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The gw method in quantum chemistry
- 3. G0W0 for solids: Bandgaps VAN SCHILFGAARDE PRL 96, 226402 (2006) HEDIN J PHYS, COND MAT. 11 R489 (SHIRLEY) (1995)
- 4. scGW for solids: Bandgaps KRESSE PRB 75, 235102 (2007) VAN SCHILFGAARDE PRL 96, 226402 (2006)
- 5. Formalism
- 6. • Electron density parameterized by KS single particle states • Greens function in spectral representation, expressed in terms of quasi- particle states leads to the quasi particle equation: • The quasi-particle energies are the electron removal and addition energies KS v.s. Quasi-Particle equation G(r, ʹr ;z) = Ψr,n qp (r, z)Ψl,n qp† ( ʹr, z) z −εn qp (z)+iηsign(εn qp (z)-µ)n ∑ − 1 2 ∇2 +VH (r)+Vext (r) ⎛ ⎝ ⎜ ⎞ ⎠ ⎟Ψr,n qp (r, z)+ d ʹr Σ(r, ʹr ;εn qp (z))∫ Ψr,n qp ( ʹr, z) =εn qp (z)Ψr,n qp (r, z) )()()()()( 2 1 )()()( KSKSKS XCextH 2 occ *KSKS rrrrr rrr nnn n nn VVV ψεψ ψψρ =⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +++∇− = ∑
- 7. • Electron density parameterized by KS single particle states • Greens function in spectral representation, expressed in terms of quasi- particle states leads to the quasi particle equation: • The quasi-particle energies are the electron removal and addition energies KS v.s. Quasi-Particle equation Vxc: exchange correlation potential HEDIN Phys Rev 139, A796, HYBERTSEN and LOUIE PRB 34, 5390 Σ: self-energy )()()()()( 2 1 )()()( KSKSKS XCextH 2 occ *KSKS rrrrr rrr nnn n nn VVV ψεψ ψψρ =⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +++∇− = ∑ G(r, ʹr ;z) = Ψr,n qp (r, z)Ψl,n qp† ( ʹr, z) z −εn qp (z)+iηsign(εn qp (z)-µ)n ∑ − 1 2 ∇2 +VH (r)+Vext (r) ⎛ ⎝ ⎜ ⎞ ⎠ ⎟Ψr,n qp (r, z)+ d ʹr Σ(r, ʹr ;εn qp (z))∫ Ψr,n qp ( ʹr, z) =εn qp (z)Ψr,n qp (r, z)
- 8. Zeroth order quasi particle equation • Full quasi particle equation • 0th order corrections (diagonal elements only) • Linearized 0th order corrections ),()(),())(;,(),()()( 2 1 qp , qpqp , qpqp ,extH 2 zzzzdzVV nrnnrnnr rrrrrrrr Ψ=ʹΨʹΣʹ+Ψ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ++∇− ∫ εε KS XC KSKSKS )(00 nnnnn WG n VZ ψεψεε −Σ+= εn G0W0 = εn KS + ψn KS Σ(εn G0W0 )−VXC ψn KS )()();,()()()( 2 1 qp , qpqp , qpqp ,extH 2 rrrrrrrr nrnnrnnr dVV Ψ=ʹΨʹΣʹ+Ψ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ++∇− ∫ εε
- 9. Advantages • There is a closed set of equations for the self-energy • Perturbative expansion in terms of the screened interaction in stead of the bare Coulomb interaction • The quasi-particle energies are the electron removal and addition energies • Extension to charge neutral excitations via the Bethe- Salpeter equation (recently implemented in Turbomole) • GW selfenergy -> RPA forces Bare Coulomb interac-on Screened Coulomb interac-on
- 10. Hedin Equations – Space time notation (the numbers indicate a contracted space, time and spin index) ∫ ∫ ∫ ∫ ∫ + + Γ−= += Γ Σ +−−=Γ Σ+= Γ=Σ )1,4()2,4,3()3,1()34()2,1( )2,4()4,3()3,1()34()2,1()2,1( )3,7,6()5,7()6,4( )5,4( )2,1( )4567()32()21()3,2,1( )2,4()4,3()3,1()34()2,1()2,1( )4,3,2()4,1()3,1()34()2,1( 00 GGdiP WPvdvW GG G d GGdGG WGdi δδ HEDIN Phys Rev 139, A796
- 11. Hedin Equations – Space time notation (the numbers indicate a contracted space, time and spin index) – Neglecting the second term in the vertex function leads to the GW approximation for the self-energy Fourier transformed to frequency domain: ωωω π ω dWEGe i E i )()( 2 )( 0 −=Σ ∫ + − )1,2()2,1()2,1( )2,4()4,3()3,1()34()2,1()2,1( )2,4()4,3()3,1()34()2,1()2,1( )2,1()2,1()2,1( 00 GGiP WPvdvW GGdGG WGi −= += Σ+= =Σ ∫ ∫ + HEDIN Phys Rev 139, A796
- 12. Full analy-c – Calculate response in spectral representa-on – Close connec-on to TDDFT (actually TDH) – Analy-c expression of Sigma as a sum over poles of G and W – Calculate Sigma analy-cally • Numerically exact except for finite basis • Full analy-c structure of Sigma • Expensive Re n Σc (εn ) n( )= in ρm( ) 2 εn −εi +Ωm εn −εi +Ωm( ) 2 +η2 i occ ∑ + an ρm( ) 2 a unocc ∑ εn −εa −Ωm εn −εa −Ωm( ) 2 +η2 ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ m ∑ van Se*en et al. JCTC 9, 232 (2013)
- 13. Plasmon pole – Approximate the response by one single pole – Calculate response at a few imaginary frequencies – Analy-c con-nua-on to the real axis – Calculate Sigma analy-cally • Cheap • Many different approaches • Jus-fica-on for complex systems
- 14. Contour deforma-on – Calculate the GW integral on the imaginary axis – Add pole contribu-ons of G in the contour • Allows for a more detailed sigma • Can be made exact • Integra-on grid parameters
- 15. Analy-c con-nua-on – Calculate the GW integral on the imaginary axis – Fit an n-pole expansion for Sigma in the en-re complex plane • Allows for a more detailed sigma • Can be made exact • May take many poles to converge • Only valid for fron-er orbitals
- 18. Benchmark set for molecules: GW100 • Original Collaboration: KIT Karlsruhe, FHI Berlin, and Berkeley Lab US DoE – TURBOMOLE: Gaussian basis sets, spectral representation via Casida – FHI-Aims: numerical local orbitals, analytic continuation – BerkeleyGW: plane waves, plasmon pole and real frequency integration • 5 different ways to evaluate the self-energy • well converged all electron reference values for IP and EA • Follow ups – CCSD(T) total energy reference (Klopper) – Plane wave results by VASP (Kresse) and WEST (Galli) – CP2k results (tes-ng their O(3) GW implementa-on) – (par-al) Stochas-c GW results (tes-ng O(1) implementa-on) – Evalua-on of 6 types of (par-al) self-consistency – Dipole moments and Densi-es (new project in process) – Core levels (new project in process) gw100.wordpress.com
- 22. Problems when solving the QPE n n making it featureless and almost constant. Then the solution is unique in the re t to us and eq. 32 may even be linearized so a single evaluation of ⌃ at the KS- cient and there is no iteration process.2 x εn G0W0 = εn KS + ψn KS Σ(εn G0W0 )−VXC ψn KS
- 23. Hard to converge ms and TURBOMOLE and the plane wave code BerkeleyGW in the results ection we will always used the extrapolated values. These will be referred to as EXTRA’. 0,001 0,01 0,1 1/Nbasis -2,00 -1,50 -1,00 -0,50 0,00 0,50ε QP H (extra)-ε QP H (basis)[eV] SVP TZVP QZVP gure 5: The deviation of the HOMO energies from the extrapolated complete basis set
- 28. GW100 Dipole moments Compared to CCSD(T) Work in progress Collabora-on with Patrick Rinke (Aalto)
- 30. Core levels Turbomole analy-c G0W0@PBE FHI aims analy-c con-nua-on G0W0@PBE Linear terms QPE -460 -560 εn G0W0 = εn KS + ψn KS Σ(εn G0W0 )−VXC ψn KS Collabora-on Francesc Illas (Barcelona)
- 32. In conclusion • GW Calcula-ons on molecules – allow for systema-c comparisons – allow for comparison to more accurate methods – improved insights • evalua-on of numerical methods • effect of self consistency • evalua-on of accuracy for densi-es • evalua-on of accuracy for core levels – efficient and accurate access to ioniza-on energies and electron affini-es
- 33. In conclusion • Open ques-ons – Strong / local correla-ons • Physics approach GW + Extended Cluster DMFT? • Chemist approach GW@mul--reference star-ng point? – Vibra-onal coupling • Orbital energy renormaliza-on • Source for finite life-mes
- 34. • Ma*eo Giantomassi • Geoffroy Hau-er • Gian-Marco Rignanese • Xavier Gonze • Ferdinand Kaplan • Florian Weigend • Ferdinand Evers • Michael Harding • Fabio Caruso • Patrick Rinke • Dorothea Golze • Sahar Sharivades • Jeffey Neaton • Georg Kresse Collaborators