Transcript Document
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:
ELDAU [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
[ H0 (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 0
i
GDFT (r, r, i)e
i 0
i
Family of Functionals
BK [G(r, r ', i)]
DFT [G(r, r, i)e
i 0
[Gloc (r, r ', i)]
r r'
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
KSDF [Gloc ] KSDF [G]
Exactly as in DFT:
kj ( r ) kj* ( r ')
GDFT ( r, r ', )
kj
kj
KDFT [ ] KDFT [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 ', )
2
e
1 ( r, r '', )
dr ''
| r '' r ' |
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 ')