.. _basis_sets:
===================
Basis sets
===================
.. _LAPW:
LAPW basis
==============
The reader is supposed to be familiar with the LAPW basis set. This section is intended as a short recapitulation and to define the notation used in the manual.
The LAPW method relies on a partitioning of space into non-overlapping MT spheres centered at the atomic nuclei and the interstitial region.
The basis functions are defined differently in the two regions, plane waves in the interstitial and numerical functions in the spheres,
consisting of a radial part :math:`{u_{lp}^a(r)}` (:math:`{a}` is the atomic index) and the spherical harmonics :math:`{Y_{lm}(\hat{\mathbf{r}})}`.
:math:`{p}` is an index to distinguish different types of radial functions: :math:`{u_{l0}^a=u_l^a}`,
:math:`{\,u_{l1}^a=\dot{u}_l^a=\partial u_l^a/\partial\epsilon}`, :math:`{\,u_{lp}^a=u_l^{a,\mathrm{LO}},\,p\ge 2}`,
where we have used the usual notation of LAPW. Augmented plane waves are formed by matching linear combinations of the
:math:`{u}` and :math:`{\dot{u}}` to the interstitial plane waves, forming the standard LAPW basis set.
Local orbitals :math:`{u^\mathrm{LO}}` can be used to extend the basis set, to enable the description of semicore and high-lying conduction states.
The plane waves and the angular part of the MT functions can be converged straightforwardly
with the reciprocal cutoff radius :math:`{g_\mathrm{max}}` and the maximal l quantum number :math:`{l_\mathrm{max}}`, respectively,
whereas the radial part of the MT functions is more difficult to improve systematically, but it is possible, see [Comput. Phys. Commun. 184, 2670 (2013)].
Running the one-shot mean-field calculation with Spex (using ``ITERATE``, see :numref:`ITERATE`) offers the possibility of
modifying the LAPW basis in "spex.inp". This can be useful for *GW* calculations, which require an accurate basis for a much larger
energy range (including many unoccupied states) than in DFT.
The respective keywords are defined in the section ``LAPW`` and are described in the following.
.. _GCUT_LAPW:
GCUT (LAPW)
-----------
The set of **G** vectors depends on the k point and is defined with the cutoff condition :math:`|\mathbf{k}+\mathbf{G}|