Electronic correlation and Hubbard approaches

• Notable failures of LDA/GGA: transition-metal oxides

• Introduction to correlation: the H2 molecule

• DFT and correlation

• DFT+U: general formulation

• Examples and applications

Matteo Cococcioni

Department of Chemical Engineering and Materials ScienceUniversity of Minnesota

Failures of LDA/GGA: transition metal oxides

• Cubic, rock-salt structure

• Antiferromagnetic (AF) ground state rhombohedral symmetry and possible structural distorsions (FeO)

• Conduction properties (exp): insulators (Mott/charge-transfer kind)

TM ion

Oxygen

FeO: GGA results

• Antiferromagnetic ground state: NO (FM)

• crystal structure (cubic): OK

• but…

We obtain a metal !!!

Fe2+

NiO: GGA results

• Antiferromagnetic ground state: OK

• crystal structure (cubic): OK

• Crystal field produces a band gap, but…

The energy gap is too small

Mott insulators: U vs W Two quantities are to be considered:

• U “on-site” electron-electron repulsion

• W bandwidth (hopping amplitude, related to kinetic energy)

Two different regimes:

• W/U >> 1: the energy is minimized making the kinetic term as small as possible through delocalization (little price is paid on the occupied atomic sites to overcome repulsion U)

• W/U << 1: the kinetic energy of electrons is not large enough to overcome the on-site repulsion. Electrons undergo a Mott localization

LDA/GGA approximations to DFT always tend to over-delocalize electrons:

• U is not well accounted for

• electronic energy functionals are affected by self-interaction

I. G. Austin and N. F. Mott, Science 168, 71 (1970)

P Gori-Giorgi et al. cond-mat/0605174

H2 molecule in DFT

12

1

r2

12 12 3 N 3 N

...

( 1)( ) | (r ,R, r ,...., r ) | R r .... r

2 4N

dN Nf r d d d

! !

"#

$%= & '

Let’s consider the spherically and system-averaged pair electronic density:

12 1 2| r r |r = ! 1 2R (r +r ) / 2=where

At very large internuclear distances each electron is still split between the two sites!

From He to 2H: abc of correlation Two protons a and b (as classical objects, no nuclear forces) and two (non-interacting) electrons 1 and 2: electronic ground state at increasing inter-proton distance.We use 1s single electron states (H) or bonding (σ) and antibonding (σ*) linear combinations (H2

+)

1) d = 0, a = b: He atom

!

"(1,2) = N #a1( )#a

2( )[ ]$%

2) d > 0: H molecule

!

a) "(1,2) = N #a1( )#b

2( ) +#b1( )#a

2( )[ ]$%

!

b) "(1,2) = N #$ 1( )#$ 2( )[ ]%&=

!

N "a1( )"b

2( ) +"b1( )"a

2( )[ ] + "a1( )"a

2( ) +"b1( )"b

2( )[ ]{ }#$

2) d >> 0: distinct H atomseach electron is on the 1s ground state of each H atom:

!

"(i) =#1si( )$

“ionic terms”

From He to 2H: continuous solution

H2 molecule: the importance of ionic terms should disappear with distance.The ground state wave function should be something like:

!

"(1,2) = N #a1( )#b

2( ) +#b1( )#a

2( )[ ] +$(d) #a1( )#a

2( ) +#b1( )#b

2( )[ ]{ }%&

This can be achieved through a linear combination of bonding (σ) and antibonding (σ*) Slater determinants:

!

"(1,2) = N a(d) #$ 1( )#$ 2( )[ ] + b(d) #$ * 1( )#$ * 2( )[ ]{ }%&=

a(d) and b(d) can be treated and computed as variational parameters. This is beyond Hartree-Fock theory because we need multi-determinantalwave-functions (multi reference CI methods etc)!

N a(d) + b(d)( ) "a1( )"a

2( ) +"b1( )"b

2( )( ) +{

!

+ a(d) " b(d)( ) #a1( )#b

2( ) +#b1( )#a

2( )( )}$%

Unrestricted (open-shell) Hartree-Fock Let the molecular orbitals for different spin have different spacial parts:

!

"#$

= cos%"# + sin%"# *

Constructing a Slater determinant for the singlet state:!

"#$

= cos%"# & sin%"# *

!

"# *$

= %sin&"# + cos&"# *

!

"# *$

= sin%"# + cos%"# *

we can use θ as a variational parameter to obtain the ground state energy and wave function at any inter-nuclear distance d

!

" = 0o

!

" = 45o

!

"UHF(1,2) = #$

%(1)#$

&(2)

!

0o

< " < 45o

!

"#$

!

"#$

θ allows for “on-site” localizationof the electrons.Note how the spin sticks to a particular atomic site.

Pictures from A. Szabo and N. S.Ostlund, “Modern Quantum Chemistry”

Unrestricted Hartree-Fock H2: results Let the molecular orbitals for different spin have different spacial parts:

For d --> we also obtain that

!

EH2

= 2EH

However the wave function is wrong and so is also E for finite d. Why? !

limR"# $UHF

(1,2) = %a

&(1)%

b

'(2)

!

limR"# $(1,2) = N %

a(1)%

b(2) +%

b(1)%

a(2)[ ] & 1( )' 2( ) (' 1( )& 2( )[ ]!

"

Small d: the “standard” RHF solution is obtained:

Large d: electrons localize on either atom: !

"UHF(1,2) = #$

%(1)#$

&(2)

“actual” wfc:

DFT H2• Despite the wave function is wrong, the charge density (in the long-distance limit) is right even within a minimal basis set representation:

!

n(r) = "ar( )

2

+ "br( )

2

• Thus DFT offers better chances than UHF to provide an accurate description

• DFT is a density method: don’t try to get Kohn-Sham orbitals close to real one-electrons wfc; density and energy will be wrong.

DFT and UHF

On a given basis of spin-orbitals UHF energy is:

!

E = hi"

i"

N"

# +1

2Jij"" $Kij

""( ) +j

N"

#i

N"

#"

#1

2Jij" $"( )

j

N$"

#i

N"

#"

#

DFT looks “similar” to UHF (we let the two spin wave-functions have differentspacial parts and solve separate Kohn-Sham equations). To see this:

!

n(r) = "kv

#(r)

k,v,#

$2

= % i

&(r)% j

ij

$ (r) "kv

# % i % j "kv

#

k,v,#

$

3) Insert in Hartree potential…

One-body: kinetic energy + external potential Coulomb Exchange

!

"kv

#(r) = $

i(r) $

i"

kv

#

i

%1) 2)

Difference: Exc. It contains exchange interactions and correlation. However it’s not exact

Idea: we can use the expansion in 2) to correct it. We will use the projectionson atomic orbitals to optimize the weights of Coulomb and exchange integrals.

Open systems in DFT: analyticity of EDFT

In open systems exchanging electrons with a reservoir E should be piece-wiselinear and show discontinuities n its first derivative1. EDFT instead is analytical

Needed correction

1Perdew et al., PRL 49, 1691 (1982)

!

EHub

nmm '

I{ }[ ] =1

2"m,m' ' |V

ee|m',m' ' '# n

mm '

I$nm''m '''

I %${m{ },I ,$

&

!

+ "m,m' ' |Vee|m',m' ' '# $ "m,m' ' |V

ee|m' ' ',m'#( )nmm'

I%nm ''m'''

I% }

!

Edc

nI"{ }[ ] =

U

2I

# nInI $1( ) $

J

2I ,"

# nI"

nI" $1( )

!

nmm'I"

= fkv

k,v

# $%kv

"|&m'

I '$&m

I|%

kv

" '

!

nI"

=m

# nmm

I"

!

nI

="

# nI"

The LDA+U energy functionalThe LDA+U method consists in a correction to the LDA (or GGA) energy functional to give a better description of electronic correlations. It is shapedon a Hubbard-like Hamiltonian including effective on-site interactions.It was introduced and developed by Anisimov and coworkers (1990-1995).

Occupations:

Fully rotationally invariant formulation (Lichtenstein et al. PRB 1995)

!

ELDA+U n(r){ }[ ] = E

LDAn(r){ }[ ] + E

Hubnm

I"{ }[ ] # EdcnI"{ }[ ]

Keep in mind:

• only occupations of “localized” orbitals included (e.g. d or f states)• no cross-site terms: integer on-site occupation are favored against hybridization

!

ak m,m',m' ',m' ' '( ) =4"

2k +1# lm |Ykq | lm'$#lm' ' |Ykq

*| lm' ' '$

q=%k

k

&

!

nmm'

I" = #mm '

1

2l +1Tr n

I"[ ]

!

U =1

2l +1( )2

"m,m' Veem,m'# = F 0

m,m'

$

!

J =1

2l 2l +1( )"m,m' |V

ee|m',m# =

F2 + F 4

14m,m'

$

Electronic interactionsHartree-Fock formalism (for d states): from the expansion of 1/|r-r’| in spherical harmonics we get:

The double counting term is evaluated as the Mean Field Approximation of the Hubbard one. So in the expression of EHub we put

and get

Keep in mind: we want screened (effective) interactions; are unscreenedkF

!

"mm' ' |Vee|m'm' ' '#= dr dr '$$

%m

*(r)%

m''

*(r ')%

m ' (r)%m''' (r ')

r & r '= a

k(m,m',m' ',m' ' ')F

k

k

'

!

EU

nmm '

I"{ }[ ] = EHub

nmm'

I"{ }[ ] # EdcnI"{ }[ ] =

A simplified approachFirst order approximation: let’s neglect the exchange interaction J:

We get:

Note: a) U is the only interaction parameter in the functional Note: b) the rotational invariance is preserved.

This is the formula implemented in PWscf. We have

!

J = F2

= F4

= 0

!

U

2nmm

I" # nmm '

I"nm'm

I"

m '

$% & '

( ) * m,"

$I

$ =U

2Tr n

I"1#nI"( )[ ]

I ,"

$

!

EDFT +U = E

DFT"[ ] + E

Unmm '

I#{ }[ ] = EDFT

"[ ] +U

2Tr n

I#1$nI#( )[ ]

I ,#

%

How does it work?Because of rotational invariance we can use a diagonal representation:

where

Potential:

} ! Partial occupations arediscouraged Gap opening! Eg U!

!

EU

=U

2"m

I#1$ "

m

I#( )[ ]m

%I ,#

%

!

nI"vm

= #m

I"vm

!

"mI#

= fkv $kv

# %mI

k,v

& %mI $kv

#

!

VU"

kv

# =$E

U

$"kv

# *=U

21% 2&

m

I#( ) 'mIm

( 'm

I "kv

#

I ,#

(

!

"m

I#>1

2$V

U< 0

!

"m

I#<1

2$V

U> 0

Potential discontinuities and energy gaps

λ

The +U correction is the one needed to recover the exact behavior of theenergy. What is the physical meaning of U?

!

EU

=U

2"m

I#1$ "

m

I#( )[ ]m

%I ,#

%

Keep in mind: the “+U” correction is only applied to localized (atomic-like) states treated as impurities exchanging electrons with the crystal

Input file for LDA+U calculation with PWscfOnly the namelist “system” is modified:

&system . . . lda_plus_u = .true., Hubbard_U(1) = $U1, Hubbard_U(2) = $U2, . . Hubbard_U(ntype) = $Untype, . ./

• There is a different U for each distinct type of “Hubbard” atom

• U is in eV

• Typical values: U is rarely larger than 7-8 eV (in most cases 0<U<5 eV)

There might be need of using finer k-points grids for a better evaluationof on-site occupations.

GGA+U FeO

M. Cococcioni and S. de Gironcoli PRB 71, 035105 (2005)

GGA GGA+U

GGA+U FeO: the Broken Symmetry Phase

M. Cococcioni and S. de Gironcoli PRB 71, 035105 (2005)

(111) Plane of Fe

New frustrated electronic GS: correlation-stabilized orbital ordering

“old” GS

“new” GS

In the new electronic ground statethe observed tetrahedral distorsionunder pressure is reproduced.

LDA+U NiOGGA GGA+U

M. Cococcioni and S. de Gironcoli PRB 71, 035105 (2005)

Mineral in the Earth’s mantle: Fe2SiO4band structure

structural parameters

M. Cococcioni and S. de Gironcoli PRB 71, 035105 (2005)

Useful references

• A. Szabo and N. S. Ostlund, “Modern quantum chemistry”, Dover Publications Inc., 1996

• Perdew et al., PRL 49, 1691 (1982)

• Anisimov, Zaanen, and Andersen, PRB 44, 943 (1991)

• Anisimov et al., PRB 48, 16929 (1993)

• Liechtenstein, Anisimov, and Zaanen, PRB 52, R5467 (1995)

• Anisimov, Aryasetiawan, and Liechtenstein, J. Phys.: Cond Matt 9, 767 (1997)

• Pickett, Erwin, and Ethridge, PRB 58, 1201 (1998)

Summary I

• Introduction to correlation: the case of H2

• H2: HF, UHF, CI

• DFT, exact exchange and correlation: between UHF and CI

• DFT and correlation: open systems and discontinuous potentials

• DFT+U: general formulation and main approximations

• Examples and applications

A consistent, linear-response approachTo DFT+U

Matteo Cococcioni

Department of Chemical Engineering and Materials ScienceUniversity of Minnesota

• DFT+U energy functional

• The meaning of U

• Linear-response calculation of U

• “self-consistent” U

• Examples and applications

!

EU

nmm '

I"{ }[ ] = EHub

nmm'

I"{ }[ ] # EdcnI"{ }[ ] =

The DFT+U energy functional

Where:

a) U is the only interaction parameter in the functional b) The functional is (on-site) rotationally invariant.

This is the formula implemented in PWscf. We have !

U

2nmm

I" # nmm '

I"nm'm

I"

m '

$% & '

( ) * m,"

$I

$ =U

2Tr n

I"1#nI"( )[ ]

I ,"

$

!

EDFT +U = E

DFT"[ ] + E

Unmm '

I#{ }[ ] = EDFT

"[ ] +U

2Tr n

I#1$nI#( )[ ]

I ,#

%

!

EDFT +U "[ ] = E

DFT"[ ] + E

Unmm'

I#{ }[ ]

What does U mean?

λ

The +U correction is the one needed to recover the exact behavior of theenergy. What is the physical meaning of U?

!

EU

=U

2"m

I#1$ "

m

I#( )[ ]m

%I ,#

%

!

U =d2EGGA

d(nI)2"d2E0

GGA

d(nI)2

Evaluation of U• U is the unphysical curvature of the DFT total energy

• We want effective interactions: we evaluate U from the DFT ground state

• A free-electron contribution (due to re-hybridization) is to be subtracted in crystals:

From self-consistent ground state(screened response)

From fixed-potential diagonalization(Kohn-Sham response)

!

E "I

{ }[ ] =minn(r )

E n r( )[ ] + "I

I

# nI

$ % &

' ( )

!

dE nJ{ }[ ]

dnI

= "#I

nJ{ }( )

!

E nI{ }[ ] =min

"I

E "I

{ }[ ] # "I

I

$ nI

% & '

( ) *

= E "I

{ }[ ] # "I

I

$ nmin

I

!

d2E n

J{ }[ ]d n

I( )2

= "d#

InJ{ }( )

dnI

Legendre transform

Second derivatives

• Second derivatives of the energy are not directly accessible

• We apply a potential shift α to the d states of each Hubbard atom I and use a Legendre transform:

J

I

IJ

d

nd

!" 00

=J

I

IJ

d

nd

!" =

!

U = "d#

I

dnI

+d#

I

dn0

I= $

0

"1 " $"1( )II

Linear responseUsing αI as perturbation parameters we can easily evaluate the response matrices:

χ0 is the bare response of the system, χ the fully interacting (screened) one

• Run a self-consistent (unperturbed) calculation.• Starting from saved potential and wavefunction add the perturbation• The response χ0 is evaluated at the first iteration (at fixed potential)• The response χ is evaluated at self consistency

The effective interaction is finally obtained as:

M. Cococcioni and S. de Gironcoli PRB 71, 035105 (2005)

What screening?

!

U = "0

#1 # "#1

!

" = "0

+ "0U"

!

" = "0

+ "0U"

0

!

U = "0

#1""0

#1 # "0

#1

is equivalent to

= U+!

0!

0!!

= U+0

!0

!0

!!

diagrammatic representation:

U is a bare interaction; U is the dressed (effective) one. We use U andlet the electrons perform the screening through the relaxation to their self-consistent ground state. But…

…weren’t we looking for screened effective interactions? What screeningare we including?

Linear response: what is inside U?

!

"Vext(r)

!

"V (r) = "Vext

+"n(r')

| r # r' |$ dr'+

%vxc(r)

%n(r')$ "n(r')dr'

!

"n(r) = #(r,r')"Vext(r')dr'$ = #

0(r,r')"V (r')dr'$

!

"Vext(r) = #$1

(r,r')"n(r')dr'%

!

"V (r) = #0

$1(r,r')"n(r')dr'%

!

"0#1(r,r') # "#1

(r,r') =1

| r # r' |+$v

xc(r)

$n(r')

!

"n(r)

See S. Baroni et al., Rev. Mod. Phys. 73, 515 (2001) and refs quoted therein

Let’s apply a perturbation to the external potential and study the responseof the electronic charge density

we can introduce response functions and write:

where:

Inverting (1) we get:

(1)

(2)

(3)

From (2) and (3) we easily obtain: xc kernel

The Coulomb interaction is “screened” by the xc kernel

Input file for computing U with PWscf“unperturbed” run

&system . . . lda_plus_u = .true., Hubbard_U(1) = 1.d-20, Hubbard_U(2) = 1.d-20, . . Hubbard_U(ntype) = 1.d-20, . ./

“perturbed” run

• Hubbard_alpha values are symmetrically distributed around 0 (typically from –0.1 to 0.1 eV)

• the perturbed atom has to be treated as of different kind

• the series of perturbed runs at different Hubbard_alpha is repeated for every different kind of Hubbard atom

• diago_thr_init is chosen close to the last ethr of the unperturbed run (ethr_conv)

foreach a (0.d0 –0.01 0.01…).&control . restart_mode = ‘from_scratch’ ./&system . lda_plus_u = .true., Hubbard_U(1) = 1.d-20, . Hubbard_U(ntype) = 1.d-20, Hubbard_alpha(1) = $a, Hubbard_alpha(2) = 0.d0, ./&electrons . startingwfc = ‘file’, startingpot = ‘file’, diago_thr_init = $ethr_conv ./.end

Constructing the response matrices• The perturbation must be fully accomodated in the considered cell (to avoid interactions with its periodic replicas): we need to run calculations in supercells

• Supercell: N Hubbard atoms of M distinct types. Response matrices: NxN

• Perturbed calculations: to be repeated for each of the M types of Hubbard atoms

• First M columns of the response matrices are calculated straightforwardly. The other N-M columns are obtained attributing the same response to equivalent shells of neighbors when perturbation is applied on equivalent atoms

• Background term (perturbation must be neutral) and other technical details: please refer to M. C. and S. d. G. PRB 71, 035105 (2005)

• Converge the computed U values with the size of the supercell

Advantages of the method

Fully ab-initio estimation of the effective interaction(no guess or semiempirical evaluation is needed)

Consistency of the effective interaction with thedefinition of the energy functional and of the on-siteoccupations;- other localized basis sets can be equivalently used:

gaussians, Wannier functions etc

Consistency with the DFT approximation

Easy implementation in different computational schemes

“Self-consistent” U

• DFT and DFT+U can lead to very different ground states

• The response of the system can be different in DFT and DFT+U

• U must be computed self-consistently from a DFT+U ground state

The energy model

• We need a model for the electronic interactions in theGGA+U ground state

• We need the second derivatives of the total energywith respect to the total on-site occupation

!

Eint =Uscf

2niI ,"

i,"

# n j

I ," '

j," '

# $1%

& ' '

(

) * *

+

, - -

.

/ 0 0

+I

#

!

+U

in

2ni

I ,"

i

#I ,"

# 1$ ni

I ,"( )

!

nI

= ni

I

i

"

Double countingfunctional

“+U” correctionfunctional

Evaluation of Uscf

• For small shifts in the on-site potential we have:

• We obtain:

where is an effective degeneracy.

• We want the value of Uscf that can be obtained from aGGA+U ground state with Uin = Uscf

!

Uout =d2EGGA+U

d nI( )2

"d2E0

GGA +U

d nI( )2

=d2Eint

d nI( )2

=Uscf "Uin

m

!

m =1/ ai

I( )2

i

"

!

d2

d nI( )2

= aiIa j

I

i, j

"d2

dniIdn j

I

!

"ni

I= a

i

I"nI and

Evaluation of Uscf in practice

• Study Uout vs Uin: identify the linear region

• Extrapolate to Uin=0

• Uscf is the one from the extrapolation of the linear region that contains it.

Importance of computing U• Consistency with the expression of the “+U” functional and with the choice of localized basis set: the computed U is “exactly” the one needed

• Sensitiveness to spin states, chemical/physical environments, structural changes (important in studying e.g. chemical reactions, phase transitions)

M. Cococcioni and S. de Gironcoli PRB 71, 035105 (2005)

FeO – rhombohedral distorsion under pressure

(MgxFe1-x)O – HS to LS transition under pressure

T. Tsuchiya, R. M. Wentzcovitch, C. R. S. da Silva andS. De Gironcoli, Phys. Rev. Lett. 96, 198501 (2006)

P (GPa)

• High voltage (3.5 V) and high theoretical capacity

• Electrochemical stability

• Thermal safety

• Non toxicity

• Low cost and easy synthesis

• Issue: low electronic conductivity

Properties Exp GGA GGA+U

Crystal structure

Magnetic structure

Formation Energy LixFePO4

Voltage

TM valence states (0< x <1)

PRB 69, 201101 (2004) PRB 70, 235121 (2004)

Similar agreement In other compounds!

Exp GGA+U

Fe2+/Fe3+

3.5 V

>0

AF

olivineolivine

2.9 V

Fe2.5+

Cathode material for Li ion batteries: LixFePO4

olivine

AFAF

<0>0

3.5 V

Fe2+/Fe3+

2 14 4 2 1

2 1

( ) ( ) ( ) ( )

( )

x xE Li FePO E Li FePO x x E Li

Vx x F

! ! !=

!

Spin states in the Heme group

Penta-coordinated iron Esa-coordinated iron (O2, CO, etc)

Fe site O2

Magnetic ground state

Exp quintuplet (S=2) singlet (S=0)

GGA triplet singlet

B3LYP triplet singlet

HF quintuplet quintuplet

GGA+U quintuplet singlet

D. Scherlis, H. L. Sit, M. Cococcioni and N. Marzari, submitted to Journ. of Phys. Chem.

H2 addition-elimination to FeO+

H. J. Kulik, M. Cococcioni, D. Scherlisand N. Marzari, PRL 97, 103001 (2006)

GGA

GGA+U

The Fe dimer

H. J. Kulik, M. Cococcioni, D.A. Scherlis, and N. Marzari,PRL 97,103001 (2006).

Conclusions

• Linear response approach to compute U: the LDA+Uas a fully ab-initio computational scheme

• Self-consistent evaluation of U

• Correct electronic ground state and structuralproperties of FeO

• Correct chemical behavior and voltage of TransitionMetal compounds for cathodes of Li batteries

• Improved description of molecules and chemicalreactions

Work in progress

• Better “+U” functionals (for better energetics)

• auto-consistent calculation of U (from a LDA+U ground state)

• inter-site interactions

• phonon+U

• covariant formulation of U

• compute J

• ………