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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, [Pg.225]

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 [Pg.226]

According to Bloch s theorem, the Kohn-Sham wave-functions, can be written as [Pg.226]

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 . [Pg.227]

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 ) [Pg.227]

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 [Pg.81]

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]. [Pg.81]

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. [Pg.81]

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. [Pg.82]

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. [Pg.82]


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. [Pg.130]

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. [Pg.663]

One can calculate the far field solution of this equation in the direction ii for an ideally impulsive plane wave insonation, with incidence n ... [Pg.744]

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]. [Pg.723]

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 ... [Pg.112]

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). [Pg.113]

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... [Pg.1277]

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) ... [Pg.1279]

Here an (undistorted) plane wave of unit amplitude is adopted for the channel wavefiinction... [Pg.2020]

A plane wave of unit amplitude can be decomposed according to... [Pg.2029]

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... [Pg.2045]

For electron-ion or ion-ion collisions, the plane waves exp(i/c. R) are simply replaced by Coulomb waves to... [Pg.2045]

Jordan oompared the use of plane wave and oonventional Gaussian basis orbitals within density funotional oaloulations in ... [Pg.2194]

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. [Pg.2210]


See other pages where Plane-waves is mentioned: [Pg.155]    [Pg.663]    [Pg.736]    [Pg.738]    [Pg.122]    [Pg.108]    [Pg.113]    [Pg.717]    [Pg.717]    [Pg.719]    [Pg.963]    [Pg.968]    [Pg.970]    [Pg.978]    [Pg.1315]    [Pg.1362]    [Pg.1365]    [Pg.1628]    [Pg.1753]    [Pg.1754]    [Pg.2012]    [Pg.2012]    [Pg.2013]    [Pg.2014]    [Pg.2015]    [Pg.2021]    [Pg.2031]    [Pg.2034]    [Pg.2043]    [Pg.2171]    [Pg.2201]    [Pg.2202]    [Pg.2210]    [Pg.2211]    [Pg.2211]    [Pg.2211]    [Pg.2212]   
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An equation of a plane traveling wave

Approximations local plane waves

Augmented plane wave

Augmented plane wave calculation

Augmented plane wave method

Augmented plane wave relativistic

Basis plane waves

Basis sets-Gaussian orbital versus plane waves

Classical Theory of Plane Detonation Wave

Contrast plane wave interference

Density functional full-potential linearized augmented plane wave method

Density plane wave

Diffraction of a plane wave

Electromagnetic waves, plane

Electronic structure augmented plane waves

Expansion of a Plane Wave in Vector Spherical Harmonics

Explosive plane-wave generator

FLAPW linear-augmented plane-wave

FLAPW plane-wave

FPLAPW augmented plane wave

Full potential augmented plane wave method

Full potential linear augmented plane wave FLAPW)

Full potential linearised augmented plane-wave

Full potential linearized augmented plane wave structures

Full-potential augmented plane-wave

Full-potential augmented plane-wave FLAPW)

Full-potential linear augmented plane wave

Full-potential linear augmented plane wave method

Full-potential linearized augmented plane wave

Full-potential linearized augmented plane wave method

Gaussian and plane waves

Herring orthogonalized plane wave

Light plane waves

Linear Augmented Plane Wave

Linear Augmented Plane Wave method

Linearized augmented plane wave method

Linearized augmented plane-wave

Linearized augmented plane-wave calculation

Local plane waves

Local plane waves evanescent fields

Local plane waves polarization

Local plane waves tunneling

Maxwell equations plane-wave solutions

Monochromatic plane waves and their polarization states

Monochromatic plane waves, quantum

Monochromatic plane waves, quantum multipole radiation

Orthogonalized plane wave

Orthogonalized plane wave method

Perturbation plane wave

Plane Detonation Wave

Plane Electromagnetic Waves in Vacuum

Plane Wave Lens

Plane Wave Optical Field

Plane Wave Pseudopotential Method

Plane Wave with a Sphere

Plane Waves and Atomic-like Basis Sets. Slater-type Functions

Plane Waves and Pseudopotentials

Plane Waves and the Brillouin Zone

Plane Waves in Anisotropic Materials without Optical Rotation

Plane Waves in Materials with Optical Rotation

Plane Waves in Nonconducting Media

Plane combustion wave

Plane combustion wave propagation

Plane detonation waves with finite reactions

Plane gravitational waves

Plane polarized electromagnetic wave

Plane wave basis functions

Plane wave generator

Plane wave generators = charges

Plane wave methods

Plane wave rocking curve

Plane wave solutions

Plane waves Point defects

Plane waves approaches

Plane waves electromagnetic theory

Plane waves form

Plane waves harmonics

Plane waves individual photons

Plane waves potential

Plane waves propagation

Plane waves recent studies

Plane waves use

Plane waves vacuum interface, reflection

Plane-Wave Expansion - The Free-Electron Models

Plane-Wave Propagation in Unbounded Media

Plane-polarized wave

Plane-wave Born Approximation

Plane-wave approximation

Plane-wave basis sets

Plane-wave calculations

Plane-wave decomposition

Plane-wave density functional theory

Plane-wave density functional theory semiconductors

Plane-wave expansion

Plane-wave first approximation

Plane-wave first approximation density

Plane-wave focusing, resolution limit

Plane-wave function

Plane-wave reflection

Plane-wave representation

Plane-wave representation generation

Plane-wave representation systems

Projector augmented plane wave

Propagation of an Electromagnetic Plane Wave

Pseudo-plane-waves

Pseudopotential plane wave

Quasi-plane waves

Rays and local plane waves

Reflected vector plane wave

Reflection and Refraction of Plane Waves

Reflection of a plane wave

Reflection of plane wave

Slater augmented plane wave

Structure of the Hydrogen Wave and Experiments on Single-Crystal Planes

Transmitted vector plane wave

Vector plane waves

Vector plane waves definition

Wavefunction augmented plane wave

Wavefunction orthogonalized plane wave

Wavefunctions plane wave

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