K-space method

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. 

Abstract We summarize an ab-initio real-space approach to electronic structure calculations based on the finite-element method. This approach brings a new quality to solving Kohn Sham equations, calculating electronic states, total energy, Hellmann-Feynman forces and material properties particularly for non-crystalline, non-periodic structures. Precise, fully non-local, environment-reflecting real-space ab-initio pseudopotentials increase the efficiency by treating the core-electrons separately, without imposing any kind of frozen-core approximation. Contrary to the variety of well established k-space methods that are based on Bloch s theorem... 

The present method focuses on solving Kohn Sham equations and calculating electronic states, total energy and material properties of non-crystalline, nonperiodic structures. Contrary to the variety of well established k-space methods that are based on Bloch s theorem and applicable to periodic structures, we don t assume periodicity in any respect. Precise ab-initio environment-reflecting pseudopotentials proven within the plane wave approach are connected with real space finite-element basis in the present approach. The main expected asset of the present approach is the combination of efficiency and high precision of ab-initio pseudopotentials with universal applicability, universal basis and excellent convergence control of finite-element method not restricted to periodic environment. 

We calculate the quantity V(k) by inverse Fourier transform, by summing V up to the six shell of neighbors. This method favorably contrasts with the evaluation of V(k) directly in k-space and is justified by the fast convergence of V with the shell number... 

The rate of convergence of the Steepest Descent method is first order. The basic difficulty with steepest descent is that the method is too sensitive to the scaling of S(k), so that convergence is very slow and oscillations in the k-space can easily occur. In general a well scaled problem is one in which similar changes in the variables lead to similar changes in the objective function (Kowalik and Osborne, 1968). For these reasons, steepest descent/ascent is not a viable method for the general purpose minimization of nonlinear functions, ft is of interest only for historical and theoretical reasons. 

Fig. 3.4.3 (a) Two-dimensional k-space acquisition using SPI or SPRITE. The k-space data acquisition is indicated numerically. High magnetic field gradient amplitudes are applied at the extremities of k-space. (b) A generic two-dimensional centric scan SPRITE method. The... 

In the present study we have extracted the EXAFS from the experimentally recorded X-ray absorption spectra following the method described in detail in Ref. (l , 20). In this procedure, a value for the energy threshold of the absorption edge is chosen to convert the energy scale into k-space. Then a smooth background described by a set of cubic splines is subtracted from the EXAFS in order to separate the non-osciHatory part in ln(l /i) and, finally, the EXAFS is multiplied by a factor k and divided by a function characteristic of the atomic absorption cross section (20). 

Appropriate methods must be used to accurately treat k space for metals. 

The third numerical detail that is important in Fig. 8.1 arises from the fact that Ag is a metal and the observation that the DOS is obtained from integrals in k space. In Section 3.1.4 we described why performing integrals in k space for metals holds some special numerical challenges. The results in Fig. 8.1 were obtained by using the Methfessel and Paxton smearing method to improve the numerical precision of integration in k space. 

All results in this chapter were obtained from calculations with the PW91 GGA functional. The Monkhorst-Pack method for sampling k space was used in all cases. 

The first few 4-dimensional hyperspherical harmonics K i, ,m(u) are shown in Table 5. Shibuya and Wulfman [19] extended Fock s momentum-space method to the many-center one-particle Schrodinger equation, and from their work it follows that the solutions can be found by solving the secular equation (63). If Fock s relationship, equation (67), is substituted into (65), we obtain ... 

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. 

As we have seen, conventional spin echo imaging (Section II.E) typically takes approximately a few minutes because an independent r.f. excitation is required for acquisition of each row of k-space data. Hence, sampling of the complete raster is limited by the repetition/recycle time of the pulse sequence used, which in turn is governed by the inherent 7) relaxation time(s) of the system under study. In general, the acquisition speed of an MR image may be improved by two basic methods ... 

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