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md2011-08-note_Part_3

md2011-08-note_Part_3 - As mentioned during the last...

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I As mentioned during the last lecture, for d transition metals (e.g, Pt), the cohesive energy is dominated by the d -band contribution I Therefore, a rectangular N i can be assumed with a width W i so that the N i per atom for a full d -band becomes 10 / W I Then, the bond energy per site i can be written as V i B - 1 20 W i N d ( 10 - N d ) , (13) where N d is the number of electrons in the d -band I W is related to the second moment of the N i via μ 2 i = Z - E 2 N i ( E ) d E = 10 12 W 2 i (14) I On the other hand, the second moment can be given directly as a sum the two-center hopping integrals h ij , which depend on the next neighbor distance r ij : μ 2 i = 10 X j 6 = i h 2 ( r ij ) (15) I Combining these, we get 1 12 W 2 i = X j 6 = i h 2 ( r ij ) (16) I Applying this to the total energy expression, we get U = 1 2 X i X j 6 = i φ ( r ij ) - D v u u u u t X j 6 = i h 2 ( r ij ) | {z } ρ i (17)
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I Indentifying the sum of the hopping integrals as the resulting electron density ρ i leads to the Rosato group potentials I Defining an embedding function F ( ρ ) = D ρ
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