Plane waves approaches

There are several advantages associated with the plane wave approach. For example ... 

Dipole scattering does not require an atomistic theory. A phenomenological theory suffices, which includes a response function dependent on dielectric constants. The cross-section for dipole scattering based on these assumptions is given in Eqs, 3.7 and 3.9 of Ibach and Mills./61/ These formulae include plane-wave reflection coefficients from the surface, which are solutions of the standard LEED problem. Since dipole scattering involves essentially only forward scattering, it is not necessary in practice to adopt the spherical-wave picture of our step 2 (cf. section 3.4.3), the plane-wave approach is adequate in this situation. 

The electronic structure of solids and surfaces is usually described in terms of band structure. To this end, a unit cell containing a given number of atoms is periodically repeated in three dimensions to account for the infinite nature of the crystalline solid, and the Schrodinger equation is solved for the atoms in the unit cell subject to periodic boundary conditions [40]. This approach can also be extended to the study of adsorbates on surfaces or of bulk defects by means of the supercell approach in which an artificial periodic structure is created where the adsorbate is translationally reproduced in correspondence to a given superlattice of the host. This procedure allows the use of efficient computer programs designed for the treatment of periodic systems and has indeed been followed by several authors to study defects using either density functional theory (DFT) and plane waves approaches [41 3] or Hartree-Fock-based (HF) methods with localized atomic orbitals [44,45]. 

J. Vandevondele, M. Krack, F. Mohamed, M. Parrinello, T. Chassaing, and J. Flutter (2005) Quickstep Fast and accurate density functional calculations using a mixed Gaussian and plane waves approach. Comp. Phys. Comm. 167, p. 103... 

Fig. 1. The convergence of the total energy (BLYP) of a single water molecule with respect to the density cutoff using a traditional plane wave approach (CPMD, dashed lines), the GPW method using Fourier space derivatives (solid lines) and spline derivatives (dash-dotted lines). An aug-TZV2P Gaussian basis has been employed for the GPW calculations. Note that the cutoff reported for CPMD calculations is usually the wavefunction cutoff, which is defined as one fourth of the density cutoff employed here...

The eqnations of motions 2.16 are solved by a plane wave approach which leads to a 2 x 2 determinant for P-waves... 

The disadvantage of the pseudopotential (plane-wave) approach can be summarized as follows. First, all information for the core-like wavefunctions and their associated electron density is lost, trivially so. Second, an essentially delocalized plane-wave basis set poses difficulties when it comes to questions of chemical interpretation in terms of atoms and bonds. In addition, there has always been some concern about the arbitrariness of pseudopotentials because there is certainly more than one pseudopotential (in fact, an infinite number) for a given atom. Recall that the partitioning of a quantum-mechanical system into subsystems is invalid because the electrons cannot be distinguished in principle, there are no "core" or "valence" electrons. Nonetheless, pseudopotential plane-wave calculations have established themselves as accurate and powerful tools for electronic-structure theory, both for molecules and, in... 

Density Functional approach (energy a function of the electron density) including electron correlation approjdmations Local basis sets implementation possible Plane wave approach also adopted Pseudopotentials applied to core electrons LDA or GGA approximations to account for electron correlation effects SIESTA DMOL CASTEP VASP " DGAUSS 5... 

Two mechanisms have been proposed for this reaction. The first includes a step in which the first methanol is protonated and dehydrated to give a reactive methoxy group attached to the framework, whereas the second involves two methanol molecules undergoing an Sn2 reaction inside the zeolite cage, catalysed by protonation of one of the methanols and with the transition state solvated by the zeolite cage. The conclusion from QM studies (using both cluster and plane wave approaches) is that the reaction proceeds through the latter pathway, in which the zeolite stabilises the transition state. 

We focus on a different approach based on real space methods, which are basis free. Real space methods have gained ground in recent years [10-13] owing in great part to their great simplicity and ease of implementation. In particular, these methods are readily implemented in parallel computing environments. A second advantage is that, in contrast with a plane wave approach, real space methods do not impose artificial periodicity in non-periodic systems. While plane wave basis... 

A first principle methodology was further rrsed in order to analyze the natrrre of irrtrirrsic acidity of Bronsted acid sites [lOSl]. Periodic models of morderrite with Si/Al = 5 were frrlly optitrrized by rising a plane-wave approach. It has been found that a property related to the electric field at the proton site correlates with intrinsic acidity. The parameter was the gradient of the electrostatic potential at the component of the electric field at the proton site in the direction of the OH bond. For the electronic stracture of mordenite, see [8811]. 

The results of HF, MP2, LSDA, and gradient-corrected DFT calculations are presented in Table 3. The DFT calculations were performed within a plane wave approach by Laasonen, Csajka, and Parrinello. i The PZ parameteriza-tion was used within the LSDA calculations, whereas the gradient-correaed DFT calculations used Becke s exchange functionaP and Perdew s correlation functional. As seen in Table 3, the LSDA does a very poor job on the water dimer. The distance is 0.26 A shorter than the experimental value... 

As we noted above, in an FD or FE solution, the exact result emerges as the grid spacing is reduced in that respect we can call these methods fully numerical in the same spirit as the plane-wave approach from physics. An added advantage... 

This book contains a detailed account of the augmented plane wave approach and the plane wave approach and their applications to the electronic structure of solids. 

Meyer, B. (2006). The pseudopotential plane wave approach. In J. Grotendorst, S. Bliigel, 8c D. Marx (Eds.), Computational nanoscience Do it yourself jUlich NIC. http //www2.fz-juelich.de/ nic-series/volume31/meyerl.pdf. Accessed 02 July 2011. 

Nuclear magnetic resonance (NMR) is one of the major experimental tools in structural chemistry and biochemistry. The prediction of NMR shifts from ab initio calculations has been demonstrated for isolated molecules (see NMR Chemical Shift Computation Ab Initio), but the development of a practical ab initio approach for the calculation on NMR shifts in solids has been accomplished only quite recently. Based on DFT-LDA and a pseudopotential plane wave approach, these authors have presented an approach which promises to be useful in the investigation of NMR shifts in crystalline solids as well as in amorphous materials and liquids. As a demonstration of this approach, Mauri et al. have calculated the H NMR shifts of LiH and HF in the state of isolated molecules and in a crystal. In the case of LiH the results show very little change between the free molecule (a = 26.6 ppm) and the crystal (cr = 26.3 ppm). However, a significant change is found for the crystal at high pressures (65 GPa), where the chemical shift increases to 31.2 ppm. A quite different picture is obtained for the HF molecule, where the theory predicts a shift of 28.4 ppm in remarkable agreement with the experimental value of 28.4 ppm. For the HF crystal, a shift of... 

The present method focuses on solving Kohn Sham equations and calculating electronic states, total energy and material properties of non-crystalline, nonperiodic structures. Contrary to the variety of well established k-space methods that are based on Bloch s theorem and applicable to periodic structures, we don t assume periodicity in any respect. Precise ab-initio environment-reflecting pseudopotentials proven within the plane wave approach are connected with real space finite-element basis in the present approach. The main expected asset of the present approach is the combination of efficiency and high precision of ab-initio pseudopotentials with universal applicability, universal basis and excellent convergence control of finite-element method not restricted to periodic environment. 

---