Electronic structure real-space methods

Electronic structure methods for studies of nanostructures can be divided broadly into supercell methods and real-space methods. Supercell methods use standard k-space electronic structure techniques separating periodically repeated nanostructures by distances large enough to neglect their interactions. Direct space methods do not need to use periodic boundary conditions. Various electronic structure methods are developed and applied using both approaches. In this section we will shortly discuss few popular but powerful electronic structure methods the pseudopotential method, linear muffin-tin orbital and related methods, and tight-binding methods. 

While the supercell approach works well for localized systems, it is typically necessary to consider a very large supercell. This results in a plane-wave basis replicating not only the relevant electronic states but also vacuum regions imposed by the supercell. A much more efficient method to implement for investigating the electronic structures of localized systems is to use real space methods such as the recursion methods [27] and the moments methods [28], These methods do not require symmetry and their cost grows linearly with the number of inequivalent atoms being considered. For these reasons, real space methods are very useful for a description of the electronic properties of complex systems, for which the usual k-space methods are either inapplicable or extremely costly. 

Real-space methods have found recent application in QM/MM methods, which couple central quantum regions with more distant molecular mechanics domains. Wavelet applications to electronic structure are at an earHer stage of development than are the methods discussed here, but they continue to hold a great deal of promise due to their inherent multiscale nature. Several recent studies suggest that wavelets will find wide application in future electronic structure codes. 

For understanding and predicting material properties such as density, elasticity, magnetization or hardness from first principles quantum mechanical calculations, a reliable and efficient tool for electronic structure calculations is necessary. The reciprocal space methods, to which most attention has been dedicated so far, are very powerful and sophisticated but by their nature are suitable mostly for crystals. For systems without translational symmetry such as metallic clusters, defects, quantum dots, adsorbates and nanocrystals, use of real-space methods is more promising. 

STM found one of its earliest applications as a tool for probing the atomic-level structure of semiconductors. In 1983, the 7x7 reconstructed surface of Si(l 11) was observed for the first time [17] in real space all previous observations had been carried out using diffraction methods, the 7x7 structure having, in fact, only been hypothesized. By capitalizing on the spectroscopic capabilities of the technique it was also proven [18] that STM could be used to probe the electronic structure of this surface (figure B1.19.3). 

Conventional HRTEM operates at ambient temperature in high vacuum and directly images the local structure of a catalyst at the atomic level, in real space. In HRTEM, as-prepared catalyst powders can be used without additional sample preparation. The method does not normally require special treatment of thin catalyst samples. In HRTEM, very thin samples can be treated as WPOs, whereby the image intensity can be correlated with the projected electrostatic potential of the crystal, leading to the atomic structural information characterizing the sample. Furthermore, the detection of electron-stimulated XRE in the EM permits simultaneous determination of the chemical composition of the catalyst. Both the surface and sub-surface regions of catalysts can be investigated. 

The method employed was similar to that of Ref. 35, but with several improvements. ab initio, norm-conserving, nonlocal pseudopotential were used to represent the metal ions. This capability enables reliably realistic representation of the metal s electronic structure. Thus the cadmium pseudopotential was able, for example, to reproduce the experimental cadmium-vacuum work function using no adjustable parameters (unlike the procedure followed in Ref. 35). Pseudopotentials of the Troullier and Martins form [53] were used with the Kleinman-Bylander [54] separable form, and a real space... 

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. 

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