md2011-06-note_Part_11

md2011-06-note_Part_11 - and the independent-particle...

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and the independent-particle kinetic energy T S is T S = - 1 2 X σ X i ± ψ σ i ² ² 2 ² ² ψ σ i ³ = 1 2 X σ X i Z d 3 r | ψ σ i ( r ) | 2 (41) I The classical Coulomb interaction energy density is defined as an electron density interacting with itself E KS = T S [ n ] + Z d r V ext ( r ) n ( r ) + E Hartree [ n ] + E II + E xc [ n ] (42) in which the terms involving V ext , E Hartree (see eq. 34) and E II form a well-defined neutral grouping. ( E II contains interactions between the nuclei and other terms independent on the electrons.) Notes
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I We can now compare this to the expression of total E HK [ n ] = T [ n ] + E int + R d 3 rV ext ( r ) n ( r ) + E II F HK [ n ] + R d 3 rV ext ( r ) n ( r ) + E II (43) I Clearly E xc [ n ] = h K i - T S [ n ] + h U int i - E Hartree [ n ] . (44) I Hence, the E xc [ n ] is just the difference of the kinetic and the internal interaction energies of the true system from those of the fictitious independent-particle system with
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md2011-06-note_Part_11 - and the independent-particle...

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