Expansion plane-wave

It is not easy to see why the authors believe that the success of orbital calculations should lead one to think that the most profound characterization of the properties of atoms implies such an importance to quantum numbers as they are claiming. As is well known in quantum chemistry, successful mathematical modeling may be achieved via any number of types of basis functions such as plane waves. Similarly, it would be a mistake to infer that the terms characterizing such plane wave expansions are of crucial importance in characterizing the behavior of atoms. 

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,... 

A first step toward quantum mechanical approximations for free energy calculations was made by Wigner and Kirkwood. A clear derivation of their method is given by Landau and Lifshitz [43]. They employ a plane-wave expansion to compute approximate canonical partition functions which then generate free energy models. The method produces an expansion of the free energy in powers of h. Here we just quote several of the results of their derivation. 

The use of a plane wave expansion implies the presence of periodic boundary conditions (PBCs). This is a natural selection in... 

Wave propagation in periodic structures can be effieiently modeled using the eoncept of Bloeh (or Floquet-Bloch) modes . This approach is also applicable for the ealeulation of band diagrams of 1 -D and 2-D photonic crystals . Contrary to classical methods like the plane-wave expansion , the material dispersion ean be fully taken into aeeount without any additional effort. For brevity we present here only the basie prineiples of the method. 

In order to calculate the band structure and the density of states (DOS) of periodic unit cells of a-rhombohedral boron (Fig. la) and of boron nanotubes (Fig. 3a), we applied the VASP package [27], an ab initio density functional code, using plane-waves basis sets and ultrasoft pseudopotentials. The electron-electron interaction was treated within the local density approximation (LDA) with the Geperley-Alder exchange-correlation functional [28]. The kinetic-energy cutoff used for the plane-wave expansion of... 

The resolution of this paradox is easily obtained once it is remembered that the NFE bands in aluminium are formed from the valence 3s and 3p electrons. These states must be orthogonal to the s and p core functions, so that they contain nodes in the core region as illustrated for the 2s wave function in Fig. 2.12. In order to reproduce these very short wavelength oscillations, plane waves of very high momentum must be included in the plane wave expansion of . Retaining only the two lowest energy plane waves in eqn (5.35) provides an extremely bad approximation. 

The calculation of the (001) cadmium-surface-water interface used a hexagonal unit cell with a c lattice constant of 135 a.u., and an a lattice constant of 33.788 a.u., corresponding to 36 cadmium atoms in the surface layer. The Cd slabs were 5 layers thick (perpendicular to the c-axis), for a total of 180 cadmium atoms per unit cell. The region between the slabs was filled with 525 water molecules. The plane wave expansion of the electron eigenstates had a cutoff of 9 Ry, giving 20,209 basis functions. Calculational details are described in reference [52]. 

We shall deal mainly with the propagation and scattering of electrons through solid surfaces. The most appropriate formalism to describe these processes will depend very much on the situation, as computational efficiency is at a premium. Spherical-wave and plane-wave expansions are frequently used, often side by side within the same theory. Spherical waves are convenient to describe the scattering or emission of electrons by an individual atom. An important quantity will then be the complex amplitude of each spherical wave leaving that atom. 

Plane waves are often used when two conditions are satisfied 1) many (but not necessarily all) atomic layers of the surface have a two-dimensional periodicity, and 2) either a plane-wave incident electron beam is present or angle-resolved electron detection is applied. Computation based on the plane-wave expansion are often much more efficient than those based on the spherical-wave expansion. This explains their frequent use even in problems that do not involve strict two-dimensional periodicity, as with disordered overlayers on an otherwise periodic substrate. 

Great computational advantage can ensue from using the plane-wave expansion between atomic layers. This occurs when such a representation converges, which requires sufficiently large spacings between the layers. Various... 

Plane-Wave Expansion - The Free-Electron Models... 

Solution of the Kohn-Sham equations as outlined above are done within the static limit, i.e. use of the Born-Oppenheimer approximation, which implies that the motions of the nuclei and electrons are solved separately. It should however in many cases be of interest to include the dynamics of, for example, the reaction of molecules with clusters or surfaces. A combined ab initio method for solving both the geometric and electronic problem simultaneously is the Car-Parrinello method, which is a DFT dynamics method [52]. This method uses a plane wave expansion for the density, and the inner ions are replaced by pseudo-potentials [53]. Today this method has been extensively used for studies of dynamic problems in solids, clusters, fullerenes etc [54-61]. We have recently in a co-operation project with Andreoni at IBM used this technique for studying the existence of different isomers of transition metal clusters [62,63]. 

The first term is calculated on the real-space grid defined by the plane wave expansion and the other two are efficiently and accurately calculated using atom centered meshes. 

Tliis theorem is, of course, valid only when i// is described exactly. Because plane waves describe all of space evenly and are not centered on atomic nuclei, Eq. (20) can be used to calculate forces accurately for approximate plane-wave expansions of lA. 1 or expansions of i/f in terms of atomic centered basis functions such as Gaussians and atomic centered grids, Eq. (19) must be used. This is because space is not covered uniformly by atomic centered basis functions, and thus, errors in i r are nonuniform, and the last two terms in Eq. (19) do not add to zero. The added expense of calculating forces via Eq. (19) often prohibits the use of atomic centered basis functions in molecular dynamics simulations. 

A combined method for solving both the geometric and electronic problem simultaneously is the Car-Parrinello method, which is a DFT dynamics method [152]. This method uses a plane wave expansion for the density, and the inner ions are replaced by pseudo-potentials. 

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