Physics 250-06 “Advanced Electronic Structure”
Lecture 3. Improvements of DFT
Contents:
1. LDA+U.
2. LDA+DMFT.
3. Supplements: Self-interaction corrections, GW
Concept of delocalized and localized states
Systems with d and f electrons show localized (atomic like)
behavior.
Examples: cuprates, manganites, lanthanides, actinides,
transition metal oxides, etc.
LDA is a static mean field theory and cannot describe many-body
features in the spectrum: example: atomic limit is with multiplets
is missing in LDA.
When magnetic order exists, LSDA frequently helps!
(Anti)ferromagnets
G  ( k ) 
1
   (k )  V 
G  ( k ) 
1
   (k )  V 
Splitting Vup-Vdn between up and down bands can be calculated
in LSDA. It always comes out small (~1 eV). In many systems,
it is of the order of 5-10 eV.
LDA+U
In LSDA splitting Vup-Vdn is controlled by Stoner parameter I
while on-site Coulomb interaction U can be much larger than
that:
H   tij ci  c j  Unˆi nˆi 
i
ij
In simplest Hartree-Fock approximation:
EC ~ Un n
i
dE
V 
 Un
dn
LDA+U functional with built-in Hubbard parameter U:
ELDAU [n( r ), nd  , nd  ]  ELDA[n( r )]  Und nd   EDC [nd ]

Paramagnetic Mott Insulators
LDA/LDA+U, other static mean field theories, cannot access paramagnetic
insulating state because spin up and spin down solutions become degenerate
GLDA ( k ) 
1
   ( k )  VLDA
How to recover the gap in the spectrum?
Frequency dependence in self-energy is required:
GDMFT (k ) 
1

   (k )  ( )
1
U2
   (k ) 
4(   d )
1/ 2
1/ 2

   (k )  U / 2    (k )  U / 2

Concept of Spectral Functions
Effective (DFT-like) single particle spectrum
always consists of delta like peaks
Real excitational spectrum
can be quite different
[  H 0 (k )  ()]G(k , )  1
Localized electrons: LDA+DMFT
Electronic structure is composed from LDA Hamiltonian for sp(d) electrons and
dynamical self-energy for (d)f-electrons extracted from solving impurity problem
ˆ dc )] ( r )    ( r )
[2  VKS ( r )  (ˆ imp
(

)

V
f
f
kj
kj kj
Poles of the Green function G(k ,  ) 
1
   kj
have information about atomic multiplets and other many body effects.
N()
dn->dn+1
dn->dn-1
0

Better description compared to LDA is obtained
Spectral Density Functional Theory
A functional method where electronic spectral function is a variable
would predict both energetics and spectra.
A DMFT based electronic structure method - an approach where local spectral
function (density of states) is at the center of interest. Can be entitled as
Spectral density functional theory
(Kotliar et.al, RMP 2006)
 Total Energy and local excitational spectrum are accessed
 Good approximation to exchange-correlation functional
is provided by local dynamical mean field theory.
 Role of Kohn-Sham potential is played by a manifestly local self-energy
operator M(r,r’,).
 Generalized Kohn Sham equations for continuous distribution of spectral
weight to be solved self-consistently.
Local Green function Functionals
 (r )  G(r, r, i)e
i
i 0
 GDFT (r, r, i)e
i 0
i
Family of Functionals
 BK [G( r, r ', i)]
[Gloc ( r, r ', i)]
r  r'
 DFT [ G( r, r, i)e
i 0
i
 DFT [  ( r )]
Gloc ( r, r ', i )  G( r, r ', i ) ( r, r ')
]
Generalization of Kohn Sham Idea
To obtain kinetic functional:
SDF [Gloc ]  KSDF [Gloc ]   SDF [Gloc ]
introduce fictious particles which describe local Green function:
 kj ( r ) kj†  ( r ')
G( r, r ',  )  
   kj
kj
K SDF [Gloc ]  K SDF [G]
Exactly as in DFT:
 kj ( r ) kj* ( r ')
GDFT ( r, r ', )  
   kj
kj
K DFT [  ]  K DFT [GDFT ]
Local Self-Energy of Spectral Density Functional
 Spectral Density Functional looks similar to DFT
[ kj ]   f kj kj    M eff ( r, r ', i )G( r, r ', i )drdr '
kj
i
i
   ( r )Vext ( r )dr EH [  ]   xc [Gloc ]
f kj 
1
(i    Ekj )
 Effective mass operator is local by construction and plays
auxiliary role exactly like Kohn-Sham potential in DFT
M eff ( r, r ',  )  [Vext ( r )  VH ( r )] ( r  r ') 
 xc
 Gloc ( r, r ',  )
 Energy dependent Kohn-Sham (Dyson) equations give
rise to energy-dependent band structure
2 kj ( r )   M eff ( r, r ', ) kj ( r ')dr '   kj kj ( r)

physical meaning in contrast to Kohn-Sham spectra.
Ekj have
are designed to reproduce local spectral density
Self-Interactions
LDA is not self-interaction free theory.
Simplest example: electron in hydrogen atom produces charge density cloud
And would have excnage correlation potential according to LDA.
Perdew and Zunger (1984) proposed to subtract spirituous self-interaction
energy for each orbital from LDA total energy by introducing
Self-Interaction Corrections (SIC)
LDA-SIC theory produces orbital-dependent potential since one needs to
define orbitals which self-interact.
SIC theory produces better total energies but wrong spectra in many cases.
GW Theory of Hedin
In GW (Hedin, 1965) spectrum is deduced from Dyson equation
with approximate self-energy:
[2  Vext (r )  VH (r )] kj (r)   GW (r, r ', ) kj ( r ')dr '   kj kj ( r)
GW theory can be viewed as perturbation theory with respect to
Coulomb interaction.
It produces correct energy gap in semiconductors which is an improvement
on top of LDA
Being a weakly coupled pertrurbation theory it also has wrong atomic limit
and does not produce atomic multiplets
Computing GW Self-Energy
Solve LDA equations and construct LDA Green functions and GW self-energy
[ 2  Vext ( r )  VH ( r )  Vxc ( r )] kj ( r )   kj kj ( r )
 kj ( r ) kj ( r ')
G ( r, r ',  )  
   kj
kj
GW ( r, r ',  )   dr '' G ( r, r '', ')W ( r '', r ',    ')dr '' d  '
W ( r, r ',  )  
e2
 ( r, r '',  )
dr ''
| r '' r ' |
1
Here, dynamically screened Coulomb interaction is calculated from the
Knowledge of the dielectric function of the material:
 1 
0 ( r, r ',  )  
q
kjj '
0
1  VC  0
f kj  f k  qj '
Ekj  Ek  qj '  
 kj ( r ) k qj ( r ) k qj ( r ') kj ( r ')