Electronic Structure Calculations, Density Functional Theory and its Modern Implementations

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1 Tutoriel Big RENOBLE Electronic Structure Calculations, Density Functional Theory and its Modern Implementations Thierry Deutsch L_Sim - CEA renoble 19 October 2011

2 Outline 1 of Atomistic calculations 2 Density Functional Theory (quick view) formalism equations 3 basis sets 4

3 of Atomistic simulations

4 of Atomistic Simulations Theory Experiment Simulation Hardware Computers Algorithms Atomistic Simulations Force fields (interatomic potentials) Tight Binding Methods Hartree-Fock Density Functional Theory Configuration interactions Quantum Monte-Carlo

5 of Atomistic Simulations Theory Experiment Simulation Hardware Computers Algorithms Atomistic Simulations Force fields (interatomic potentials) Tight Binding Methods Hartree-Fock Density Functional Theory Configuration interactions Quantum Monte-Carlo

6 Algorithms Search of efficient algorithms: Fast Fourier Transform (FFT) : T CPU n log(n) ; Order N atoms methods Electrostatic (Hartree) potential of an electronic density ρ(r): V(r) = V Mesh: n = = 10 6 Direct: n 2 = : 100 s FFT: n log(n) 10 6 : 0.1 ms ρ(r ) r r dr

7 Algorithms Search of efficient algorithms: Fast Fourier Transform (FFT) : T CPU n log(n) ; Order N atoms methods Electrostatic (Hartree) potential of an electronic density ρ(r): V(r) = V Mesh: n = = 10 6 Direct: n 2 = : 100 s FFT: n log(n) 10 6 : 0.1 ms ρ(r ) r r dr

8 Atomistic Simulations Two intrinsic difficulties for numerical atomistic simulations, related to complexity: Interactions The way that atoms interact is known: i ħ Ψ t = H Ψ H ψ = E 0 ψ Exploration of the configuration space (ART, minima hopping, spline search)

9 Atomistic Simulations Two intrinsic difficulties for numerical atomistic simulations, related to complexity: Interactions The way that atoms interact is known: i ħ Ψ t = H Ψ H ψ = E 0 ψ Exploration of the configuration space (ART, minima hopping, spline search) E pot R1

10 Atomistic Simulations Two intrinsic difficulties for numerical atomistic simulations, related to complexity: Interactions The way that atoms interact is known: i ħ Ψ t = H Ψ H ψ = E 0 ψ Exploration of the configuration space (ART, minima hopping, spline search) E pot R1 R2

11 Atomistic Simulations Two intrinsic difficulties for numerical atomistic simulations, related to complexity: Interactions The way that atoms interact is known: i ħ Ψ t = H Ψ H ψ = E 0 ψ Exploration of the configuration space (ART, minima hopping, spline search) E pot R1 R3 R2

12 Atomistic Simulations Two intrinsic difficulties for numerical atomistic simulations, related to complexity: Interactions The way that atoms interact is known: i ħ Ψ t = H Ψ H ψ = E 0 ψ Exploration of the configuration space (ART, minima hopping, spline search) E pot R1000 R41 R3 Rn R2 R1

13 oals of Criteria eneral (enerality) Precision geometry atomic positions lengths of bonds angles of bonds energy of bonds vibration frequencies dipolar moments UV and visible spectra photo-emission magnetic moments System size Time scale oals all atoms of the periodic table all kinds of bonds (transition state) ±0.001 angström ±0.001 angström ±1 degree ev/atom (0.1 kj/mol) 1 Hz 0.01Debye (1 Debye = C.m 0.01 ev 0.01 ev 0.01µ B 10 4 atoms for ab initio methods 10 6 atoms for parametrized methods 1 s

14 Criteria for classification 3 criteria 1 enerality (elements, alloys) 2 Precision ( r, E) 3 System size (N, t) P S

15 Quantum Mechanics Schrödinger equation for N electrons in an external potential V ext (r): antisymmetric Ψ(r 1,r 2,...,r N ) = Ψ(r 2,r 1,...,r N ) H = 2 2m N i 2 r i + 1 4πε 0 i j 1 r i r j + N i V ext (r i ) H Ψ(r 1,r 2,...,r N ) = EΨ(r 1,r 2,...,r N ) Ion-electron interactions: V ext (r) = 1 Z α 4πε 0 R α r α

16 Hartree-Fock and Post Hartree-Fock We build the most simple antisymmetric wavefunction for N electrons from N wavefunctions at 1 electron (Slater determinant): φ 1 (x 1 )... φ 1 (x N ) φ 2 (x 1 )... φ 2 (x N ) Φ(x 1,...,x N ) = φ N (x 1 )... φ N (x N ) The Hamiltonian are exact. Many Slater determinants: Configuration interaction, MP2, coupled clusters,... P P S S

17 Hartree-Fock and Post Hartree-Fock We build the most simple antisymmetric wavefunction for N electrons from N wavefunctions at 1 electron (Slater determinant): φ 1 (x 1 )... φ 1 (x N ) φ 2 (x 1 )... φ 2 (x N ) Φ(x 1,...,x N ) = φ N (x 1 )... φ N (x N ) The Hamiltonian are exact. Many Slater determinants: Configuration interaction, MP2, coupled clusters,... P P S S

18 Quantum Monte Carlo Very complicated wavefunctions for N electrons (Jastrow factor): φ 1 (x 1 )... φ 1 (x N ) Ψ(x 1,...,x N ) = φ N (x 1 )... φ N (x N ) i j J (r ij ) Integration of the total energy: E = < Ψ H Ψ > < Ψ Ψ > Use of Monte-Carlo methods for the integration. P S Diffusive Monte Carlo (fixed node, sign problem)

19 Density Functional Theory () Theorem: The total energy of the ground state in an external potential V ext is a functional of the electronic density ρ(r): E [ρ(r)] = F [ρ(r)] + V ext (r)ρ(r)dr r Variational principles: E [ρ 0 (r)] E [ρ(r)] ρ(r) Equation: (approximation) Having a one-electron hamiltonian in a mean field. [ V eff (r) ] ψ i = ε i ψ i P occupied ρ(r) = i ψ i (r) 2 S

20 Density Functional Theory () Theorem: The total energy of the ground state in an external potential V ext is a functional of the electronic density ρ(r): E [ρ(r)] = F [ρ(r)] + V ext (r)ρ(r)dr r Variational principles: E [ρ 0 (r)] E [ρ(r)] ρ(r) Equation: (approximation) Having a one-electron hamiltonian in a mean field. [ V eff (r) ] ψ i = ε i ψ i P occupied ρ(r) = i ψ i (r) 2 S

21 Tight Binding Methods Semi-empirical quantum methods We solve a one-electron parametrized hamiltonian (Slater-Koster parameter): H ψ i >= ε i ψ i > occupied Band term: E band = ions Repulsive term: E rep = I,J i ε i f (R I,J ) P E total = E rep + E band S Mainly used for electronic properties (10 6 atoms) or transport

22 Choice of Chemistry Physics P P Compromise between accuracy, system size and CPU time. Force Fields Tight Binding Hartree-Fock S S P S S P Ab initio () Configuration Interactions P P Quantum Monte-Carlo S S

23 Choice of Chemistry Physics P P Compromise between accuracy, system size and CPU time. Force Fields Tight Binding Hartree-Fock S S P S S P Ab initio () Configuration Interactions P P Quantum Monte-Carlo S S

24 Choice of Chemistry Physics P P Compromise between accuracy, system size and CPU time. Force Fields Tight Binding Hartree-Fock S S P S S P Ab initio () Configuration Interactions P P Quantum Monte-Carlo S S

25 Choice of Chemistry Physics P P Compromise between accuracy, system size and CPU time. Force Fields Tight Binding Hartree-Fock S S P S S P Ab initio () Configuration Interactions P P Quantum Monte-Carlo S S

26 Choice of Chemistry Physics P P Compromise between accuracy, system size and CPU time. Force Fields Tight Binding Hartree-Fock S S P S S P Ab initio () Configuration Interactions P P Quantum Monte-Carlo S S

27 Choice of Chemistry Physics P P Compromise between accuracy, system size and CPU time. Force Fields Tight Binding Hartree-Fock S S P S S P Ab initio () Configuration Interactions P P Quantum Monte-Carlo S S

28 Density Functional Theory equations

29 Ab initio calculations with Several advantages Ab initio: No adjustable parameters : Quantum mechanical (fundamental) treatment Main limitations Approximated approach Requires high computer power, limited to few hundreds atoms in most cases Wide range of applications: nanoscience, biology, materials

30 Quantum Mechanics Schrödinger equation for N electrons in an external potential V ext (r): antisymmetric Ψ(r 1,r 2,...,r N ) = Ψ(r 2,r 1,...,r N ) H = 2 2m N i 2 r i + 1 4πε 0 i j 1 r i r j + N i V ext (r i ) H Ψ(r 1,r 2,...,r N ) = EΨ(r 1,r 2,...,r N ) Ion-electron interactions: V ext (r) = 1 Z α 4πε 0 R α r α

31 The Hohenberg-Kohn theorem Schrödinger equation H = N i=1 ( r i + V ext (r i,{r}) ) i j Very difficult to solve for more than two electrons! 1 r i r j The fundamental variable of the problem is however not the wavefunction, but the electronic density ρ(r) = N dr 2 dr N ψ (r,r 2,,r N )ψ(r,r 2,,r N ) Hohenberg-Kohn theorem (1964) The ground state density ρ(r) of a many-electron system uniquely determines (up to a constant) the external potential. The external potential is a functional of ρ: V ext = V ext [ρ]

32 The Hohenberg-Kohn theorem Schrödinger equation H = N i=1 ( r i + V ext (r i,{r}) ) i j Very difficult to solve for more than two electrons! 1 r i r j The fundamental variable of the problem is however not the wavefunction, but the electronic density ρ(r) = N dr 2 dr N ψ (r,r 2,,r N )ψ(r,r 2,,r N ) Hohenberg-Kohn theorem (1964) The ground state density ρ(r) of a many-electron system uniquely determines (up to a constant) the external potential. The external potential is a functional of ρ: V ext = V ext [ρ]

33 formalism H-K theorem: E is an unknown functional of the density E = E[ρ] Density Functional Theory approach Mapping of an interacting many-electron system into a system with independent particles moving into an effective potential. Find a set of orthonormal orbitals Ψ i (r) that minimizes: E = 1 2 N i=1 Ψ i (r) 2 Ψ i (r)dr + 1 ρ(r)v H (r)dr 2 + E xc [ρ(r)] + V ext (r)ρ(r)dr ρ(r) = N i=1 Ψ i (r)ψ i (r) 2 V H (r) = 4πρ(r)

34 formalism H-K theorem: E is an unknown functional of the density E = E[ρ] Density Functional Theory approach Mapping of an interacting many-electron system into a system with independent particles moving into an effective potential. Find a set of orthonormal orbitals Ψ i (r) that minimizes: E = 1 2 N i=1 Ψ i (r) 2 Ψ i (r)dr + 1 ρ(r)v H (r)dr 2 + E xc [ρ(r)] + V ext (r)ρ(r)dr ρ(r) = N i=1 Ψ i (r)ψ i (r) 2 V H (r) = 4πρ(r)

35 formalism H-K theorem: E is an unknown functional of the density E = E[ρ] Density Functional Theory approach Mapping of an interacting many-electron system into a system with independent particles moving into an effective potential. Find a set of orthonormal orbitals Ψ i (r) that minimizes: E = 1 2 N i=1 Ψ i (r) 2 Ψ i (r)dr + 1 ρ(r)v H (r)dr 2 + E xc [ρ(r)] + V ext (r)ρ(r)dr ρ(r) = N i=1 Ψ i (r)ψ i (r) 2 V H (r) = 4πρ(r)

36 Performing a calculation A self-consistent equation ρ(r) = Ψ i (r)ψ i (r), where ψ i satisfies i ( 12 ) 2 + V H [ρ] + V xc [ρ] + V ext + V pseudo ψ i = E i ψ i, () Ingredients An XC potential, functional of the density several approximations exists (LDA,A,... ) A choice of the pseudopotential (if not all-electrons) (norm conserving, ultrasoft, PAW,... ) A basis set for expressing the ψ i An (iterative) algorithm for finding the wavefunctions ψ i A (good) computer...

37 Exchange-Correlation Energy E xc [ρ] = K exact [ρ] K KS [ρ] + V e e exact [ρ] V Hartree [ρ] K KS [ρ]: Kinetic energy for a non-interaction electron gas is an exact reformulation of the many-body problem formalism is exact (mapping of an interacting electron system into a non-interacting system Many types: Local Density Approximation (LDA) eneralized gradient approximation (A), meta-a, Hybrid functional (a part of exchange energy) Random Phase Approximation (RPA),...

38 Exchange-Correlation Energy E xc [ρ] = K exact [ρ] K KS [ρ] + V e e exact [ρ] V Hartree [ρ] K KS [ρ]: Kinetic energy for a non-interaction electron gas is an exact reformulation of the many-body problem formalism is exact (mapping of an interacting electron system into a non-interacting system Many types: Local Density Approximation (LDA) eneralized gradient approximation (A), meta-a, Hybrid functional (a part of exchange energy) Random Phase Approximation (RPA),...

39 Local Density Approximation (LDA) Assumption: For each point r in space with a density ρ(r), the exchange and correlation energy at that point is the same as for an uniform electron gas with that density: Exc LDA = ρ(r)ε xc(ρ)dr V V LDA xc (r) = E LDA xc ρ(r) = ε xc (ρ) + ρ(r) ε xc(ρ) ρ(r) ε xc (ρ): Exchange-correlation energy for a uniform electron gas of ( density (analytic): Exc LDA = 3 3π ) 1/3 4 ρ(r) 4/3 dr Correlation energy (interpolated analytically): Based on Quantum Monte Carlo calculations (Ceperley and Alder (1980)) for the exchange and correlation of the electron gas. Based on uniform gas of electron: ood cancellation of error

40 eneralized radient Approximations (A) Use the gradient of the density to have a more accurate exchange energy Many generalised gradient approximations: Perdew (1985), Lee, Yang and Parr (1988), Perdew and Wang (1991), PBE (1996),HCTH (1998),... Perdew and Yue (1986) functional (which is relatively simple): s = E A xc [ρ] = 3 4 ρ(r) 2k F (ρ(r)) ( 3 π ) 1/3 ρ(r) 4/3 F(s)dr k F = ((3π 2 ρ(r)) 1/3 F(s) = ( s s s 6 ) 1/15 LDA: Overbind (energies too high) underestimate bond lengths A: In general better energies specially PBE

41 eneralized radient Approximations (A) Use the gradient of the density to have a more accurate exchange energy Many generalised gradient approximations: Perdew (1985), Lee, Yang and Parr (1988), Perdew and Wang (1991), PBE (1996),HCTH (1998),... Perdew and Yue (1986) functional (which is relatively simple): s = E A xc [ρ] = 3 4 ρ(r) 2k F (ρ(r)) ( 3 π ) 1/3 ρ(r) 4/3 F(s)dr k F = ((3π 2 ρ(r)) 1/3 F(s) = ( s s s 6 ) 1/15 LDA: Overbind (energies too high) underestimate bond lengths A: In general better energies specially PBE

42 Hybrid functionals Add a small part of exchange interaction (25% to 50%) E Exchange = i<j V Many hybrid functionals: B3LYP, HSE,... φ i (r)φ j(r)φ i (r )φ j (r ) r r drdr The gap is (really) better and also cohesion energies Very time-consuming (10) Not the right physics and sometimes not so accurate...

43 Hybrid functionals Add a small part of exchange interaction (25% to 50%) E Exchange = i<j V Many hybrid functionals: B3LYP, HSE,... φ i (r)φ j(r)φ i (r )φ j (r ) r r drdr The gap is (really) better and also cohesion energies Very time-consuming (10) Not the right physics and sometimes not so accurate...

44 Performing a calculation A self-consistent equation ρ(r) = Ψ i (r)ψ i (r), where ψ i satisfies i ( 12 ) 2 + V H [ρ] + V xc [ρ] + V ext + V pseudo ψ i = E i ψ i, () Ingredients An XC potential, functional of the density several approximations exists (LDA,A,... ) A choice of the pseudopotential (if not all-electrons) (norm conserving, ultrasoft, PAW,... ) A basis set for expressing the ψ i An (iterative) algorithm for finding the wavefunctions ψ i A (good) computer...

45 s smoothening of wavefunctions For chemical properties only the valence electrons are relevant: Eliminate the chemically inactive core electrons Reduce the number of electron orbitals The pseudo-wavefunctions of the valence electrons are smooth Eliminate the rapid variations of the valence wavefunction in the core region A reasonable approximation V ext (r) = 1 4πε 0 R α r The pseudopotential approximation is less severe than the approximate nature of the exchange correlation functional α Z α

46 Non-Local s V PP (r,r ) = Y lm (r)v l (r)δ r,r Y lm (r ) l,m where Y l,m are spherical harmonics. V NL = φ I lm> V l <φ I lm I,l,m where I is the index of atom, l and m and quantum numbers. Use ab initio atomic wavefunctions to obtain pseudopotentials. The (potential) energy contribution due to the valence charge density must be substracted away to give an ionic pseudopotential: V ion l,ps(r) = V l,ps (r) + (V Hartree [ρ(r)] + V xc [ρ(r)]) free atom

47 aussian type separable s (HH) Local part V loc (r) = Z [ ] [ ion r erf + exp 1 ( ) ] 2 r r 2rloc 2 r loc { ( ) 2 ( ) 4 ( r r r C 1 + C 2 + C 3 + C 4 r loc r loc Nonlocal (separable) part H( r, r ) H sep ( r, r ) = p l 1(r) = 2 r l+ 3 2 l 2 i=1 r loc Y s,m (ˆr) p s i (r) h s i p s i (r ) Ys,m(ˆr ) m + Y p,m (ˆr) p p 1 (r) hp 1 pp 1 (r ) Yp,m(ˆr ) m r l+ 7 2 l ) 6 } r l e 1 2 ( r ) 2 r l p Γ(l l r l+2 e 1 2 ( r ) 2 r l 2(r) = ) Γ(l + 7) 2

48 Performing a calculation A self-consistent equation ρ(r) = Ψ i (r)ψ i (r), where ψ i satisfies i ( 12 ) 2 + V H [ρ] + V xc [ρ] + V ext + V pseudo ψ i = E i ψ i, () Ingredients An XC potential, functional of the density several approximations exists (LDA,A,... ) A choice of the pseudopotential (if not all-electrons) (norm conserving, ultrasoft, PAW,... ) A basis set for expressing the ψ i An (iterative) algorithm for finding the wavefunctions ψ i A (good) computer...

49 for electronic structure calculation How can we express the wavefunctions? Plane Waves Localization in Fourier space, efficient preconditioning Systematic convergence properties No localization in real space. Empty regions must be filled with PW. Non adaptive, Slater type Orbitals localized, well suited for molecules and other open structures Small number of basis functions for moderate accuracy Many different recipes for generating basis sets Over-completeness before convergence. Non systematic basis set.

50 for electronic structure? Wavelets A basis set both adaptive and systematic, real space based Wavelet basis sets Localized both in real and in Fourier space Allow for adaptivity (for internal electrons) Systematic basis set

51 Performing a calculation A self-consistent equation ρ(r) = Ψ i (r)ψ i (r), where ψ i satisfies i ( 12 ) 2 + V H [ρ] + V xc [ρ] + V ext + V pseudo ψ i = E i ψ i, () Ingredients An XC potential, functional of the density several approximations exists (LDA,A,... ) A choice of the pseudopotential (if not all-electrons) (norm conserving, ultrasoft, PAW,... ) A basis set for expressing the ψ i An (iterative) algorithm for finding the wavefunctions ψ i A (good) computer...

52 Equations: Operators Apply different operators Having a one-electron hamiltonian in a mean field. [ 1 ] V eff (r)(r) ψ i = ε i ψ i V eff (r) = V ext (r) + V ρ(r ) r r dr + µ xc (r) E[ρ] can be expressed by the orthonormalized states of one particule: ψ i (r) with the fractional occupancy number f i (0 f i 1) : ρ(r) = f i ψ i (r) 2 i

53 Equations: Computing Energies Calculate different integrals E[ρ] = K [ρ] + U[ρ] U[ρ] = K [ρ] = m e i V dr ψ i 2 ψ i dr V ext(r)ρ(r)+ 1 ρ(r)ρ(r ) dr dr V 2 V r r }{{} Hartree We minimise with the variables ψ i (r) and f i with the constraint dr ρ(r) = N el. V + E xc [ρ] }{{} exchange correlation

54 KS Equations: Self-Consistent Field Set of self-consistent equations: { 1 } V eff ψ i = ε i ψ i 2 m e with an effective potential: V eff (r) = V ext (r)+ ) dr ρ(r V r r }{{} Hartree δe xc + δρ(r) }{{} exchange correlation and: ρ(r) = i f i ψ i (r) 2

55 KS Equations: Self-Consistent Field Set of self-consistent equations: { 1 } V eff ψ i = ε i ψ i 2 m e with an effective potential: V eff (r) = V ext (r)+ ) dr ρ(r V r r }{{} Hartree δe xc + δρ(r) }{{} exchange correlation and: ρ(r) = i f i ψ i (r) 2 Poisson Equation: V Hartree = ρ (Laplacian: = 2 x y z 2 ) Real Mesh (100 3 = 10 6 ): = evaluations!

56 KS Equations: Self-Consistent Field Set of self-consistent equations: { 1 } V eff ψ i = ε i ψ i 2 m e with an effective potential: V eff (r) = V ext (r)+ ) dr ρ(r V r r }{{} Hartree δe xc + δρ(r) }{{} exchange correlation and: ρ(r) = i f i ψ i (r) 2 Poisson Equation: V Hartree = ρ (Laplacian: = 2 x y z 2 ) Real Mesh (100 3 = 10 6 ): = evaluations!

57 KS Equations: Self-Consistent Field Set of self-consistent equations: { 1 } V eff ψ i = ε i ψ i 2 m e with an effective potential: V eff (r) = V ext (r)+ ) dr ρ(r V r r }{{} Hartree δe xc + δρ(r) }{{} exchange correlation and: ρ(r) = i f i ψ i (r) 2 Poisson Equation: V Hartree = ρ (Laplacian: = 2 x y z 2 ) Real Mesh (100 3 = 10 6 ): = evaluations!

58 KS Equations: Self-Consistent Field Set of self-consistent equations: { 1 } V eff ψ i = ε i ψ i 2 m e with an effective potential: V eff (r) = V ext (r)+ ) dr ρ(r V r r }{{} Hartree δe xc + δρ(r) }{{} exchange correlation and: ρ(r) = i f i ψ i (r) 2 Poisson Equation: V Hartree = ρ (Laplacian: = 2 x y z 2 ) Real Mesh (100 3 = 10 6 ): = evaluations!

59 KS Equations: Self-Consistent Field Set of self-consistent equations: { 1 } V eff ψ i = ε i ψ i 2 m e with an effective potential: V eff (r) = V ext (r)+ ) dr ρ(r V r r }{{} Hartree δe xc + δρ(r) }{{} exchange correlation and: ρ(r) = i f i ψ i (r) 2 Poisson Equation: V Hartree = ρ (Laplacian: = 2 x y z 2 ) Real Mesh (100 3 = 10 6 ): = evaluations!

60 Ab Initio Methods: [ ] V(r) +µ xc (r) ψ k i = εk i ψ k i

61 Ab Initio Methods: [ ] V(r) +µ xc (r) ψ k i = ε k i ψ k i atomic orbitals plane waves augmented Slater numerical real space finite difference Wavelet

62 Ab Initio Methods: self-consistent (SCF) Harris-Foulkes functional pseudopotential all electrons norm-conserving PAW [ V(r) ] +µ xc (r) ψ k i = ε k i ψ k i atomic orbitals plane waves augmented Slater numerical real space finite difference Wavelet

63 Ab Initio Methods: self-consistent (SCF) Harris-Foulkes functional pseudopotential all electrons norm-conserving PAW [ V(r) periodic non-periodic ] +µ xc (r) ψ k i atomic orbitals plane waves augmented real space = ε k i ψ k i Slater numerical finite difference Wavelet

64 Ab Initio Methods: self-consistent (SCF) Harris-Foulkes functional pseudopotential norm-conserving all electrons PAW Hybrid functionals beyond LDA W method [ V(r) periodic non-periodic LDA,A ] +µ xc (r) ψ k i atomic orbitals plane waves augmented real space LDA+U = ε k i ψ k i Slater numerical finite difference Wavelet

65 Ab Initio Methods: self-consistent (SCF) Harris-Foulkes functional pseudopotential norm-conserving all electrons PAW Hybrid functionals beyond LDA W [ V(r) periodic non-periodic non-collinear collinear non-spin polarized spin polarized LDA,A ] +µ xc (r) ψ k i atomic orbitals plane waves augmented real space LDA+U = ε k i ψ k i Slater numerical finite difference Wavelet

66 Ab Initio Methods: self-consistent (SCF) Harris-Foulkes functional relativistic pseudopotential norm-conserving all electrons PAW Hybrid functionals beyond LDA W non-relativistic [ periodic non-periodic non-collinear collinear +V(r) non-spin polarized spin polarized LDA,A ] +µ xc (r) ψ k i = ε k i ψ k i atomic orbitals plane waves augmented real space LDA+U Slater numerical finite difference Wavelet

67 Ab Initio Methods: self-consistent (SCF) Harris-Foulkes functional relativistic pseudopotential norm-conserving all electrons PAW Hybrid functionals beyond LDA W non-relativistic [ N 3 scaling O(N) methods periodic non-periodic non-collinear collinear +V(r) non-spin polarized spin polarized LDA,A ] +µ xc (r) ψ k i = ε k i ψ k i atomic orbitals plane waves augmented real space LDA+U Slater numerical finite difference Wavelet

68 List of ab initio Codes Plane Waves ABINIT Louvain-la-Neuve CPMD Zurich, Lugano PWSCF Italy VASP Vienna aussian aussian DeMon CP2K Siesta Madrid (numerical basis sets) Wien Vienna (FPLAPW, all electrons) basis set ONETEP Big

69 Minimisation of the electronic density

70 Plane Wave basis sets Natural basis set for electronic structure calculations in periodic solids. 1 V exp(i r) where V is the volume of the periodicity volume and is a multiple of the reciprocal lattice vectors, i.e. there are integers l 1,l 2,l 3 such that = l 1 b 1 + l 2 b 2 + l 3 b 3.

71 Plane Wave basis sets Matrix elements of the kinetic energy part of the Hamiltonian calculated analytically exp( i r) ( 12 ) 2 exp(i r) = 2 δ, The potential energy part obtained numerically by calculating the wave-functions in real space Ψ(r) = exp(i r) and then integrating numerically in real space. The transformation on a grid in real space is done with the help of the Fast Fourier transformation at a cost of N log 2 (N) operations for a basis set of N plane waves.

72 : Planes Waves Operator approach: no need to set up Hamiltonian matrix FFT allows for quasi linear M log(m) scaling where M is number of plane waves Localization in Fourier space allows for efficient preconditioning techniques. Hence number of iterations independent of M Systematic convergence propertie No localization in real space. Empty regions have to be filled with plane waves. Only supercell approach. O(N) scaling hard to achieve Resolution cannot adaptively be refined around the nucleus. All electron calculations impossible, hard pseudopotentials difficult.

73 : Planes Waves Operator approach: no need to set up Hamiltonian matrix FFT allows for quasi linear M log(m) scaling where M is number of plane waves Localization in Fourier space allows for efficient preconditioning techniques. Hence number of iterations independent of M Systematic convergence propertie No localization in real space. Empty regions have to be filled with plane waves. Only supercell approach. O(N) scaling hard to achieve Resolution cannot adaptively be refined around the nucleus. All electron calculations impossible, hard pseudopotentials difficult.

74 : Planes Waves self-consistent (SCF) Harris-Foulkes functional relativistic pseudopotential norm-conserving all electrons PAW Hybrid functionals beyond LDA W non-relativistic [ N 3 scaling O(N) methods periodic non-periodic non-collinear collinear +V(r) non-spin polarized spin polarized LDA,A ] +µ xc (r) ψ k i = ε k i ψ k i atomic orbitals plane waves augmented real space LDA+U Slater numerical finite difference Wavelet

75 : List of Codes ABINIT Louvain-la-Neuve PAW, linear response, W, many features, good // CPMD Zurich, Lugano USP, good //, QM/MM with romacs PWSCF Italy USP, linear response, // VASP Vienna robust, fast, hybrid functional, W, //

76 : Flowchart { } ψ j = c j ei.r Basis (N ): 0,..., Orthonormalized

77 : Flowchart { inv FFT } ψ j = c j ei.r Basis (N ): 0,..., Orthonormalized ρ(r) = f j ψ j (r) 2 j Basis (2 3 N ): 0,...,2

78 : Flowchart { inv FFT } ψ j = c j ei.r Basis (N ): 0,..., Orthonormalized ρ(r) = f j ψ j (r) 2 j Basis (2 3 N ): 0,...,2 FFT V 2 xc [ρ(r)] V H () = ρ() V NL ({ψ j }) V effective

79 : Flowchart { inv FFT } ψ j = c j ei.r Basis (N ): 0,..., Orthonormalized ρ(r) = f j ψ j (r) 2 j Basis (2 3 N ): 0,...,2 FFT V 2 xc [ρ(r)] V H () = ρ() V NL ({ψ j }) V effective c j Kinetic Term

80 : Flowchart { inv FFT } ψ j = c j ei.r Basis (N ): 0,..., Orthonormalized ρ(r) = f j ψ j (r) 2 j Basis (2 3 N ): 0,...,2 FFT V 2 xc [ρ(r)] V H () = ρ() V NL ({ψ j }) V effective c j FFT Kinetic Term δc j = E total c j () + l Λ jl c l Λ jl =< ψ j H ψ l >

81 : Flowchart { inv FFT } ψ j = c j ei.r Basis (N ): 0,..., Orthonormalized ρ(r) = f j ψ j (r) 2 j Basis (2 3 N ): 0,...,2 FFT V 2 xc [ρ(r)] V H () = ρ() V NL ({ψ j }) V effective c j FFT Kinetic Term δc j = E total c new,j c j () + l Λ jl c l = c j + h stepδc j Λ jl =< ψ j H ψ l > Steepest Descent, Conjugate radient, Direct Inversion of the Iterative Subspace

82 : Flowchart { inv FFT } ψ j = c j ei.r Basis (N ): 0,..., Orthonormalized ρ(r) = f j ψ j (r) 2 j Basis (2 3 N ): 0,...,2 FFT V 2 xc [ρ(r)] V H () = ρ() V NL ({ψ j }) V effective c j FFT Kinetic Term δc j = E total Stop when δc j small c new,j c j () + l Λ jl c l = c j + h stepδc j Λ jl =< ψ j H ψ l > Steepest Descent, Conjugate radient, Direct Inversion of the Iterative Subspace

83 Direct Minimization: List of Codes Plane Waves aussian CPMD Zurich, Lugano CP2K basis set ONETEP PARATEC Big

84 Scheme: Flowchart { ψ j = c j }, ei.r {f j } Basis (N ): 0,..., Orthonormalized

85 Scheme: Flowchart { inv FFT ψ j = c j }, ei.r {f j } Basis (N ): 0,..., Orthonormalized ρ(r) = f j ψ j (r) 2 j Basis (2 3 N ): 0,...,2

86 Scheme: Flowchart { inv FFT ψ j = c j }, ei.r {f j } Basis (N ): 0,..., Orthonormalized ρ(r) = f j ψ j (r) 2 j FFT Basis (2 3 N ): 0,...,2 V 2 xc [ρ(r)] V H () = ρ() V NL ({ψ j }) V effective

87 Scheme: Flowchart { inv FFT ψ j = c j }, ei.r {f j } Basis (N ): 0,..., Orthonormalized ρ(r) = f j ψ j (r) 2 j FFT Basis (2 3 N ): 0,...,2 V 2 xc [ρ(r)] V H () = ρ() V NL ({ψ j }) V effective c j Kinetic Term

88 Scheme: Flowchart { inv FFT ψ j = c j }, ei.r {f j } Basis (N ): 0,..., Orthonormalized ρ(r) = f j ψ j (r) 2 j FFT Basis (2 3 N ): 0,...,2 V 2 xc [ρ(r)] V H () = ρ() V NL ({ψ j }) V effective c j FFT Kinetic Term [ V eff (r) ] ψ j = ε j ψ j

89 Scheme: Flowchart { inv FFT ψ j = c j }, ei.r {f j } Basis (N ): 0,..., Orthonormalized ρ(r) = f j ψ j (r) 2 j FFT Basis (2 3 N ): 0,...,2 V 2 xc [ρ(r)] V H () = ρ() V NL ({ψ j }) V effective c j FFT Kinetic Term [ V eff (r) ] ψ j = ε j ψ j ρ new (r) = αρ out + (1 α)ρ in ρ out (r) = f j ψ j (r) 2 j Mixing Density: Anderson, Broyden,DIIS

90 Scheme: Flowchart { inv FFT ψ j = c j }, ei.r {f j } Basis (N ): 0,..., Orthonormalized ρ(r) = f j ψ j (r) 2 j FFT Basis (2 3 N ): 0,...,2 V 2 xc [ρ(r)] V H () = ρ() V NL ({ψ j }) V effective c j FFT Kinetic Term [ V eff (r) ] ψ j = ε j ψ j Stop when ρ new ρ in small ρ new (r) = αρ out + (1 α)ρ in ρ out (r) = f j ψ j (r) 2 j Mixing Density: Anderson, Broyden,DIIS

91 Iterative diagonalization Standard diagonalization routines, such as found in software packages like LAPACK, are designed to find all the eigenvalues and eigenvectors of a matrix. Scale like n 3, where n is the dimension of the matrix We need to calculate only a small number of eigenvalues and eigenvectors of matrices of a very large dimension, another type of diagonalization, namely iterative diagonalization is used: Lanczos, Davidson, RM-DIIS, Conjugate radient, LOBPC,...

92 Diagonalization scheme: List of Codes ABINIT Louvain-la-Neuve Diagonalization: Conjugate-radient, LOBPC Mixing: Broyden, Pulay, Anderson, others based on dielectric constant CPMD Zurich, Lugano Diagonalization: Lanczos, Davidson Mixing: Broyden, Pulay, Anderson PWSCF Italy VASP Vienna Diagonalization: RM-DIIS Big

93 Atom centered are the most popular basis functions for electronic structure calculations in the quantum chemistry community. (x X i ) l 1 (y Y i ) l 2 (z Z i ) l 3 exp ( α((x X i ) 2 + (y Y i ) 2 + (z Z i ) 2 ) ) The different functions associated to R i = (X i,y i,z i ) differ by their extent, specified by α and by the exponents l 1, l 2, l 3. Because the have qualitatively the shape of atomic orbitals one needs for moderate accuracy only a small number of basis function per atom (typically 10 to 20 per atom).

94 : localization makes them well suited for molecules and other open structures. Kinetic and overlap matrix elements can be calculated analytically. ood description of core electrons. Small number of basis functions necessary for moderate accuracy No systematic convergence. Over-completeness before convergence. This leads also to problems with O(N). Many different recipes for generating basis sets. Pulay forces have been implemented.

95 : localization makes them well suited for molecules and other open structures. Kinetic and overlap matrix elements can be calculated analytically. ood description of core electrons. Small number of basis functions necessary for moderate accuracy No systematic convergence. Over-completeness before convergence. This leads also to problems with O(N). Many different recipes for generating basis sets. Pulay forces have been implemented.

96 : self-consistent (SCF) Harris-Foulkes functional relativistic pseudopotential norm-conserving all electrons PAW Hybrid functionals beyond LDA W non-relativistic [ N 3 scaling O(N) methods periodic non-periodic non-collinear collinear +V(r) non-spin polarized spin polarized LDA,A LDA+U ] +µ xc (r) ψ= k i ε k i S ψ k i atomic orbitals plane waves augmented real space Slater numerical finite difference Wavelet

97 basis set: wavelets (Big) self-consistent (SCF) Harris-Foulkes functional relativistic pseudopotential norm-conserving all electrons PAW Hybrid functionals beyond LDA W method non-relativistic [ N 3 scaling O(N) methods periodic non-periodic non-collinear collinear +V(r) +V xc (r) non-spin polarized spin polarized LDA,A ] ψ k i atomic orbitals plane waves augmented real space LDA+U = ε k i ψ k i Slater numerical finite difference Wavelet

98 : List of Codes ONETEP sinc function, O(N) method, inside Material Studio, good // Big wavelets, good // PARATEC finite difference, good // OCTOPUS finite difference, TD- PAW rid-based projector-augmented wave method, finite difference

99 : List of Codes ONETEP sinc function, O(N) method, inside Material Studio, good // Big wavelets, good // PARATEC finite difference, good // OCTOPUS finite difference, TD- PAW rid-based projector-augmented wave method, finite difference

100 : List of Codes ONETEP sinc function, O(N) method, inside Material Studio, good // Big wavelets, good // PARATEC finite difference, good // OCTOPUS finite difference, TD- PAW rid-based projector-augmented wave method, finite difference

101 : List of Codes ONETEP sinc function, O(N) method, inside Material Studio, good // Big wavelets, good // PARATEC finite difference, good // OCTOPUS finite difference, TD- PAW rid-based projector-augmented wave method, finite difference

102 : List of Codes ONETEP sinc function, O(N) method, inside Material Studio, good // Big wavelets, good // PARATEC finite difference, good // OCTOPUS finite difference, TD- PAW rid-based projector-augmented wave method, finite difference

103

104 Functionals: Mixed between physics and chemistry approach QM/MM (Quantum Mechanics/Molecular Modeling) Multi-scale approach (more than 1000 atoms feasible for a better parametrization) Order N (real space basis set as wavelets) Numerical experience (high performance computing): One-day simulation Better exploration of atomic configurations Molecular Dynamics (4s per step for 32 water molecules)