Self-Consistent Implementation of Self-Interaction Corrected DFT and of the Exact Exchange Functionals in

Plane-Wave DFT

Kiril Tsemekhman (a), Eric Bylaska (b), Hannes Jonsson (a,c)

(a) Department of Chemistry, University of Washington, Seattle(b) Pacific Northwest National Laboratory, Richland, USA(c) Faculty of Science, University of Iceland, Reykjavik, Iceland

Self-Interaction in DFT

• Total energy of electronic system in the DFT:

( ) ( ) ( ) ( ) [ ]2, , 1 2

,

1 1 ,2 2DFT i ext i xc

iE V d d d E

′⎛ ⎞ ′= Ψ − ∇ + Ψ + +⎜ ⎟ ′−⎝ ⎠∑∫ ∫ ∫

r rr r r r r

r rσ σσ

ρ ρρ ρ

( ) ( ), ,, ,

1 ,0 02

i iSI i xc iE d d E

′′⎡ ⎤ ⎡ ⎤= + ≠⎣ ⎦ ⎣ ⎦′−∫ ∫

r rr r

r rσ σ

σ σ

ρ ρρ ρ

• Interaction of the density of the state with itself (“self-interaction”):, ( )i rσΨ

for any approximate exchange-correlation functional. NON-PHYSICAL!

• In Hartree-Fock approximation, exchange interaction is evaluated exactly, correlation energy is absent, and self-interaction is cancelled out.

( ) ( ) ( ) ( ) ( )* *

,12

occ occn m n mHF

Exch in m

E drdrr rσ

′ ′Ψ Ψ Ψ Ψ⎡ ⎤ ′Ψ = −⎣ ⎦ ′−∑∑∫∫

r r r rr

In this talk:

1. Why self-interaction corrections?2. Solving of DFT-SIC functional. Wannier functions.3. Direct minimization scheme and extension to periodic systems.4. Exact exchange (hybrid) functional in plane-wave DFT.5. Applications.

Why Apply Self-Interaction?

1. Some problems of DFT known (or suspected) to be caused by the SI: • Size consistency (dissociation of H2

+)• Tendency towards charge and spin de-localization

2. If treated carefully, self-interaction corrections can help calculate accurate band gaps in semiconductors and insulators.

H H+e-

Examples of the phenomena studied with DFT-SIC where DFT fails:

• Band gaps in semiconductors and insulators: Ge becomes a narrow-gap semiconductor; band gaps are generally improved.

• Qualitatively (and quantitatively) different description of the systems:a) Localization of a vacancy state at TiO2 rutile (110) surface. b) Complex spin structure of hematite Fe2O3 : formation of a small polaron.c) Localization of spin and charge densities around the structural defects.d) Self-trapping of excitons and holes in SiO2.

DFT-SIC functionalRemove self-interaction in a very straightforward way (Perdew & Zunger, ’82):

[ ] ( ) ( ), ,- , ,

,

1( ) - ,02 -

i iDFT SIC i DFT xc i

iE E d d Eσ σ

σ σσ

ρ ρρ ρ ρ

⎧ ⎫′⎪ ⎪⎡ ⎤ ′ ⎡ ⎤= +⎨ ⎬⎣ ⎦⎣ ⎦ ′⎪ ⎪⎩ ⎭∑ ∫

r rr r r

r r

SICE

• In (1) we get an orbital-dependent potential, i.e different Hamiltonians for different states! Although it is an “unusual” set of Schrodinger equations, we will try to get around having to solve them directly.

• Scheme (2) turns out to be much more difficult if at all possible.

is an orbital-dependent functional. It can be minimized in two different ways:-DFT SICE

( ) ( )( )

( ),

,

SIC iiSIC

i

EV σ

σ

δ Ψ

δΨ

⎡ ⎤⎣ ⎦=r

rr

( )( )

( ),SIC i

SIC

EV σδ ρ

δρ

⎡ ⎤⎣ ⎦=r

rr

(1) Variation wrt orbitals (2) “OEP”

Calculating the SIC energy( ) ( ), ,

, ,1 ,02

i iSIC i xc iE d d E

′′⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦′−∫ ∫

r rr r

r rσ σ

σ σ

ρ ρψ ρ

• SIC term is not invariant with respect to the unitary transformations between occupied states• Calculating SIC terms with Bloch functions produces zero: SIC scale as , where V

is a system volume• Orbitals which minimize DFT-SIC functional are known to be spatially localized;

For periodic systems, such localized orbitals are Wannier functions. SIC should be calculated with the Wannier functions.

V −α

Bloch functions vs. Wannier functionsdelocalized over entire system localized (hopefully with a ‘small’ unit cell)

( )( )

( )3,32

i

n nBZ

Vw d eπ

− ⋅− = ⋅ Ψ∫

k R

kr R k r

The inverse transformation:( ) ( ),

i

i

in n ie w⋅Ψ = −∑ k R

kR

r r R

( ) ( )in neΨ + = ΨkR

k kr R r

( ) ( ) ( ), ,n nm mm

UΨ → Ψ∑ kk kr r

A general unitary transformation mixing Bloch functions with the same crystal momentum but from different bands is used to localize them inside each unit cell:

Marzari and Vanderbilt (’97) and Silvestrelli (2000) developed a scheme for construction of the transformation producing maximally localized Wannier functions. ( )

nmU k

In practical implementations, only few different k-vectors are used for Bloch functions.

( ) ( ),i

n nBZ

w e− ⋅

∈

− = Ψ∑ k Rk

kr R r%

Γ-point k=0 and k=π/R

Identify each WF with the piece-wise defined BF

w(r-R0) w(r-R6)

Calculating the SIC energy with Wannier functions• Wannier functions constructed from single Γ-point (or just limited number of)

Kohn-Sham Bloch states, have infinitely many artificial periodic replicas.• To use the plane-wave method to calculate SIC, one has to limit integration in

the Coulomb SIC term to a single unit cell.

To handle extended systems, we replace the Coulomb interaction by a screened Coulomb interaction. A practical example of such screening:

( ) ( ) ( )r

RrrfNN/exp11 −−−

=

This kernel is nearly equal to 1/r for r<Rand rapidly decays to 0 for r>R.

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

(1-(1-exp(-(x/8)**8))**8)/abs(x)1/abs(x)

Just a useful guessDirect Minimization of DFT-SIC functional

DFT K-S orbitals (Maximally) localizedWannier functions

Calculation of SIC terms

( )( )

DFT SIC i

i

E − ⎡ ⎤⎣ ⎦rr

δ ΨδΨ

Total Energy,Geometric Structure,

( )iΨ r

( )ˆ iij i jH H= Ψ Ψ iε

Hermitian at the minimum

Self-consistent calculation of the density and wave functions appears to be very important: DFT-SIC total density is typically different from the DFT density; the self-consistent wave functions are well but not necessarily maximally localized.

Self-consistent direct minimization

( ) ( )i ji SIC j j SIC iV VΨ Ψ = Ψ Ψ

Unitary transformation

FAQ:

No. LDA total energy is always higher than LDA-SIC; GGA is typically lower than GGA-SIC.

2. If DFT-SIC raises the DFT total energy why aren’t the Bloch functions (giving vanishing SIC) better solutions?DFT-SIC functional calculated with the Bloch functions is NOT size consistent.

∗SIC SICE = N εSICε

Wannier functions: N →∞ SIC SICE /N = ε

Bloch functions: →SICE /N 0 size inconsistent

Expect:

3. If Bloch functions are excluded from search from the start, how does this formalismdescribe the metal-insulator transition?The localization character of the Wannier functions appears as part of the solution: exponentially localized WF’s indicate insulator, presence of the power-law decayingWF’s speaks of a metal; for the latter, SIC is almost zero.

4. What about the OEP for DFT-SIC?As in any KS scheme, OEP solutions will be Bloch functions => standard OEP for DFT-SIC is at least size inconsistent (and has to give zero SIC for periodic systems).

1. Does SIC always lower the total energy?

• Using full SIC strongly overestimates the corrections. On a set of various systems we saw that, in most cases, one needs to use a damping factor of 0.4:

• All qualitative effects such as charge/spin localization and opening of the band gaps appear to be insensitive to the value of this factor.

- 0.4DFT SIC DFT SICE E E= − ∗

Zero-gap semiconductor in LDA Semiconductor in DFT+SICGe

SIC-driven metal-insulator transition as transition from algebraically to exponentially localized Wannier functions

Band gaps in several systems.

Single-Particle Band Gap (eV)SystemExperiment DFT DFT-0.4*SIC

HfO2 5.6 3.6 5.7

Fe2O3 2.1 0.8 2.3HfSiO4 ? 4.8 7.2

Al2O3 8.9 5.9 8.4TiO2 (rutile) 3.0 1.8 3.0SiO2 8.9 6.1 8.6

Si 1.12 0.7 2.2Ge 0.66 0. 1.2

Why Si and Ge are so much overcorrected?

Small polarons in hematite (Fe2O3)

Antiferromagnetic ground state Additional electron densityLDA LDA - 0.4 SIC

Localization of the vacancy state on the rutile TiO2 (110) surface

N(ε)Conduction band

Vacancy state3.2 eV

N(ε)

Valence band top

Conduction band

Vacancy state1.3 eV

N(ε)

Valence band top

Conduction band

Vacancy state3 eV

0.8 eV

Valence band top

1.0 eV

DFT (PBE96) Experiment PBE96-0.4•SIC

What we learned:1. DFT-SIC is capable of solving at least some serious problems of DFT.

2. Using full SIC strongly overcorrects the DFT results in all systems studied and observed that one should use the factor of 0.4 instead in front of the SIC term.

3. Self-consistent SIC are implemented in the NWChem plane-wave package and can be used to study complex systems.

4. Computationally DFT-SIC is about 3 times more expensive than standard DFT (for GGA; the factor is significantly smaller for LDA). SIC part cost scales as N2 rather than N3.

Exact exchange functionals in plane-wave DFT

( ) ( ) ( ) ( ) ( )* *

,12

occ occn m n m

Exch in m

E drdrr rσ

′ ′Ψ Ψ Ψ Ψ⎡ ⎤ ′Ψ = −⎣ ⎦ ′−∑∑∫∫

r r r rr

This looks much more difficult than

[ ] ( ) ( ) ( ) ( )* * ( ) ( )( )occ occ

n n m mCoul

n m

r rE r drdr drdrr r r r

ρ ρρ′ ′Ψ Ψ Ψ Ψ ′

′ ′= =′ ′− −∑∑∫∫ ∫∫

r r r r

• Even for finite systems, EExch is very involved, poorly scales with the size, and is a bottleneck in applications to larger molecules or clusters.• For periodic systems, are the Bloch functions, and the sums include all occupied bands as well as integration over k-vectors.• Exact exchange term is invariant with respect to the unitary transformation between occupied states.

( )n rΨ

( ) ( ) ( ) ( )* *

,

occ occ

m m m mm m

w w′ ′Ψ Ψ = − −∑ ∑R

r r r R r R

( ) ( ) ( ) ( )* *

, ,

occ occn n m m

E xchn m

w w w wE drdr

r r′

′ ′ ′ ′− − − −′= −

′−∑ ∑∫∫R R

r R r R r R r R

Why is it any better the with the Bloch functions? It gets better if each Wannier function is localized in one unit cell: = ′R R

( )nw −r R

( )*mw −r R ( )*

mw ′−r R

( )nw ′−r R

R ′R ′′R

( ) ( ) ( ) ( )* *1

2

occ occn n m m

Exchn m

w w w wE drdr

r r

′ ′′= −

′−∑∑∫∫

r r r r - no k-vector integration

( )1w −r R

( )2w −r R ( )3w −r R

( )4w −r RBut we can do even better:

If it is possible to create localized Wannier functions for a given system,( ) ( )*

,( , )

occn n

n

w wF r r

r r′− −

′ = −′−∑

R

r R r R

is short-range ( )r r L′− <

( ) ( ) ( )*

,( , )

occ

n n Ln

F r r w w f r r′ ′ ′= − − − −∑R

r R r R

Furthemore, because of the invariance, Γ-point Bloch functions can be used instead of WF’s (only if Coulomb interaction is replaced with the screened Coulomb kernel).

Some details of the implementation of exact exchange functionals (HF and hybrids) in the plane-wave NWChem code:• Start with maximally localized WF’s (for maximum gain)• Calculate exact exchange term using these WF’s and screened Coulomb

kernel• Perform direct minimization with respect to Wannier orbitals. WF’s do not

delocalize during this minimization, possibly because of the effect of the screened kernel.

Band gaps in several systems calculated with the hybrid functional PBE0

Single-Particle Gap (eV)SystemExperiment PBE96 PBE0

Al2O3 8.9 5.9 8.4TiO2 (rutile) 3.0 1.8 3.1SiO2 8.9 6.1 8.5Si 1.12 0.7 1.8Ge 0.66 0. 1.0

N(ε)Conduction band

Vacancy state3.1 eV

N(ε)

Valence band top

Conduction band

Vacancy state1.3 eV

N(ε)

Valence band top

Conduction band

Vacancy state3 eV

0.8 eV

Valence band top

0.9 eV

DFT (PBE96) Experiment PBE0

PBE0 functional

( ) [ ] [ ], 1 2 1 2, ,hybridxc Exch i x cE E E Eσα β ρ ρ ρ ρ⎡ ⎤= Ψ + +⎣ ⎦r

0.250.75

96XC PBE

αβ===

Similar to B3LYP functional.

Localization of spin and charge densities by SIC(Al,Si)O2 system: Al substitutional defect in silica

Experiment: Structural symmetry breaking; spin density strongly localized on a single O atom; corresponding bond is longer than the bonds with three other O atoms.

PBE96 calculation Cluster calculation in Hartree-Fock

Bulk periodic system calculation with DFT - 0.4*SIC

(PBE96-0.4SIC) Calculation of Exciton in Rutile (TiO2)(PBE96-0.4SIC) Calculation of Exciton in Rutile (TiO2)

state ground singlet ⇔↑↓ρstate excited triplet ⇔↑↑ρ

singlet21triplet ρρ −↑

singlet21triplet ρρ −↓

(electron)

(hole)

relaxed structureunrelaxed structure

not a self-trapped exciton!

Band Gaps in Semiconductors and Insulators

( 1) 2 ( ) ( 1)gap E N E N E N∆ = + − + −True band gap

KS c v∆ = ε − εKohn-Sham band gap

Discontinuity in VXC

gap KSδ = ∆ −∆

LDA or GGA0δ =

DFT-SIC0δ ≠

( ) 0vSICV r ≠

( ) 0cSICV r =

Triplet-Singlet splitting( ) ( )triplet singlet

gap E N E N∆ = −

SIC SIC SICgap c v∆ = ε − ε

Total energy difference between the ground (singlet) state and the lowest triplet state, with theexception of the systems with strongly bound triplet excitons, is a good approximation to a single-particle band gap: for sufficiently large unit cells excitation of ONE electron changesthe potential only insignificantly; the lowest excited single-particle state in the absence of excitons, Is at the bottom of conduction band, and has vanishing SIC (as do conduction all band states).

In practical implementations, only few different k-vectors are used for Bloch functions. Discretized transformation to Wannier functions becomes:

( ) ( ), ( )i

i

ii in n n i

BZ BZw e e e w⋅− ⋅ − ⋅

∈ ∈

− = Ψ = −∑ ∑ ∑ k Rk R k Rk

k k Rr R r r R%

For central cell R=0 and in the extreme case of single (Γ-point) k=0 the ‘Wannier’function is not at all localized and has the periodic images in every cell. Ifvectors k=0 and k=π/R are included, the periodicity of becomes 2R.

( )nw r%( )nw r%

Γ-point k=0 and k=π/R

• What determines the choice of ?( )nmU k

Depends on the type of the problem and on the goals. Heuristic arguments show that the orbitals maximizing the SIC are maximally localized Wannier functions (MLWF). We will try to obtain MLWF as a starting point of the minimization procedure. Once again:

( ) ( ),i

i

in n ie w⋅Ψ = −∑ k R

kR

r r R

If we manage to make localized within a cell ⇒only one contributes into each ⇒

( )n iw −r R( ),nΨ k r

( ),nΨ k r is also “maximally localized” – determined almost piecewise (although it is still a delocalized function).

( )nmU k ( ),nΨ k rOur goal is then to find which produces such

( )n iw −r R

Marzari-Vanderbilt MLWF2 2

nnn

r⎡ ⎤Ω = −⎣ ⎦∑ rMinimize the spread functional:

2 2 ,n

r n r n= 0 0 22 ;n n n=r 0 r 0

( )nn w= −R r R

(PBE96-0.4SIC) Calculation of Exciton in Rutile, and Electron, and Hole In Anatase (TiO2)

(PBE96-0.4SIC) Calculation of Exciton in Rutile, and Electron, and Hole In Anatase (TiO2)

•Self-interaction creates a Coulomb barrier to charge localization:

•DFT functionals that are not self-interaction free predict either completely delocalized or only partially localized densities (Stokbro et al, Pacchione et al, Jonsson et al).

•Calculations performed by M. Gabriel

doubletdoublet

2

1qTiO Anatase

↓↑ −

−=

ρρdoubletdoublet

2

1qTiO Anatase

↓↑ −

+=

ρρ

How much SIC is enough?• We found that using full SIC strongly overestimates the corrections. On a set of various systems we saw that, in most cases, one needs to use a damping factor of 0.4:

• This observation is consistent with the discussion by Perdew et al (’96) of the exact exchange contribution which is based on the adiabatic connection formula.• Qualitative effects such as charge/spin localization and opening of the band gaps appear to be insensitive to the value of this factor.

- 0.4DFT SIC DFT SICE E E= − ∗

Single-particle energies in DFT-SIC formalism

1. For the orbitals minimizing DFT-SIC functional( ) ( )i j

i SIC j j SIC iV VΨ Ψ = Ψ Ψ

and Hamiltonian matrix is hermitian (Pederson et al, ’84)2. Eigenvalues of this hermitian matrix are also the solutions of a reformulated

eigenvalue problem with Hamiltonian with SIC terms (Pederson et al, ’84)

One can identify the diagonalized Hamiltonian matrix with the single particle energies

Atomization energies of selected dimers (kcal/mol).Plane-Wave

PBE96 PBE96-0.4*SIC PBE96-1.0*SIC

114 85

207

108

NO 150 148 138 126

224

117

O2 118 31 135

N2 225 235

P2 116 36 123

Experiment Hartree-Fock

Reaction and transition state energies (eV) in silanes.

HHSiHSi 24262 +→ SiHSiHHSi 2462 +→ SiH2HSi 242 →

-----2.403.80-2.30-2.40-2.702.60-2.60QCISD(T)*)

0.082.343.67-2.24-2.38-2.652.44-2.51B3LYP*)

0.363.044.56-2.70-2.24-2.173.22-2.63HF0.112.543.72-2.43-2.64-2.902.69-2.70PBE96-0.4*SIC0.563.204.32-2.92-2.73-2.683.35-2.87PBE96-1.0*SIC0.212.073.29-2.09-2.58-3.062.21-2.56PBE96

2.61

2.06

Ets

2.52

1.98

Ets2

3.70

3.18

Ets1Erxn

0.32-2.56-2.93-3.43-3.06LDA-0.4*SIC

0.45-2.26-2.86-3.49-2.89LDA

Plane-Wave

ErxnErxnErxn∆∆E

HSiHSiH 224 +→ 2 4 2Si H 2SiH → 2 6 2 4 2Si H Si H H → +2 6 4 2Si H SiH SiH → +

*) P.Nachtigall, K. Jordan, A.Smith, and H.Jonsson (1996)