Computational Methods and Algorithms

Density functional theory (DFT) has proven to be an efficient tool to estimate the material properties with acceptable accuracy and thus, revolutionized the research in computational materials science. The efficiency and accuracy of DFT in exploring materials’properties depend on the basis sets used in solving the Kohn-Sham equation. Some well-known basis sets that are generally suggested for solving Kohn-Sham equation include (i) Plane waves (ii) Augmented plane wave (APW), (iii) Linearized augmented-plane-wave (LAPW), (iv) LAPW + Local Orbitals (LOs), (v) APW + local orbitals (los), (vi) Muffin-Tin Orbitals (MTO), and (vii) Linear Muffin-Tin Orbital (LMTO), etc. Each of them has their own pros and cons. Presently, the popular and efficiently employed basis set out of these are APW+lo (in DFT packages like WIEN2k and Elk), LMTO (in packages like LmtART, RSPT, and Questal), and plane waves basis set (in Abinit and Quantum Expresso). Various DFT packages use these basis sets to self-consistently solve the Kohn-Sham equation to obtain the ground state charge density, eigenstates, and eigenvalues. While solving the Kohn-Sham equation, usually the core states are not considered as they have negligible effect on most of the transport and related properties. For considering the valence states, the regions around an atom in solid is divided into two parts (I) spherical muffin-tin ( \(R_{MT}\) ) part around the nucleus of atoms where the states have steep variation and (II) interstitial regions between the neighboring spheres where there is relatively smoother variation of states of atoms forming the solid. Different types of functions are used to solve the Kohn-Sham equation in these regions. Based on this approach basis sets like APW, LAPW, and LMTO are designed. These basis sets intrinsically considers the spherical symmetric approximation of potential in the \(R_{MT}\) regions. However, these basis sets are sometimes modified to removes the spherical symmetry approximation, thus making the method more accurate [1]. This improved method is referred as full-potential method. There is yet another improvement in DFT implementation, where not only the valence states/electrons are treated self-consistently but also the core states/electrons. This approach of additional self-consistent treatment of DFT is the most accurate approach which is popularly referred as all-electrons method. Apart from these methods, there is another popular approach called Pseudopotentials method. This uses the fact that states in the regions closer to the nucleus are almost shielded from the outer portions, and act like the one in free atoms. Thus, they don’t affect the regions of atoms in solids where bonding happens. This encourages to approximate the core regions by a smooth varying functions called as Pseudopotentials. Thus, the method does not explicitly treat the regions closer to the nucleus while solving the Kohn-Sham equation. This increases its computational efficiency while compromising its overall accuracy to some extent when compared with full potential approach. Present chapter will deal with these aspects along with the method to obtain converged ground state using DFT approach. The chapter will also shed light on aspects that must be taken care of while setting up any DFT calculations.