By Hugues Dreysse
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Extra resources for Electronic Structure and Physical Properties of Solids: The Uses of the LMTO Method
Next, choose the physically and chemically motivated screening (β) and rescreen the Green matrix to the downfolded representation, Gβ (ε) or Gb (ε) , using the scaling relations (53) or (55) derived below. As will be explained in the following Sect. 3, this should be done for a number of energies. In addition, one will need the ﬁrst energy derivatives G˙ b (ε) . The latter may be obtained from K˙ a (ε) via numerical diﬀerentiation of the weakly energy dependent structure matrix, B a (ε) , and s s 2 2 a (ε, r) r2 dr for the energy derivative of calculation of 0 ϕa (ε, r) r2 dr − a ϕ◦RL the logarithmic derivative function in (51), as will be shown in (61)-(63) below.
Hence, we have found the following simple and practical scaling relation for re-screening of the Green matrix: Gb (ε) = ϕ◦ a (ε, b) Ga (ε) ϕ◦ a (ε, b) + j a (ε, b) ϕ◦ a (ε, b) . 5 (55) Green Functions, Matrix Elements, and Charge Density The kinked partial wave is the solution of the inhomogeneous Schr¨ odinger equation: (H − ε) φaR L (ε, r) = − a δ (rR − aRL ) YL (ˆ rR ) KRL,R L (ε) , (56) RL provided that we deﬁne the MTO (36) the 3-fold way indicated in Figs. 2 – 4, and therefore – for the MT-Hamiltonian H (4) – use the radial Schr¨ odinger equation (2) channel-wise.
But this solution is useless, because it yields: χ(N ) (r) = 0. K. Andersen et al. we can write down the corresponding expression for the set χ(N ) (r) without (N ) 2 explicitly solving for the (N + 1) matrices An (εm ) , and then prove afterwards that each basis function has its triple-valuedness reduced consistently with the remaining error ∝ (εi − ε0 ) (εi − ε1 ) ... (εi − εN ) of the set. Since we want χ(N ) (εn , r) to be independent of n for 0 ≤ n ≤ N, all its divided diﬀerences on the mesh – up to and including the divided diﬀerence of order N – vanish, with the exception of the 0th divided diﬀerence, which is χ(N ) (r).