Biomedical Engineering Reference
In-Depth Information
  
 
 
 
E
n r
( )
F
n r
( )
v
( ) ( )
r
n r
dr
 
(8.1)
HK
ext
2
with
n r
( )
 f
( )
r
( ( )
i
f
r
being the electron wave functions);
i
i
is an external potential (i.e., the ionic potential) and the
universal Hohenberg-Kohn functional
v
( )
r
ext
n  
F
( )
casts into:
HK
     
F
n r
( )
T
n r
( )
V
n r
( )
(8.2)
     
HK
ee
     
     
T
are, respectively, the electron kinetic energy and
the electron-electron interaction potential energy that contains
the Coulomb repulsive energy and the exchange correlation energy
between the valence electrons.
The total energy functional is variational with respect to the
electron density and therefore the ground state electron density is
the one that minimizes the total energy functional:
n r
( )
,
V
n r
( )
     
     
ee
  
E
E
n
( )
r
E
n r
( )
(8.3)
  
0
0
The above variational problem is usually recast into an
independent electrons eigenvalue problem through the Kohn-Sham
(KS) equations:
H
f i
=
e i f i
(8.4)
KS
with the Kohn-Sham Hamiltonian defined as

 

  
 
dE
n r
( )
2
n r
( )
¢
XC
(8.5)
¢
H
-
v
( )
r
dr
KS
ion
2
| - |
r
r
¢
dn r
( )
V
( )
r
V
( )
r
H
XC
where
are the Coulomb and the exchange correlation
potentials, respectively.
The KS equations can be solved self-consistently to find the
ground state electron density and total energy [27].
The key factor affecting the accuracy of the DFT total energy
calculations is the approximation used to treat the exchange
correlation potential that is unknown. Several approximations have
been developed to treat the exchange correlation energy functional,
some of them being very close to the chemical accuracy such as PBE,
B3LYP, etc. [9, 10, 51, 82, 83]. The ionic potentials are usually replaced
V
( ), ( )
r
V
r
H
XC
 
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