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where ul(r,El) is the (at the origin) regular solution of the radial Schroedinger equation for energy El (chosen normally at the center of the corresponding band with l-like character) and the spherical part of the potential inside sphere t $\dot u_l(r,E_l)$ is the energy derivative of ul taken at the same energy El. A linear combination of these two functions constitute the linearization of the radial function; the coefficients Alm and Blm are functions of kn (see below) determined by requiring that this basis function matches (in value and slope) the corresponding basis function of the interstitial region; $u_l$ and $\dot u_l$ are obtained by numerical integration of the radial Schroedinger equation on a radial mesh inside the sphere.
As shown by Madsen et al. (2001) this new scheme converges practically to identical results as the LAPW method, but allows to reduce ``RKmax'' by about one, leading to significantly smaller basis sets (up to 50 %) and thus the corresponding the computational time is drastically reduced (up to an order of magnitude). Within one calculation a mixed ``LAPW and APW+lo'' basis can be used for different atoms and even different l-values for the same atom (Madsen et al. 2001). In general one describes by APW+lo those orbitals which converge most slowly with the number of PWs (such as TM 3d states) or the atoms with a small sphere size, but the rest with ordinary LAPWs. One can also add a second ``lo'' at a different energy so that both, semicore and valence states, can be described simultaneously.
where kn=k+Kn; Kn are the reciprocal lattice vectors and k is the wave vector inside the first Brillouin zone. Each plane wave is augmented by an atomic-like function in every atomic sphere.
A very recent extensive overview is given in:
WIEN2k: An APW+lo program for calculating the properties of solids.
P. Blaha, K.Schwarz, F. Tran, R. Laskowski, G.K.H. Madsen and L.D. Marks,
J. Chem. Phys. 152, 074101 (2020)
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