The Augmented Spherical Wave Method by Volker Eyert

Author:Volker Eyert
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

(4.6.22)

Here, as before, c and M(m) denote integration over the unit cell and the muffin-tin sphere centered at site τ m , respectively. Note that we have suppressed the spin index since spin-flip transitions were ruled out. Like (4.1.12) and (4.2.16) for the products of basis functions and the Hamiltonian matrix, respectively, (4.6.22) contains the decomposition into smooth, local, and cross terms. As already outlined in the context of the construction of the Hamiltonian matrix the cross terms, i.e. the last two terms in the square brackets, vanish due to the orthonormality of the spherical harmonics.

In evaluating the remaining terms we start with the first term on the right-hand side of (4.6.22), namely, the integral over the unit cell, which has to be built with the pseudo functions. However, in the same manner as in Sect. 4.2 for the secular matrix, we formally replace the pseudo functions by the original envelope functions. Yet, since the envelope and the pseudo functions differ only inside the on-center sphere and the contributions from inside this sphere will be compensated by the second term in the square brackets of (4.6.22), the final result will be the same. Next, defining in complete analogy to (2.2.10)

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