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Exact Exchange in Density Functional Theory

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ABDERRAHMANE REGGAD

This document discusses density functional theory (DFT) and exact exchange methods. It provides background on DFT, the Kohn-Sham equations, and common exchange-correlation functionals like the local density approximation (LDA) and generalized gradient approximations (GGA). It then introduces exact exchange (EXX) methods, which neglect correlation and use the Hartree-Fock exchange energy. Calculating the functional derivative of the exchange energy is discussed to obtain the exchange potential within the Kohn-Sham scheme for EXX.

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Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Exact Exchange in Density Functional Theory S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Institut fu¨rPhysik Karl-Franzens-Universit¨ta Graz Universita¨tsplatz 5 A-8010Graz, Austria 19th January 2005 S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Density functional theory Many-body theory involves solving for functions in 3N coordinates: Ψ(r1,r2,... ,rN ) Density functional theory (DFT) requires solving for functions in 3 coordinates: n(r) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Density functional theory Many-body theory involves solving for functions in 3N coordinates: Ψ(r1,r2,... ,rN ) Density functional theory (DFT) requires solving for functions in 3 coordinates: n(r) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Density functional theory Hohenberg-Kohn Theorems (1964) 1 2 External potential v(r) is uniquely determined by n(r) The variational principle holds E0 = Ev0 [n0] < Ev0 [n] 3 Ev0[n] = F [n]+ ¸ dr v0(r)n(r) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Density functional theory Hohenberg-Kohn Theorems (1964) 1 2 External potential v(r) is uniquely determined by n(r) The variational principle holds E0 = Ev0 [n0] < Ev0 [n] 3 Ev0[n] = F [n]+ ¸ dr v0(r)n(r) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Density functional theory Hohenberg-Kohn Theorems (1964) 1 2 External potential v(r) is uniquely determined by n(r) The variational principle holds E0 = Ev0 [n0] < Ev0 [n] 3 Ev0[n] = F [n]+ ¸ dr v0(r)n(r) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Density functional theory DFT is an exact theory for interacting systems in the ground state S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Kohn-Sham equations Find set of auxilliary single particle orbitals such that 1 2 . . − 2 ∇ + vs(r) φi(r) = siφi(r) and occ n(r) = . |φi(r)|2. i If 1 ¸ F [n] =Ts[n] + 2 dr drt n(r)n(rt) |r −rt| + Exc[n] S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Kohn-Sham equations Find set of auxilliary single particle orbitals such that 1 2 . . − 2 ∇ + vs(r) φi(r) = siφi(r) and occ n(r) = . |φi(r)|2. i If 1 ¸ F [n] =Ts[n] + 2 dr drt n(r)n(rt) |r −rt| + Exc[n] S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Kohn-Sham equations Then ¸ vs[n](r) = v(r) + drt n(rt) |r− rt| + vxc[n](r) where vxc[n](r) = δExc[n] . δn(r) Many sins hidden in Exc[n] ! S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Kohn-Sham equations Then ¸ vs[n](r) = v(r) + drt n(rt) |r− rt| + vxc[n](r) where vxc[n](r) = δExc[n] . δn(r) Many sins hidden in Exc[n] ! S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals First generation : Local density approximation (LDA) ELDA xc ¸ unif[n] = dr n(r)exc (n(r)) Second generation : Generalised gradient approximations EGGA xc ¸ [n] = dr f (n(r),∇n(r)) EMeta−GGA xc ¸ [n] = dr g(n(r),∇n(r),τ (r)) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals First generation : Local density approximation (LDA) ELDA xc ¸ unif[n] = dr n(r)exc (n(r)) Second generation : Generalised gradient approximations EGGA xc ¸ [n] = dr f (n(r),∇n(r)) EMeta−GGA xc ¸ [n] = dr g(n(r),∇n(r),τ (r)) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals First generation : Local density approximation (LDA) ELDA xc ¸ unif[n] = dr n(r)exc (n(r)) Second generation : Generalised gradient approximations EGGA xc ¸ [n] = dr f (n(r),∇n(r)) EMeta−GGA xc ¸ [n] = dr g(n(r),∇n(r),τ (r)) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals Development of newfunctionals leads not only to improved accuracy but also correct qualitative features S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals Third generation: Exact exchange (EXX) Neglect correlation and use the Hartree-Fock exchange energy Ex[n] = − 2 . i,j 1 occ ¸ dr drt φ∗ ∗ t t i (r)φj (r )φj(r)φi(r ) |r− rt| To solve the Kohn-Sham system we require δEx[n] vx[n](r) = δn(r) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals Third generation: Exact exchange (EXX) Neglect correlation and use the Hartree-Fock exchange energy Ex[n] = − 2 . i,j 1 occ ¸ dr drt φ∗ ∗ t t i (r)φj (r )φj(r)φi(r ) |r− rt| To solve the Kohn-Sham system we require δEx[n] vx[n](r) = δn(r) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals Using the functional derivative chain rule: i occ ¸ vx[n](r) = . dr drtt . δEx δφi(rtt) + i i δφi(rtt) δvs(rt) δφ∗(rtt) δvs(rt) δEx δφ∗(rtt) . δvs(rt) δn(r) ¸ = drt occ unocc . . i j i NL x j(φ |vˆ |φ ) j iφ∗(rt)φ (rt) si −sj + c.c. δvs(rt) δn(r) , where ik NL x (φ |vˆ | occ. lkt φ ) = wjk kt ¸ dr drt t ik lktφ∗ (r)φlk (r)φ∗ (rt)φjk(rt) |r− rt| . LR S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals Using the linear-response operator tχ(r,r ) ≡ δn(r) s tδv (r ) = occ unocc . . i j ∗ i j i ∗ t t φ (r)φ (r)φ (r )φ (r )j si − sj + c.c. we have δn(r) δvs(rt) = χ˜−1(r,rt). basis S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals Exact exchangeis NOT Hartree-Fock a mean-field theory an attempt to solve the quasi-particle equation a parameterised correction to LDA easyto implement S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals Exact exchangeis NOT Hartree-Fock a mean-field theory an attempt to solve the quasi-particle equation a parameterised correction to LDA easyto implement S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals Exact exchangeis NOT Hartree-Fock a mean-field theory an attempt to solve the quasi-particle equation a parameterised correction to LDA easyto implement S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals Exact exchangeis NOT Hartree-Fock a mean-field theory an attempt to solve the quasi-particle equation a parameterised correction to LDA easyto implement S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals Exact exchangeis NOT Hartree-Fock a mean-field theory an attempt to solve the quasi-particle equation a parameterised correction to LDA easyto implement S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Pseudopotentials (PP) atomic coreis frozen and representedby a non-local potential doesnot react properly to the solid state environment: no relaxation of core states planewavesare used as the basis S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Pseudopotentials (PP) atomic coreis frozen and representedby a non-local potential doesnot react properly to the solid state environment: no relaxation of core states planewavesare used as the basis S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Pseudopotentials (PP) atomic coreis frozen and representedby a non-local potential doesnot react properly to the solid state environment: no relaxation of core states planewavesare used as the basis S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Atomic sphereapproximation (ASA) includes core orbitals potential is spherically symmetric muffin-tins are space-filling S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Atomic sphereapproximation (ASA) includes core orbitals potential is spherically symmetric muffin-tins are space-filling S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Atomic sphereapproximation (ASA) includes core orbitals potential is spherically symmetric muffin-tins are space-filling S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Full-potential linearised augmented planewaves(FP-LAPW) includes core orbitals potential is fully described space divided into muffin-tin and interstitial regions most precise method available MT I II I S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Full-potential linearised augmented planewaves(FP-LAPW) includes core orbitals potential is fully described space divided into muffin-tin and interstitial regions most precise method available MT I II I S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Full-potential linearised augmented planewaves(FP-LAPW) includes core orbitals potential is fully described space divided into muffin-tin and interstitial regions most precise method available MT I II I S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Full-potential linearised augmented planewaves(FP-LAPW) includes core orbitals potential is fully described space divided into muffin-tin and interstitial regions most precise method available MT I II I S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Full-potential linearised augmented planewaves(FP-LAPW) includes core orbitals potential is fully described space divided into muffin-tin and interstitial regions most precise method available MT I II I S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Basis set An efficient basis set is required so that the response t χ(r,r ) ≡ δn(r) s tδv (r ) can be inverted Basis set should not contain constant functions response S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Basis set An efficient basis set is required so that the response t χ(r,r ) ≡ δn(r) s tδv (r ) can be inverted Basis set should not contain constant functions response S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Choose the overlap densities ik ρα(r) ≡ φ∗ (r)φjk(r), and complex conjugates, where α ≡ (ik, jk). Diagonalise ¸ α Oαβ ≡ drρ∗(r)ρβ (r), and eliminate eigenvectorswith small eigenvalues. S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Choose the overlap densities ik ρα(r) ≡ φ∗ (r)φjk(r), and complex conjugates, where α ≡ (ik, jk). Diagonalise ¸ α Oαβ ≡ drρ∗(r)ρβ (r), and eliminate eigenvectorswith small eigenvalues. S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Find transformation matrix C such that if α β ρ˜ (r) = Cβ . . γ α β γ γv ρ (r) then ¸ αdr ρ˜∗(r)ρ˜β(r) = δαβ. The matrix equation CC† = . Ov .−1 v† is solved by Cholesky decomposition. S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Find transformation matrix C such that if α β ρ˜ (r) = Cβ . . γ α β γ γv ρ (r) then ¸ αdr ρ˜∗(r)ρ˜β(r) = δαβ. The matrix equation CC† = . Ov .−1 v† is solved by Cholesky decomposition. S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities By construction ¸ dr ρ˜α(r)= 0, so {ρ˜α} form an ideal basis for inversion of χ S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities NL Lastly, given q ≡ k − kt, the long range coulomb term of the NL matrix elements NL φik x jk LR occ. lq (φ |vˆ | ) = wq 4πΩ q2 ilρ∗(q)ρlj(q), where ρil(q) and ρlj (q) are the pseudo-charge densities. Poor convergence with respect to the number of q-points. Approximate it by an integral over a sphereof volume equivalent to that of the BZ NL ik x jk LR(φ |vˆ |φ ) c 2 π . 6Ω5 .1/3 occ. lq q ∗ il ljw ρ (q)ρ (q). S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities NL Lastly, given q ≡ k − kt, the long range coulomb term of the NL matrix elements NL φ occ. lq (φ |vˆ | ) = wik x jk LR q 4πΩ q2 ilρ∗(q)ρlj(q), where ρil(q) and ρlj (q) are the pseudo-charge densities. Poor convergence with respect to the number of q-points. Approximate it by an integral over a sphereof volume equivalent to that of the BZ NL ik x(φ |vˆ | jk LRφ ) c 2 π . 6Ω5 .1/3 occ. lq q ∗ il ljw ρ (q)ρ (q). S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities NL Lastly, given q ≡ k − kt, the long range coulomb term of the NL matrix elements NL φik x jk LR occ. lq (φ |vˆ | ) = wq 4πΩ q2 ilρ∗(q)ρlj(q), where ρil(q) and ρlj (q) are the pseudo-charge densities. Poor convergence with respect to the number of q-points. Approximate it by an integral over a sphereof volume equivalent to that of the BZ NL ik x(φ |vˆ | jk LRφ ) c 2 π . 6Ω5 .1/3 occ. lq q ∗ il ljw ρ (q)ρ (q). S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Applications EXX Applied to 1 2 Magnetic metals Semiconductors and insulators S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook DOS Potential Magnetic metals: introduction to the problem Magnetic moment in Bohr magneton Compound FP-LDA Experiment FeAl 0.71 0.0 1 2 P. Mohn et al. Phys. Rev. Lett. 87 196401 (2001): LDA+U Petukhov et al. Phys. Rev. B 67 153106 (2003): LDA+U +DMFT S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook DOS Potential Magnetic metals: introduction to the problem Magnetic moment in Bohr magneton Compound FP-LDA Experiment FeAl 0.71 0.0 1 2 P. Mohn et al. Phys. Rev. Lett. 87 196401 (2001): LDA+U Petukhov et al. Phys. Rev. B 67 153106 (2003): LDA+U +DMFT S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook DOS Potential Magnetic metals: FeAl -6 -4 -2 2 4 60 Energy [eV] 0 3 2 1 DOS(states/eV/spin) LDA Expt S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook DOS Potential Magnetic metals: FeAl -6 -4 -2 2 4 60 Energy [eV] 0 3 2 1 DOS(states/eV/spin) LDA EXX Expt S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook DOS Potential Magnetic metals: FeAl S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook DOS Potential Magnetic metals: FeAl -400 -300 -200 100 0 -100 Energy[eV] LDA EXX FeAl Al S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook DOS Potential Magnetic metals: FeAl -6 -4 -2 2 4 60 Energy [eV] 0 1 2 3 4 DOS(states/eV/spin) LDA EXX-sph EXX S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook DOS Potential Magnetic metals: stringent tests for EXX Magnetic moment in Bohr magneton Compound FP-LDA FP-EXX Experiment FeAl 0.71 0.0 0.0 Ni3Ga 0.79 ? 0.0 Ni3Al 0.70 ? 0.23 http://arxiv.org/abs/cond-mat/0501258 S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook DOS Potential Magnetic metals: FeAl, Ni3Ga and Ni3Al Magnetic moment in Bohr magneton Compound FP-LDA FP-EXX Experiment FeAl 0.71 0.0 0.0 Ni3Ga 0.79 0.0 0.0 Ni3Al 0.70 0.20 0.23 http://arxiv.org/abs/cond-mat/0501258 S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition -2 Ge GaAs CdS 2 KS Eg-Eg(eV) PP-EXX FP-LDA Si ZnS C BN Ar Ne Kr Xe 0 1 -2 0 -4 -1 -6 Semiconductors and insulators: band-gaps PP-EXX : Städele et al. PRB 59 10031 PP-EXX: Magyar et al. PRB 69045111 S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition The fundamental band-gap for materials: Eg = A−I = (EN+1−EN )−(EN −EN −1) δE δE Eg = δn+ − δn− E[n] = T[n] + Us[n]+ Exc[n] δT δT δvxc δvxc Eg = ( δn+ − δn− ) + ( δn+ − δn− ) gEg = EKS + ∆xc S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition The fundamental band-gap for materials: Eg = A−I = (EN+1−EN )−(EN −EN −1) δE δE Eg = δn+ − δn− E[n] = T[n] + Us[n]+ Exc[n] δT δT δvxc δvxc Eg = ( δn+ − δn− ) + ( δn+ − δn− ) gEg = EKS + ∆xc S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition The fundamental band-gap for materials: Eg = A−I = (EN+1−EN )−(EN −EN −1) δE δE Eg = δn+ − δn− E[n] = T[n] + Us[n]+ Exc[n] δT δT δvxc δvxc Eg = ( δn+ − δn− ) + ( δn+ − δn− ) gEg = EKS + ∆xc S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition The fundamental band-gap for materials: Eg = A−I = (EN+1−EN )−(EN −EN −1) δE δE Eg = δn+ − δn− E[n] = T[n] + Us[n]+ Exc[n] δT δT δvxc δvxc Eg = ( δn+ − δn− ) + ( δn+ − δn− ) gEg = EKS + ∆xc S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition The fundamental band-gap for materials: Eg = A−I = (EN+1−EN )−(EN −EN −1) δE δE Eg = δn+ − δn− E[n] = T[n] + Us[n]+ Exc[n] δT δT δvxc δvxc Eg = ( δn+ − δn− ) + ( δn+ − δn− ) gEg = EKS + ∆xc S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition Semiconductors and insulators: band-gaps 2 1 KS Eg-Eg(eV) PP-EXX LMTO-ASA-EXX FP-LDA 0 -4 -2 0 Kotani PRL 742989 -1 -6 -2 Ge GaAs CdS Si ZnS C BN Ar Ne Kr Xe S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition Semiconductors and insulators: band-gaps 2 1 KS Eg-Eg(eV) FP-EXX PP-EXX LMTO-ASA-EXX FP-LDA 0 -4 -2 0 -1 -6 -2 Ge GaAs CdS Si ZnS C BN Ar Ne Kr Xe S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition Semiconductors and insulators: band-gaps 2 1 KS Eg-Eg(eV) FP-EXX PP-EXX LMTO-ASA-EXX FP-EXX-ncv FP-LDA 0 -4 -2 0 http://arxiv.org/abs/cond-mat/0501353 -1 -6 -2 Ge GaAs CdS Si ZnS C BN Ar Ne Kr Xe S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition Semiconductors and insulators: d-band position -4 -2 2 0 4 Ge GaAs InP ZnS CdS KS Ed-Ed(eV) PP-EXX SIRC PP-SIC FP-LDA SIC: Vogle et al. PRB 54 5495 PP-EXX: Rinke et al.(unpublished) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition Semiconductors and insulators: d-band position -4 -2 2 0 4 Ge GaAs InP ZnS CdS KS Ed-Ed(eV) PP-EXX GW-FPLMTO GW-LMTO-ASA GW-PP-EXX SIRC PP-SIC FP-LDA GW LMTO-ASA: Aryasetiawan et al. PRB 54 17564 FPLMTO: Kotani et al. SSC121 S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition Semiconductors and insulators: d-band position -4 -2 2 0 4 Ge GaAs InP ZnS CdS KS Ed-Ed(eV) FP-EXX PP-EXX GW-FPLMTO GW-LMTO-ASA GW-PP-EXX SIRC PP-SIC FP-EXX-ncv FP-LDA http://arxiv.org/abs/cond-mat/0501353 S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Conclusions Outlook Conclusions 1 2 3 4 5 EXX within an all electron full potential method is implemented within EXC!TING code. Right nowthis is the only FP code to be able to do EXX. A newand one of the most optimal basis is proposed for calculating and inverting the response.This basis may be useful for future TD-DFT and GW calculations. Magnetic metals: Asymmetry in exchangepotential is very important to get the correct ground-state. Semiconductors and insulators: Core-valence interaction is crucial for correct treatment of the EXX. Lack of Asymmetry and/or core-valence interaction could lead to spurious agreement with experiments. S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Conclusions Outlook Conclusions 1 2 3 4 5 EXX within an all electron full potential method is implemented within EXC!TING code. Right nowthis is the only FP code to be able to do EXX. A newand one of the most optimal basis is proposed for calculating and inverting the response.This basis may be useful for future TD-DFT and GW calculations. Magnetic metals: Asymmetry in exchangepotential is very important to get the correct ground-state. Semiconductors and insulators: Core-valence interaction is crucial for correct treatment of the EXX. Lack of Asymmetry and/or core-valence interaction could lead to spurious agreement with experiments. S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Conclusions Outlook Conclusions 1 2 3 4 5 EXX within an all electron full potential method is implemented within EXC!TING code. Right nowthis is the only FP code to be able to do EXX. A newand one of the most optimal basis is proposed for calculating and inverting the response.This basis may be useful for future TD-DFT and GW calculations. Magnetic metals: Asymmetry in exchangepotential is very important to get the correct ground-state. Semiconductors and insulators: Core-valence interaction is crucial for correct treatment of the EXX. Lack of Asymmetry and/or core-valence interaction could lead to spurious agreement with experiments. S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Conclusions Outlook Conclusions 1 2 3 4 5 EXX within an all electron full potential method is implemented within EXC!TING code. Right nowthis is the only FP code to be able to do EXX. A newand one of the most optimal basis is proposed for calculating and inverting the response.This basis may be useful for future TD-DFT and GW calculations. Magnetic metals: Asymmetry in exchangepotential is very important to get the correct ground-state. Semiconductors and insulators: Core-valence interaction is crucial for correct treatment of the EXX. Lack of Asymmetry and/or core-valence interaction could lead to spurious agreement with experiments. S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Conclusions Outlook Conclusions 1 2 3 4 5 EXX within an all electron full potential method is implemented within EXC!TING code. Right nowthis is the only FP code to be able to do EXX. A newand one of the most optimal basis is proposed for calculating and inverting the response.This basis may be useful for future TD-DFT and GW calculations. Magnetic metals: Asymmetry in exchangepotential is very important to get the correct ground-state. Semiconductors and insulators: Core-valence interaction is crucial for correct treatment of the EXX. Lack of Asymmetry and/or core-valence interaction could lead to spurious agreement with experiments. S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Conclusions Outlook Outlook 1 2 EXX can be generalised to handle non-collinear magnetism (derivatives w.r.t. nσσt (r) ). Future inclusion of exact correlation may be possible (multiconfiguration approach or adiabatic fluctuation dissipation) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Conclusions Outlook Outlook 1 2 EXX can be generalised to handle non-collinear magnetism (derivatives w.r.t. nσσt (r) ). Future inclusion of exact correlation may be possible (multiconfiguration approach or adiabatic fluctuation dissipation) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Conclusions Outlook Acknowledgements Prof. P. Mohn Magnetic metals workdone in collaboration with Dr. C. Persson. Austrian Science Fund (project P16227) EXCITING network funded by the EU (HPRN-CT-2002-00317) Code available at: http://p h y si k . kf u n i g raz.ac.at/ ∼ k de/ secret garden/exciting.html S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory

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Exact Exchange in Density Functional Theory

  • 1. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Exact Exchange in Density Functional Theory S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Institut fu¨rPhysik Karl-Franzens-Universit¨ta Graz Universita¨tsplatz 5 A-8010Graz, Austria 19th January 2005 S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 2. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Density functional theory Many-body theory involves solving for functions in 3N coordinates: Ψ(r1,r2,... ,rN ) Density functional theory (DFT) requires solving for functions in 3 coordinates: n(r) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 3. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Density functional theory Many-body theory involves solving for functions in 3N coordinates: Ψ(r1,r2,... ,rN ) Density functional theory (DFT) requires solving for functions in 3 coordinates: n(r) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 4. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Density functional theory Hohenberg-Kohn Theorems (1964) 1 2 External potential v(r) is uniquely determined by n(r) The variational principle holds E0 = Ev0 [n0] < Ev0 [n] 3 Ev0[n] = F [n]+ ¸ dr v0(r)n(r) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 5. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Density functional theory Hohenberg-Kohn Theorems (1964) 1 2 External potential v(r) is uniquely determined by n(r) The variational principle holds E0 = Ev0 [n0] < Ev0 [n] 3 Ev0[n] = F [n]+ ¸ dr v0(r)n(r) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 6. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Density functional theory Hohenberg-Kohn Theorems (1964) 1 2 External potential v(r) is uniquely determined by n(r) The variational principle holds E0 = Ev0 [n0] < Ev0 [n] 3 Ev0[n] = F [n]+ ¸ dr v0(r)n(r) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 7. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Density functional theory DFT is an exact theory for interacting systems in the ground state S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 8. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Kohn-Sham equations Find set of auxilliary single particle orbitals such that 1 2 . . − 2 ∇ + vs(r) φi(r) = siφi(r) and occ n(r) = . |φi(r)|2. i If 1 ¸ F [n] =Ts[n] + 2 dr drt n(r)n(rt) |r −rt| + Exc[n] S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 9. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Kohn-Sham equations Find set of auxilliary single particle orbitals such that 1 2 . . − 2 ∇ + vs(r) φi(r) = siφi(r) and occ n(r) = . |φi(r)|2. i If 1 ¸ F [n] =Ts[n] + 2 dr drt n(r)n(rt) |r −rt| + Exc[n] S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 10. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Kohn-Sham equations Then ¸ vs[n](r) = v(r) + drt n(rt) |r− rt| + vxc[n](r) where vxc[n](r) = δExc[n] . δn(r) Many sins hidden in Exc[n] ! S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 11. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Kohn-Sham equations Then ¸ vs[n](r) = v(r) + drt n(rt) |r− rt| + vxc[n](r) where vxc[n](r) = δExc[n] . δn(r) Many sins hidden in Exc[n] ! S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 12. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals First generation : Local density approximation (LDA) ELDA xc ¸ unif[n] = dr n(r)exc (n(r)) Second generation : Generalised gradient approximations EGGA xc ¸ [n] = dr f (n(r),∇n(r)) EMeta−GGA xc ¸ [n] = dr g(n(r),∇n(r),τ (r)) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 13. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals First generation : Local density approximation (LDA) ELDA xc ¸ unif[n] = dr n(r)exc (n(r)) Second generation : Generalised gradient approximations EGGA xc ¸ [n] = dr f (n(r),∇n(r)) EMeta−GGA xc ¸ [n] = dr g(n(r),∇n(r),τ (r)) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 14. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals First generation : Local density approximation (LDA) ELDA xc ¸ unif[n] = dr n(r)exc (n(r)) Second generation : Generalised gradient approximations EGGA xc ¸ [n] = dr f (n(r),∇n(r)) EMeta−GGA xc ¸ [n] = dr g(n(r),∇n(r),τ (r)) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 15. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals Development of newfunctionals leads not only to improved accuracy but also correct qualitative features S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 16. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals Third generation: Exact exchange (EXX) Neglect correlation and use the Hartree-Fock exchange energy Ex[n] = − 2 . i,j 1 occ ¸ dr drt φ∗ ∗ t t i (r)φj (r )φj(r)φi(r ) |r− rt| To solve the Kohn-Sham system we require δEx[n] vx[n](r) = δn(r) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 17. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals Third generation: Exact exchange (EXX) Neglect correlation and use the Hartree-Fock exchange energy Ex[n] = − 2 . i,j 1 occ ¸ dr drt φ∗ ∗ t t i (r)φj (r )φj(r)φi(r ) |r− rt| To solve the Kohn-Sham system we require δEx[n] vx[n](r) = δn(r) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 18. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals Using the functional derivative chain rule: i occ ¸ vx[n](r) = . dr drtt . δEx δφi(rtt) + i i δφi(rtt) δvs(rt) δφ∗(rtt) δvs(rt) δEx δφ∗(rtt) . δvs(rt) δn(r) ¸ = drt occ unocc . . i j i NL x j(φ |vˆ |φ ) j iφ∗(rt)φ (rt) si −sj + c.c. δvs(rt) δn(r) , where ik NL x (φ |vˆ | occ. lkt φ ) = wjk kt ¸ dr drt t ik lktφ∗ (r)φlk (r)φ∗ (rt)φjk(rt) |r− rt| . LR S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 19. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals Using the linear-response operator tχ(r,r ) ≡ δn(r) s tδv (r ) = occ unocc . . i j ∗ i j i ∗ t t φ (r)φ (r)φ (r )φ (r )j si − sj + c.c. we have δn(r) δvs(rt) = χ˜−1(r,rt). basis S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 20. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals Exact exchangeis NOT Hartree-Fock a mean-field theory an attempt to solve the quasi-particle equation a parameterised correction to LDA easyto implement S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 21. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals Exact exchangeis NOT Hartree-Fock a mean-field theory an attempt to solve the quasi-particle equation a parameterised correction to LDA easyto implement S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 22. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals Exact exchangeis NOT Hartree-Fock a mean-field theory an attempt to solve the quasi-particle equation a parameterised correction to LDA easyto implement S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 23. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals Exact exchangeis NOT Hartree-Fock a mean-field theory an attempt to solve the quasi-particle equation a parameterised correction to LDA easyto implement S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 24. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Exchange-correlation functionals Exact exchangeis NOT Hartree-Fock a mean-field theory an attempt to solve the quasi-particle equation a parameterised correction to LDA easyto implement S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 25. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Pseudopotentials (PP) atomic coreis frozen and representedby a non-local potential doesnot react properly to the solid state environment: no relaxation of core states planewavesare used as the basis S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 26. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Pseudopotentials (PP) atomic coreis frozen and representedby a non-local potential doesnot react properly to the solid state environment: no relaxation of core states planewavesare used as the basis S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 27. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Pseudopotentials (PP) atomic coreis frozen and representedby a non-local potential doesnot react properly to the solid state environment: no relaxation of core states planewavesare used as the basis S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 28. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Atomic sphereapproximation (ASA) includes core orbitals potential is spherically symmetric muffin-tins are space-filling S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 29. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Atomic sphereapproximation (ASA) includes core orbitals potential is spherically symmetric muffin-tins are space-filling S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 30. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Atomic sphereapproximation (ASA) includes core orbitals potential is spherically symmetric muffin-tins are space-filling S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 31. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Full-potential linearised augmented planewaves(FP-LAPW) includes core orbitals potential is fully described space divided into muffin-tin and interstitial regions most precise method available MT I II I S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 32. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Full-potential linearised augmented planewaves(FP-LAPW) includes core orbitals potential is fully described space divided into muffin-tin and interstitial regions most precise method available MT I II I S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 33. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Full-potential linearised augmented planewaves(FP-LAPW) includes core orbitals potential is fully described space divided into muffin-tin and interstitial regions most precise method available MT I II I S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 34. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Full-potential linearised augmented planewaves(FP-LAPW) includes core orbitals potential is fully described space divided into muffin-tin and interstitial regions most precise method available MT I II I S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 35. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Full-potential linearised augmented planewaves(FP-LAPW) includes core orbitals potential is fully described space divided into muffin-tin and interstitial regions most precise method available MT I II I S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 36. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Basis set An efficient basis set is required so that the response t χ(r,r ) ≡ δn(r) s tδv (r ) can be inverted Basis set should not contain constant functions response S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 37. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Basis set An efficient basis set is required so that the response t χ(r,r ) ≡ δn(r) s tδv (r ) can be inverted Basis set should not contain constant functions response S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 38. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Choose the overlap densities ik ρα(r) ≡ φ∗ (r)φjk(r), and complex conjugates, where α ≡ (ik, jk). Diagonalise ¸ α Oαβ ≡ drρ∗(r)ρβ (r), and eliminate eigenvectorswith small eigenvalues. S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 39. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Choose the overlap densities ik ρα(r) ≡ φ∗ (r)φjk(r), and complex conjugates, where α ≡ (ik, jk). Diagonalise ¸ α Oαβ ≡ drρ∗(r)ρβ (r), and eliminate eigenvectorswith small eigenvalues. S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 40. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Find transformation matrix C such that if α β ρ˜ (r) = Cβ . . γ α β γ γv ρ (r) then ¸ αdr ρ˜∗(r)ρ˜β(r) = δαβ. The matrix equation CC† = . Ov .−1 v† is solved by Cholesky decomposition. S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 41. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities Find transformation matrix C such that if α β ρ˜ (r) = Cβ . . γ α β γ γv ρ (r) then ¸ αdr ρ˜∗(r)ρ˜β(r) = δαβ. The matrix equation CC† = . Ov .−1 v† is solved by Cholesky decomposition. S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 42. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities By construction ¸ dr ρ˜α(r)= 0, so {ρ˜α} form an ideal basis for inversion of χ S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 43. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities NL Lastly, given q ≡ k − kt, the long range coulomb term of the NL matrix elements NL φik x jk LR occ. lq (φ |vˆ | ) = wq 4πΩ q2 ilρ∗(q)ρlj(q), where ρil(q) and ρlj (q) are the pseudo-charge densities. Poor convergence with respect to the number of q-points. Approximate it by an integral over a sphereof volume equivalent to that of the BZ NL ik x jk LR(φ |vˆ |φ ) c 2 π . 6Ω5 .1/3 occ. lq q ∗ il ljw ρ (q)ρ (q). S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 44. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities NL Lastly, given q ≡ k − kt, the long range coulomb term of the NL matrix elements NL φ occ. lq (φ |vˆ | ) = wik x jk LR q 4πΩ q2 ilρ∗(q)ρlj(q), where ρil(q) and ρlj (q) are the pseudo-charge densities. Poor convergence with respect to the number of q-points. Approximate it by an integral over a sphereof volume equivalent to that of the BZ NL ik x(φ |vˆ | jk LRφ ) c 2 π . 6Ω5 .1/3 occ. lq q ∗ il ljw ρ (q)ρ (q). S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 45. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Practicalities NL Lastly, given q ≡ k − kt, the long range coulomb term of the NL matrix elements NL φik x jk LR occ. lq (φ |vˆ | ) = wq 4πΩ q2 ilρ∗(q)ρlj(q), where ρil(q) and ρlj (q) are the pseudo-charge densities. Poor convergence with respect to the number of q-points. Approximate it by an integral over a sphereof volume equivalent to that of the BZ NL ik x(φ |vˆ | jk LRφ ) c 2 π . 6Ω5 .1/3 occ. lq q ∗ il ljw ρ (q)ρ (q). S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 46. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Density functional theory Kohn-Sham equations Exchange-correlation functionals Practicalities Applications EXX Applied to 1 2 Magnetic metals Semiconductors and insulators S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 47. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook DOS Potential Magnetic metals: introduction to the problem Magnetic moment in Bohr magneton Compound FP-LDA Experiment FeAl 0.71 0.0 1 2 P. Mohn et al. Phys. Rev. Lett. 87 196401 (2001): LDA+U Petukhov et al. Phys. Rev. B 67 153106 (2003): LDA+U +DMFT S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 48. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook DOS Potential Magnetic metals: introduction to the problem Magnetic moment in Bohr magneton Compound FP-LDA Experiment FeAl 0.71 0.0 1 2 P. Mohn et al. Phys. Rev. Lett. 87 196401 (2001): LDA+U Petukhov et al. Phys. Rev. B 67 153106 (2003): LDA+U +DMFT S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 49. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook DOS Potential Magnetic metals: FeAl -6 -4 -2 2 4 60 Energy [eV] 0 3 2 1 DOS(states/eV/spin) LDA Expt S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 50. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook DOS Potential Magnetic metals: FeAl -6 -4 -2 2 4 60 Energy [eV] 0 3 2 1 DOS(states/eV/spin) LDA EXX Expt S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 51. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook DOS Potential Magnetic metals: FeAl S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 52. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook DOS Potential Magnetic metals: FeAl -400 -300 -200 100 0 -100 Energy[eV] LDA EXX FeAl Al S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 53. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook DOS Potential Magnetic metals: FeAl -6 -4 -2 2 4 60 Energy [eV] 0 1 2 3 4 DOS(states/eV/spin) LDA EXX-sph EXX S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 54. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook DOS Potential Magnetic metals: stringent tests for EXX Magnetic moment in Bohr magneton Compound FP-LDA FP-EXX Experiment FeAl 0.71 0.0 0.0 Ni3Ga 0.79 ? 0.0 Ni3Al 0.70 ? 0.23 http://arxiv.org/abs/cond-mat/0501258 S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 55. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook DOS Potential Magnetic metals: FeAl, Ni3Ga and Ni3Al Magnetic moment in Bohr magneton Compound FP-LDA FP-EXX Experiment FeAl 0.71 0.0 0.0 Ni3Ga 0.79 0.0 0.0 Ni3Al 0.70 0.20 0.23 http://arxiv.org/abs/cond-mat/0501258 S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 56. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition -2 Ge GaAs CdS 2 KS Eg-Eg(eV) PP-EXX FP-LDA Si ZnS C BN Ar Ne Kr Xe 0 1 -2 0 -4 -1 -6 Semiconductors and insulators: band-gaps PP-EXX : Städele et al. PRB 59 10031 PP-EXX: Magyar et al. PRB 69045111 S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 57. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition The fundamental band-gap for materials: Eg = A−I = (EN+1−EN )−(EN −EN −1) δE δE Eg = δn+ − δn− E[n] = T[n] + Us[n]+ Exc[n] δT δT δvxc δvxc Eg = ( δn+ − δn− ) + ( δn+ − δn− ) gEg = EKS + ∆xc S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 58. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition The fundamental band-gap for materials: Eg = A−I = (EN+1−EN )−(EN −EN −1) δE δE Eg = δn+ − δn− E[n] = T[n] + Us[n]+ Exc[n] δT δT δvxc δvxc Eg = ( δn+ − δn− ) + ( δn+ − δn− ) gEg = EKS + ∆xc S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 59. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition The fundamental band-gap for materials: Eg = A−I = (EN+1−EN )−(EN −EN −1) δE δE Eg = δn+ − δn− E[n] = T[n] + Us[n]+ Exc[n] δT δT δvxc δvxc Eg = ( δn+ − δn− ) + ( δn+ − δn− ) gEg = EKS + ∆xc S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 60. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition The fundamental band-gap for materials: Eg = A−I = (EN+1−EN )−(EN −EN −1) δE δE Eg = δn+ − δn− E[n] = T[n] + Us[n]+ Exc[n] δT δT δvxc δvxc Eg = ( δn+ − δn− ) + ( δn+ − δn− ) gEg = EKS + ∆xc S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 61. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition The fundamental band-gap for materials: Eg = A−I = (EN+1−EN )−(EN −EN −1) δE δE Eg = δn+ − δn− E[n] = T[n] + Us[n]+ Exc[n] δT δT δvxc δvxc Eg = ( δn+ − δn− ) + ( δn+ − δn− ) gEg = EKS + ∆xc S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 62. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition Semiconductors and insulators: band-gaps 2 1 KS Eg-Eg(eV) PP-EXX LMTO-ASA-EXX FP-LDA 0 -4 -2 0 Kotani PRL 742989 -1 -6 -2 Ge GaAs CdS Si ZnS C BN Ar Ne Kr Xe S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 63. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition Semiconductors and insulators: band-gaps 2 1 KS Eg-Eg(eV) FP-EXX PP-EXX LMTO-ASA-EXX FP-LDA 0 -4 -2 0 -1 -6 -2 Ge GaAs CdS Si ZnS C BN Ar Ne Kr Xe S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 64. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition Semiconductors and insulators: band-gaps 2 1 KS Eg-Eg(eV) FP-EXX PP-EXX LMTO-ASA-EXX FP-EXX-ncv FP-LDA 0 -4 -2 0 http://arxiv.org/abs/cond-mat/0501353 -1 -6 -2 Ge GaAs CdS Si ZnS C BN Ar Ne Kr Xe S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 65. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition Semiconductors and insulators: d-band position -4 -2 2 0 4 Ge GaAs InP ZnS CdS KS Ed-Ed(eV) PP-EXX SIRC PP-SIC FP-LDA SIC: Vogle et al. PRB 54 5495 PP-EXX: Rinke et al.(unpublished) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 66. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition Semiconductors and insulators: d-band position -4 -2 2 0 4 Ge GaAs InP ZnS CdS KS Ed-Ed(eV) PP-EXX GW-FPLMTO GW-LMTO-ASA GW-PP-EXX SIRC PP-SIC FP-LDA GW LMTO-ASA: Aryasetiawan et al. PRB 54 17564 FPLMTO: Kotani et al. SSC121 S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 67. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Band-gap Band-gap problem d-bandposition Semiconductors and insulators: d-band position -4 -2 2 0 4 Ge GaAs InP ZnS CdS KS Ed-Ed(eV) FP-EXX PP-EXX GW-FPLMTO GW-LMTO-ASA GW-PP-EXX SIRC PP-SIC FP-EXX-ncv FP-LDA http://arxiv.org/abs/cond-mat/0501353 S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 68. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Conclusions Outlook Conclusions 1 2 3 4 5 EXX within an all electron full potential method is implemented within EXC!TING code. Right nowthis is the only FP code to be able to do EXX. A newand one of the most optimal basis is proposed for calculating and inverting the response.This basis may be useful for future TD-DFT and GW calculations. Magnetic metals: Asymmetry in exchangepotential is very important to get the correct ground-state. Semiconductors and insulators: Core-valence interaction is crucial for correct treatment of the EXX. Lack of Asymmetry and/or core-valence interaction could lead to spurious agreement with experiments. S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 69. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Conclusions Outlook Conclusions 1 2 3 4 5 EXX within an all electron full potential method is implemented within EXC!TING code. Right nowthis is the only FP code to be able to do EXX. A newand one of the most optimal basis is proposed for calculating and inverting the response.This basis may be useful for future TD-DFT and GW calculations. Magnetic metals: Asymmetry in exchangepotential is very important to get the correct ground-state. Semiconductors and insulators: Core-valence interaction is crucial for correct treatment of the EXX. Lack of Asymmetry and/or core-valence interaction could lead to spurious agreement with experiments. S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 70. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Conclusions Outlook Conclusions 1 2 3 4 5 EXX within an all electron full potential method is implemented within EXC!TING code. Right nowthis is the only FP code to be able to do EXX. A newand one of the most optimal basis is proposed for calculating and inverting the response.This basis may be useful for future TD-DFT and GW calculations. Magnetic metals: Asymmetry in exchangepotential is very important to get the correct ground-state. Semiconductors and insulators: Core-valence interaction is crucial for correct treatment of the EXX. Lack of Asymmetry and/or core-valence interaction could lead to spurious agreement with experiments. S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 71. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Conclusions Outlook Conclusions 1 2 3 4 5 EXX within an all electron full potential method is implemented within EXC!TING code. Right nowthis is the only FP code to be able to do EXX. A newand one of the most optimal basis is proposed for calculating and inverting the response.This basis may be useful for future TD-DFT and GW calculations. Magnetic metals: Asymmetry in exchangepotential is very important to get the correct ground-state. Semiconductors and insulators: Core-valence interaction is crucial for correct treatment of the EXX. Lack of Asymmetry and/or core-valence interaction could lead to spurious agreement with experiments. S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 72. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Conclusions Outlook Conclusions 1 2 3 4 5 EXX within an all electron full potential method is implemented within EXC!TING code. Right nowthis is the only FP code to be able to do EXX. A newand one of the most optimal basis is proposed for calculating and inverting the response.This basis may be useful for future TD-DFT and GW calculations. Magnetic metals: Asymmetry in exchangepotential is very important to get the correct ground-state. Semiconductors and insulators: Core-valence interaction is crucial for correct treatment of the EXX. Lack of Asymmetry and/or core-valence interaction could lead to spurious agreement with experiments. S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 73. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Conclusions Outlook Outlook 1 2 EXX can be generalised to handle non-collinear magnetism (derivatives w.r.t. nσσt (r) ). Future inclusion of exact correlation may be possible (multiconfiguration approach or adiabatic fluctuation dissipation) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 74. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Conclusions Outlook Outlook 1 2 EXX can be generalised to handle non-collinear magnetism (derivatives w.r.t. nσσt (r) ). Future inclusion of exact correlation may be possible (multiconfiguration approach or adiabatic fluctuation dissipation) S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory
  • 75. Theory and motivation Magnetic metals Semiconductors and insulators Conclusions and outlook Conclusions Outlook Acknowledgements Prof. P. Mohn Magnetic metals workdone in collaboration with Dr. C. Persson. Austrian Science Fund (project P16227) EXCITING network funded by the EU (HPRN-CT-2002-00317) Code available at: http://p h y si k . kf u n i g raz.ac.at/ ∼ k de/ secret garden/exciting.html S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory