Density functional theory in the solid state

Transcription

1 Density functional theory in the solid state Ari P Seitsonen IMPMC, CNRS & Universités 6 et 7 Paris, IPGP Department of Applied Physics, Helsinki University of Technology Physikalisch-Chemisches Institut der Universität Zürich September 24, 2009

2 Summary DFT 1 DFT 2 DFT in the solid state September 24, / 42

3 Kohn-Sham method: Total energy Let us write the total energy as: E tot [n] = E kin [n] + E ext [n] + E H [n] + E xc [n] E kin [n] = QM kinetic energy of electrons E ext [n] = energy due to external potential (usually ions) E H [n] = classical Hartree repulsion (e e ) E xc [n] = exchange-correlation energy DFT in the solid state September 24, / 42

4 Total energy expression Kohn-Sham (total 1 ) energy: E KS [n] = 1 2 f i i r ψ i 2 ψ i + n (r) n (r ) r r r r dr dr + E xc n (r) V ext (r) dr 1 without ion-ion interaction DFT in the solid state September 24, / 42

5 Kohn-Sham equations DFT Vary the Kohn-Sham energy E KS with respect to ψ j (r ): Kohn-Sham equations { 1 } V KS (r) ψ i (r) = ε i ψ i (r) δe KS δψ j (r ) n (r) = i f i ψ i (r) 2 V KS (r) = V ext (r) + V H (r) + V xc (r) V xc (r) = δexc δn(r) DFT in the solid state September 24, / 42

6 Kohn-Sham equations: Notes { 1 } V KS (r) ψ i (r) = ε i ψ i (r) ; n (r) = i f i ψ i (r) 2 Equation looking like Schrödinger equation The Kohn-Sham potential, however, depends on density The equations are coupled and highly non-linear Self-consistent solution required ε i and ψ i are in principle only help variables (only ε HOMO has a meaning) The potential V KS is local The scheme is in principle exact DFT in the solid state September 24, / 42

7 Kohn-Sham equations: Self-consistency 1 Generate a starting density n init 2 Generate the Kohn-Sham potential V init KS 3 Solve the Kohn-Sham equations ψ init i 4 New density n 1 5 Kohn-Sham potential V 1 KS 6 Kohn-Sham orbitals ψ 1 i 7 Density n until self-consistency is achieved (to required precision) DFT in the solid state September 24, / 42

8 Kohn-Sham energy: Alternative expression Take the Kohn-Sham equation, multiply from the left with f i ψi and integrate: 1 2 f i ψ i (r) 2 ψ i (r) dr + f i V KS (r) ψ i (r) 2 dr = f i ε i r Sum over i and substitute into the expression for Kohn-Sham energy: E KS [n] = f i ε i E H + E xc n (r) V xc dr i r r DFT in the solid state September 24, / 42

9 DFT in the solid state September 24, / 42

10 The Kohn-Sham scheme is in principle exact however, the exchange-correlation energy functional is not known explicitly DFT in the solid state September 24, / 42

11 The Kohn-Sham scheme is in principle exact however, the exchange-correlation energy functional is not known explicitly Exchange is known exactly, however its evaluation is usually very time-consuming, and often the accuracy is not improved over simpler approximations (due to error-cancellations in the latter group) DFT in the solid state September 24, / 42

12 The Kohn-Sham scheme is in principle exact however, the exchange-correlation energy functional is not known explicitly Exchange is known exactly, however its evaluation is usually very time-consuming, and often the accuracy is not improved over simpler approximations (due to error-cancellations in the latter group) Many exact properties, like high/low-density limits, scaling rules etc are known DFT in the solid state September 24, / 42

13 The Kohn-Sham scheme is in principle exact however, the exchange-correlation energy functional is not known explicitly Exchange is known exactly, however its evaluation is usually very time-consuming, and often the accuracy is not improved over simpler approximations (due to error-cancellations in the latter group) Many exact properties, like high/low-density limits, scaling rules etc are known Famous approximations: DFT in the solid state September 24, / 42

14 The Kohn-Sham scheme is in principle exact however, the exchange-correlation energy functional is not known explicitly Exchange is known exactly, however its evaluation is usually very time-consuming, and often the accuracy is not improved over simpler approximations (due to error-cancellations in the latter group) Many exact properties, like high/low-density limits, scaling rules etc are known Famous approximations: Local density approximation, LDA DFT in the solid state September 24, / 42

15 The Kohn-Sham scheme is in principle exact however, the exchange-correlation energy functional is not known explicitly Exchange is known exactly, however its evaluation is usually very time-consuming, and often the accuracy is not improved over simpler approximations (due to error-cancellations in the latter group) Many exact properties, like high/low-density limits, scaling rules etc are known Famous approximations: Local density approximation, LDA Generalised gradient approximations, GGA DFT in the solid state September 24, / 42

16 The Kohn-Sham scheme is in principle exact however, the exchange-correlation energy functional is not known explicitly Exchange is known exactly, however its evaluation is usually very time-consuming, and often the accuracy is not improved over simpler approximations (due to error-cancellations in the latter group) Many exact properties, like high/low-density limits, scaling rules etc are known Famous approximations: Local density approximation, LDA Generalised gradient approximations, GGA Hybrid functionals DFT in the solid state September 24, / 42

17 The Kohn-Sham scheme is in principle exact however, the exchange-correlation energy functional is not known explicitly Exchange is known exactly, however its evaluation is usually very time-consuming, and often the accuracy is not improved over simpler approximations (due to error-cancellations in the latter group) Many exact properties, like high/low-density limits, scaling rules etc are known Famous approximations: Local density approximation, LDA Generalised gradient approximations, GGA Hybrid functionals... DFT in the solid state September 24, / 42

18 Exchange-correlation hole The exchange-correlation energy can be expressed using the exchange-correlation hole n xc (r, r ) E xc = 1 n xc (r, r r) n (r) 2 r r r r dr dr = 1 n (r) R 2 1 n xc (r, R)dΩ dr dr 2 R=0 R r Ω DFT in the solid state September 24, / 42

19 Exchange-correlation hole The exchange-correlation energy can be expressed using the exchange-correlation hole n xc (r, r ) E xc = 1 n xc (r, r r) n (r) 2 r r r r dr dr = 1 n (r) R 2 1 n xc (r, R)dΩ dr dr 2 R=0 R r Thus E xc only depends on the spherical average of n xc (r, r ) Ω DFT in the solid state September 24, / 42

20 Exchange-correlation hole The exchange-correlation energy can be expressed using the exchange-correlation hole n xc (r, r ) E xc = 1 n xc (r, r r) n (r) 2 r r r r dr dr = 1 n (r) R 2 1 n xc (r, R)dΩ dr dr 2 R=0 R r Thus E xc only depends on the spherical average of n xc (r, r ) Sum rule Ω r n xc ( r, r r ) dr = 1 DFT in the solid state September 24, / 42

21 Exchange-correlation hole: Silicon PP Rushton, DJ Tozer SJ Clark, Phys Rev B 65 (2002) DFT in the solid state September 24, / 42

22 : LDA Local density approximation: Use the exchange-correlation energy functional for homogeneous electron gas at each point of space: E xc r n (r) eheg xc [n(r)]dr DFT in the solid state September 24, / 42

23 : LDA Local density approximation: Use the exchange-correlation energy functional for homogeneous electron gas at each point of space: E xc r n (r) eheg xc [n(r)]dr Works surprisingly well even in inhomogeneous electron systems, thanks fulfillment of certain sum rules DFT in the solid state September 24, / 42

24 : LDA Local density approximation: Use the exchange-correlation energy functional for homogeneous electron gas at each point of space: E xc r n (r) eheg xc [n(r)]dr Works surprisingly well even in inhomogeneous electron systems, thanks fulfillment of certain sum rules Energy differences over-bound: Cohesion, dissociation, adsorption energies DFT in the solid state September 24, / 42

25 : LDA Local density approximation: Use the exchange-correlation energy functional for homogeneous electron gas at each point of space: E xc r n (r) eheg xc [n(r)]dr Works surprisingly well even in inhomogeneous electron systems, thanks fulfillment of certain sum rules Energy differences over-bound: Cohesion, dissociation, adsorption energies Lattice constants somewhat (1-3 %) too small, bulk moduli too large DFT in the solid state September 24, / 42

26 : LDA Local density approximation: Use the exchange-correlation energy functional for homogeneous electron gas at each point of space: E xc r n (r) eheg xc [n(r)]dr Works surprisingly well even in inhomogeneous electron systems, thanks fulfillment of certain sum rules Energy differences over-bound: Cohesion, dissociation, adsorption energies Lattice constants somewhat (1-3 %) too small, bulk moduli too large Asymptotic potential decays exponentially outside charge distribution, leads to too weak binding energies for the electrons, eg. ionisation potentials DFT in the solid state September 24, / 42

27 : LDA Local density approximation: Use the exchange-correlation energy functional for homogeneous electron gas at each point of space: E xc r n (r) eheg xc [n(r)]dr Works surprisingly well even in inhomogeneous electron systems, thanks fulfillment of certain sum rules Energy differences over-bound: Cohesion, dissociation, adsorption energies Lattice constants somewhat (1-3 %) too small, bulk moduli too large Asymptotic potential decays exponentially outside charge distribution, leads to too weak binding energies for the electrons, eg. ionisation potentials Contains self-interaction in single-particle case DFT in the solid state September 24, / 42

28 : LDA The LDA exchange energy is due to Slater and Dirac, ɛ x (n) = C x n 1/3, where C x = 3 4 ( 3 π )1/3 The exact analytical form of the correlation energy ɛ c (n) of the homogeneous electron gas is not known. The correlation energy is therefore fitted to quantum Monte-Carlo results (Ceperley & Alder) and analytical highand low-density limits There are several parametrisations: Vosko, Wilk & Nusair (1980); Perdew & Zunger (1981); Perdew & Wang (1992)) Pade-interpolation reproducing the Perdew-Wang data S Goedecker, M Teter & J Hutter, Phys Rev B 54 (1996) 1703 DFT in the solid state September 24, / 42

29 : LDA Correlation energy function ɛ c (n) DFT in the solid state September 24, / 42

30 : LDA Why does it albeit work so well? The XC contribution is the smallest (that s why it was packed aside in the first place) LDA, despite is simplicity, still fulfills many important requirements set for the exact functional; scaling relations, sum rules,... There is a major error cancellation between the exchange and correlation (Warning: Same occurs with many other functionals also; thereby best keeping the same level of sophistication in both parts) DFT in the solid state September 24, / 42

31 : LDA The hole (top) is badly described, however the spherical average (bottom), which is the property needed, agrees reasonably LDA fulfills several sum rules DFT in the solid state September 24, / 42

32 : LDA Why does it not work so well? Missing integer discontinuity in the potential; thus bad excitation energies; the first place) Self-interaction not excluded: The electron interacts directly with itself Thus for example d and f functions underbound (eg density of states in fcc Cu) Wrong tail: no Rydberg states, image states at metal surfaces DFT in the solid state September 24, / 42

33 : GGA Generalised gradient approximation: The gradient expansion of the exchange-correlation energy does not improve results; sometimes leads to divergences DFT in the solid state September 24, / 42

34 : GGA Generalised gradient approximation: The gradient expansion of the exchange-correlation energy does not improve results; sometimes leads to divergences Thus a more general approach is taken, and there is room for several forms of GGA: E xc r n (r) e xc[n(r), n(r) 2 ]dr DFT in the solid state September 24, / 42

35 : GGA Generalised gradient approximation: The gradient expansion of the exchange-correlation energy does not improve results; sometimes leads to divergences Thus a more general approach is taken, and there is room for several forms of GGA: E xc r n (r) e xc[n(r), n(r) 2 ]dr Works reasonably well, again fulfilling certain sum rules DFT in the solid state September 24, / 42

36 : GGA Generalised gradient approximation: The gradient expansion of the exchange-correlation energy does not improve results; sometimes leads to divergences Thus a more general approach is taken, and there is room for several forms of GGA: E xc r n (r) e xc[n(r), n(r) 2 ]dr Works reasonably well, again fulfilling certain sum rules Energy differences are improved DFT in the solid state September 24, / 42

37 : GGA Generalised gradient approximation: The gradient expansion of the exchange-correlation energy does not improve results; sometimes leads to divergences Thus a more general approach is taken, and there is room for several forms of GGA: E xc r n (r) e xc[n(r), n(r) 2 ]dr Works reasonably well, again fulfilling certain sum rules Energy differences are improved Lattice constants somewhat, 1-3 % too large, bulk moduli too small DFT in the solid state September 24, / 42

38 : GGA Generalised gradient approximation: The gradient expansion of the exchange-correlation energy does not improve results; sometimes leads to divergences Thus a more general approach is taken, and there is room for several forms of GGA: E xc r n (r) e xc[n(r), n(r) 2 ]dr Works reasonably well, again fulfilling certain sum rules Energy differences are improved Lattice constants somewhat, 1-3 % too large, bulk moduli too small Contains self-interaction in single-particle case DFT in the solid state September 24, / 42

39 : GGA Generalised gradient approximation: The gradient expansion of the exchange-correlation energy does not improve results; sometimes leads to divergences Thus a more general approach is taken, and there is room for several forms of GGA: E xc r n (r) e xc[n(r), n(r) 2 ]dr Works reasonably well, again fulfilling certain sum rules Energy differences are improved Lattice constants somewhat, 1-3 % too large, bulk moduli too small Contains self-interaction in single-particle case Again exponential asymptotic decay in potential DFT in the solid state September 24, / 42

40 : GGA Generalised gradient approximation: The gradient expansion of the exchange-correlation energy does not improve results; sometimes leads to divergences Thus a more general approach is taken, and there is room for several forms of GGA: E xc r n (r) e xc[n(r), n(r) 2 ]dr Works reasonably well, again fulfilling certain sum rules Energy differences are improved Lattice constants somewhat, 1-3 % too large, bulk moduli too small Contains self-interaction in single-particle case Again exponential asymptotic decay in potential Negative ions normally not bound DFT in the solid state September 24, / 42

41 : GGA Generalised gradient approximation: The gradient expansion of the exchange-correlation energy does not improve results; sometimes leads to divergences Thus a more general approach is taken, and there is room for several forms of GGA: E xc r n (r) e xc[n(r), n(r) 2 ]dr Works reasonably well, again fulfilling certain sum rules Energy differences are improved Lattice constants somewhat, 1-3 % too large, bulk moduli too small Contains self-interaction in single-particle case Again exponential asymptotic decay in potential Negative ions normally not bound Usually the best compromise between speed and accuracy in large systems DFT in the solid state September 24, / 42

42 : GGA-PBE J Perdew, KBurke & M Ernzerhof, Phys Rev Lett (1996): Like Perdew-Wang 91: Analytical function, only natural constants E PBE xc = E PBE x + E PBE c E PBE x (n, n ) = r n ε LDA x F PBE x (s) = 1 + κ ( ) π 2 µ = β 3 (n) Fx PBE (s) dr, κ 1 + µs2 κ, DFT in the solid state September 24, / 42

43 : GGA-PBE E PBE c (n, n ) = H PBE c (r s, η, t) = γφ 3 ln r drn [ ε LDA xc (n) + H PBE [1 + βγ ( t2 c (r s, η, t) ], 1 + At At 2 + A 2 t 4 )], A (r s, η) = β 1 γ e εlda c /γφ 3 1, φ (η) = 1 [ (1 + η) 2/3 + (1 η) 2/3], 2 γ = 1 ln 2 [ ] 3 1/3 π 2 ; r s = local Wigner-Seitz radius 4πn DFT in the solid state September 24, / 42

44 : GGA-PBE Here s(r) = n n 2k F n and t(r) = 2φk, k sn s = 4k F /π, are dimensionless density gradients and β comes from the generalised gradient expansion for the correlation (Perdew et al, 1992) and (Wang & Perdew, 1991). κ is formally set by the Lieb-Oxford bound (1981) for the exchange energy E x [n] E xc [n] n 4/3 (r) dr. Note: revpbe plays exactly with this parameter r DFT in the solid state September 24, / 42

45 : LDA+U/GGA+U Add on-site Hubbard term: E U = U 2 I,σ [ λ Iσ m ( )] 1 λ Iσ m Improves description (energetics, magnetic moments,... ) in many cases but not always DFT in the solid state September 24, / 42

46 : Meta-GGA Meta-Generalised gradient approximation: In addition to the gradient, add also orbital kinetic energy τ (r) = i 1 2 ψ i (r) 2 E xc n (r) e xc [n(r), n(r) 2, τ(r)]dr r DFT in the solid state September 24, / 42

47 : SIC The usual functionals include self-interaction: The electron interacts with itself because the functionals contain the full density, thus also that of a given orbital ψ i DFT in the solid state September 24, / 42

48 : SIC The usual functionals include self-interaction: The electron interacts with itself because the functionals contain the full density, thus also that of a given orbital ψ i Different schemes to correct for this: DFT in the solid state September 24, / 42

49 : SIC The usual functionals include self-interaction: The electron interacts with itself because the functionals contain the full density, thus also that of a given orbital ψ i Different schemes to correct for this: JP Perdew & A Zunger, Phys Rev B 23 (1981) 5048 E SIC,PZ = E KS i E Hxc [n i ] DFT in the solid state September 24, / 42

50 : SIC The usual functionals include self-interaction: The electron interacts with itself because the functionals contain the full density, thus also that of a given orbital ψ i Different schemes to correct for this: JP Perdew & A Zunger, Phys Rev B 23 (1981) 5048 E SIC,PZ = E KS i E Hxc [n i ] M d Avezac, M Calandra & F Mauri, Phys Rev B 71 (2005) E SIC,SPZ = E KS E H [m] E xc [m, 0] DFT in the solid state September 24, / 42

51 : SIC The usual functionals include self-interaction: The electron interacts with itself because the functionals contain the full density, thus also that of a given orbital ψ i Different schemes to correct for this: JP Perdew & A Zunger, Phys Rev B 23 (1981) 5048 E SIC,PZ = E KS i E Hxc [n i ] M d Avezac, M Calandra & F Mauri, Phys Rev B 71 (2005) E SIC,SPZ = E KS E H [m] E xc [m, 0] M d Avezac, M Calandra & F Mauri, Phys Rev B 71 (2005) E SIC,US = E KS E H [m] E xc [n, n ] + E xc [n m, n ] DFT in the solid state September 24, / 42

52 : Hybrid functionals Hybrid functionals Include partially the exact (Hartree-Fock) exchange: E xc αe HF + (1 α)ex GGA + Ec GGA ; again many variants DFT in the solid state September 24, / 42

53 : Hybrid functionals Hybrid functionals Include partially the exact (Hartree-Fock) exchange: E xc αe HF + (1 α)ex GGA + Ec GGA ; again many variants Works in general well DFT in the solid state September 24, / 42

54 : Hybrid functionals Hybrid functionals Include partially the exact (Hartree-Fock) exchange: E xc αe HF + (1 α)ex GGA + Ec GGA ; again many variants Works in general well Energy differences are still improved DFT in the solid state September 24, / 42

55 : Hybrid functionals Hybrid functionals Include partially the exact (Hartree-Fock) exchange: E xc αe HF + (1 α)ex GGA + Ec GGA ; again many variants Works in general well Energy differences are still improved Improved magnetic moments in some systems DFT in the solid state September 24, / 42

56 : Hybrid functionals Hybrid functionals Include partially the exact (Hartree-Fock) exchange: E xc αe HF + (1 α)ex GGA + Ec GGA ; again many variants Works in general well Energy differences are still improved Improved magnetic moments in some systems Partial improvement in asymptotic form DFT in the solid state September 24, / 42

57 : Hybrid functionals Hybrid functionals Include partially the exact (Hartree-Fock) exchange: E xc αe HF + (1 α)ex GGA + Ec GGA ; again many variants Works in general well Energy differences are still improved Improved magnetic moments in some systems Partial improvement in asymptotic form Usually the best accuracy if the computation burden can be handled DFT in the solid state September 24, / 42

58 : Hybrid functionals Hybrid functionals Include partially the exact (Hartree-Fock) exchange: E xc αe HF + (1 α)ex GGA + Ec GGA ; again many variants Works in general well Energy differences are still improved Improved magnetic moments in some systems Partial improvement in asymptotic form Usually the best accuracy if the computation burden can be handled Calculations for crystals appearing DFT in the solid state September 24, / 42

59 : Hybrid functionals Hybrid functionals Include partially the exact (Hartree-Fock) exchange: E xc αe HF + (1 α)ex GGA + Ec GGA ; again many variants Works in general well Energy differences are still improved Improved magnetic moments in some systems Partial improvement in asymptotic form Usually the best accuracy if the computation burden can be handled Calculations for crystals appearing The method is no longer pure Kohn-Sham method: Fock operator is non-local; a mixed DFT-KS/Hartree-Fock scheme DFT in the solid state September 24, / 42

60 : OEP/OPM/EXX The Hartree-Fock energy is the exact exchange energy DFT in the solid state September 24, / 42

61 : OEP/OPM/EXX i r { The Hartree-Fock energy is the exact exchange energy This can be varied with respect to potential, leading to an integral equation for the exchange potential ψ i ( r ) [ V OEP xc ( r ) 1 δexc OEP ψ p (r ) δψ i (r ) ] ( G i r, r ) } ψ i (r) + cc dr = 0 G i ( r, r ) = j ψ j (r) ψ j (r ) ε i ε j DFT in the solid state September 24, / 42

62 : ACDFT E c = 1 λ=0 ω=0 1 π Tr {V bare [χ KS (iω) χ λ (iω)]} dω dλ DFT in the solid state September 24, / 42

63 : Observations The accuracy can not be systematically improved! DFT in the solid state September 24, / 42

64 : Observations The accuracy can not be systematically improved! van der Waals interactions still a problem (tailored approximations in sight, like Langreth-Lundqvist functional) DFT in the solid state September 24, / 42

65 : Observations The accuracy can not be systematically improved! van der Waals interactions still a problem (tailored approximations in sight, like Langreth-Lundqvist functional) Most widely used parametrisations: DFT in the solid state September 24, / 42

66 : Observations The accuracy can not be systematically improved! van der Waals interactions still a problem (tailored approximations in sight, like Langreth-Lundqvist functional) Most widely used parametrisations: GGA DFT in the solid state September 24, / 42

67 : Observations The accuracy can not be systematically improved! van der Waals interactions still a problem (tailored approximations in sight, like Langreth-Lundqvist functional) Most widely used parametrisations: GGA Perdew-Burke-Ernzerhof, PBE (1996); among physicists DFT in the solid state September 24, / 42

68 : Observations The accuracy can not be systematically improved! van der Waals interactions still a problem (tailored approximations in sight, like Langreth-Lundqvist functional) Most widely used parametrisations: GGA Perdew-Burke-Ernzerhof, PBE (1996); among physicists Beck Lee-Yang-Parr, BLYP (early 1990 s); among chemists DFT in the solid state September 24, / 42

69 : Observations The accuracy can not be systematically improved! van der Waals interactions still a problem (tailored approximations in sight, like Langreth-Lundqvist functional) Most widely used parametrisations: GGA Perdew-Burke-Ernzerhof, PBE (1996); among physicists Beck Lee-Yang-Parr, BLYP (early 1990 s); among chemists Hybrid functionals DFT in the solid state September 24, / 42

70 : Observations The accuracy can not be systematically improved! van der Waals interactions still a problem (tailored approximations in sight, like Langreth-Lundqvist functional) Most widely used parametrisations: GGA Perdew-Burke-Ernzerhof, PBE (1996); among physicists Beck Lee-Yang-Parr, BLYP (early 1990 s); among chemists Hybrid functionals PBE0; among physicists DFT in the solid state September 24, / 42

71 : Observations The accuracy can not be systematically improved! van der Waals interactions still a problem (tailored approximations in sight, like Langreth-Lundqvist functional) Most widely used parametrisations: GGA Perdew-Burke-Ernzerhof, PBE (1996); among physicists Beck Lee-Yang-Parr, BLYP (early 1990 s); among chemists Hybrid functionals PBE0; among physicists B3LYP; among chemists DFT in the solid state September 24, / 42

72 : Jacob s ladder DFT in the solid state September 24, / 42

73 : Results Small molecules DFT in the solid state September 24, / 42

74 : Results Water clusters DFT in the solid state September 24, / 42

75 : Simple reactions DFT in the solid state September 24, / 42

76 : Results Solids J Chem Phys 123 (2005) DFT in the solid state September 24, / 42

77 : Results Magnetic oxides Phys Rev B 73 (2006) DFT in the solid state September 24, / 42

78 Functionals: Chemical shifts Mean absolute deviation [ppm] method absolute relative HF MP LDA BLYP B3LYP PBE CH 4, NH 3, H 2 O, C 2 H 6, C 2 H 4, C 2 H 2, N 2, CF 4,... DFT in the solid state September 24, / 42

79 Functionals: Chemical shifts in ozone O 3 ; [ppm] method O terminal O central HF MP CCSD(T) LDA BLYP B3LYP PBE expr 1290, , 688 DFT in the solid state September 24, / 42

80 : Scaling DFT DFT in the solid state September 24, / 42

81 Summary DFT 1 DFT 2 DFT in the solid state September 24, / 42

82 Input for ld1.x DFT &input title = O prefix = O zed = 8.0 rel = 1 config = 1s2 2s2 2p4 3s-1 3p-1 3d-1 iswitch = 3 dft = PBE / &inputp pseudotype = 1 tm =.true. lloc = 2 file_pseudopw = O.pbe-tm-gipaw.UPF lgipaw_reconstruction =.true. / 3 2S P D &test / 4 2S P S P DFT in the solid state September 24, / 42

83 Wave functions DFT DFT in the solid state September 24, / 42

84 Wave functions: Scaled DFT in the solid state September 24, / 42