Introduction to Pseudopotentials and Electronic Structure Philip B. Allen Stony Brook University Nov. 3, 2014 Brookhaven National Laboratory You searched for: TOPIC: (TiO2 110 surface) – Results: 3,551 (from Web of Science Core Collection) Example: rutile (TiO2) Hybrid func,onal studies of the oxygen 6-atom tetragonal unit cell vacancy in TiO2, A. Jano, J. B. Varley, P. Rinke, N. Umezawa, G. Kresse, and C. G. Van de Walle, PRB 81, 085212 (2010) Ti: 4s23d2 (4 electrons, 6 orbitals) O: 2s22p4 (6 electrons, 4 orbitals) 2(TiO2) – 32 valence electrons 28 minimal basis orbitals ~1130 plane waves DFT hybrid 12 empty bands Perdew, Burke, Ernzerhof Heyd, Scuseria, and Ernzerhof 12 filled bands 4 oxygen 2s bands lower in energy Single-particle ideas and philosophies – converging or diverging? 1. Drude, Sommerfeld, Bloch. Classical, Quantum, free to bound in solid. ! 1/ 3 2. Hartree-Fock; Slater ρ ( r ) (average exchange). 3. Interacting many-body theory: Single-particle Green’s function; Landau theory; “GW” approximation 4. Hohenberg-Kohn-Sham density functional theory (DFT) € Single-particle Methods of Solution 1. Local orbitals; LCAO-Hückel 2. Plane waves 3. Augmented plane waves (APW) 4. Othogonalized plane waves (OPW) 5. Green’s function methods (KKR) 6. Pseudopotentials 7. PAW Walter Kohn Pseudopotentials (~1959) were born before DFT (1964) and remain partially independent. DFT “does” ρ(r). What do pseudopotentials do? Bloch’s theorem: can choose eigenstates of a periodic Hamiltonian in the form ! ! ! ! ! ! ! ˆ ! T (R )ψ(r ) ≡ ψ(r + R ) = exp(ik ⋅ R )ψ(r ) (label as ψk ) ˆ ! (eigenvalue of H labeled as εk ) ! ! ! (eigenvalue of translation Tˆ (R ) is exp(ik ⋅ R ) ! ! ! k and k + G give the same translation eigenvalue. € 2 2 1d free electrons ε ! = " k /2m k 1d free electrons with periodicity of k-space (the reciprocal lattice) 1d single-orbital tight-binding band LCAO: ψk=Σexp(ikR)φ(r-R) εk = -2tcos(ka) € volume 51 (1937) Augmented Plane Waves (APW) The wave funcPon is expanded in spherical harmonics and radial soluPons of the wave equaon within the spheres, and in plane waves outside the spheres, joining students: conPnuously at the surface. A single unperturbed funcPon consists of a single plane Leland Allen wave outside the spheres, together with the Don Ellis necessary spherical funcPons within the Art Freeman spheres. ... It is hoped that the method will be Frank Herman useful for comparavely low energy excited George Koster electrons, for which the usual method of Len Mattheiss expansion in plane waves converges too slowly. Dick Watson John Wood Birth of computational electronic structure theory? The pseudopotential emerged ~1959 as a related way of reducing the number of plane waves needed. Richard M. Martin, Electronic Structure: Basic Theory and Practical Methods Cambridge (2004) 3 How many plane waves are needed? ~(Gmax/Gmin) Crude estimate for s-orbitals: λmin ~ 2aB/Z 3 3 (Gmax/Gmin) ~ (Za/2√3aB) ~ 71,400 for Si. With a smooth pseudopotential, λmin ~ aB works. ~ 200 plane waves for Si. But it depends on the kind of pseudopotential. Bloch’s theorem ! “k” is short for k , n , σ (all the quantum numbers of the state). Let there be a “single-particle” Hamiltonian. Then: Need to solve (in a primitive cell): Two€ issues: 1. What is H? 2. How to solve? Most methods of solution expand ψ in a “basis set.” The simplest basis set adapted to Bloch’s theorem is plane-waves. ! ! This is a Bloch wave, since exp( iG ⋅ R ) =1 They are complete, orthogonal, and bias-free. € Dimensionless units “Rydberg atomic units” “Hartree atomic units” In both systems, the unit of length is the Bohr radius: Simplified notation; drop the label “k”. Determine the expansion coefficients cg (secular equation.) This is only one of many matrix versions of the Schrödinger equation ∞ where is a complete, orthonormal set of states. ψ = ∑ α n n { n } n=1 ∞ ∞ ˆ ˆ Hψ = ∑ α n H n = ∑ αn E n = Eψ € € n=1 n=1 ∞ ∞ Left project by m ∑ α n m H n = ∑ α n E m n = Eα m n=1 n=1 € ⎛ 1 H 1 ! 1 H N !⎞⎛ α1 ⎞ ⎛ α1 ⎞ Exact version of ⎜ ⎟⎜ ⎟ ⎜ ⎟ one-electron ⎜ " € # €" !⎟⎜ " ⎟ ⎜ " ⎟ Schroedinger eqn. = E (assuming we know H!) ⎜ N H 1 ! N H N !⎟⎜ α N ⎟ ⎜α N ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎝ " ! ! #⎠⎝ " ⎠ ⎝ " ⎠ € Truncated version of the matrix formulation – variational accuracy Theorem: if Hψ=Eψ and we expand ψ in a (finite, truncated) orthonormal basis, N ψ = ∑ α n n n=1 then the states that are stationary under variation of all coefficients δ * [ ψ H ψ − λ ψ ψ ] = 0 € δαn obey the matrix eigenvalue equation ⎛ ⎞⎛ ⎞ ⎛ ⎞ € 1 H 1 ! 1 H N α1 α1 ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ " # " ⎟⎜ " ⎟ = λ⎜ " ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ N H 1 N H N ⎟ α α ⎝ ! ⎠⎝ n ⎠ ⎝ n ⎠ € ⎛ α1 ⎞ ⎛ α1 ⎞ Basis functions are often not ⎜ ⎟ ⎜ ⎟ orthogonal or normalized. ! ! This creates a “generalized” Hˆ ⎜ ⎟ = ESˆ ⎜ ⎟ Hermitean eigenvalue problem. ⎜α N ⎟ ⎜α N ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ! ⎠ ⎝ ! ⎠ ⎛ 1 H 1 ! 1 H N !⎞ ⎛ 1 1 ! 1 N !⎞ ⎜ ⎟ ⎜ ⎟ " # " ! " # " ! Hˆ = ⎜ € ⎟ Sˆ = ⎜ ⎟ ⎜ N H 1 ! N H N !⎟ ⎜ N 1 ! N N !⎟ ⎜ ⎟ ⎜ ⎟ " ! ! # " ! ! # ⎝ ⎠ ⎝ ⎠ Linear algebra methods solve this with little added difficulty. One approach: € Hˆ ψ = ESˆ ψ and φ€= Sˆ 1/ 2ψ ˆ ˆ ˆ −1/ 2 ˆ ˆ −1/ 2 → H effψ = Eψ where H eff = S HS € More annoyingly, efficient basis functions (or a “pseudopotential”) tend to be energy-dependent. Then it is not a linear problem any more. ⎛ α1 ⎞ ⎛ α1 ⎞ ⎜ ⎟ ⎜ ⎟ ˆ ⎜ ! ⎟ ˆ ⎜ ! ⎟ Solve iteratively? H (E) = ES (E) Lose nice linear algebra theorems. ⎜α N ⎟ ⎜α N ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ! ⎠ ⎝ ! ⎠ Hˆ (E) ψ = ESˆ (E) ψ € € Hˆ (E) ψ = ESˆ (E) ψ Linearized approximations: LAPW (O. K. Andersen), etc. ˆ ˆ ˆ H (E) ≈ H (E0 ) + (E − E0 )H Eʹ € 0 Sˆ (E) ≈ Sˆ (E ) + (E − E )Sˆ ʹ 0 0 E0 This leads to a generalized linear Hermitean eigenvalue problem: ˆ ˆ ˆ H (E0 ) − E0 ψ = (E − E0 ) 1+ E0S Eʹ − H Eʹ ψ €[ ] [ 0 0 ] and works only in an energy interval around E0. € The soft letter ell (L): “linearized” (LAPW) or “local” (LDA) Janotti, Varley, Rinke, Umezawa, Kresse, and Back to rutile Van de Walle, PRB 81, 085212 (2010) 4 x 3 oxygen 2p levels p(x), p(y), p(z) bonded with 2 x 2 Ti Eg levels d(3z2-r2), d(3x2-r2), d(3y2-r2) Local orbital picture necessary to describe covalent orbital mixing chemistry. covalent bonding/anti-bonding splitting. “Oxygen vacancies in rutile TiO2 were simulated by removing one O atom from a supercell with 72 atoms.” 12 primitive cells ! 336 orbitals in a minimal basis “We used a plane-wave basis set with a cutoff of 400 eV” 2 2 400 eV = kmax = (2π/λmin) corresponds to a minimum wave-length of 0.6 Å 3 ! kmax=5.42 a.u. ! Vk-space (4π/3)kmax = 667 a.u. The super-cell has dimensions V = 2a x 2a x 3c = 748 Å3 = 5,055 a.u. 3 The Brillouin zone volume VBZ = (2π) /V = 0.049 a.u. The matrix dimension is Vk-space/VBZ = 13,600 plane waves Plane waves enlarge the basis set by 40 and the computer time by (40)3 = 64,000 “... integrations over the Brillouin zone were perfomed using a 2 x 2 x 2 mesh of special k points.” Self-consistent potential V(r) depends on charge density ρ(r) 0 (π/a)(1,0,0) Discrete k-mesh of special points used to evaluate ρ(r). Birth of the pseudopotential Phys. Rev. 116, 287 (1959) Conyers Herring ! " real wave - function : ψ(r ) = r ψ ˜ ! " ˜ pseudo wave - function : ψ (r ) = r ψ ψ˜ ≡ ψ − ∑ i i ψ i € i i is the projection operator that orthogonalizes ∑ ψ to the (somewhat arbitrary?) states |i>. i H ψ =€E ψ defines an effective Hamiltonian H˜ ψ ˜ = Eψ˜ . If H = T +V (V = potential), then H˜ = T +V˜ (V˜ = pseudopotential). € DFT is not yet born. People were guessing that V “existed” (Slater showed how to “guess” it quite well). Then V ˜ is a weaker potential, a non-local € operator, that also exists and has pseudo-wavefunctions for its solution. So, brave computational scientist, proceed to guess V ˜ ! € € R. M. Martin, Electronic Structure ... Pseudopotential for a problem where the 1-electron potential is “known”: – real pseudo – Phillips-Kleinman-Antoncik The full self-consistent treatment was not an option until 15 years later. The full apparatus of non-local (and energy-dependent) potentials was over the top. Simpler approaches were encouraged. “SL” = semi-local Phys. Rev. 141, 789–796 (1966) [cited 1336 times] Empirical pseudopotentials “Empirical pseudo-potential method” local version ! 1 1 ! 2 a a 0, , A π (1 11) = ( 2 2 ) = a ! ! fcc structure: b a 1 ,0, 1 B 2π (11 1) = ( 2 2 ) = a ! 1 1 ! 2 c = a , ,0 C = π (111 ) ( 2 2 ) a 5 smallest reciprocal lattice vector (RLV) types ! ! ! ! ! G = 2π (111), G = 2π (200), G = 2π (220), G = 2π (311), G = 2π (222) 3 a 4 a 8 a 11 a 12 a € ! ! ! ! matrix elements k V k + G 3 = V(| G 3 |) ≡ V3 are the same for all 12 vectors € Diamond structure: fcc with a basis of two identical atoms at locations τ1 and τ2.. ! ! ! ! ! ! ! ! ! ! k V k + G = V(G )S(G ) where S(G ) = eiG ⋅τ 1 + eiG ⋅τ 2 /2 € ( ) Note – destructive interference kills structure factor S(G) for certain G’s. ! ! S(G ) S(G ) 0 Three numbers (V3, V8, V11) give a “good” 4 = 12 = band structure for C, Si, Ge, Sn. € € Empirical pseudopotentials are adjusted until theory “agrees” with optical experiment. There are aspects of experiment lying outside “quasi-particle theory:” excitonic effects, for example. Two particle Green’s function, Bethe-Salpeter equation.