Plane waves use

The plane-wave basis has the advantage that it can be systematically improved by simply increasing the number of plane-waves used. The basis set is usually truncated, based on an energy cut off, Bcut. which defines the kinehc energy limit for the plane-waves. The corresponding maximum G vector is related to the energy cut off via. 

If we transform ef into a Taylor s series as a regular function, we can prove Eq. (1.6). This lecture on analytic functions went on like this The polar form of z with z = r and arg(z) = 0 is z = reie." Here we transform Eq. (1.5) by using the polar form. If we overlook the strictly critical study of the argument 6, we obtain the general formula of a plane wave, using the correspondence r = A and d=(kr- (at). In physics, the following equation is always used as the wave formula. This is done to take advantage of the ease with which complex exponentials can be manipulated. Only if we want to represent the actual wave must we take the real part into account. 

Plane waves used historically in the theory of the solid state, these functions are being used increasingly in molecular theories in conjunction with the density functional method discussed in Chapter 32. These functions are not dependent on the positions of the nuclei and offer considerable simplifications in gradient calculations. 

As can be seen from Table I the convergence was found to be relatively slow, and one has to use all conduction bands calculated. The number of conduction bands of course is determined by the number of plane waves used in the basis in which the electron wave functions are expanded. Using all calculated conduction bands means that the first order wave function is expanded in exactly the same number of plane waves as the zeroth order wave function. 

As can be seen from the table the convergence is relatively slow. Therefore, as a first approximation, an extrapolation is made based on the hypothesis that the phonon frequencies are inversely proportional to the number of plane waves used in the Hamiltonian. The results of this extrapolation based on the values obtained with 137 and 150 plane waves are also shown in Table III. 

To solve the Kohn-Sham equations with pseudopotentials, the standard approach is to expand the electron wavefunctions by a plane wave set in reciprocal space lattice vectors. The electron structure is obtained by diagonalization of the Hamiltonian matrix. This basis set has been mostly employed for semiconductor studies because of the relatively smooth pseudopotentials and delocalized electron wavefunctions of these systems. There are several advantages for using plane waves. The Hamiltonian matrix elements are simple to evaluate. Test of convergence in the basis expansion can be done by simply increasing the number of plane waves used. Moreover, the calculation of Hellmann-Feynman forces is the less involved in a plane wave basis. 

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. 

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 ... 

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). 

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

Figure B3.2.4. A schematic illustration of an energy-independent augmented plane wave basis fimction used in the LAPW method. The black sine fimction represents the plane wave, the localized oscillations represent the augmentation of the fimction inside the atomic spheres used for the solution of the Sclirodinger equation. The nuclei are represented by filled black circles. In the lower part of the picture, the crystal potential is sketched.

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].)... 

Figure 3. Floquet band structure for a threefold cyclic barrier (a) in the plane wave case after using Eq. (A.l 1) to fold the band onto the interval —I < and (b) in the presence of a threefold potential barrier. Open circles in case (b) mark the eigenvalues at = 0, 1, consistent with periodic boundary conditions. Closed circles mark those at consistent with sign-changing...

Plane waves are often considered the most obvious basis set to use for calculations on periodic sy stems, not least because this representation is equivalent to a Fourier series, which itself is the natural language of periodic fimctions. Each orbital wavefimction is expressed as a linear combination of plane waves which differ by reciprocal lattice vectors ... 

---