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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 (

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:
Next: A..2 Fully relativistic case
Up: A. Atomic Calculations
Previous: A. Atomic Calculations
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Layla Martin-Samos Colomer
2012-11-21