Non Bonded Interactions in cylindrical capacitor of ( m , n ) @ ( m ’ , n ’ ) @ ( m ” , n ” ) Three Walled Nano Carbon Nanotubes

In this study we have exhibited that there are regular non-bonded interactions among the layers of Three walled Carbon nanotubes (TWCNTs) and (10, 10), (8, 8) DWCNTs & (6,6) SWBNNTs. It has been observed the energies between three layers are sensitive to chirality dramatically. In other hands the external fields of the inner tubes make strongest fields for the outer tubes. It has been shown the non-bonded quantum mechanics in terms of wave functions and charges densities between three layers. For the systems including charges Q, the densities (|ψ(x)|2 multiplied the atomic charges, yielding toward the charges densities of |Q(x)|2.


INTRODUCTION
Three walled nanotubes (TWCNTs) can be considered as a tube twisted from hexagon sheets of graphene or boron nitrides plates which have been synthesized 1 in 1995 firstly.Boron nitrides tubes have structural analogsas carbon nano tubes.SWBNNTs, in contrast of CNTs is metallic or semiconductors depends on its chirality 1 , boron nitrides tube are occasionally believed 2 to be a1-dimension insulators regarding of their helicity 3 , tube's diameter and the number of tubes wall 4 Since BNNTswere synthesized 1 via arc 5 discharges successfully, the mechanism 6 for its formationsare obscure 7 yet, and the behaviors of the metallic catalyst remain an unknown question 7 .
Internal-layers couplings play a critical 19 function for determining properties 21 of layered three-dimensional 22 material.Truly, the novel studies have exhibited which mechanical 20 coupling among the outer and inner shells of TWCNT sleads 19,20 for characterizing radials-breathing 20 vibration that corresponds to oscillated modes 21 with concerted 22 inner-and outer-shells motion.
It has been exhibited which TWCNTs are having high oxidativere sistance 21 ,which were later confirmed 23 , by other scientists.Recently, some result sindicate which purified 22 SWBNNTs are resist for oxidase up to 1,100°C 23 .Further more, SWBNNTs are predicted to have piezoelectricity 24 character and are applicable 25 for room-temperatures hydrogen 26 storages.
An elastic 26 property of an individual SWBNNT has been clearly studied through thermal vibration 27 methods.
The piezoelectricity approaches originated 34 from the deformations effects because of twisting of the h-BNplanes for forming the tubes structures.It is an important behavior in BNNTs 24 .It has been exhibited that BNNTcan be amazing piezoelectric 24 system while the piezoelectric constants for different zigzags BNNT was defined for increasing along 25 with the decreases of the radiuses of BNNTs 25 .One of the most important features of BNNTs is that BNNTs possess large band gaps between 5.0 and 6.0 electron volteirrespective 29 of the number of BNNT's wall, diameter and chirality 2,[16][17][18][19][20][21][22][23][24][25][26][27] .
In this study we have exhibited that there are regular 18 non-bonded interactions among the layers of TWCNTs and it has been observed the energies between three layers are sensitive to chirality dramatically 12 .In other hands the external fields of the inner tubes make strongest fields for the outer tubes.
It has been shown the non-bonded quantum mechanics in terms of wave functions and charges densities 16 between three layers [17][18][19] .For the systems including chargesQ, the densities (|ψ(x)| 2 multiplied the atomic charges, yielding toward the charges densities of |Q(x)| 2 .Therefore, there are good relationship samong charge and wave functions of those inner tubes rather than outers.Although the wave functionsof each molecule in the systems for Φ n Innertube and Φ m Innertube under the permutations of electrons coordinate is antisymmetric, the product of those non-bonded molecule under intermolecular exchanges of the electron couldnot be anti-symmetric.
For non-covalent interactions,the BLYP methods are unable for describing van der Waals [51][52][53] TWCNTs systems by medium-ranges interaction such as interaction between three tubes of TWCNTs.The B3LYP and some other functional are not suitable for illustrating the exchanges and correlations energies indistant medium-range non-bonded system obviously.In addition, some novel studies have exhibited that inaccuracies of the medium-ranges exchanges energies lead to asystematic error to predictof molecular properties [53][54][55][56][57][58] .
Geometry optimization and electronic structures have been accomplished using the semi empirical methods such as CNDO, MINDO, and ZINDO which their approach are based on an semi empirical solution ab-initio and the Kohn-Sham equation 57 also.The density function theories in the plane-waves sets with the projector-augmented wave pseudo-potential are calculated.The Perdew-Burke-Ernzerhof 58 exchange-correlations functional for the generalized gradient approximations or GGA are adopted 58 .The optimization of the lattices constant and the atomics coordinate are made through the minimization of those total energies.The charge stransferring and electrostatic potentials-derived were also calculated using the Merz-Kollman-Singh 59 , chelp 60 , or chelpg 61 .
The charges calculations method based on molecular electrostatics potentials fitting are not complete-suited for estimating of bigger systems which some of those inner-most atom are located 62 far away from that point at which the MESP 62 is computed [60][61][62] .In these conditions, variation of the inner-most atomics charge would not leads to a significant change of the MESP 61 outside of the molecules, meaning where the accurate 62 value for the innermost atomics charge are not well-determined by MESP 62 outside 63 of those molecules.The representatives atomics charge for molecule must be computed as averages value over various molecules and conformation.A detailed overview of those effect of those basis sets on the charges distribution might be found in 63,64 .It has been extracted the charges densities prole from the rst-principle calculations through the average process described 65,66 .

RESULT AND DISCUSSION
The results are plotted in 6 curves and HOMO LUMO is listed in Table1.
We have estimated the electron densities and gradient norms and Laplacians, values of orbitals wave-functions, electron spin densities, electrostatic potentials from nuclear / atomics charge, electron localization functions or ELF [64][65][66] , localized orbitals locator or LOL [64][65][66] defined via Becke andTsirelson, total electrostatics potentials, and the exchange-correlations densities, correlations hole and correlations factor, Averages local ionization energiesusing the Multifunctional 66 Wave-functions Analyzer 65,66 Through density-of-state DOS 66 it has been represented the number of state in the units energies intervals, since energies levelare contiguous, so DOS 66 has drawn as curves map.The systemsare isolated, so the energies levels are discrete 66 .
We have investigated the DOS 66 graphs as the tools for analyzing the natures of electron structures in the nanotubes The original total DOS (TDOS) of our system have calculated based on)(Ε)=Σ i δ(Ε−ε i ) where eigen value set of single-particle Hamilton is, cs Dirac Delta function which after replacing Gaussian, it can be yielded 63 .Where FWHM 63 is acronyms of "full width at half maximum 63 , it is an adjustable parameter in multi-wave function, the larger FWHM the TDOS analysis is easier for performing.The normalized Lorentzian function is defined as L(χ)= Pseudo-Voigt function is weighted linear combination for Gaussian function and Lorentzian formula:P(χ)=w gauss G(χ)+(1−w gauss )L(χ) The curve map of broadened partial DOS (PDOS) and overlap DOS (OPDOS) are valuable for visualizing orbital composition analysis, PDOS function of fragment A is defined as: where Ξ i,A is the composition of fragment "A" in orbital i.The OPDOS among fragments A and B are defined as) where X i A,B A,B is the composition of total cross term between fragment A and B in orbital i.
Both original and broadened TDOS/PDOS/ OPDOS are shown in the graph below.The original DOS graphs are discrete comb-like lines, from which we cannot obtain any additional information other than energies level distributions, it is impossible for distinguishing different types of a line and degenerate energies level sowing for over lapping.
From the height of black curves (TDOS) we can clearly know how dense the energies levels are distributed every where.Besides, the curves corresponding to TDOS, PDOS (red line for fragment 1, blue line for fragment 2) and OPDOS (green line) no longer overlap, we can clearly identifies characters of each orbitals through observing those curves 67 .

CONCLUSION
In conclusion it has been found that the external fields of the inner tubes make strongest fields for the outer tubes.It has been shown the non-bonded quantum mechanics in terms of wave functions and charges densities between three layers.It is necessary for investigating the density functional theories including theory of van-der-Waals densities for modeling exchange-correlation energies in the nanotubes for any further discussion.