Electronic structure methods local density approximation

In tins study, we report an evaluation of structural, mechanical, and vibrational prqierties of SWCNTs [C(n,n), and C(n,0), n=4,6,8,10], applying a FP-LCAO method (26,27), as it has previously been pointed out tiuit all-electron full-potential local density approximation (LDA)-based calculations are appropriate for treating van der Waals interactions in systems such as fiillerenes, gr hene and graphite (28,29). Although the full-potential all-electron scheme is conqiutationally intensive, our work contributes to the further validation of previous tiieoretical studies. Large-scale pseudopotential DFT calculations were also recently performed by Yoon et al. (30), in order to study the effects of intertube interactions on the transport properties of SWCNT junctions. 

Our work demonstrates that EELS and in particular the combination of this technique with first principles electronic structure calculations are very powerful methods to study the bonding character in intermetallic alloys and study the alloying effects of ternary elements on the electronic structure. Our success in modelling spectra indicates the validity of our methodology of calculating spectra using the local density approximation and the single particle approach. 

The relaxation of the structure in the KMC-DR method was done using an approach based on the density functional theory and linear combination of atomic orbitals implemented in the Siesta code [97]. The minimum basis set of localized numerical orbitals of Sankey type [98] was used for all atoms except silicon atoms near the interface, for which polarization functions were added to improve the description of the SiOx layer. The core electrons were replaced with norm-conserving Troullier-Martins pseudopotentials [99] (Zr atoms also include 4p electrons in the valence shell). Calculations were done in the local density approximation (LDA) of DFT. The grid in the real space for the calculation of matrix elements has an equivalent cutoff energy of 60 Ry. The standard diagonalization scheme with Pulay mixing was used to get a self-consistent solution. In the framework of the KMC-DR method, it is not necessary to perform an accurate optimization of the structure, since structure relaxation is performed many times. 

For direct Af-electron variational methods, the computational effort increases so rapidly with increasing N that alternative simplified methods must be used for calculations of the electronic structure of large molecules and solids. Especially for calculations of the electronic energy levels of solids (energy-band structure), the methodology of choice is that of independent-electron models, usually in the framework of density functional theory [189, 321, 90], When restricted to local potentials, as in the local-density approximation (LDA), this is a valid variational theory for any A-electron system. It can readily be applied to heavy atoms by relativistic or semirelativistic modification of the kinetic energy operator in the orbital Kohn-Sham equations [229, 384],... 

Recent calculations of the band structure of polyethylene have employed variations of the ab initio method incorporating electron correlation. Sun and Bartlett (1996) utilised many-body perturbation theory to encompass electron correlation in the ab initio framework. Siile et al. (2000) and Serra et al. (2000) have employed variants of DFT. These calculations involved the optimisation of local effective potentials and a local-density approximation respectively. Figure 4.13 shows a comparison of the band structure obtained by Siile et al., Fig. 4.13(d), with those obtained by other ab initio DFT calculations using the Hartree-Fock (HF), Fig 4.13(a), and Slater approaches, Figs. 4.13(b) and (c). 

Most of the cluster calculations for molecule/surface interactions are currently performed at the ab initio SCF or DFT levels of theory. (In the following we will use the abbreviations SCF= self consistent field and HF= Hartree-Fock synonymously. DFT stands for density functional theory .) We will not describe these methods here, but refer the reader to the text books on electronic structure theory in which these methods are treated in detail [59-61 ]. Some of the characteristic features of these approaches, as far as molecule/surface interactions are concerned, are summarized in Table 2. We have included three types of DFT functionals which are used in calculations for adsorption properties local density functionals (local density approximation, the LDA... 

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