 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,
φ_{i},
can be described as
a superposition
(linear combination)
of atomic orbitals
χ,
as:
φ_{i}=c_{1}χ_{1}+c_{2}
χ_{2}+...+c_{n}χ_{n}.
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,
c_{i},
can be determined to
minimize the
total energy of the
system. Common and
reliable
representatives of
atomic orbitals are
Slatertype and
Gaussiantype
orbitals. Depending
on the accuracy in
need, one may
also combine two or
more functions to
represent individual
atomic orbital. In
contrast to
such atomcentered
orbitals, the choice
of basis functions
within solidstate
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
basisset
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
planewave basis
functions
and lead to enormous
computational costs.
Therefore, some
modern solidstate
ab
initio
codes
combine delocalized
planewave 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
tightbinding
approaches) or
modify the core
functions by using
pseudopotentials [2]
or projector
augmeted waves (PAW)
techniques.

Picture / Figure / Diagram: 

Fig. 1: Schematic of a wave function of a crystal, which displays the different behaviors of inside and outside of atomiclike regions.


SFBLink:  The most common ab initio code used in SFB 761 is the VASP code from Vienna University. It is based on planewave 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. 

