Plane density waves

There are a variety of other approaches to understanding the electronic structure of crystals. Most of them rely on a density functional approach, with or without the pseudopotential, and use different bases. For example, instead of a plane wave basis, one might write a basis composed of atomic-like orbitals ... 

Jordan oompared the use of plane wave and oonventional Gaussian basis orbitals within density funotional oaloulations in ... 

Floizwarth N A W, Matthews G E, Dunning R B, Tackett A R and Zeng Y 1997 Comparison of the pro]ector augmented-wave, pseudopotentlal and linearized augmented-plane-wave formalisms for density-functional calculations of solids Phys. Rev. B 55 2005... 

The pseudopotential density-functional technique is used to calculate total energies, forces on atoms and stress tensors as described in Ref. 13 and implemented in the computer code CASTEP. CASTEP uses a plane-wave basis set to expand wave-functions and a preconditioned conjugate gradient scheme to solve the density-functional theory (DFT) equations iteratively. Brillouin zone integration is carried out via the special points scheme by Monkhorst and Pack. The nonlocal pseudopotentials in Kleynman-Bylander form were optimized in order to achieve the best convergence with respect to the basis set size. 5... 

How must this theory be modified to describe the effect of the optical excitation The incident electric and magnetic X-ray fields are now pulses Ex(r, t) = Exo(t) exp[j(q r - Oxt)] and Hx(r, t) = Hxo(t) exp[/(q/r - Oxt)]. They still are plane waves with a carrier frequency Ctx, but their amphtudes Exo(t) and Hxo(t) vary with time. The same statement applies to the electron density n r, t), which also is time dependent. However, these variations are all slow with time scales on the order of 1/Ox, and one can neglect 5Exo(0/ 8Hxo(t)/8t as compared to iOxExo(t) and iTlxHxo(0- Detailed calculations then show that [17]... 

The symmetry properties of the density show up experimentally as properties of its Fourier components p. If those components vanish except when the wave vector k equals one of the lattice vectors K of a certain reciprocal lattice, the general plane wave expansion of the density,... 

In molecular DFT calculations, it is natural to include all electrons in the calculations and hence no further subtleties than the ones described arise in the calculation of the isomer shift. However, there are situations where other approaches are advantageous. The most prominent situation is met in the case of solids. Here, it is difficult to capture the effects of an infinite system with a finite size cluster model and one should resort to dedicated solid state techniques. It appears that very efficient solid state DFT implementations are possible on the basis of plane wave basis sets. However, it is difficult to describe the core region with plane wave basis sets. Hence, the core electrons need to be replaced by pseudopotentials, which precludes a direct calculation of the electron density at the Mossbauer absorber atom. However, there are workarounds and the subtleties involved in this subject are discussed in a complementary chapter by Blaha (see CD-ROM, Part HI). 

We should also mention that basis sets which do not actually comply with the LCAO scheme are employed under certain circumstances in density functional calculations, i. e., plane waves. These are the solutions of the Schrodinger equation of a free particle and are simple exponential functions of the general form... 

Hamada, N. and Ohnishi, S. (1986) Self-interaction correction to the local-density approximation in the calculation of the energy band gaps of semiconductors based on the full-potential linearized augmented-plane-wave method, Phys. Rev., B34,9042-9044. 

An expression for e(k) in the case of a Fermi gas of free electrons can be obtained by considering the effect of an introduced point charge potential, small enough so the arguments of perturbation theory are valid. In the absence of this potential, the electronic wave functions are plane waves V 1/2exp(ik r), where V is the volume of the system, and the electron density is uniform. The point charge potential is screened by the electrons, so that the potential felt by an electron, O, is due to the point charge and to the other electrons, whose wave functions are distorted from plane waves. The electron density and the potential are related by the Poisson equation,... 

Newns40 calculated the response to an external electric field of an idealized metal, plane waves at an infinite barrier, with density profile [see Eq. (12)]... 

Based on the same underlying principles as the molecular-based quantum methods, solid-state DFT represents the bulk material using periodic boundary conditions. The imposition of these boundary conditions means that it becomes more efficient to expand the electron density in periodic functions such as plane waves, rather than atom-based functions as in the molecular case. The efficiency of the calculations is further enhanced by the use of pseudo-potentials to represent the core electrons and to make the changes in the electron density... 

A later chapter treats local-density methods in more detail. As currently applied they are frequently incorporated into supercell models of the defect, with extended basis sets that include many plane waves. 

All calculations presented here are based on density-functional theory [37] (DFT) within the LDA and LSD approximations. The Kohn-Sham orbitals [38] are expanded in a plane wave (PW) basis set, with a kinetic energy cutoff of 70 Ry. The Ceperley-Alder expression for correlation and gradient corrections of the Becke-Perdew type are used [39]. We employ ah initio pseudopotentials, generated by use of the Troullier-Martins scheme [40], The coreradii used, in au, were 1.23 for the s, p atomic orbitals of carbon, 1.12 for s, p of N, 0.5 for the s of H, and 1.9, 2.0, 1.5, 1.97,... 

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