Basis set
Definition:The entire mathematical functions used for solving the Schrödinger’s equation and constructing/representing the wave function (or density) of an atomistic system.
Explanation:There are several approaches to choose proper basis functions that are suitable for approximating the wave function of a molecule or crystal. For example, the linear combination atomic orbital (LCAO) method assumes that any molecular orbital, φi, can be described as a superposition (linear combination) of atomic orbitals χ, as:
φi=c1χ1+c2 χ2+...+cnχn.

Using the variational principle and a reliable Hamiltonian operator (such as the one in the Hartree‒Fock method or in density functional theory (→DFT)), the coefficients, ci, can be determined to minimize the total energy of the system. Common and reliable representatives of atomic orbitals are Slater-type and Gaussian-type orbitals. Depending on the accuracy in need, one may also combine two or more functions to represent individual atomic orbital. In contrast to such atom-centered orbitals, the choice of basis functions within solid-state materials is inspired by Bloch’s theorem, where plane wave functions (e.g., sine and cosine functions) are introduced. Since these functions intrinsically reflect the translational symmetry, they have enormous mathematical advantages; they do not suffer from basis-set superposition errors and properly describe the interatomic regions. However, they are difficult (if not impossible) to interpret in a chemical point of view. In addition, the high kinetic energy in the proximity of the atomic cores would need an extremely large number of plane-wave basis functions and lead to enormous computational costs. Therefore, some modern solid-state ab initio codes combine delocalized plane-wave basis sets in the bonding region with localized basis sets in the core region (Fig. 1). There are other and simpler approaches, which either ignore the core functions completely (empirical tight-binding approaches) or modify the core functions by using pseudopotentials [2] or projector augmeted waves (PAW) techniques.
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Fig. 1: Schematic of a wave function of a crystal, which displays the different behaviors of inside and outside of atomic-like regions.
SFB-Link:The most common ab initio code used in SFB 761 is the VASP code from Vienna University. It is based on plane-wave basis sets combined with either pseudopotentials or PAW to deal with the core region.
References:[1] F. Bloch, Z. Phys. 1928, 52, 555.
[2] H. Hellmann, J. Chem. Phys. 1935, 3, 61; Acta Physicochim. URSS 1934, 1, 913.