Plane Wave Pseudopotential Method

From the above brief overview one can conclude that the development of density functional theory and the demonstration of the tractability and accuracy of the local density approximation to it constitutes an important milestone in soUd-state physics and chemistry. 

We have considered the electron-electron interaction in the previous sections. Here we discuss the electron-ion interaction. 

The one-electron Schrodinger equation and consequently, the Kohn-Sham equations, still pose substantial calculational difficulties [17]. 

In the bonding region between the atoms the situation is opposite. The kinetic energy is small and the eigenfunction is smooth. However, the eigenfunction is flexible and responds strongly to the environment. This requires large and nearly complete basis sets. 

That is of the essence that the resulting pseudoeigenfunction V pseudo may be well represented by plane waves. The most general solution has to satisfy the Bloch theorem (Section 5.2) and boundary conditions. Each electronic wave function in a periodic crystal lattice can be written as the product of a cell-periodic part and a wave-like part. 

---

The local density approximation (LDA) and GGA within a plane-wave pseudopotential method was used in Ishibashi and Kohyama (2000) while DFT within the linearized augmented plane wave (LAPW) approach was employed in Sing et al. (2003b). 

Note A/B implies A grown or strained to B and vice versa. A B implies no growth direction or explicit strain dependence, i.e. natural. ) T = theoretical E = experimental AVL = average lattice XPS = X-ray photoelectron spectroscopy PL = photoluminescence CL = cathodoluminescence UPS = ultraviolet photoelectron spectroscopy LMTO = linear muffin tin orbital method LAPW = linearised augmented plane wave method PWP = plane wave pseudopotential method VCA = virtual crystal approximation. 

PWP = plane wave pseudopotential method Au p = Au Schottky barrier theory. 

The EDA analysis is performed with the geometry optimized by the plane-wave pseudopotential method. For binary hydrides, MH , the respective atomic energy densities of M and H are related closely to the nature of the chemical bond... 

In the LDA, Adolph and Bechstedt [157,158] adopted the approach of Aspnes [116] with a plane-wave-pseudopotential method to determine the dynamic x of the usual IB V semiconductors as well as of SiC polytypes. They emphasized (i) the difficulty to obtain converged Brillouin zone integration and (ii) the relatively good quality of the scissors operator for including quasiparticle effects (from a comparison with the GW approximation, which takes into account wave-vector- and band-dependent shifts). Another implementation of the SOS x —2 ffi, ffi) expressions at the independent-particle level was carried out by Raskheev et al. [159] by using the linearized muffin-tin orbital (LMTO) method in the atomic sphere approximation. They considered... 

The objective of this article is to expose the chemical engineering community to Car-Parrinello methods, what they have accomplished, and what their potential is for chemical engineering. Consistent with this objective, in Section IV, I give an overview of the most widely used quantum mechanical method for solving the many-body electronic problem, density-functional theory, but describe other methods only cursorily. I also describe the practical solution of the equations of density-functional theory for molecular and extended systems via the plane-wave pseudopotential method, mentioning other methods only cursorily. Finally, I end this section with a description of the Car-Parrinello method itself. 

By far the major computational quantum mechanical method used to compute the electronic state in Car-Parrinello simulations is density-functional theory (DFT) (Hohenberg and Kohn, 1964 Kohn and Sham, 1965 Parr and Yang, 1989). It is the method used originally by Roberto Car and Michele Parrinello in 1985, and it provides the highest level of accuracy for the computational cost. For these reasons, in this section the only computational quantum mechanical method discussed is DFT. Section A consists of a brief review of classical molecular dynamics methods. Following this is a description of DFT in general (Section B) and then a description of practical DFT computations of chemical systems using the plane-wave pseudopotential method (Section C). The section ends with a description of the Car-Parrinello method and some basic issues involved in its use (Section D). 

Gas-phase systems, if they are small enough, are convenient to study, because of the relatively small computational time. To study such systems using the plane-wave pseudopotential method, large supercells must be chosen to avoid spurious interactions from periodic potentials. Of course, if localized basis functions are chosen, such as Gaussians, there is no periodicity. 

The experimental methods have made it possible to study materials over large ranges of pressures, which can change the properties of materials completely. Figure 7.2 shows the dependence of energy on volume for silicon calculated using the plane wave pseudopotential method and the local density approximation (these methods are described in Chapter 8). The calculated E(V) curve fits with experiment... 

Figure 5 (a) Structures and (b) relative energies of FeOOH polymorphs calculated with parameterized model, plane-wave pseudopotential methods, and experiment (Laberty and Navrotsky, 1998). 

The development of plane-wave pseudopotential methods for electronic structure calculations of solids (e.g., Payne et al. 1992) has also opened the door to real first-principles molecular dynamics simulations using the algorithm of Car and Parinello (1985). Here, we let the wavefimctions become part of the dynamics of the system. To do this, we introduce a fictitious kinetic energy associated with a dynamical motion of the wavefunction ... 

---