1.3 Description du montage exp´ erimental
1.3.4 D´ etecter
1.3.4.4 Limites d’utilisation
Um conjunto completo de funções de base precisa ser usado para representar os spin-orbitais corretamente. O uso de um número in nito de funções de base resultaria na energia de Hartree- Fock. Este limite, conhecido como limite de Hartree-Fock, não é exatamente a energia do estado fundamental da molécula, pois o efeito de correlação eletrônica continua não sendo levado em conta. No entanto, um conjunto in nito de funções de base é computacionalmente impossível e uma base nita precisa ser usada, o que acarreta um erro conhecido. Este erro é chamado de erro de truncamento do conjunto de base. A medida desse erro é a diferença entre o limite de Hartree-Fock e o valor computado de menor energia.
A escolha do conjunto de base deve ter o cuidado de manter um número de funções pequeno (para evitar o número de integrais de dois elétrons), minimizar o esforço computacional em calcular cada integral, e manter um pequeno erro de truncamento.
A base escolhida é frequentemente real. Uma escolha de base são os orbitais de Slater, ou Slater-type orbitals (STO)2. Para cálculos atômicos, as funções de base STO estão centradas em cada átomo. No entanto para cálculos com moléculas de três ou mais átomos, o cálculo das integrais de mais do que dois elétrons (ab|cd) é impraticável.
A introdução dos orbitais do tipo gaussiana, ou (Gaussian-type orbitals GTO) por S.F. Boys em 1950foi uma importante contribuição no sentido que tornou cálculos ab initio computacionalmente possíveis. Gaussianas são funções da forma:
θijk(r1− rc) = (x1− xc)i(y1− yc)j(z1− zc)ke−α|r1−rc|
2
(A.167) onde (xc, yc, zc) são coordenadas cartesianas do centro das funções gaussianas em rc; (x1, y1, z1) são coordenadas cartesianas de um elétron em r1; i, j e k são inteiros não negativos e α é um expoente positivo. Quando i = j = k = 0, as Gaussianas são do tipo s; quando i + j + k = 1 é do tipo p; quando i + j + k = 2 é do tipo d.
A principal vantagem do GTOs é que o produto de duas gaussianas centradas em diferentes átomos é equivalente a uma gaussiana centrada no ponto médio entre estes dois átomos. Portanto, integrais de dois, três e quatro centros podem ser reduzidas a integrais de dois centros, tornando o cálculo computacionalmente mais rápido.
No entanto, uma desvantagem deste tipo de função, é que as GTOs dão uma representação muito pobre do orbital próximo ao núcleo, e por isso é necessário uma base grande de funções pra atingir a acurácia obtida pelas funções STOs. Para amenizar esse problema, algumas GTOs são frequentemente agrupadas para formar o que é conhecido como funções gaussianas contraídas. Cada gaussiana, χ é tomada com uma combinação linear xa das gaussianas originais, ou primitivas, g, centrada no mesmo núcleo atômico:
χj = X
i
dijgi (A.168)
com os coe cientes de contração dij e os parâmetros g xos durante os cálculos. Os orbitais são
então expressados através de uma combinação linear de Gaussianas contraídas. ψi =
X
j
cjiχj (A.169)
O uso de funções contraídas no lugar de Gaussianas primitivas reduz o número de coe cientes cji desconhecidos a serem determinados no cálculo HF. Por exemplo, se cada Gaussiana contraída é composta de três primitivas de um conjunto de 30 funções de bases primitivas, então a expansão que envolveria 30 coe cientes cji desconhecidos, com a expansão correspondente usando funções contraídas envolve somente 10 coe cientes, o que economiza esforço computacional, com pouca perda de acurácia se as funções contraídas forem bem escolhidas.
As gaussianas primitivas e contraídas são construídas através do cálculo atômico com um con- junto de funções previamente escolhidas. Através desse resultado um conjunto de expoentes α otimizados é usado para os cálculos moleculares.
O conjunto mais simples de funções de base é chamado de conjunto mínimo de base e é aquele no qual uma função é usada para representar cada orbital da teoria de valência. No entanto, um conjunto mínimo de base fornecem energia e funções de onda que não são muito próximas do limite de Hartree-Fock e por isso cálculos mais precisos precisam de bases mais extensas.
Uma melhora signi cativa é alcançada adotando um conjunto de base double-zeta (DZ) no qual cada função de base do conjunto mínimo é substituído por duas funções de base, o número de funções de base é duplicado e com ele o número de coe cientes a serem determinados. Na base triple-zeta(TZ), três funções de base são usadas para representar cada orbital da teoria de valência. Entre a inadequação da base mínima e a demanda computacional das bases DZ e TZ surgem o conjunto de base split-valence (SV). Cada orbital atômico de valência é representado por duas funções de base enquanto cada orbital atômico da camada interna é representado por apenas uma função de base.
As bases descritas até aqui ignoram contribuições de funções de base que representem orbitais para quais o valor do número quântico l é maior que o valor máximo considerado pela teoria de valência. Porém, quando ligações são formadas em moléculas, os orbitais atômicos são distorcidos (ou polarizados) pelos átomos adjacentes. Esta distorção pode ser representada incluindo funções que representem valor mais alto de l. Por exemplo, a inclusão de um orbital do tipo p pode modelar razoavelmente bem a distorção de um orbital 1s. A adição dessas funções polarizadas à base DZ resulta na double-zeta plus polarization basis (DZP).
Existem inúmeras maneiras de construir o conjunto de base com gaussianas contraídas. Uma abordagem é ajustar os quadrados mínimos das N gaussianas primitivas para um conjunto de STOs otimizados através de cálculos atômicos para o carbono. A expansão de STO em termos de N gaussianas primitivas é designado STO-NG. Uma escolha comum é a de N=3 e designada STO-3G. Uma outra abordagem consiste em realizar cálculos atômicos usando uma base relativamente grande de primitivas. Este procedimento resulta em coe cientes cji determinados através de um cálculo SCF para as primitivas de cada orbital, que serão usados para obter as gaussianas contraídas para o cálculo molecular.
Outro comum esquema de contração é o 3 − 21G, onde uma gaussiana contraída composta de três primitivas é usada para representar os orbitais da camada interna e cada orbital de valência é representado por duas funções, uma contraída de duas primitivas e uma primitiva não contraída. As
FUNÇÕES DE BASE 87
primitivas são otimizadas anteriormente através de cálculos SCF atômico e os conjuntos contraídos são usados no cálculo molecular. A base 6 − 31G∗ é a base 6 − 31G com a adição de funções polarizadas na forma de funções do tipo d para cada átomo que não seja o H. Já a base 6 − 31G ∗ ∗ é adicionada um conjunto de três funções do tipo p, para cada átomo de H.
Artigo
Physics Procedia 00 (2010) 000–000
www.elsevier.com/locate/procedia
15
thBrazilian Workshop on Semiconductor Physics
Theoretical study of atomic arrangement in BXCYNZ nanotubular
structures
T.A. Souza
a, M.R.A. Silva
a, A.C.M. Carvalho
a*
a
GDENB: Grupo de Desenvolvimento de Estruturas Nanométricas e Materiais Biocompatíveis
Departamento de Física e Química – ICE – Universidade Federal de Itajubá, CEP 37500-970, Itajubá, MG, Brazil
Abstract
A theoretical approach was used to study the atomic arrangements of boron carbonitride nanotubes with diameters from 4 to 16 Å. The role played by nitrogen and boron doping in the structural stabilization of these molecular systems was evaluated, and the geometry of carbon and boron carbonitride nanotubes was investigated using quantum chemical methods. The atomic arrangements and chemical compositions of B-C-N tubular structures proposed in the literature (BCN, B3C2N3,and BC2N) were
analyzed. The results showed that the energy associated with boron and nitrogen incorporation depends strongly on tube diameter and the B-C-N atomic distribution in the tubular structures.
© 2010 Published by Elsevier B.V.
Keywords: boron carbonitride nanostructures, nanotubes, semi-empirical, structural stabilization, enthalpy of formation
1. Introduction
The outstanding mechanical and electronic properties of carbon nanotubes (CNT) [1] have led to the investigation of analogous materials such as boron nitride (BN), boron carbonitride (BXCYNZ), and boron carbide
(BC) nanotubes. The electronic properties of carbon nanotubes depend only on their diameter and chirality [1]. These systems are usually treated as one-dimensional semiconductors or metals, depending on the geometry of the tubes [1]. The classification of single-walled carbon nanotubes (SWNTs) is based on two chiral indices (n, m), which give the geometry of the graphene ribbon that is rolled to form a nanotube. According to the usual nomenclature, nanotubes are said to be achiral when one of the indices is zero (zigzag) or when n = m (armchair), while all the other variants are chiral. It has been proposed that CNTs behave as 1-D conductors when the difference between the chiral indices is a multiple of 3: n – m = 3q, where q is an integer [1].
* Corresponding author. Tel.: +55-35-3629-1138; fax: +55-35-3629-1140.
Semiconducting nanotubes are of interest in the fabrication of electronic devices as they offer the mechanical properties of small band gap semiconductors in systems of nanoscopic dimensions. Metallic nanotubes are the prototypes of mechanically robust molecular wires. However, the development of experimental techniques to precisely synthesize carbon nanotubes with uniform helicity and electronic properties remains a challenge, and imposes limitations on the technological applications of these nanostructures. Theoretical [2] and experimental[3-5] studies have shown that it is possible to modify the electronic properties of the nanotubes by replacing some of the carbon atoms with heteroatoms [6]. Since the incorporation of these heteroatoms can also alter the structure [7, 8], chemical reactivity [9], and mechanical characteristics [10] of the nanotubes, it follows that it should be possible to control nanotube properties.
Following the discovery of BXCYNZ nanotubes in 1994 [11], various methods to synthesize ternary boron
carbonitride nanotubes have been reported, including arc discharge [12-16], laser ablation [17], pyrolysis [18, 19], hot-filament chemical vapor deposition [20], plasma rotating electrode process [21], and others. Quantum chemical calculations of the structural stability and electronic properties of boron carbonitride systems have been reported by several authors [11, 22-26]. BN nanotubes are semiconductors characterized by wide band gap energy, of about 5.5 eV, that is independent of radius and helicity [27]. Theoretical studies have revealed that the electronic properties of BXCYNZ nanotubes can be tuned simply by changing their atomic compositions and configurations [28-32]. This
characteristic means that BXCYNZ nanotubes could be useful in technological applications where carbon and BN
nanotubes are unsuitable.
Goldberg et al. [5] reported that multi-walled BN nanotubes have preferentially zigzag type chirality along their circumference, based on their diffraction patterns. In the case of BXCYNZ nanotubes, experimental measurements
have revealed B-C-N compounds with distinct stoichiometries [22-41]. An important member of the BXCYNZ family
is the BC2N nanotube [29-31, 40, 41]. Experimental and theoretical results have indicated that the BC2N nanotube is
the most probable structure [31], although other possible atomic arrangements and chemical compositions (BCN, B3C2N3, BC4N, and B5CN5) of B-C-N tubular structures have also been studied. Although the general chemical
characterization of B-C-N materials is well established, and routinely performed using electron energy loss spectroscopy, determining the spatial distributions of B, C, and N species in the structures still remains problematic. In the present work, we report a quantum chemical study of BXCYNZ nanotubes with diameters varying from 4 to 16
Å. We analyze the role played by boron and nitrogen doping (BN-pair doping) in the energetic stabilization of these molecular systems. In our previous theoretical work [42, 43], it was analyzed nitrogen incorporation energy in carbon nitride nanotubes with diameters from 5 to 10 Å. It was concluded that carbon atoms are more easily substituted by nitrogen atoms in small diameter nanotubes than in larger ones. In the case of BXCYNZ
nanostructures, our theoretical results revealed that the energy of incorporation of the BN-pair also depends on nanotube diameter. Determination of the heat of formation after BN doping showed that the BN-pair incorporation energy depended on tube diameter, as well as on the atomic arrangement within the tubular structure.
2. Computational details
The geometries of tubular structures composed of carbon and boron-carbon-nitrogen (B-C-N) atoms were fully optimized using the semi-empirical Austin Method 1 (AM1) quantum chemical technique [44], which is based on Hartree-Fock theory. The advantages of semi-empirical calculations are that they are much faster than ab initio calculations, and that they can be used for large organic molecules. A disadvantage of semi-empirical methods is that certain properties cannot be confidently predicted. In the case of the properties analyzed in this study, the AM1 technique is very reliable for prediction of the molecular geometries and heats of formation of carbon materials. The error in heats of formation determined using AM1 is about 8.0 Kcal/mol [45], compared to experimental values. The average error in estimation of bond length is 0.05 Å [45].
Terminal bonds at tube ends were saturated with hydrogen atoms. Certain carbon atoms were substituted by boron and nitrogen atoms, and the geometries were re-optimized. In contrast to our earlier work, boron-nitrogen pair (BN-pair) substitution was not random, with sites being chosen in order to give various stoichiometric configurations.
The energy associated with the incorporation of boron and nitrogen was calculated as the difference between the formation enthalpies of BN-doped and pure carbon systems, divided by the number of BN-pairs. These calculations were performed within the Spartan quantum chemical package [46].
T.A. Souza, M.R.A. Silva, A.C.M. Carvalho / Physics Procedia 00 (2012) 000–000
3. Results and Discussion
Schematic models of possible BXCYNZ nanotube structures are shown in Figure 1. These correspond to the
optimized conformations of tubes with B3C2N3, B4CN3, BCN, B3CN4, B5C2N5, and BC2N stoichiometries (other
BXCYNZ stoichiometries were also investigated). The diameters of these model molecules range from 4 to 16 Å.
After BN-doping, the theoretical results showed some distortion in the tube walls, which generated variations in the BXCYNZ tube diameter. The diameters considered corresponded to the average values calculated for nanostructures
with the same chirality. Over 50 model molecules were constructed, considering armchair ((4,4), (6,6), (8,8), and (12,12)) and zigzag ((6,0), (8,0), (9,0), (10,0), (12,0), (16,0), and (18,0)) nanotube helicitie.
The graphene sheet formed the basis for construction of the model molecules, with BN-pair substitution being dependent on the desired BXCYNZ stoichiometry. This approach enabled analysis of the structural stabilization of
different BN-pair tubular distributions, for the same nanotube chirality and/or BXCYNZ stoichiometry.
Figure 1. Fully relaxed model molecules studied in this work: (a) B3C2N3, (b) B4CN3, (c) BCN, (d) B3CN4, (e)
B2xCx-2Nx+2, (f) BC2N, (g) B5C2N5, (h) B3C2N3, and (i) B3C2N3. Other BXCYNZ stoichiometries, not shown in this
picture, were also considered. In this ball-tube scheme, yellow balls represents boron atoms, grey balls are carbon, blue balls are nitrogen, and white balls are hydrogen atoms.
Theoretical studies concerning the energetic stability and electronic structures of BXCYNZ nanotubes and
nanojunctions have been performed using ab initio density functional theory (DFT) and semi-empirical methods [23, 28, 35, 40, 47, 48]. The geometries and electronic structures of double-walled boron carbonitride nanotubes have also been calculated [26]. In these studies, the formation energy of the nanotubes was determined according to an earlier model[29] that employed a zero-temperature thermodynamic approach based on the prior determination of the chemical potentials of BB- and CC-pairs [49, 50]. In the present work, determination of the formation energy of BXCYNZ nanotubes, and the incorporation energy of BN-pairs, was based on calculation of Enthalpy of Formation
using the semi-empirical AM1 method.
Figures 2 and 3 show the evolution of the enthalpy associated with BN-pair incorporation, as a function of nanotube diameter, for BXCYNZ zigzag (dark symbols) and armchair (open symbols) nanotubes, respectively. The
enthalpy of incorporation was calculated as the relative enthalpy per added BN-pair ([ Hf(BXCYNZ) -
Hf(C)]/n(BN)), where Hf is the heat of formation obtained from AM1 calculations. Analysis of the energy of
incorporation of BN-pairs in the model molecules suggests that small diameter nanotubes are more easily doped by BN-pairs than large nanotubes. Theoretical comparisons of the energy of incorporation for zigzag boron carbonitride nanotubes showed that, for the same BXCYNZ stoichiometry, the atomic arrangement affected the formation enthalpy
of the model molecules. In summary, it can be concluded that the energy associated with the incorporation of BN- pairs is influenced by both tube diameter and the atomic arrangement.
(g)
(h)
(i)
Figure 2. Heat of Formation results for zigzag BXCYNZ nanotubes with different diameters. The B3C2N3 and B5C2N5
stoichiometries correspond to the model molecules shown in Figure 1 (see text). B3C2N3 (1) and B3C2N3 (2) have
different atomic arrangements for the same stoichiometry. The black and white symbols correspond to zigzag tubes with BCN and BC2N stoichiometry, as proposed by Azevedo et al. [31]
Figure 3. Heat of Formation results for armchair BXCYNZ nanotubes with different diameters. B3C2N3, BCN, and
T.A. Souza, M.R.A. Silva, A.C.M. Carvalho / Physics Procedia 00 (2012) 000–000
A summary of the results obtained for several of the BXCYNZ nanotubes analyzed in this work is presented in
Figure 4. It can be concluded from comparison of the calculated energies of incorporation for these model molecules that the BC2N nanotube is not the most stable structure in terms of its chirality. It has been suggested that B and N
atoms have a strong preference to exist as neighbors in boron carbonitride nanotubes. The B3C2N3 armchair
nanotubes showed the lowest formation energy because there are a greater number of B-N bonds in this atomic arrangement [27]. In the case of zigzag nanotubes, the incorporation energy results indicated that tubes with B5C2N5
stoichiometry are the most energetically stable. This configuration also maximizes the number of B-N bonds.
Figure 4. Heat of Formation results for zigzag (dark symbols) and armchair (open symbols) BXCYNZ nanotubes of
different diameters and stoichiometries. The BXCYNZ stoichiometries indicated in the legend correspond to the
model molecules shown in Figure 1 (see text).
Although BC2N stoichiometry is considered to be the most probable nanotubular structure [31], it can be
concluded from the results shown in Figures 2 and 4 that BXCYNZ nanotubes with B3C2N3 and B5C2N5
stoichiometries can be also stable structures as BC2N. These stoichiometries result in the lowest BN-pair
incorporation energies, because the atomic arrangements minimize the number of B-C and C-N bonds.
4. Conclusions
Semi-empirical methods were used to investigate the energetic and structural stability of BXCYNZ nanotubes. The
results of formation enthalpy analyses suggested that for zigzag nanotubes, the energy of incorporation of BN-pairs depends more on atomic arrangements than on nanotube diameter and helicity. As with carbon nitride nanotubes, small diameter boron carbonitride nanotubes are more easily doped by BN-pairs than large diameter tubes. The most stable structures are those that minimize the number of B-C and C-N bonds, such as B3C2N3 and B5C2N5 structures.
5. Acknowledgements
The authors acknowledge financial support from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).
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