon atoms, ris set equal to a constant, and for all carbon-carbon bonds, rsis assumed to be equal,
24、.For B and N atoms, the parameters can be adjusted as listed in Table 1.By solving Eq. (4) , the energies of molecular orbitals are obtained in the formTable 1 Atom and bond parameters in Hckel theory 下载原表 Substituting Eq. (5) into Eq. (3) , the set of atomic orbitals coefficentscirare found, and th
25、en the molecular orbitals at the energy of Eican be determined in Eq. (1) .The secular determinants were established according to the calculation models in Fig.1.All above equations were solved with Matlab programming.1.3 Determination of HOMO and LUMOAll-electrons in calculation models are assigned
26、 to the calculated molecular orbitals based on general rules of electron configuration, including minimum total potential energy principle, Paulis exclusion principle, and Hunds rules.Firstly, -electrons prefer to occupy empty orbitals at the lowest energy;Secondly, one orbital can accommodate up to
27、 two-electrons;For degenerate orbitals, finally, -electrons must first occupy empty orbitals before double occupying them.The assignment was accomplished by running a programmed Matlab script.The last occupied molecule orbital was considered as HOMO, and the nearest unoccupied molecule orbital was r
28、egarded as LUMO.1.4 Calculation of-electrons density and-bonder orderThe-electrons density on atom site r can be calculated by the equationwhere vldenotes the number of-electrons assigned to the molecular orbital l.The corresponding net charge can be calculated by subtracting the-electrons density f
29、rom the number of-electrons provided by the atom (Table 1) .The-bond order between two bonded atomic sites r, s, can be calculated by the equationThe charge densities and bond orders are both calculated with Matlab programming.2 Result and discussion2.1 Pristine and defect-doped CNT (5, 5) The Hckel
30、 results of CNT are depicted on the topological diagram of CNT in Fig.2, including-electron density (Fig.2 (a) ) , and atomic orbital coefficients in HOMO (Fig.2 (b) ) and LUMO (Fig.2 (c) ) .All of carbon atoms exhibit the unique-electron density of single electron, showing the homogeneous distribut
31、ion ofelectrons.The orbital patterns of HOMO and LUMO well coincide with the periodicity and symmetry of tube structure.The energies of HOMO and LUMO are listed in Table 1, and the HOMO-LUMO gap is calculated to be 0.3383.In the Hckel results of v-CNT in Fig.3, the-electrons density on all carbon at
32、oms are also single electron (Fig.3 (a) ) , keeping the homogeneous distribution of-electrons of CNT, while the HOMO (Fig.3 (b) ) and LUMO (Fig.3 (c) ) are changed.For both orbits, the greater orbital coefficients are observed on some edge carbon atoms of the vacancy.Note that the orbital coefficien
33、ts mentioned later specially refer to their absolute value unless otherwise stated.Generally, a greater atomic orbital coefficient in HOMO and LUMO is the indications of preferred reactivity to electrophile and nucleophile, respectively21.The results are in agreement with the work of Robinson et al.
34、5.They reported that the defect structure played an important role in chemical sensing properties of CNTs for the adsorption of electrophile vapor such as acetone and trichloroethylene, and the adsorption of nucleophile vapor such as ethanol and ammonia.Fig.2 Hckel results of CNT (a) -electron densi
35、ty; (b) atomic orbital coefficients in HOMO; (c) LUMO 下载原图(The circle area at atom site reflects absolute value while the line width between atom sites represents the level of-bond order.) Fig.3 Hckel results of v-CNT (a) -electron density; (b) atomic orbital coefficients in HOMO; (c) LUMO 下载原图(The
36、circle area at atom site reflects absolute value while the line width between atom sites represents the level of-bond order.) For SW-CNT (Fig.4) , the distribution of-electrons is greatly changed compared with that of CNT.The carbon atoms on the pentagon pairs except the two central atoms exhibit hi
37、gher-electron densities than the others, implying the electron-accepting character of these atoms.As a result, the-electron densities on the adjacent carbon atoms are decreased.For the HOMO and LUMO, the greater atomic orbital coefficients on some edge atoms of the SW defect are observed, while the
38、coefficients on the two central atoms are close to zero.This is in good agreement with the DFT calculation by Lu et al.22that the central C-C band of SW defects in CNT (5, 5) is chemically less reactive while the peripheral 5-6 and 6-7 ring fusion are much more reactive.Fig.4 Hckel results of SW-CNT
39、 (a) -electron density; (b) atomic orbital coefficients of HOMO; (c) LUMO 下载原图(The circle area at atom site reflects absolute value while the line width between atom sites represents the level of-bond order.) Table 2 Energies of HOMO and LUMO and HOMO-LUMO gap of calculation models 下载原表 Both HOMO-LU
40、MO gaps of v-CNT (0.384 8) and SW-CNT (0.386 4) are higher that of CNT (0.338 3) as listed in Table 2.A larger gap implies higher barrier for electron transport and thus the worse electrical conductivity23.Suzukiet et al.24observed that the defect formation decreased electrical conductivity of CNTs.
41、The HOMO-LUMO gaps for v-CNT and SW-CNT are close to each other, which may suggest that the defects without introducing doping atoms may lead to similar lowering effect on electrical conduction.2.2 N-doped CNT (5, 5) The Hckel results of g N-CNT are shown in Fig.5.The graphitic N atom exhibits a-ele
42、ctron density of 1.5e with the positive net charge of 0.5e if assuming the-electron on neutral N atom is 2.0e, whereas the three adjacent carbon atoms possess the-electron density of 0.9e, apparently lower than other carbon atoms.The three atoms exhibit a positive net charge of 0.1e, consistent with
43、 the DFT calculation of Gong et al.7Since the net positive charge on N and a few adjacent carbon atoms are found, the other carbon atoms in CNT all have some negative net charges to neutralize the system.This shows the N atom in g N-CNT causes an obvious delocalized effect on the electron density di
44、stribution of g N-CNT25.For both HOMO and LUMO, greater atomic orbital coefficients are observed on carbon atoms near to the N atom, corresponding to the reactivity of graphitic N species to electrophile such as oxygen26-28, and to nucleophile such as methane28.Therefore, our calculations may provid
45、e a clue on novel chemistry that graphitic N brings.For example, Ni et al.26reported that the graphitic N could efficiently decrease the energy barrier of oxygen molecule dissociation on CNTs for ORR.Chen et al.27found that graphitic N could speed up the activation of oxygen for oxidative dehydrogen
46、ation of propane.Furthermore, the DFT calculation of Hu et al.28revealed that the graphitic N can activate the C-H bond of methane.Fig.5 Hckel results of g N-CNT (a) -electron density; (b) atomic orbital coefficients of HOMO; (c) LUMO 下载原图(The circle area at atom site reflects absolute value while t
47、he line width between atom sites represents the level of-bond order.) For p N-CNT (Fig.6) , the N atom exhibits-electron density at 1.34e with lower-electrons density on two adjacent carbon atoms.In the HOMO, much greater atomic orbital coefficients are seen on two carbon atoms near the N atom as we
48、ll as N atom itself, suggesting the Lewis base properties and high reactivity of pyridinic N species to electrophile.Fig.6 Hckel results of p N-CNT (a) -electron density; (b) atomic orbital coefficients of HOMO; (c) LUMO 下载原图(The circle area at atom site reflects absolute value while the line width
49、between atom sites represents the level of-bond order.) Van Dommele et al.6found that the reactivity of N-doped CNTs as solid base catalysts for Knoevenagel condensation was related to the amount of pyridinic N.Pyridinic N was also reported to be responsible to the high ORR activity of CNTs29.In the LUMO, relative great coefficients are also found on the carbon atoms near to the N atom.The predicted r
