Density Functional Theory and the Local Density Approximation
Density Functional Theory and the Local Density Approximation
Density Functional Theory and the Local Density Approximation
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The Kohn-Sham Equations<br />
Vary <strong>the</strong> energy with respect to <strong>the</strong> orbitals <strong>and</strong> ….<br />
( 1 n(<br />
r')<br />
%<br />
&) * 2 + V ( r)<br />
r ' ( r ) ( r ) ( r<br />
ext<br />
+ d V xc #! i<br />
= "<br />
i!<br />
i<br />
)<br />
' 2<br />
+ +<br />
r ) r'<br />
$<br />
where:<br />
V<br />
xc<br />
( r)<br />
=<br />
! Exc[<br />
n]<br />
! n(<br />
r)<br />
KS equations are solved via an iterative procedure until<br />
self-consistency is reached<br />
No approximations, So…<br />
If we knew E xc [n] we could solve for <strong>the</strong> exact ground state<br />
energy <strong>and</strong> density !<br />
Kohn & Sham, PRA 140, 1133 (1965)<br />
The Non-interacting System<br />
There exists an effective mean field potential which, when<br />
applied to a system of non-interacting fermions, will generate<br />
<strong>the</strong> exact ground state energy <strong>and</strong> charge density !!!<br />
1<br />
| r " r |<br />
V xc ( r)<br />
i j<br />
! ( r)<br />
!(<br />
r 1 , r 2 ,...)<br />
i<br />
E[n], n(r)
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