The electronic structure of materials 2 - DFT

Transcription

1 Quantum mechanics 2 - Lecture 9 December 19, 2012

2 1 Density functional theory (DFT) 2 Literature

3 Contents 1 Density functional theory (DFT) 2 Literature

4 Historical background The beginnings: L. de Broglie (1923) E. Schrödinger (1925) W. Pauli (1925) L. Thomas & E. Fermi (1927) P. A. M. Dirac (1928)

5 Historical background Quantum mechanics technology transition: F. Bloch (1928) R. Wilson (1931) - implications of band theory - insulators/metals J. Slater ( ) - bands of Na E. Wigner & F. Seitz (1935) - quantitative calcs on Na J. Bardeen (1935) - Fermi surface first understanding of semiconductors (1930 s) transistor (1940 s)

6 Historical background Electronic structure methods: E. Hylleraas (1929) - numerically exact solution for H 2 J. Slater (1937) - augmented plane waves (APW) C. Herring (1940) - orthogonalized plane waves (OPW) S. F. Boys (1950 s) - Gaussian orbitals J. C. Phillips & L. Kleinman (1950 s) - pseudopotentials O. K. Andersen (1975) - linear muffin tin orbitals (LMTO)

7 Historical background Density functional theory: P. Hohenberg & W. Kohn (1964) W. Kohn & L. J. Sham (1965) R. Car & M. Parrinello (1985) - CPMD improved approximations for functionals evolution of computer power W, Kohn (1998) - Nobel Prize for Chemistry widely used codes - abinit, quantum espresso, vaps, castep, wien2k, cpmd, fhi98md, siesta, crystal, fplo, elk,...

8 Historical background From Physics Today, June (2005).

9 Hohenberg-Kohn ansatz Starting ideas of Hohenberg & Kohn create an exact many-body theory around the electronic ground-state density n(r) start from the system of N interacting particles (B-O approx.): H elec = N i=1 1 2 i N M i=1 A=1 Z A r ia + N i=1 N 1 j>i r ij A question What do you think, why did they want n(r) as the main variable?

10 Hohenberg-Kohn ansatz Reduction of the number of variables Example: system of 100 particles positions (r 1,..., r 100), r k = (x k, y k, z k ) Schrödinger theory: ψ = ψ (r 1,..., r 100) Questions 1 ψ (r 1,..., r 100) depends on how many variables? 2 if you want to calculate the energy from the variational principle, and guess the initial w.f. with p = 3 parameters, on how many variables will the minimization of energy depend?

11 Hohenberg-Kohn ansatz Reduction of the number of variables Example: system of 100 particles positions (r 1,..., r 100), r k = (x k, y k, z k ) Schrödinger theory: ψ = ψ (r 1,..., r 100) Questions 1 ψ (r 1,..., r 100) depends on how many variables? if you want to calculate the energy from the variational principle, and guess the initial w.f. with p = 3 parameters, on how many variables will the minimization of energy depend? ! Exponential wall!

12 Hohenberg-Kohn ansatz Reduction of the number of variables Example: system of 100 particles positions (r 1,..., r 100), r k = (x k, y k, z k ) Schrödinger theory: ψ = ψ (r 1,..., r 100) DFT: ψ = ψ [n (r)], n = n (n x, n y, n z) Questions 1 ψ [n (r)] depends on how many variables? 2 on how many variables now does the minimization of energy E 0 = min n(r) E [n (r)] depend?

13 Hohenberg-Kohn ansatz

14 Hohenberg-Kohn ansatz 1st HK theorem The ground-state density n(r) of a bound system of interacting electrons in some external potential V ext(r) determines this potential uniquely.

15 Hohenberg-Kohn ansatz 2nd HK theorem A universal functional E[n] can be defined for any external potential V ext(r). The exact ground state energy of the system is the global minimum of this functional, and the density n(r) that minimizes the functional is the exact ground state density n 0(r). E 0 = E[n 0(r)] = min E [n (r)] n(r) E [n (r)] = V ext(r)n (r) d 3 r + F [n] F [n] = T [n] + E int [n] A question What about F [n]? Do we know it?

16 Hohenberg-Kohn ansatz HK DFT pros an exact theory for ineracting particle systems provides a fundamental understanding of physical quantities like electron density and response functions adds a practical contribution via lowering the number of variables HK DFT cons we don t know F [n]

17 Kohn-Sham ansatz Motivation Hartree s single particle self consistent equations

18 Kohn-Sham ansatz Motivation Hartree s single particle self consistent equations = ignore interaction: V int = 0 = E [n (r)] = V ext(r)n (r) d 3 r + T s[n] T s[n] = kinetic energy of the ground state of noninteracting electrons with density n (r)

19 Kohn-Sham ansatz From this assumption, they got: ( 1 ) 2 + Vext(r) ɛ j ϕ j (r) = 0, E = n (r) = N ɛ j, j=1 N ϕ j (r) 2 j=1

20 Kohn-Sham ansatz To obtain the connection between HK and KS ansatz, rewrite: E HK = V ext(r)n (r) d 3 r + T [n] + E int [n] }{{} E KS = V ext(r)n (r) d 3 r + T s[n] + E Hartree [n] + E xc[n] }{{} Definition of exchange-correlation energy E xc[n] = T [n] + E int [n] (T s[n] + E Hartree [n])

21 Kohn-Sham ansatz Rewriting E KS as E KS = V eff (r)d 3 r + T s[n(r)] gives the equations ( 12 ) + V eff (r) ɛ j ϕ j (r) = 0, n (r) = N ϕ j (r) 2, j=1 V eff = V ext(r) + V Hartree (r) + V xc(r)

22 Kohn-Sham ansatz Noninteracting system ( 1 ) 2 + Vext(r) ɛ j ϕ j (r) = 0 Interacting system ( 12 ) + V eff (r) ɛ j ϕ j (r) = 0 single particle equations for the system of interacting particles = noninteracting particles in an effective potential A question What about V eff? (hint: it depends on V xc)

23 Kohn-Sham ansatz KS DFT pros one N-particle problem N one-particle problems if E xc was known = exact E 0 and n 0 KS DFT cons an exact form of E xc is unknown A question What do you think is the meaning of: a) ϕ j (use an analogy with Schrödinger theory)? b) ɛ j?

24 Kohn-Sham ansatz KS DFT pros one N-particle problem N one-particle problems if E xc was known = exact E 0 and n 0 KS DFT cons an exact form of E xc is unknown A question What do you think is the meaning of: a) ϕ j? n (r) = N j=1 ϕ j(r) 2 b) ɛ j? ɛ highest j = E ionization. Starting approx. for more precise calculations.

25 Kohn-Sham ansatz

26 Approximation to E xc - LDA LDA = Local Density Approximation Exc LDA = n(r) ɛ hom xc [n(r)] }{{} d3 r A question xc density at each point same as that in a homogenous electron gas with that density Where do you think LDA performs well and where not?

27 Approximation to E xc - LDA LDA precision Quantity Deviation from exp. atomic & molecular ground state energies < 0.5% molecular equilibrium distances < 5% band structures of metals few % band gap < 100% lattice constants < 2% E x O(10%) E c 2 Eionization atom & E dissoc & E coh 10 20%

28 Approximation to E xc - GGA GGA = General Gradient Approximation E xc (0) = ɛ hom xc [n(r)]n(r)d 3 r (LDA) E xc (1) = f (1) [n(r), n(r) ]n(r)d 3 r (GGA) E xc (2) = f (2) [n(r), n(r) ] 2 n(r)d 3 r A question What do you think why GGA opened DFT to chemists?

29 Approximation to E xc - GGA GGA precision Quantity B valence bandwidth lattice constants TO(Γ) ν phon E binding E atom ionization & E tot Deviation wrt LDA underestimate narrows corrects or overcorrects underestimate corrects the overestimation corrects

30 Plane waves core states strongly bound to nuclei atomic-like valence states changes in the materials bonding, electric & optic properties, magnetism,...

31 Plane waves A question What would be the most appropriate basis for representing the wave functions in solids?

32 Plane waves A question What would be the most appropriate basis for representing the wave functions in solids? Possibilities: 1 plane waves e ikr 2 localized orbitals (gaussians,...) 3 augmented functions (LAPW, PAW,...)

33 Plane waves Period solid simulation Bloch s theorem: ψ m,k (r + a i ) = e ik a i u m,k (r) Born-von Karman boundary condition: ψ m,k (r + N i a i ) = ψ m,k (r)

34 Plane waves How to make a plane wave basis? Plane wave kin. energy Plane wave sphere E pw kin = 1 k + G k + G 2 < E cut 2

35 Plane waves

36 Plane waves But, there are problems! Si core and valence electron w.f.

37 Pseudopotentials Basic assumption ψ core are the same in atomic or bounding conditions = n(r) = n core(r) + n val (r) Atomic Si electron energy levels and core w.f.

38 Pseudopotentials Basic assumption ψ core are the same in atomic or bounding conditions = n(r) = n core(r) + n val (r) A question What do you think, is this core/valence partitioning obvious for all elements?

39 Pseudopotentials Basic assumption ψ core are the same in atomic or bounding conditions = n(r) = n core(r) + n val (r) Examples F atom: (1s) 2 + (2s) 2 (2p) 5 IP 1 kev ev E exp ion 18 ev Ti atom: (1s) 2 (2s) 2 (2p) 6 + (3s) 2 (3p) 6 (4s) 2 (3d) 2 small core IP 99.2 ev E exp ion 7 ev (1s) 2 (2s) 2 (2p) 6 (3s) 2 (3p) 6 + (4s) 2 (3d) 2 large core IP 43.3 ev

40 Pseudopotentials Basic idea Freeze core electrons = effective potential Si 3s and 3s with frozen 3s electron w.f.

41 Pseudopotentials

42 Pseudopotentials

43 Pseudopotentials Sometimes the core correction is needed. (a) Na as semi-metal (no core correction). (b) Na as insulator (with core correction).

44 Brillouin zone integration In order to calculate something in a period case, you have to: 1 sum over bands 2 integrate over the BZ Expressions for T and n T = n n(r) = n 1 Ω 0k 1 Ω 0k (ɛ F ɛ nk ) ψ nk 1 Ω 0k 2 ψ nk dk (ɛ F ɛ nk ) ψnk(r)ψ nk (r)dk Ω 0k

45 Brillouin zone integration How to calculate an integral over the BZ? 1 Ω 0k Ω 0k X k dk {k} w k X k How to choose {k} and {w k }? Monkhorst-Pack grids tetrahedron methods. Homogenous sampling of the BZ.

46 Contents 1 Density functional theory (DFT) 2 Literature

47 Literature 1 R. M. Martin, Electronic Structure - Basic Theory and Practical Methods, Cambridge University Press, Cambridge, W. Kohn Electronic structure of matter - Nobel Lecture