Microscopic Modelling of Correlated Low-dimensional Systems

Chapter 2: Method 21
The simplest version of MTO equations results when replacing the muffin tin spheres by
space filling spheres, this is justified by the fact that if there is only a short distance between
the spheres, as in a close-packed solid, the wavefunction will be nearly correct because it
has the correct value and slope at the sphere boundary. This is associated with the atomic
sphere approximation (ASA), in which the Wigner-Seitz sphere around each atom is replaced
by a sphere as shown schematically in Fig. 2.2. It is evident that for closed-packed cases
the distances between spheres are indeed short.
Figure 2.2: Atomic Sphere Approximation (ASA) in which the muffin tin spheres are chosen
to have the same volume as the Wigner-Seitz cell, which leads to overlapping spheres [77].
Within the ASA approximation, the LMTOs are written as
where,
||χ α RL(E)〉[N α RL] −1 = |ψ(Eν)〉 − �
h α R ′ L ′| ˙ ψ(E)〉 (2.33)
R ′ L ′
h α = −( ˙
P α ) −1/2 [P α − S α ]( ˙
P α ) −1/2
(2.34)
Eq. (2.33) expresses the tail of an LMTO (function outside a sphere) extending into another
sphere in terms of functions centered on that sphere. Defining a function ϕ, |φRL(Eν)〉 =
N(E)N −1 |ϕ(E)〉, so that |φ〉 = |ϕ〉 and | ˙ φ〉 = ˙ φ + o|φ〉, this gives
||χ〉 = (I + ho)φ + h ˙ φ (2.35)
where I is the identity matrix. Finally, by orthogonalizing the LMTOs ||˜χ〉 = (I +ho) −1 ||χ〉,
gives the Hamiltonian form (neglecting few small terms)
H = Eν + h(I + ho) −1 = Eν + h − hoh − ... (2.36)