As an aid towards improving the treatment of exchange and correlation effects in electronic structure calculations, it is desirable to have a clear picture of concepts of exchange-correlation functionals in computational calculations. For achieving this aim, it is necessary to perform different theoretical methods for many groups of materials. We have performed hybrid density functional theory (DFT) methods to investigate the density charges of atoms in rings and cages of carbon nanotube. DFT methods are engaged and compared their results. We have also been inclined to see the impression of exchange and correlation on nuclearnuclear energy and electron-nuclear energy and kinetic energy. With due attention to existence methods, B3P86, B3PW91, B1B96, BLYP and B3LYP have used in this work. Carbon nanotubes were discovered in 1991 by Iijima of NEC Corporation. 1 Then, efforts in synthesis, characterization and theoretical investigation on nanotubes has grown, rapidly. In 1993, the simplest kind of carbon nanotubes, single walled carbon nanotubes, SWNTs, were discovered independently by Iijima group 2 and Bethune IBM team. 3 Thess and coworkers 4 later produced carbon nanotubes with 100 to 500 SWNTs bundled into a 2-D triangular lattice. Classical molecular mechanics (MM), lattice dynamics (LD), molecular dynamics (MD), tight binding and ab initio level Quantum Mechanical (QM) methods 5 - 22 are used for theoretical calculation. Also, Density functional theory (DFT) is one of the methods is used in theoretical calculation successfully. In this paper, we present a detailed study of the energetic, structures, and mechanical properties for zigzag (4, 0) carbon nanotubes. We used an accurate force field, with using of exchange and correlation effect derived through quantum mechanical calculations, DFT, to calculate density charges between the carbon atoms and N-N energy (between nuclear and nuclear)and E-N energy (between nuclear and electrons) and KE (kinetic energy). Here, we have presented the results of a theoretical study by density functional theory (DFT), in this work we have used Gaussian 98 code for our calculations and applied different DFT methods, B3P86, B3PW91, B1B96, BLYP and B3LYP. The basis set in our calculation is 6-31G*. We have shown the effect of exchange and correlation on density charges between the carbon atoms and N-N energy (between nuclear and nuclear) and E-N energy (between nuclear and electrons) and KE (kinetic energy). We selected zigzag (4, 0) carbon nanotubes. This nanotube has some rings and cages. Gaussian 98 offers a wide variety of Density Functional Theory (DFT). 23 - 26 Energies, 27 analytic gradients, and true analytic frequencies 28 - 35 are available for all DFT models. In Hartree-Fock theory, the energy has the form: Where: V is the nuclear repulsion energy, ρ is the density matrix, < h_p > is the one electron (kinetic plus potential) energy, is the classical coulomb repulsion of the electrons, and is the exchange energy resulting from the quantum (fermion) nature of electrons. In density functional theory, the exact exchange (HF) for a single determinant is replaced by a more general expression, the exchange-correlation functional, which can include terms accounting for both exchange energy and the electron correlation which is omitted from Hartree-Fock theory: Where E_X [ρ] is the exchange functional, and E_C [ ρ ] is the correlation functional. Hartree-Fock theory is really a special case of density functional theory, with E_X [ ρ ] given by the exchange integral and E_C = 0. The functionals normally used in density functional are integrals of some function of the density and possibly the density gradient: Where the methods differ in which function f is used for E_X and which (if any) f is used for E_C . In addition to pure DFT methods, Gaussian 98 supports hybrid methods in which the exchange functional is a linear combination of the Hartree-Fock exchange and a functional integral of the above form. Proposed functionals lead to integrals which cannot be evaluated in closed form and are solved by numerical quadrature. There are many correlation functional in DFT. VWN, Vosko, Wilk, and Nusair 1980 correlation functional fitting the RPA solution to the uniform electron gas, often referred to as Local Spin Density (LSD) correlation. 36 LYP the correlation functional of Lee, Yang, and Parr, which includes both local and non-local terms. 37 , 38 P86 the gradient corrections of Perdew, along with his 1981 local correlation functional 39 and other functionals. In DFT exchange functional combine with correlation functional and create hybrid Functionals, for example B3LYP, BLYP, B3P86, B3PW91, B1B96 and other hybrid functionals. In this work we have calculated some properties of zigzag carbon nanotube (4, 0) in . 1 . The results are summarized in 1 . Where N-N is the nuclear repulsion energy, E-N is the attraction energy between electrons and nuclear and KE is the kinetic energy. Some conclusions can be drawn from these results. First, the repulsion energy between nuclears in B3PW91, BVWN and LSDA methods is maximum and equal. Second, we can also consider kinetic and attraction energy, the BVWN has a maximum values in repulsion and kinetic energy, 5521.016 and 1816.157 hartrees, respectively and has minimum SCF and attractive energy, -1841.101 and -15271.485 hartrees, because of the variety of attractive energy and SCF energy is same. We have shown it in . 2 . Finally, the LSDA (SVWN) has a maximum Dipole moment, 0.8173. Focusing on dipole moment result, will lead us this fact that any methods has B3 exchange functional, yields a small dipole moment. VWN as a correlation functional and S as an exchange functional yields a large dipole moment value. We have shown it in 1 . Difference energy among B3P86, B3PW91 and B3LYP demonstrates the correlation effect. They have same exchange functional, B3. This is Becke’s 3 parameters functional, 40 which has the form: Difference energy between B3P86 and B3PW91 methods (5.875 hartrees) is related to correlation energy. It is 5.208 hartrees for B3P86 and B3LYP. If we repeat it for B3LYP and B3PW91, the result will be 0.667 hartrees. We can conclude the correlation functionals of the B3PW91 and B3LYP have a same effect on SCF energy of system nearly, but it is not true about B3P86. We have also calculated the difference energy between B1 and B3 by comparison of energy B1LYP and B3LYP, 0.492 hartrees for obtaining the exchange effect. This result points out that the difference exchange energy between B1, B3 is not large. It was repeated for LSDA and BVWN. Difference energy between them demonstrates difference exchange energy between Becke and Slater functionals. In other word, it is 24.511 hartrees. This value is large. The charges of atoms of carbon nanotube have been collected in 2 . We have shown and compared the variety charges of atoms among B3LYP, B3PW91 and B3P86 methods at 6-31G* basis set and it was also repeated for LSDA and BVWN in . 3 . From this Figure find that if the exchange functional fixes, the charge of atoms will not change. For example in . 3 the carbon atoms with 2, 6, 10, 14, 35, 39, 43 and 47 numbers have the same variety in charge when B3P86, B3PW91 and B3LYP methods with B3 exchange functional were used. It has repeated for SVWN (LSDA) and BVWN with the same correlation functional. The importance of electron exchange for the density of rings is demonstrated by a comparison of SVWN and BVWN results. We summarized the total density of rings for zigzag nanotube in 3 and . 4 . The results calculated for rings of nanotube shows that the poetry in . 4 is less than of . 3 , because the rings are composed of some carbon atom. . 4 shows that the rings 13 and 15 numbers have the same variety in all methods. Then total density of cages depends on exchange functional. . 5 shows that B3P86, B3LYP and B3PW91 because of having the same exchange functional (B3) point out the same variety. By changing the exchange functional for example SVWN and BVWN, we have proved it in 4 and . 5 . While DFT in principle gives a good description of ground state properties, practical applications of DFT are based on approximations for the so-called exchange-correlation potential. The exchange-correlation potential describes the effects of the Pauli principle and coulomb potential beyond a pure electrostatic interaction of the electrons. Possessing the exact exchange-correlation potential means that we solved the many-body problem exactly. The effect of the theoretical treatment of electron correlation and exchange for the description of the varying charges in cages and rings and atoms of carbon nanotube has been investigated by using DFT methods at 6-31G* basis set. In this work, it has been shown that the inclusion of electron exchange-correlation is essential for the description of these properties of carbon nanotube. In present case, we have shown the effect of correlation energy by fixing the exchange functional. For achieving pure correlation energy, it is necessary to combine the exact Hartree-Fock expression for the exchange with an approximate of electron correlation functional.