Pseudo-plane-waves

The projector augmented-wave (PAW) DFT method was invented by Blochl to generalize both the pseudopotential and the LAPW DFT teclmiques [M]- PAW, however, provides all-electron one-particle wavefiinctions not accessible with the pseudopotential approach. The central idea of the PAW is to express the all-electron quantities in tenns of a pseudo-wavefiinction (easily expanded in plane waves) tenn that describes mterstitial contributions well, and one-centre corrections expanded in tenns of atom-centred fiinctions, that allow for the recovery of the all-electron quantities. The LAPW method is a special case of the PAW method and the pseudopotential fonnalism is obtained by an approximation. Comparisons of the PAW method to other all-electron methods show an accuracy similar to the FLAPW results and an efficiency comparable to plane wave pseudopotential calculations [, ]. PAW is also fonnulated to carry out DFT dynamics, where the forces on nuclei and wavefiinctions are calculated from the PAW wavefiinctions. (Another all-electron DFT molecular dynamics teclmique using a mixed-basis approach is applied in [84].)... 

For the sake of simplicity, we consider an example of a one-dimensional periodic system of length L with N atoms with one core electronic state per atom. The interatom space is a. The pseudo-valence electron is assumed to be in a single plane wave... 

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... 

The wave functions are expended in a plane wave basis set, and the effective potential of ions is described by ultrasoft pseudo potential. The generalized gradient approximation (GGA)-PW91, and local gradient-corrected exchange-correlation functional (LDA)-CAPZ are used for the exchange-correlation functional. 

Fig. 11.4. Velocities of bulk and surface waves in an (001) plane the angle of propagation in the plane is relative to a [100] direction, (a) Zirconia, anisotropy factor Aan = 0.36 (b) gallium arsenide, anisotropy factor Aan = 1.83 material constants taken from Table 11.3. Bulk polarizations L, longitudinal SV, shear vertical, polarized normal to the (001) plane SH, shear horizontal, polarized in the (001) plane. Surface modes R, Rayleigh, slower than any bulk wave in that propagation direction PS, pseudo-surface wave, faster than one polarization of bulk shear wave propagating in...

Figure 5.5 The figure shows the projection of a box normalized pseudo-continuum state onto rotated analytical continuum states with outgoing, plane wave asymptotic behavior, i.e., F0 in Eq. (31). The x-axis shows the energy of the analytical continuum states. The half width of the distribution increases with increasing 9. More specifically, the width of F0 m 2 coincides quite well with twice the absolute value of the imaginary part of the energy of the box normalized pseudo-continuum state, lm(En). This particular case corresponds to the i = 0 channel with the scaling angle 0 = 5° and a box state with Re(En) = 2.0 a.u.

The calculated and measured electron effective mass m c and its k-dependency for WZ and ZB GaN and AIN are summarised in TABLES 1 and 2, respectively. Suzuki et al derived them with a full-potential linearised augmented plane wave (FLAPW) band calculation [4,5], Miwa et al used a pseudopotential mixed basis approach to calculate them [6]. Kim et al [7] determined values for WZ nitrides by the full-potential linear muffin-tin orbital (FP-LMTO) method. Majewski et al [8] and Chow et al [9,10] used the norm-conserving pseudo-potential plane-wave (PPPW) method. Chen et al [11] also used the FLAPW method to determine values for WZ GaN, and Fan et al obtained values for ZB nitrides by their empirical pseudo-potential (EPP) calculation [12],... 

P PAE PD PDS PEC PL PLE PMBE PPC PPPW PR PV PWP PWPP pi-MODFET precipitate power added efficiency photodetector photothermal deflection spectroscopy photoelectrochemical photoluminescence photoluminescence excitation spectroscopy plasma-assisted molecular beam epitaxy persistent photoconductivity pseudo-potential plane-wave photoreflectance photovoltage plane-wave pseudo-potential plane-wave pseudo-potential piezoelectric modulation doped field effect transistor... 

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]. 

If not mentioned otherwise, all the calculations presented in the next sections use the original Car-Parrinello scheme based on (gradient-corrected) density functional theory in the framework of a pseudo potential approach and a basis set of plane waves. 

Much effort has been put to improve the pseudopotentials. [71,72] The most successful pseudopotential model is the so called ultra-soft pseudopotentials, proposed by Vanderbilt. [73] The model allows one to work with optimally smooth pseudopotentials. Thus the number of plane waves needed to express the pseudo-wavefunctions can be greatly reduced. In the model, pseudo-wavefunctions ipiif match the true orbitals outside a given core radius re, within Tc, are al-... 

Figure 4.7. Observed and curve-fitted Si spectra of three erystalline forms of the silica polymorph tridymite. A. Room-temperature ordered monoclinic form, showing resolution of nine of the twelve Si sites, fitted to 12 pseudo-Voight lines of equal area with a Gaussian Lorentzian ratio of 0.3. B. Orthorhombic form at 142°C, fitted to six lines broadened by an ineommensurate plane-wave modulation. C. Orthorhombic form at 202°C, fitted to a single line simulated with a non-linear incommensurate modulation. Adapted from Kitchin et al. (1996), by permission of the...

For a wide range of chemically interesting events, such as bond breaking and formation, an accurate description is required only for the valence electrons. Such an accurate description can be obtained using a pseudo potential description of the nuclei. This technique is well established in the plane wave community. We take advantage of the experience with this scheme using the pseudo potentials of Goedecker et al. (GTH) [12,13]. These accurate and transferable pseudo potentials have an analytic form that allows for an efficient treatment of all terms within the GPW method. 

Similar performance tests have been carried out using the GAPW method for both the pseudo potential and the all electrons implementation. For these methods, the TZV2P and the TZVP basis sets have been employed respectively. The smaller basis set used for the all electron calculations is due to memory constraints. We ran single point energy calculations on an IBM Regatta node (32 CPUs) with a plane wave cutoff of 200 Ry and the BLYP functional. As expected, also in this case by increasing the size of the box from 32 up to 1024 (512 for the all electron case) water molecules, the cpu... 

---