Microscopic Modelling of Correlated Low-dimensional Systems

Chapter 2: Method 19
derivative is discontinuous. Using this approach means that per atom 2ℓ + 1 orbitals are
added to the basis set. This makes the basis considerably larger than the LAPW basis set.
This is compensated by the fact that a lower R min
α Kmax is needed for accurate results 6 . A
mixed LAPW/APW+lo mixed basis set can be used for different atoms and even different
ℓ-values for the same atom. In general, APW+lo is used for describing those orbitals which
converge most slowly with the number of plane waves (like d- or f-orbitals) or the atoms
with a small sphere size, and the rest with ordinary LAPW’s.
MTO and LMTO
Muffin tin orbitals (MTO) form a basis of localized augmented orbitals introduced by An-
dersen [2] in 1971. It describes the electronic states in a small number of meaningful orbitals
that can be accurate because the MTOs are generated from the Kohn-Sham Hamiltonian
itself. An MTO can be understood in terms of a single atomic sphere with a flat potential
in all space outside the sphere. It is defined to be a localized basis function continuous in
value and derivative at the sphere boundary. The MTO basis functions are given by
||χ α RL(E)〉 = |K 0 RL〉 + |χ α RL(E)〉 i
where |K 0 RL 〉 is defined inside the sphere and |χα RL (E)〉i in the interstitial region.
|K 0 RL〉 = |ψRL(E)〉N α RL(E) + �
R ′ L ′
(2.26)
|J α R ′ L ′〉[P α R ′ L ′(E)δR ′ L ′ ,RL − S α R ′ L ′ ,RL ] (2.27)
where |ψRL(E)〉 are the solutions of the radial Schrödinger equation inside a muffin tin
sphere, for certain energy E and angular momentum ℓ. |K0 RL 〉 fulfill the conditions of
matching continuously and smoothly the angular momentum components at the surface of
the sphere. SR ′ L ′ ,RL are the structure factors which depend only on the lattice structure and
are characterized by an energy κ 2 (E − V0). For κ = 0, the wavefunction in the interstitial
region, is equivalent to the electrostatic potential due to a multi-pole moment and therefore
satisfies the Laplace equation. The N(E) and P (E) are the normalization function and
the potential function respectively, which are chosen in such a way that make the join
smooth. They are defined in terms of the logarithmic derivative of the spherical Bessel
J and Neumann functions K. α is the screening constant, which is introduced due to
6 the reduction of RKmax leads to smaller basis sets and thus the computational time is drastically reduced,
all this with the same efficiency as with only LAPW basis set