Electronic structure LSDA

In order to perform the calculation., of the conductivity shown here we first performed a calculation of the electronic structure of the material using first-principles techniques. The problem of many electrons interacting with each other was treated in a mean field approximation using the Local Spin Density Approximation (LSDA) which has been shown to be quite accurate for determining electronic densities and interatomic distances and forces. It is also known to reliably describe the magnetic structure of transition metal systems. 

The LSDA approach requires simultaneous self-consistent solutions of the Schrbdinger and Poisson equations. This was accomplished using the Layer Korringa-Kohn-Rostoker technique which has many useful features for calculations of properties of layered systems. It is, for example, one of only a few electronic structure techniques that can treat non-periodic infinite systems. It also has the virtue that the computational time required for a calculation scales linearly with the number of different layers, not as the third power as most other techniques. 

A different approach to treat correlation effects which are not well described within the LSDA consists in incorporating self-interaction corrections (SIC) [111-114] in electron structure methods for solids, Svane et al. [115-120]. In the Hartree-Fock (HF) theory the electron-electron interactions are usually divided into two contributions, the Coulomb term and the exchange term although they both are Coulomb interactions. The separation though, is convenient because simplifications of self-consistent-field calculations can be obtained by including in both terms the interaction of the electron itself. In the HF theory this has no influence on the solutions because these selfinteractions in the Coulomb and exchange terms exactly cancel each other. However, when the exchange term is treated... 

The strong fluctuation of IP or of the mass abundance is an electronic-structure effect, reflecting the global shape of the cluster, but not necessarily its detailed ionic structure. This is demonstrated in Fig. 6, where the ionization potentials of sodium clusters obtained by the spheroidal jellium model [32] are compared with their experimental values [46]. The odd-even oscillation of IP for low N is reproduced well. The amplitude of these oscillations is exaggerated, but this is corrected by using the spin-dependent LSDA, instead of the simple LDA [47]. The same occurs for the staggering of d2 N) [48]. 

As most of the electronic structure simulation methods, we start with the Born-Oppenheimer approximation to decouple the ionic and electronic degrees of freedom. The ions are treated classically, while the electrons are described by quantum mechanics. The electronic wavefunctions are solved in the instantaneous potential created by the ions, and are assumed to evolve adiabatically during the ionic dynamics, so as to remain on the Born-Oppenheimer surface. Beyond this, the most basic approximations of the method concern the treatment of exchange and correlation (XC) and the use of pseudopotentials. XC is treated within Kohn-Sham DFT [3]. Both the local (spin) density approximation (LDA/LSDA) [16] and the generalized gradients approximation (GGA) [17] are implemented. The pseudopotentials are standard norm-conserving [18, 19], treated in the fully non-local form proposed by Kleinman and Bylander [20]. 

Independent electronic structure calculations on YbPtBi were performed by Oppeneer et al, (1997) on the basis of density-functional theory in the local-spin-density approximation (LSDA), generalized with additional intra-atomic Coulomb correlations between 4f electrons. These calculations show that the Yb 4f level is pinned at the Fermi energy. This pinning is a generic property. Furthermore the hybridized 4f level is split into two van Hove-like side maxima. 

While these methods provide some useful insight into Gd and Ce, they yield unrealistic results for any other lanthanide material as the f-bands bimch at the Fermi level leading to unphysically large densities of states at the Fermi energy and disagreement with the de Haas van Alphen measurements. It is clear that a satisfactory theory of lanthanide electronic structures requires a method that treats all electrons on an equal footing and from which both localized and itinerant behaviour of electrons may be derived. SIC to the LSDA provide one such theory. 

A SIC-LSDA calculation, however, where all seven majority 4f-states are corrected, opens up a Hubbard gap between the occupied and unoccupied f-states. The majority f-states are pushed down to approximately 16 eV below the Fermi energy and the minority f-states are moved away from the conduction bands, reducing the f contribution to the DOS at the Fermi energy. The SIC-LSDA electronic structure thus describes the experimental picture well, with the f-states playing little role in conduction. 

Finally, a very interesting set of applications was that of Hagler and co-workers. o Employing the LSDA, they studied the changes in the electronic structure of folate, dihydrofolate, and NADPH upon binding to the enzyme... 

Most calculations of the electronic structure of solids, including those performed for f electron metals, are based on density functional theory within the Kohn-Sham ansatz to which, in turn, a local density (LDA) or local spin density (LSDA) approximation is applied. Each part of this characterization has it own implications and it is important to understand which is relevant when judging the validity of results. 

Torsion While the torsional electromechanical response was successfully explained via the deformation of the BriUuoin zone, another possible explanation for the bandgap oscillations may be the periodic formation of Moire patterns due to registry mismatch between the different walls of the multi-waUed tubes, which have a different curvature. In order to rule out this possibility, DFT calculations within the LSDA were used to show that the interlayer coupling is weak and the electronic structure of the individual walls resembles that of single-waUed nanotubes for different intertube orientations (Nagapriya et al. 2008). 

The simplest approximation to the complete problem is one based only on the electron density, called a local density approximation (LDA). For high-spin systems, this is called the local spin density approximation (LSDA). LDA calculations have been widely used for band structure calculations. Their performance is less impressive for molecular calculations, where both qualitative and quantitative errors are encountered. For example, bonds tend to be too short and too strong. In recent years, LDA, LSDA, and VWN (the Vosko, Wilks, and Nusair functional) have become synonymous in the literature. 

Fig. 9. Electronic band structure of spin polarized fee Ni. The majority spin character is indicated by the thickness of the lines. The calculation was carried out with RFPLO in LSDA, Perdew-Wang 92 [25].

The first term results from the Fermi contact interaction, while the second represents the long-range dipole-dipole interaction. In the equations above, ge is the free-electron g factor, /Xe the Bohr magneton, gi the nuclear gyromagnetic ratio, and /xi the nuclear moment. Moreover, the nucleus is located at position R, and the vector r has the nuclear position as its origin. Finally, p (r) = p (r) — p (r) is the electron spin density. The only nontrivial input into these equations is precisely this last quantity, i.e. Ps(r), which can be computed in the LSDA or another DFT approximation. The resulting Hamiltonian can be used to interpret the hyperfine structure measured in experiments. A recent application to metal clusters is reported in Ref. [118]. 

Early LSDA static pseudopotential approaches to sodium microclusters date back approximately 20 years [122], see Appendix C. It would be misleading to consider LDA calculations as the natural extension of jellium models. However, the global validity of the latter cannot but anticipate the success of the former. Clearly, these should also clarify the role of the atomic structure in determining the electronic behavior of the clusters and the extent to which the inhomogeneity of the electron distribution is reflected in the measurable properties. Many structural determinations are by now available for the smaller aggregates, made at different levels of approximation and of accuracy (e.g. [110, 111], see Appendix C). The most extensive investigation of sodium clusters so far is the LDA-CP study of Ref. [123] (see Appendix C), which makes use of all the features of the CP method. Namely, it uses dynamical SA to explore the potential-energy surface, MD to simulate clusters at different temperatures, and detailed analysis of the one-electron properties, which can be compared to the predictions of jellium-based models. 

DFT computations using localized basis functions have sporadically been applied to silver and gold microclusters. The early LSDA approach [226] (see Appendix C) to silver dimer and trimer (neutral and cationic) was a comparative study of results obtained with both le- and 11 e-pseudopotentials. It clarified the importance of the 4d-electron states in determining structural and bonding properties, which is partly related to their polarization. 

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