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Subsections
Let us assume that the charge density n(r) and the potential V(r)
are spherically symmetric. The Kohn-Sham (KS) equation:
can be written in spherical coordinates. We write the wavefunctions as
where n is the main quantum 
number 
l = n - 1, n - 2,..., 0 is angular momentum, 
m = l, l - 1,..., - l + 1, - l
is the projection of the angular momentum on some axis. 
The radial KS equation becomes:
|  -    + (V(r) -  )  Rnl(r)  Ylm(  ) |  |  |  | 
| -      sin    +     Rnl(r) = 0. |  |  | (3) | 
 
This yields an angular equation for the spherical harmonics 
Ylm( ):
):
| -     sin    +    = l (l + 1)Ylm(  ) | (4) | 
 
and a radial equation for the radial part Rnl(r):
| -   +    + V(r) -   Rnl(r) = 0. | (5) | 
 
The charge density is given by
where 
 are the occupancies (
 are the occupancies (

 2l + 1)
and it is assumed that the occupancies of m are such as to yield
a spherically symmetric charge density (which is true only for closed
shell atoms).
2l + 1)
and it is assumed that the occupancies of m are such as to yield
a spherically symmetric charge density (which is true only for closed
shell atoms).
Gradient in spherical coordinates 
(r, ,
, ):
):
Laplacian in spherical coordinates:
 
 
 
 
 
 
 
  
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Filippo Spiga
2016-04-24