Plane-waves

To calculate the total energy of solids, a plane-wave expansion of the Kohn-Sham wave-functions is very useful, as it takes advantage of the periodicity of the crystal [14,15,16]. For finite systems, such as atoms, molecules and clusters, plane-waves can also be used in a super-cell approach . In this method, 

When using the pseudo-potential approximation, the external potential, Uext, is simply the sum of the pseudo-potentials of all the atoms in the system. If atom a. is located in the unit cell at Tq, and its pseudo-potential is Wa r, r ), the external potential is 

According to Bloch s theorem, the Kohn-Sham wave-functions, can be written as 

The sums over k are performed over all Brillouin zone vectors, but can be reduced to sums on the irreducible Brillouin zone by taking advantage of the space group of the lattice . 

There are thus two convergence parameters that need to be fine-tuned for every calculation the Brillouin zone sampling and a cutoff radius in reciprocal space to truncate the sums over reciprocal lattice vectors (we cannot perform infinite summations ) 

The second route to solving Eq. (17), developed mainly by the solid-state community, relies on simple functions to express the KS orbitals. By far the most common choice is plane-waves. In this scheme, the orbitals are expressed by a finite and discrete Fourier series 

The sum includes all the G vectors belonging to some simple and regular lattice in reciprocal space, up to a maximum modulus G max determined by the spatial resolution required by i/ /(r) [62]. 

Besides these positive points, which led to the remarkable success of plane-wave implementations of DFT, there are also important drawbacks, in addition to the sheer dimension of the basis set required for many applications. First of all, the discretization of the Fourier expansion in reciprocal space implicitly defines a periodicity in real space by using Eq. (20) we assume that the system under study is periodically repeated in space. 

Another technical difficulty is that of describing electron states that are highly localized, such as the 3d states in transition metals, by means of plane-waves. This problem and currently adopted solutions are discussed in the following sections. 

The choice of the basis set is strictly related to another ingredient of several implementations of the DFT scheme, i.e. pseudopotentials. As is well known, the chemistry of the elements depends predominantly on the valence electrons, i.e. on those in the highest-energy incomplete atomic shell. It is natural, therefore, to include only the chemically active electrons explicitly in the computation. Two different but closely related approaches have been introduced to exploit this basic simplification (1) the frozen-core approximation, which assumes that the core is not modified by the formation of chemical bonds, and (2) the pseudopotential formalism, which replaces the interaction between valence and core electrons by an external potential acting on the former and does not explicitly include the latter. 

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In this section, two illustrative numerical results, obtained by means of the described reconstruction algorithm, are presented. Input data are calculated in the frequency range of 26 to 38 GHz using matrix formulas [8], describing the reflection of a normally incident plane wave from the multilayered half-space. 

Let us consider the scheme showed in Fig. I to calculate the field scattered by a rough cylindrical surface (i.e. a wire). The wire is illuminated by a monochromatic, linearly polarized plane wave at an angle of incidence a with its axis of symmetry. The surface is described, in a system fixed to the wire, by p = h (cylindrical coordinates. We shall denote the incident wave vector lying on the x-z plane as kj and the emergent wave vector simply as k. 

One can calculate the far field solution of this equation in the direction ii for an ideally impulsive plane wave insonation, with incidence n ... 

A catalyst may play an active role in a different sense. There are interesting temporal oscillations in the rate of the Pt-catalyzed oxidation of CO. Ertl and coworkers have related the effect to back-and-forth transitions between Pt surface structures [220] (note Fig. XVI-8). See also Ref. 221 and citations therein. More recently Ertl and co-workers have produced spiral as well as plane waves of surface reconstruction in this system [222] as well as reconstruction waves on the Pt tip of a field emission microscope as the reaction of H2 with O2 to form water occurred [223]. Theoretical simulations of these types of effects have been reviewed [224]. 

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 ... 

Other methods for detennining the energy band structure include cellular methods. Green fiinction approaches and augmented plane waves [2, 3]. The choice of which method to use is often dictated by die particular system of interest. Details in applying these methods to condensed matter phases can be found elsewhere (see section B3.2). 

Flere we model the pump beams associated with fields E(a> ) and (102) as plane waves with wavevectors Jti = and Jt, = feiwiv/fii (wil/r - The directions of tlie reflected and transmitted beams can... 

The higher-order bulk contribution to the nonlmear response arises, as just mentioned, from a spatially nonlocal response in which the induced nonlinear polarization does not depend solely on the value of the fiindamental electric field at the same point. To leading order, we may represent these non-local tenns as bemg proportional to a nonlinear response incorporating a first spatial derivative of the fiindamental electric field. Such tenns conespond in the microscopic theory to the inclusion of electric-quadnipole and magnetic-dipole contributions. The fonn of these bulk contributions may be derived on the basis of synnnetry considerations. As an example of a frequently encountered situation, we indicate here the non-local polarization for SFIG in a cubic material excited by a plane wave (co) ... 

Here an (undistorted) plane wave of unit amplitude is adopted for the channel wavefiinction... 

A plane wave of unit amplitude can be decomposed according to... 

Here the distortion (diagonal) and back coupling matrix elements in the two-level equations (section B2.2.8.4) are ignored so that = exp(ik.-R) remains an imdistorted plane wave. The asymptotic solution for ij-when compared with the asymptotic boundary condition then provides the Bom elastic ( =f) or inelastic scattering amplitudes... 

For electron-ion or ion-ion collisions, the plane waves exp(i/c. R) are simply replaced by Coulomb waves to... 

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

We now discuss the most important theoretical methods developed thus far the augmented plane wave (APW) and the Korringa-Kolm-Rostoker (KKR) methods, as well as the linear methods (linear APW (LAPW), the linear miiflfm-tin orbital [LMTO] and the projector-augmented wave [PAW]) methods. 

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