Plane Waves and Pseudopotentials

The most common form of AIMD simulation employs DFT (see section First Principles Electronic Structure Methods ) to calculate atomic forces, in conjimction with periodic boundary conditions and a plane wave basis set. Using a plane wave basis has two major advantages over atom-centered basis functions (1) there is no basis set superposition error (Boys and Bernardi 1970 Marx and Hutter 2000) and (2) the Pulay correction (Pulay 1969,1987) to the HeUmann-Feynman force, due to basis set incompleteness, vanishes (Marx and Hutter 2000, 2009). 

As a consequence of Bloch s theorem, in a periodic lattice, the Kohn-Sham orbitals (see O Eq. 7.59) can be expanded in a set of plane waves (Ashcroft and Mermin 1976 Meyer 2006), 

Ni being integer numbers, and ai, a2, as the vectors defining the periodically repeated simulation box. 

Molecular Dynamics Simulation From Ab Initio to Coarse Grained  

In O Eq. 7.67, the summation is over aU reciprocal lattice vectors G which fulfill the condition G T = 27tM, M being an integer number. In practice, this plane-wave expansion of the Kohn-Sham orbitals is truncated such that the individual terms all yield kinetic energies lower than a specified cutoff value, Ecut, 

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X. Gonze, "First-principle responses of solids to atomic displacements and homogeneous electric fields a conjugate-gradient algorithm, used with plane waves and pseudopotentials," Phys. Rev. B 55 (1997), 10337-10354. 

One interesting scheme based on density functional theory (DFT) is particularly appealing, because with the current power of the available computational facilities it enables the study of reasonably extended systems. DFT has been applied with a variety of basis sets (atomic orbitals or plane-waves) and potential formulations (all-electron or pseudopotentials) to complex nu-cleobase assemblies, including model systems [90-92] and realistic structures [58, 93-95]. DFT [96-98] is in principle an ab initio approach, as well as MP2//HF. However, its implementation in manageable software requires some approximations. The most drastic of all the approximations concerns the exchange-correlation (xc) contribution to the total DFT functional. 

In the present calculation, we used two ab initio methods to investigate structural stability and the potential for carrier generation of X B6 and X Bi2 clusters in c-Si. Here X is from H to Br in the periodic table. The following two methodsused were (i) plane wave ultrasoft pseudopotential method for the optimization of atomic structures and (ii) discrete variational-Xa (DV-Xa) molecular orbital method for the analysis of the fine electronic structures and activation energies of the clusters. 

Alternatively, standard electron band calculations like augmented plane waves, linearized augmented plane waves, or pseudopotential methods can be employed. Pseudopotential methods using plane waves and a smooth crystal potential are quick and sufficiently... 

VASP uses the Density Functional Theory method on periodical systems, with plane waves and ultrasoft pseudopotentials (US-PP)[4, 5]. The functional from the Generalized Gradient Approximation (GGA) of Perdew and Wang[6] has been chosen because of its good description of chemical bond energies. 

The PW basis set is universal, in the sense that it does not depend on the positions of the atoms in the unit cell, nor on their nature [458]. One does not have to construct a new basis set for every atom in the periodic table nor modify them in different materials, as is the case with locahzed atomic-hke functions and the basis can be made better (and more expensive) or worse (and cheaper) by varying a single parameter -the number of plane waves defined by the cutoff energy value. This characteristic is particularly valuable in the molecular-dynamics calculations, where nuclear positions are constantly changing. It is relatively easy to compute forces on atoms. Finally, plane-wave calculations do not suffer from the basis-set superposition error (BSSE) considered later. In practice, one must use a finite set of plane waves, and this in fact means that well-localized core electrons cannot be described in this manner. One must either augment the basis set with additional functions (as in linear combination of augmented plane waves scheme), or use pseudopotentials to describe the core states. Both AS and PW methods, developed in solid-state physics are used to solve Kohn-Sham equations. We refer the reader to recently published books for the detailed description of these methods [9-11]. 

Within DFT quantum mechanics, first-principles GPT provides a fundamental basis for ab initio interatomic potentials in metals and alloys. In the GPT apphed to transition metals [49], a mixed basis of plane waves and localized d-state orbitals is used to self-consistently expand the electron density and total energy of the system in terms of weak sp pseudopotential, d-d tight-binding, and sp-d hybridization matrix elements, which in turn are all directly calculable from first principles. For a bulk transition metal, one obtains the real-space total-energy functional... 

The recent study of CoSi2 by Stadler et al. illustrates the current capabilities to calculate the cohesive, elastic, and dynamical properties of a material by first-principles pseudopotential and all-electron techniques based on DFT. On the LDA level, the lattice constant is slightly smaller than experiment, namely 5.292 A (FLAPW), 5.283 A (pseudopotential plane wave) and 5.365 A (experiment). On the GGA level, the calculated lattice constant is closer to experiment, namely 5.350 A. The difference in the calculated bulk modulus between LDA (168.4 GPa) and GGA (169.0 GPa) is smaller than that between the LDA FLAPW (171.6 GPa) and LDA pseudopotential plane wave (168.4 GPa) calculations. All results are in excellent agreement with the experimental value of 171.5 3.4-GPa. 

J.H. Cho, K. Terakura, Plane-wave-basis pseudopotential calculations of the surface relaxations of Ti(OOOl) and Zr(OOOl). Phys. Rev. B 56(15), 9282-9285 (1997)... 

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 ... 

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].)... 

Pulci O, Onida G, Shkrebtii A I, Del Sole R and Adolph B 1997 Plane-wave pseudopotential calculation of the optical properties of GaAs Phys. Rev. B 55 6685... 

In practice, therefore, a pseudopotential is invariably employed and only plane waves with a kinetic energy (= /2m) k -t Gp) less than some cutoff are included in the calculation. The... 

The pseudopotential density-functional technique is used to calculate total energies, forces on atoms and stress tensors as described in Ref. 13 and implemented in the computer code CASTEP. CASTEP uses a plane-wave basis set to expand wave-functions and a preconditioned conjugate gradient scheme to solve the density-functional theory (DFT) equations iteratively. Brillouin zone integration is carried out via the special points scheme by Monkhorst and Pack. The nonlocal pseudopotentials in Kleynman-Bylander form were optimized in order to achieve the best convergence with respect to the basis set size. 5... 

The Car-Parrinello simulations were performed using the MOTECC-90 computer code [13]. All considered systems consist of 64 atoms in a cubic unit cell with a length of 23.4 a.u. and periodic boundary conditions. The plane-wave cut-off was chosen to be 6 Ryd. The atomic cores were described by the pseudopotentials of Bachelet et al. [14]. 

In molecular DFT calculations, it is natural to include all electrons in the calculations and hence no further subtleties than the ones described arise in the calculation of the isomer shift. However, there are situations where other approaches are advantageous. The most prominent situation is met in the case of solids. Here, it is difficult to capture the effects of an infinite system with a finite size cluster model and one should resort to dedicated solid state techniques. It appears that very efficient solid state DFT implementations are possible on the basis of plane wave basis sets. However, it is difficult to describe the core region with plane wave basis sets. Hence, the core electrons need to be replaced by pseudopotentials, which precludes a direct calculation of the electron density at the Mossbauer absorber atom. However, there are workarounds and the subtleties involved in this subject are discussed in a complementary chapter by Blaha (see CD-ROM, Part HI). 

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