Institut Néel Institut Laue Langevin. Introduction to electronic structure calculations
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1 Institut Néel Institut Laue Langevin Introduction to electronic structure calculations 1 Institut Néel - 25 rue des Martyrs - Grenoble - France 2 Institut Laue Langevin - 71 avenue des Martyrs - Grenoble - France Ecole GdR Matériaux et Interactions en COmpétitions ILL, May 25 th 31 th 2014
2 The Born-Oppenheimer approximation System : No nucleus and N electrons
3 The Born-Oppenheimer approximation System : No nucleus and N electrons Ĥ = Nucl. kin. energy ˆT n + + Elec.-nucl. attraction ˆ ne + Elec. kin. energy ˆT e Elec.-elec repulsion ˆ ee + Nucl.-nucl repulsion ˆ nn
4 The Born-Oppenheimer approximation System : No nucleus and N electrons Ĥ = Nucl. kin. energy ˆT n + + Elec.-nucl. attraction ˆ ne + Elec. kin. energy ˆT e Elec.-elec repulsion ˆ ee + Nucl.-nucl repulsion ˆ nn ˆT n = No 2 a=1 2 M a 2 Ra
5 The Born-Oppenheimer approximation System : No nucleus and N electrons Ĥ = Nucl. kin. energy ˆT n + + Elec.-nucl. attraction ˆ ne + Elec. kin. energy ˆT e Elec.-elec repulsion ˆ ee + Nucl.-nucl repulsion ˆ nn No ˆT 2 n = 2 ˆTe 2 M = a Ra a=1 N i=1 2 2 m e 2 r i
6 The Born-Oppenheimer approximation System : No nucleus and N electrons Ĥ = Nucl. kin. energy ˆT n + + Elec.-nucl. attraction ˆ ne + Elec. kin. energy ˆT e Elec.-elec repulsion ˆ ee + Nucl.-nucl repulsion ˆ nn ˆ ne = e2 4πǫ 0 No N a=1 i=1 Z a Ra r i
7 The Born-Oppenheimer approximation System : No nucleus and N electrons Ĥ = Nucl. kin. energy ˆT n + + Elec.-nucl. attraction ˆ ne + Elec. kin. energy ˆT e Elec.-elec repulsion ˆ ee + Nucl.-nucl repulsion ˆ nn ˆ ne = e2 4πǫ 0 No N a=1 i=1 Z a Ra r i ˆ nn = e2 4πǫ 0 No a 1 a=1 b=1 Z a Z b Ra Rb ˆ ee = e2 4πǫ 0 N i 1 i=1 j=1 1 r i r j
8 The Born-Oppenheimer approximation System : No nucleus and N electrons Ĥ = Nucl. kin. energy ˆT n + + Elec.-nucl. attraction ˆ ne + Elec. kin. energy ˆT e Elec.-elec repulsion ˆ ee + Nucl.-nucl repulsion ˆ nn Electron mass mē = 9, kg Proton mass m p = 1, kg Neutron mass m n = 1, kg
9 The Born-Oppenheimer approximation System : No nucleus and N electrons Ĥ = Nucl. kin. energy ˆT n + + Elec.-nucl. attraction ˆ ne + Elec. kin. energy ˆT e Elec.-elec repulsion ˆ ee + Nucl.-nucl repulsion ˆ nn Approximation : Mobile electrons Fixed nucleus
10 The Born-Oppenheimer approximation System : No nucleus and N electrons Ĥ = Nucl. kin. energy ˆT n + + Elec.-nucl. attraction ˆ ne + Elec. kin. energy ˆT e Elec.-elec repulsion ˆ ee + Nucl.-nucl repulsion ˆ nn Approximation : Mobile electrons Fixed nucleus First step : Ĥ e = ˆTe + ˆ ne + ˆ ee + K with K = nn
11 The Born-Oppenheimer approximation System : No nucleus and N electrons Ĥ = Nucl. kin. energy ˆT n + + Elec.-nucl. attraction ˆ ne + Elec. kin. energy ˆT e Elec.-elec repulsion ˆ ee + Nucl.-nucl repulsion ˆ nn Approximation : Mobile electrons Fixed nucleus First step : Ĥ e = ˆTe + ˆ ne + ˆ ee + K Ĥ e Ψ e = E e Ψ e
12 The Born-Oppenheimer approximation System : No nucleus and N electrons Ĥ = Nucl. kin. energy ˆT n + + Elec.-nucl. attraction ˆ ne + Elec. kin. energy ˆT e Elec.-elec repulsion ˆ ee + Nucl.-nucl repulsion ˆ nn Approximation : Mobile electrons Fixed nucleus First step : Ĥ e = ˆTe + ˆ ne + ˆ ee + K Ĥ e Ψ e = E e Ψ e Ĥ n = ˆTn + E e Ĥ n ξ ν = (E ν + E e ) ξ ν
13 E = E e +E ν Ψ = Ψ e ξ ν The Born-Oppenheimer approximation System : No nucleus and N electrons Ĥ = Nucl. kin. energy ˆT n + + Elec.-nucl. attraction ˆ ne + Elec. kin. energy ˆT e Elec.-elec repulsion ˆ ee + Nucl.-nucl repulsion ˆ nn Approximation : Mobile electrons Fixed nucleus First step : Ĥ e = ˆTe + ˆ ne + ˆ ee + K Ĥ e Ψ e = E e Ψ e Ĥ n = ˆTn + E e Ĥ n ξ ν = (E ν + E e ) ξ ν
14 What type of information can we get? The electronic part The ground state energy (E e ( Ra )) The ground state wave-function (charge, orders, etc...) Structural optimization (min Ra E e ( Ra )) The electronic excitation spectrum dots The nuclear part Phonons spectrum (E ν ) Dynamics (E (k) e ( Ra )) Diffusion processes (impurity, vacancies etc...) dots
15 Solving the electronic problem Ĥ e = ˆTe + ˆ ne + ˆ ee + nn N = 2 2 e2 2 m r e i 4πǫ 0 = i=1 N ĥ 1 ( r i ) + i=1 } {{ } simple No a=1 N i 1 ĥ 2 ( r i, r j ) i=1 j=1 } {{ } difficult Z N i 1 a Ra r i + + nn i=1 j=1 e 2 4πǫ 0 1 r i r j + nn
16 Solving the electronic problem Ĥ e = ˆTe + ˆ ne + ˆ ee + nn N = 2 2 e2 2 m r e i 4πǫ 0 = i=1 N ĥ 1 ( r i ) + i=1 } {{ } dominant No a=1 N i 1 ĥ 2 ( r i, r j ) i=1 j=1 } {{ } dominated by average Z N i 1 a Ra r i + + nn i=1 j=1 e 2 4πǫ 0 1 r i r j + nn
17 Solving the electronic problem Ĥ e = ˆTe + ˆ ne + ˆ ee + nn N = 2 2 e2 2 m r e i 4πǫ 0 = i=1 N ĥ 1 ( r i ) + i=1 } {{ } dominant No a=1 N i 1 ĥ 2 ( r i, r j ) i=1 j=1 } {{ } dominated by average Z N i 1 a Ra r i + + nn i=1 j=1 e 2 4πǫ 0 1 r i r j + nn N i=1 i 1 j=1 ĥ2( r i, r j ) N i=1 δĥ1( r i )
18 Solving the electronic problem The one-particle approximation N ĥ 1 ( r i ) + i=1 N i 1 ĥ 2 ( r i, r j ) i=1 j=1 N [ĥ1 ( r i ) +δĥ 1 ( r i ) ] i=1
19 Solving the electronic problem The one-particle approximation N ĥ 1 ( r i ) + i=1 N i 1 ĥ 2 ( r i, r j ) i=1 j=1 N [ĥ1 ( r i ) +δĥ 1 ( r i ) ] i=1 From a N electrons problem to a 1 electron problem From Ĥ e Ψ e ( r 1,..., r N ) = E e Ψ e ( r 1,..., r N ) ) to (ĥ1 + δĥ 1 ϕk ( r) = ε k ϕ k ( r)
20 Solving the electronic problem The one-particle approximation N ĥ 1 ( r i ) + i=1 N i 1 ĥ 2 ( r i, r j ) i=1 j=1 N [ĥ1 ( r i ) +δĥ 1 ( r i ) ] i=1 From the 1 electron system to the N electrons one.. ε F Filling following Hund s rules Ψ e ( r 1,..., r N ) = ϕ 1 ( r 1 )...ϕ N ( r N ) n E e = i=1 ε i + nn
21 Solving the electronic problem for the ground state The Hartree-Fock approximation Mean-field approximation on electron positions N i 1 i=1 j=1 e 2 4πǫ 0 1 r i r j N i=1 e 2 4πǫ 0 1 r i r j r j The average is taken over the ground-state wave-function
22 Solving the electronic problem for the ground state The Hartree-Fock approximation Mean-field approximation on electron positions N i 1 i=1 j=1 e 2 4πǫ 0 1 r i r j N i=1 e 2 4πǫ 0 1 r i r j r j The average is taken over the ground-state wave-function Fock operator ˆF = ĥ1 +δĥ 1 depends on its solution
23 Solving the electronic problem for the ground state The Hartree-Fock approximation Mean-field approximation on electron positions N i 1 i=1 j=1 e 2 4πǫ 0 1 r i r j N i=1 e 2 4πǫ 0 1 r i r j r j The average is taken over the ground-state wave-function Fock operator ˆF = ĥ1 +δĥ 1 depends on its solution Lowest energy Slater determinant
24 Solving the electronic problem for the ground state The Hartree-Fock approximation Mean-field approximation on electron positions N i 1 i=1 j=1 e 2 4πǫ 0 1 r i r j N i=1 e 2 4πǫ 0 1 r i r j r j The average is taken over the ground-state wave-function Fock operator ˆF = ĥ1 +δĥ 1 depends on its solution Lowest energy Slater determinant ˆF treats exactly ˆF does not contain Kinetic energy Nuclear attraction Coulomb repulsion energy Exchange energy Correlation energy (correlation between electrons positions)
25 Solving the electronic problem for the ground state The Hartree-Fock approximation Advantages Well defined method Much less costly than N-body methods Easy to improve on it (perturbation theory etc...) Drawbacks A tendency to pair localize the electrons Over-estimate the gaps Still too costly for very large systems
26 Solving the electronic problem for the ground state The Density Functional Theory The fundamental theorems There is a one to one correspondence between nuclear positions and GS electronic density ˆ ne n( r) The GS density is the N-representable density associated with the lowest energy Min n( r), N rep. E[n( r)]
27 Solving the electronic problem for the ground state The Density Functional Theory The fundamental theorems There is a one to one correspondence between nuclear positions and GS electronic density ˆ ne n( r) The GS density is the N-representable density associated with the lowest energy A major problem Nobody knows how to get An exact functional for E[n( r)] Min n( r), N rep. E[n( r)] Even for the kinetic energy part
28 Solving the electronic problem for the ground state The Density Functional Theory Back to a one-electron pb : the Kohn-Sham approximation A N-electron pb : Ĥ( r 1... r N ) Ψ( r 1... r N ) Density n( r) An effective 1-electron pb : Ĥ eff = N i=1 ĥks( r i ) ϕ 1 ( r 1 )...ϕ N ( r N ) Density n KS ( r) = n( r)
29 Solving the electronic problem for the ground state The Density Functional Theory Back to a one-electron pb : the Kohn-Sham approximation A N-electron pb : Ĥ( r 1... r N ) Ψ( r 1... r N ) Density n( r) Kinetic energy ˆTe An effective 1-electron pb : Ĥ eff = N i=1 ĥks( r i ) ϕ 1 ( r 1 )...ϕ N ( r N ) Density n KS ( r) = n( r) Kinetic ener. of 1ē pb : ˆTKS
30 Solving the electronic problem for the ground state The Density Functional Theory Back to a one-electron pb : the Kohn-Sham approximation A N-electron pb : Ĥ( r 1... r N ) Ψ( r 1... r N ) Density n( r) Kinetic energy ˆTe ˆ ne An effective 1-electron pb : Ĥ eff = N i=1 ĥks( r i ) ϕ 1 ( r 1 )...ϕ N ( r N ) Density n KS ( r) = n( r) Kinetic ener. of 1ē pb : ˆTKS ˆ ext [n] = e2 No Z a n( r) 4πǫ 0 a=1 Ra r
31 Solving the electronic problem for the ground state The Density Functional Theory Back to a one-electron pb : the Kohn-Sham approximation A N-electron pb : Ĥ( r 1... r N ) Ψ( r 1... r N ) Density n( r) Kinetic energy ˆTe ˆ ne ˆ ee An effective 1-electron pb : Ĥ eff = N i=1 ĥks( r i ) ϕ 1 ( r 1 )...ϕ N ( r N ) Density n KS ( r) = n( r) Kinetic ener. of 1ē pb : ˆTKS ˆ ext [n] = e2 No Z a n( r) 4πǫ 0 a=1 Ra r ˆ Hartree [n]+ˆ xc [n] ˆ Hartree [n] = e2 n( r ) 4πǫ 0 r r d r n( r) ˆ xc [n] =??
32 Solving the electronic problem for the ground state The Density Functional Theory Theˆ xc [n] functionals LDA/LSDA Local Density Approximation ˆ xc [n] = homogeneous ē gas Perdew et Zunger (PZ), Vosko Wilkes and Nusiar (VWN)
33 Solving the electronic problem for the ground state The Density Functional Theory Theˆ xc [n] functionals LDA/LSDA Local Density Approximation ˆ xc [n] = homogeneous ē gas Perdew et Zunger (PZ), Vosko Wilkes and Nusiar (VWN) Easy, fast OK for Fermi ē of metals Bad for core ē Only local f(n( r)) exch.-corr. : non local f( r, r ) Self-inter. pb Overdelocalize Underestimates gaps No dispersion forces
34 Solving the electronic problem for the ground state The Density Functional Theory Theˆ xc [n] functionals GGA Generalized Gradient Approximation Add semi-local terms : f(n( r), r (n( r)) Perdew-Burke-Ernzerhof (PBE, PBESOL), Perdew-Wang 91 (PW91), BLYP, second order (SOGGA), Perdew-Wang (PW), Wu-Cohen (WC) A little better than LDA Still same pb
35 Solving the electronic problem for the ground state The Density Functional Theory Theˆ xc [n] functionals GGA Generalized Gradient Approximation Add semi-local terms : f(n( r), r (n( r)) Perdew-Burke-Ernzerhof (PBE, PBESOL), Perdew-Wang 91 (PW91), BLYP, second order (SOGGA), Perdew-Wang (PW), Wu-Cohen (WC) A little better than LDA Still same pb
36 Solving the electronic problem for the ground state The Density Functional Theory Theˆ xc [n] functionals Hybrids functionals Mix of LDA - GGA - HF exact exchange B3LYP, B3PW, PBE0 and PBESOL0, B1PW, B1WC Best GS energies Good structures Part self-inter. pb Gaps Bad for core ē No dispersion forces More costly Cooking recipe
37 Solving the electronic problem for the ground state The Density Functional Theory Theˆ xc [n] functionals Hybrids functionals Mix of LDA - GGA - HF exact exchange B3LYP, B3PW, PBE0 and PBESOL0, B1PW, B1WC Best GS energies Good structures Part self-inter. pb Gaps Bad for core ē No dispersion forces More costly Cooking recipe Double Hybrid functionals Dispersion forces Mix LDA - GGA - HF exchange - 2 nd order pert. (Grimme)
38 Solving the electronic problem for the ground state The SCF procedure 1ē Hamiltonian ˆF depends on its solution SCF Guess n( r) and/orϕ k No 1ē Hamiltonian ˆF New n( r) and/orϕ k Convergence test yes Converged
39 Solving the electronic problem for the ground state The basis sets 1ē Hamiltonian projected on a finite basis set Plane waves (CASTEP, AbInit...) Atom centered gaussians (radial) + Y lm (angular) CRYSTAL Atom centered staler (radial) + Y lm (angular) SIESTA Atom centered numerical basis sets SIESTA, AdfBand Numerical grids Wavelets on grids BigDFT
40 Solving the electronic problem for the ground state The basis sets 1ē Hamiltonian projected on a finite basis set Plane waves (CASTEP, AbInit...) Atom centered gaussians (radial) + Y lm (angular) CRYSTAL Atom centered staler (radial) + Y lm (angular) SIESTA Atom centered numerical basis sets SIESTA, AdfBand Numerical grids Wavelets on grids BigDFT Inner electrons All electrons Pseudopotentials Atomic functions within muffin-tin spheres
41 Solving the electronic problem for the ground state Do not forget Test numerical parameters Type of pseudopotential Heavy elements : relativistic corrections Plane waves : energy cutoff Atomic basis sets : basis set quality k points convergence Metals : smearing temperature Phonons : geom. opt. convergence (negative phonons)
42 Electronic solution yield many properties DOSS and PDOSS 800 DENSITY OF STATES (STATES/HARTREE/CELL) ,5-0,4-0,3-0,2-0,1 0 0,1 ENERGY (HARTREE)
43 Electronic solution yield many properties DOSS and PDOSS DENSITY OF STATES (STATES/HARTREE/CELL) DENSITY OF STATES (STATES/HARTREE/CELL) V 1 - z 2 V 1 - xz V 1 - yz V 1 - x 2 -y 2 V 1 - xy -0,2-0,1 0 0,1 0,2 ENERGY (HARTREE) V 2 - z 2 V 2 - xz V 2 - yz -0,2-0,1 0 0,1 0,2 ENERGY (HARTREE) V 2 - x 2 -y 2 V 2 - xy
44 Conclusion Useful methods Easy to interpret (1ē picture) Access many properties Do not use as a black box Works well for weakly correlated systems Works well for prop. slightly dependent on Fermi level ē (geometries, phonons, polarisation, etc...) Careful with magnetism (may work reasonably but also be all wrong) Remember to test for validity and computational parameters
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