We have developed and implemented a selfconsistent density functional method using standard norm-conserving pseudopotentials and a flexible, numerical linear combination of atomic orbitals basis set, which includes multiple-zeta and polarization orbitals. Exchange and correlation are treated with the local spin density or generalized gradient approximations. The basis functions and the electron density are projected on a real-space grid, in order to calculate the Hartree and exchange-correlation potentials and matrix elements, with a number of operations that scales linearly with the size of the system. We use a modified energy functional, whose minimization produces orthogonal wavefunctions and the same energy and density as the Kohn-Sham energy functional, without the need for an explicit orthogonalization. Additionally, using localized Wannier-like electron wavefunctions allows the computation time and memory required to minimize the energy to also scale linearly with the size of the system. Forces and stresses are also calculated efficiently and accurately, thus allowing structural relaxation and molecular dynamics simulations.

We describe an ab initio method for calculating the electronic structure, electronic transport, and forces acting on the atoms, for atomic scale systems connected to semi-infinite electrodes and with an applied voltage bias. Our method is based on the density-functional theory ͑DFT͒ as implemented in the well tested SIESTA approach ͑which uses nonlocal norm-conserving pseudopotentials to describe the effect of the core electrons, and linear combination of finite-range numerical atomic orbitals to describe the valence states͒. We fully deal with the atomistic structure of the whole system, treating both the contact and the electrodes on the same footing. The effect of the finite bias ͑including self-consistency and the solution of the electrostatic problem͒ is taken into account using nonequilibrium Green's functions. We relate the nonequilibrium Green's function expressions to the more transparent scheme involving the scattering states. As an illustration, the method is applied to three systems where we are able to compare our results to earlier ab initio DFT calculations or experiments, and we point out differences between this method and existing schemes. The systems considered are: ͑i͒ single atom carbon wires connected to aluminum electrodes with extended or finite cross section, ͑ii͒ single atom gold wires, and finally ͑iii͒ large carbon nanotube systems with point defects.

We present a method to perform fully selfconsistent density-functional calculations, which scales linearly with the system size and which is well suited for very large systems. It uses strictly localized pseudoatomic orbitals as basis functions. The sparse Hamiltonian and overlap matrices are calculated with an O(N ) effort. The long range selfconsistent potential and its matrix elements are computed in a real-space grid. The other matrix elements are directly calculated and tabulated as a function of the interatomic distances. The computation of the total energy and atomic forces is also done in O(N ) operations using truncated, Wannier-like localized functions to describe the occupied states, and a band-energy functional which is iteratively minimized with no orthogonality constraints. We illustrate the method with several examples, including carbon and silicon supercells with up to 1000 Si atoms and supercells of β-C3N4. We apply the method to solve the existing controversy about the faceting of large icosahedral fullerenes by performing dynamical simulations on C60, C240, and C540.A large effort has been devoted in the last few years to develop approximate methods to solve the electronic structure of large systems with a computational cost proportional to its size. 1 Several approaches are now sufficiently accurate and robust to obtain reliable results for systems with thousands of atoms. So far, however, most of these schemes have been useful only with simple Hamiltonians, like empirical tight-binding models, which provide an ideal setting for order-N calculations. First-principles order-N calculations have been performed mainly in the non-selfconsistent Harris functional version 2 of the local density approximation (LDA) for electronic exchange and correlation (XC) using minimal bases. 1,3 Linear scaling algorithms in fully selfconsistent LDA have also been tried, 4 but the results are far from the linear scaling regime, due to the relatively small number of manageable atoms in those simulations. Hernandez et al. 5 have successfully produced LDA results in large silicon systems using a real-space grid method. The computational requirements that this kind of approach demands are, however, extremely large, and calculations must be performed in massive computational platforms.We have developed a selfconsistent density-functional formulation with linear scaling, capable of producing results for very large systems, whose computational demands are not overwhelmingly large, so that systems with many hundreds of atoms can be treated in modest computational platforms like workstations, and much larger systems can be treated in massive platforms. The method is based on the linear combination of atomic orbitals (LCAO) approximation as basis of expansion of the electronic states. Non-orthogonal LCAO bases are very efficient, reducing the number of variables dramatically, compared to plane-wave (PW) or real-space-grid approaches, so that larger systems can be studied. Also, LCAO can provide up to extremely accurate base...

We investigate the tight-binding approximation for the dispersion of the and * electronic bands in graphene and carbon nanotubes. The nearest-neighbor tight-binding approximation with a fixed ␥ 0 applies only to a very limited range of wave vectors. We derive an analytic expression for the tight-binding dispersion including up to third-nearest neighbors. Interaction with more distant neighbors qualitatively improves the tight-binding picture, as we show for graphene and three selected carbon nanotubes.The band structure of carbon nanotubes is widely modeled by a zone-folding approximation of the graphene and à electronic states as obtained from a tight-binding Hamiltonian. [1][2][3][4][5] The large benefit of this method is the very simple formula for the nanotube electronic bands. While the tight-binding picture provides qualitative insight into the one-dimensional nanotube band structure, it is more and more being used for quantitative comparisons as well. For instance, attempts to assign diameters and chiralities of carbon nanotubes based on optical absorption and Raman data rely heavily on the assumed transition energies. 2,6 Differences between the zone-folding, tight-binding -orbital description and experiment, as observed, e.g., in scanning tunneling measurements, are usually ascribed to ''curvature effects.'' 1 However, the common -orbital tight-binding approach for the nanotube band structure involves two approximations: ͑i͒ zone folding, which neglects the curvature of the wall; and ͑ii͒ the tight-binding approximation to the graphene bands including only first-neighbor interaction. Whereas the first point received some attention in the literature, 7-9 the second approximation is usually assumed to be sufficient.In this paper we compare the tight-binding approximation of the graphene orbitals to first-principles calculations. We show that the nearest-neighbor tight-binding Hamiltonian does not accurately reproduce the and * graphene bands over a sufficiently large range of the Brillouin zone. We derive an improved tight-binding electronic dispersion by including up to third-nearest-neighbor interaction and overlap. The formula for the electronic states we present may readily be used, e.g., in combination with zone folding to obtain the band structure of nanotubes.The first tight-binding description of graphene was given by Wallace in 1947. 10 He considered nearest-and nextnearest-neighbor interaction for the graphene p z orbitals, but neglected the overlap between wave functions centered at different atoms. The other-nowadays better known-tightbinding approximation was nicely described by Saito et al. 4 It considers the nonfinite overlap between the basis functions, but includes only interactions between nearest neighbors within the graphene sheet. To study the different levels of approximation we start from the most general form of the secular equation, the tight-binding Hamiltonian H, and the overlap matrix S, 4where E(k) are the electronic eigenvalues. We used the equivalence of the A and B carbon ...