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

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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)

1 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

2 Self-Interaction in DFT Total energy of electronic system in the DFT: ( r) ρ( r ) 1 1 ρ E = Ψ ( r) + V Ψ ( r) dr+ drdr + E [ ρ, ρ ] DFT σ, i ext σ, i xc 1 σ, i r r Interaction of the density of the state Ψ σ, i () r with itself ( self-interaction ): ( r) ρ ( r ) 1 ρσ, i σ, i ESI ρ, i d d σ = + Exc, i,0 0 rr r r ρσ 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. E HF Exch * * ( r) ( r) ( r ) ( r ) occ occ 1 Ψ Ψ Ψ Ψ { σ, i ( )} Ψ r = drdr r r n m n m n m

3 In this talk: 1. Why self-interaction corrections?. 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.

4 Why Apply Self-Interaction? e - H H + 1. Some problems of DFT known (or suspected) to be caused by the SI: Size consistency (dissociation of H + ) Tendency towards charge and spin de-localization. If treated carefully, self-interaction corrections can help calculate accurate band gaps in semiconductors and insulators. 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 TiO rutile (110) surface. b) Complex spin structure of hematite Fe O 3 : formation of a small polaron. c) Localization of spin and charge densities around the structural defects. d) Self-trapping of excitons and holes in SiO.

5 DFT-SIC functional Remove self-interaction in a very straightforward way (Perdew & Zunger, 8): ( r) ρ ( r ) 1 ρ σ, i σ, i E DFT - SIC { ρ, i} σ = EDFT [ ρ()- r ] d d + Exc ρσ, i,0 σ, i r r - r r E SIC E DFT - SIC is an orbital-dependent functional. It can be minimized in two different ways: (1) Variation wrt orbitals () OEP δe () SIC { Ψ, i ( ) i σ } V r δe SIC, i ( ) SIC ( r) = V ρ σ r SIC ( r) = δψ ( r) δρ ( r) σ, i { } 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 () turns out to be much more difficult if at all possible.

6 Calculating the SIC energy ( r) ρ ( r ) 1 ρσ, i σ, i ESIC ψ, i d d σ = + Exc, i,0 rr r r ρσ 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. Bloch functions vs. Wannier functions delocalized over entire system localized (hopefully with a small unit cell) ikr ( r R) e ( r) Ψ + = Ψ nk nk ( ) The inverse transformation: i i Ψ r = e w r R n, kr k n i R ( ) ( ) i ikr 3 wn d e 3 BZ ( π ) V α V r R = k Ψ r n, k ( )

7 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: ( ) ( r) U k ( r) Ψ Ψ n, k nm m, k m Marzari and Vanderbilt ( 97) and Silvestrelli (000) developed a scheme for construction of ( ) the transformation producing maximally localized Wannier functions. U k nm In practical implementations, only few different k-vectors are used for Bloch functions. n kr n, k k BZ i ( r R) = Ψ ( r) w% e w(r-r 0 ) w(r-r Γ-point 6 ) k=0 andk=π/r Identify each WF with the piece-wise defined BF

8 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: f () r N ( 1 exp{ ( r / R) }) 1 = r N 1 (1-(1-exp(-(x/8)**8))**8)/abs(x) 1/abs(x) This kernel is nearly equal to 1/r for r<r and rapidly decays to 0 for r>r

9 Direct Minimization of DFT-SIC functional DFT K-S orbitals Unitary transformation (Maximally) localized Wannier functions Just a useful guess Calculation of SIC terms Self-consistent direct minimization i ( r) ( r) δedft SIC Ψ i δψ Total Energy, Geometric Structure, Ψ r i ( ) Ψ V Ψ = Ψ V Ψ () i ( j) i SIC j j SIC i H ˆ () i ij i H j = Ψ Ψ { } 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. ε i

10 FAQ: 1. Does SIC always lower the total energy? No. LDA total energy is always higher than LDA-SIC; GGA is typically lower than GGA-SIC.. 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. ε SIC Expect: E =N ε SIC SIC N Wannier functions: Bloch functions: E SIC /N=ε SIC E /N 0 size inconsistent SIC 3. If Bloch functions are excluded from search from the start, how does this formalism describe 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 decaying WF 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).

Using full SIC strongly overestimates the corrections.

damping factor of 0.4: EDFT -SIC = EDFT 0.

the band gaps appear to be insensitive to the value of this factor.

11 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: EDFT -SIC = EDFT 0.4 ESIC All qualitative effects such as charge/spin localization and opening of the band gaps appear to be insensitive to the value of this factor. Zero-gap semiconductor in LDA Ge Semiconductor in DFT+SIC SIC-driven metal-insulator transition as transition from algebraically to exponentially localized Wannier functions

12 Band gaps in several systems. System Single-Particle Band Gap (ev) Experiment DFT DFT-0.4*SIC Al O TiO (rutile) SiO HfO HfSiO 4? Fe O Si Ge Why Si and Ge are so much overcorrected?

13 Small polarons in hematite (Fe O 3 ) Antiferromagnetic ground state LDA Additional electron density LDA SIC Localization of the vacancy state on the rutile TiO (110) surface

14 1.3 ev N(ε) Conduction band Vacancy state 3 ev N(ε) Conduction band 0.8 ev Vacancy state 3. ev N(ε) Conduction band 1.0 ev Vacancy state Valence band top Valence band top Valence band top DFT (PBE96) Experiment PBE SIC What we learned: 1. DFT-SIC is capable of solving at least some serious problems of DFT.. 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 N rather than N 3.

15 Exact exchange functionals in plane-wave DFT E Exch * * ( r) ( r) ( r ) ( r ) occ occ 1 Ψ Ψ Ψ Ψ { σ, i ( )} Ψ r = drdr r r This looks much more difficult than n m n m n m * * ( r) ( r) ( r ) ( r ) occ occ Ψn Ψn Ψm Ψ m ρ() r ρ( r ) ECoul [ ρ() r ] = drdr = drdr r r r r n m Even for finite systems, E Exch is very involved, poorly scales with the size, and is a bottleneck in applications to larger molecules or clusters. For periodic systems, Ψ () r n 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. occ m occ * * ( r) ( r ) w ( r R) w ( r R) Ψ Ψ = m m m m m, R

16 E Exch = occ occ drdr n, R m, R * * ( r R) ( r R) ( r R ) ( r R ) w w w w r r n n m m 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 * w m * ( r R ) w ( r R ) m w n ( r R) n ( r R ) w R R R E Exch = 1 drdr occ occ n m * * ( r) ( r ) ( r) ( r ) w w w w n n m m r r - no k-vector integration w ( r R ) w ( r R) 3 But we can do even better: w1 ( r R) w4 ( r R)

17 If it is possible to create localized Wannier functions for a given system, occ * wn( r R) wn( r R) Frr (, ) = r r is short-range ( r r < L) occ n, R n, R * ( r R) ( r R) ( ) Frr (, ) = w w f r r n n L 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.

18 Band gaps in several systems calculated with the hybrid functional PBE0 System Single-Particle Gap (ev) Experiment PBE96 PBE0 Al O TiO (rutile) SiO Si Ge ev N(ε) Conduction band Vacancy state 3 ev N(ε) Conduction band 0.8 ev Vacancy state 3.1 ev N(ε) Conduction band 0.9 ev Vacancy state Valence band top Valence band top Valence band top DFT (PBE96) Experiment PBE0

21 PBE0 functional { σ, ( r) } [ 1, ] [ 1, ] hybrid Exc = α E Exch Ψ i + βex ρ ρ + Ec ρ ρ α = 0.5 β = 0.75 XC = PBE96 Similar to B3LYP functional.

density strongly localized on a single O atom; corresponding bond is longer

22 Localization of spin and charge densities by SIC (Al,Si)O 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

23 Bulk periodic system calculation with DFT - 0.4*SIC

24 (PBE96-0.4SIC) Calculation of Exciton in Rutile (TiO ) ρ ρ singlet triplet ground state excited state unrelaxed structure relaxed structure ρ triplet (electron) 1 ρ singlet not a selftrapped exciton! ρ triplet (hole) 1 ρ singlet

25 Band Gaps in Semiconductors and Insulators True band gap gap = EN ( + 1) EN ( ) + EN ( 1) Kohn-Sham band gap =ε ε KS c v Triplet-Singlet splitting triplet singlet gap = E ( N) E ( N) Discontinuity in V XC δ = gap KS SIC SIC SIC gap =εc εv V DFT-SIC δ 0 v SIC c SIC () r 0 V () r = 0 LDA or GGA δ = 0 Total energy difference between the ground (singlet) state and the lowest triplet state, with the exception 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 changes the 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).

26 In practical implementations, only few different k-vectors are used for Bloch functions. Discretized transformation to Wannier functions becomes: kr i ( ) kr ( ) i i i w% n r R = e Ψ n, = e kr k r e wn( r Ri) k BZ k BZ R For central cell R=0 and in the extreme case of single (Γ-point) k=0 the Wannier function w% ( r n ) is not at all localized and has the periodic images in every cell. If vectors k=0 and k=π/r are included, the periodicity of w% ( r) becomes R. i n Γ-point k=0 and k=π/r

27 ( ) What determines the choice of? U k nm 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: n, kr k n i R i i ( r) e w ( r R ) Ψ = ( ) i If we manage to make wn r Ri localized within a cell only one w r R contributes into each Ψ n, k ( r) ( ) n ( ) i Ψn, k r is also maximally localized determined almost piecewise (although it is still a delocalized function). ( ) U k ( r) Our goal is then to find which produces such nm Marzari-Vanderbilt MLWF Minimize the spread functional: Ω= r r n n n Ψ n, k r = 0n r 0n n, r 0 r 0 n = n n ( ) Rn = w n r R ;

Coulomb barrier to charge localization: DFT functionals that are not self-interaction free predict either

28 (PBE96-0.4SIC) Calculation of Exciton in Rutile, and Electron, and Hole In Anatase (TiO ) 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 Anatase TiO q = + 1 ρ doublet ρ doublet Anatase TiO q = 1 ρ doublet ρ doublet

29 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: EDFT -SIC = EDFT 0.4 ESIC 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. Single-particle energies in DFT-SIC formalism 1. For the orbitals minimizing DFT-SIC functional Ψ V Ψ = Ψ V Ψ () i ( j) i SIC j j SIC i and Hamiltonian matrix is hermitian (Pederson et al, 84). 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

30 Atomization energies of selected dimers (kcal/mol). Experiment Hartree-Fock Plane-Wave PBE96 PBE96-0.4*SIC PBE96-1.0*SIC O N P NO Reaction and transition state energies (ev) in silanes. Plane-Wave SiH4 SiH + H SiH4 SiH SiH6 SiH4 + SiH SiH6 SiH4 + H E E Si H SiH Si H 6 SiH 4 + SiH 4 rxn ts E rxn E rxn Si H 6 Si H 4 + H E rxn E ts1 E ts E LDA LDA-0.4*SIC PBE PBE96-1.0*SIC PBE96-0.4*SIC HF B3LYP *) QCISD(T) *) *) P.Nachtigall, K. Jordan, A.Smith, and H.Jonsson (1996)

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