The FP-LAPW and APW+lo methods Peter Blaha Institute of Materials Chemistry TU Wien
Concepts when solving Schrödingers-equation Treatment of spin Non-spinpolarized Spin polarized (with certain magnetic order) Form of potential Relativistic treatment of the electrons non relativistic semi-relativistic fully-relativistic “Muffin-tin” MT atomic sphere approximation (ASA) pseudopotential (PP) Full potential : FP exchange and correlation potential Hartree-Fock (+correlations) Density functional theory (DFT) Local density approximation (LDA) Generalized gradient approximation (GGA) Beyond LDA: e. g. LDA+U Schrödinger - equation non periodic (cluster) periodic (unit cell) Representation of solid Basis functions plane waves : PW augmented plane waves : APW atomic oribtals. e. g. Slater (STO), Gaussians (GTO), LMTO, numerical basis
Unitcells, Supercells • Describe crystal by small unit cell, which is repeated in all 3 dimensions infinitely. • No surface, no defects, no impurities ! • Create “supercells” to simulate surfaces, impurities, …. Rh N B Rh BN
Concepts when solving Schrödingers-equation Form of potential Relativistic treatment of the electrons non relativistic semi-relativistic fully-relativistic “Muffin-tin” MT atomic sphere approximation (ASA) Full potential : FP pseudopotential (PP) exchange and correlation potential Hartree-Fock (+correlations) Density functional theory (DFT) Local density approximation (LDA) Generalized gradient approximation (GGA) Beyond LDA: e. g. LDA+U Schrödinger - equation non periodic (cluster) periodic (unit cell) Representation of solid Treatment of spin Non-spinpolarized Spin polarized (with certain magnetic order) Basis functions plane waves : PW augmented plane waves : APW atomic orbitals. e. g. Slater (STO), Gaussians (GTO), LMTO, numerical basis
DFT Density Functional Theory Hohenberg-Kohn theorem: (exact) The total energy of an interacting inhomogeneous electron gas in the presence of an external potential Vext(r ) is a functional of the density Kohn-Sham: (still exact!) Ekinetic non interacting Ene Ecoulomb Eee Exc exchange-correlation In KS the many body problem of interacting electrons and nuclei is mapped to a one-electron reference system that leads to the same density as the real system.
Kohn Sham equations LDA, GGA vary 1 -electron equations (Kohn Sham) -Z/r LDA treats both, exchange and GGA correlation effects approximately Lagrange multiplier !
Walter Kohn, Nobel Prize 1998 Chemistry
Success and failure of “standard” DFT in solids Standard LDA (GGA) gives good description of structural and electronic properties of most solids (lattice parameters within 1 -2%, at least qualitatively correct bandstructure, metalinsulator, magnetism, …) n Problems: “localized” (correlated) electrons n n late 3 d transition metal oxides/halides n n n 4 f, 5 f electrons n n n metals instead of insulators (Fe. O, Fe. F 2, cuprates, …) nonmagnetic instead of anti-ferromagnetic (La 2 Cu. O 4, YBa 2 Cu 3 O 6) all f-states pinned at the Fermi energy, “always” metallic orbital moments much too small “weakly” correlated metals n n Fe. Al is ferromagnetic in theory, but nonmagnetic experimentally 3 d-band position, exchange splitting, …
Is LDA repairable ? ab initio methods n GGA: usually improvement, but often too small. n Hartree-Fock: completely neglects correlation, very poor in solids n Exact exchange: imbalance between exact X and approximate (no) C n Hybrid-Functionals: mix of HF + LDA; good for insulators, poor for metals GW: gaps in semiconductors, but groundstate? expensive! n Quantum Monte-Carlo: very expensive not fully ab initio n Self-interaction-correction: vanishes for Bloch states n Orbital polarization: Hund’s 2 nd rule by atomic Slater-parameter n LDA+U: strong Coulomb repulsion via external Hubbard U parameter n DMFT: extension of LDA+U for weakly correlated systems n
Concepts when solving Schrödingers-equation Form of potential Relativistic treatment of the electrons non relativistic semi-relativistic fully-relativistic “Muffin-tin” MT atomic sphere approximation (ASA) Full potential : FP pseudopotential (PP) exchange and correlation potential Hartree-Fock (+correlations) Density functional theory (DFT) Local density approximation (LDA) Generalized gradient approximation (GGA) Beyond LDA: e. g. LDA+U Schrödinger - equation non periodic (cluster) periodic (unit cell) Representation of solid Treatment of spin Non-spinpolarized Spin polarized (with certain magnetic order) Basis functions plane waves : PW augmented plane waves : APW atomic orbitals. e. g. Slater (STO), Gaussians (GTO), LMTO, numerical basis
APW Augmented Plane Wave method The unit cell is partitioned into: atomic spheres Interstitial region unit cell Rmt Basisset: PW: Atomic partial waves ul(r, e) are the numerical solutions join of the radial Schrödinger equation in a given spherical potential for a particular energy e Alm. K coefficients for matching the PW
APW based schemes n APW (J. C. Slater 1937) Non-linear eigenvalue problem n Computationally very demanding n n LAPW (O. K. Anderssen 1975) Generalized eigenvalue problem n Full-potential n n Local orbitals (D. J. Singh 1991) n n treatment of semi-core states (avoids ghostbands) APW+lo (E. Sjöstedt, L. Nordstörm, D. J. Singh 2000) Efficience of APW + convenience of LAPW n Basis for n K. Schwarz, P. Blaha, G. K. H. Madsen, Comp. Phys. Commun. 147, 71 -76 (2002)
Slater‘s APW (1937) Atomic partial waves Energy dependent basis functions lead to Non-linear eigenvalue problem H Hamiltonian S overlap matrix One had to numerically search for the energy, for which the det|H-ES| vanishes. Computationally very demanding. “Exact” solution for given MT potential!
Linearization of energy dependence O. K. Andersen, Phys. Rev. B 12, 3060 (1975) expand ul at fixed energy El and add LAPW suggested by Almk, Blmk: join PWs in value and slope additional constraint requires more PWs than APW basis flexible enough for single diagonalization Atomic sphere LAPW PW
Full-potential in LAPW n Sr. Ti. O 3 Full potential n (A. Freeman etal. ) The potential (and charge density) can be of general form (no shape approximation) Inside each atomic sphere a local coordinate system is used (defining LM) Muffin tin approximation Ti. O 2 rutile Ti O
Core, semi-core and valence states For example: Ti Valences states n High in energy n Delocalized wavefunctions n Semi-core states n Medium energy n Principal QN one less than valence (e. g. in Ti 3 p and 4 p) n not completely confined inside sphere n Core states n Low in energy n Reside inside sphere n
Problems of the LAPW method: EFG Calculation for Rutile Ti. O 2 as a function of the Ti-p linearization energy Ep exp. EFG „ghostband“ region P. Blaha, D. J. Singh, P. I. Sorantin and K. Schwarz, Phys. Rev. B 46, 1321 (1992).
Problems of the LAPW method Problems with semi-core states
Extending the basis: Local orbitals (LO) n LO n n n is confined to an atomic sphere has zero value and slope at R can treat two principal QN n for each azimuthal QN (3 p and 4 p) corresponding states are strictly orthogonal (no “ghostbands”) tail of semi-core states can be represented by plane waves only slight increase of basis set (matrix size) D. J. Singh, Phys. Rev. B 43 6388 (1991)
The LAPW+LO Method D. Singh, Phys. Rev. B 43, 6388 (1991). Cubic APW La RMT = 3. 3 a 0 QAPW RKmax LAPW+LO converges like LAPW. The LO add a few basis functions (i. e. 3 per atom for p states). Can also use LO to relax linearization errors, e. g. for a narrow d or f band. Suggested settings: Two “energy” parameters, one for u and ů and the other for u(2). Choose one at the semi-core position and the other at the valence.
New ideas from Uppsala and Washington E. Sjöstedt, L. Nordström, D. J. Singh, An alternative way of linearizing the augmented plane wave method, Solid State Commun. 114, 15 (2000) • Use APW, but at fixed El (superior PW convergence) • Linearize with additional lo (add a few basis functions) optimal solution: mixed basis • use APW+lo for states which are difficult to converge: (f or d- states, atoms with small spheres) • use LAPW+LO for all other atoms and angular momenta
Convergence of the APW+LO Method E. Sjostedt, L. Nordstrom and D. J. Singh, Solid State Commun. 114, 15 (2000). x 100 Ce
Improved convergence of APW+lo K. Schwarz, P. Blaha, G. K. H. Madsen, Comp. Phys. Commun. 147, 71 -76 (2002) Force (Fy) on oxygen in SES (sodium electro sodalite) vs. # plane waves changes sign and converges slowly in LAPW n better convergence in APW+lo n
Relativistic treatment For example: Ti n Valence states n Scalar relativistic n n mass-velocity Darwin s-shift Spin orbit coupling on demand by second variational treatment Semi-core states n Scalar relativistic n No spin orbit coupling n on demand n n n spin orbit coupling by second variational treatment Additional local orbital (see Th-6 p 1/2) Core states n Fully relativistic n Dirac equation
Relativistic semi-core states in fcc Th additional local orbitals for 6 p 1/2 orbital in Th n Spin-orbit (2 nd variational method) n J. Kuneš, P. Novak, R. Schmid, P. Blaha, K. Schwarz, Phys. Rev. B. 64, 153102 (2001)
Atomic forces (Yu et al. ; Kohler et al. ) n n Total Energy: n Electrostatic energy n Kinetic energy n XC-energy Force on atom a: n n Hellmann-Feynman-force Pulay corrections Core n Valence expensive, contains a summation of matrix elements over all occupied states n n
Quantum mechanics at work
WIEN 2 k software package An Augmented Plane Wave Plus Local Orbital Program for Calculating Crystal Properties Peter Blaha Karlheinz Schwarz Georg Madsen Dieter Kvasnicka Joachim Luitz WIEN 97: ~500 users WIEN 2 k: ~1030 users mailinglist: 1500 users November 2001 Vienna, AUSTRIA Vienna University of Technology http: //www. wien 2 k. at
Development of WIEN 2 k n Authors of WIEN 2 k P. Blaha, K. Schwarz, D. Kvasnicka, G. Madsen and J. Luitz Other contributions to WIEN 2 k n C. Ambrosch-Draxl (Univ. Graz, Austria), optics n D. J. Singh (NRL, Washington D. C. ), local oribtals (LO), APW+lo n U. Birkenheuer (Dresden), wave function plotting n T. Charpin (Paris), elastic constants n R. Dohmen und J. Pichlmeier (RZG, Garching), parallelization n P. Novák and J. Kunes (Prague), LDA+U, SO n C. Persson (Uppsala), irreducible representations n M. Scheffler (Fritz Haber Inst. , Berlin), forces n E. Sjöstedt and L Nordström (Uppsala, Sweden), APW+lo n J. Sofo and J. Fuhr (Barriloche), Bader analysis n B. Yanchitsky and A. Timoshevskii (Kiev), spacegroup n R. Laskowski (Vienna), non-collinear magnetism n B. Olejnik (Vienna), non-linear optics n and many others …. n
International co-operations n More than 500 user groups worldwide Industries (Canon, Eastman, Exxon, Fuji, A. D. Little, Mitsubishi, Motorola, NEC, Norsk Hydro, Osram, Panasonic, Samsung, Sony, Sumitomo). n Europe: (EHT Zürich, MPI Stuttgart, Dresden, FHI Berlin, DESY, TH Aachen, ESRF, Prague, Paris, Chalmers, Cambridge, Oxford) n America: ARG, BZ, CDN, MX, USA (MIT, NIST, Berkeley, Princeton, Harvard, Argonne NL, Los Alamos Nat. Lab. , Penn State, Georgia Tech, Lehigh, Chicago, SUNY, UC St. Barbara, Toronto) n far east: AUS, China, India, JPN, Korea, Pakistan, Singapore, Taiwan (Beijing, Tokyo, Osaka, Sendai, Tsukuba, Hong Kong) n n Registration at www. wien 2 k. at 400/4000 Euro for Universites/Industries n code download via www (with password), updates, bug fixes, news n usersguide, faq-page, mailing-list with help-requests n
WIEN 2 k- hardware/software WIEN 2 k runs on any Unix/Linux platform from PCs, workstations, clusters to supercomputers n Fortran 90 (dynamical allocation) n many individual modules, linked together with C-shell or perl-scripts n f 90 compiler, BLAS-library, perl 5, ghostview, gnuplot, Tcl/Tk (Xcrysden), pdf-reader, www-browser n • web-based GUI – w 2 web • real/complex version (inversion) • 10 atom cells on 128 Mb PC • 100 atom cells require 1 -2 Gb RAM • k-point parallel on clusters with common NFS (slow network) • MPI/Scalapack parallelization for big cases (>50 atoms) and fast network • installation support for most platforms
How to run WIEN 2 k consists of many independent F 90 programs, which are linked together via C-shell scripts. n Each „case“ runs in his own directory. /case n The „master input“ is called case. struct n Initialize a calculation: init_lapw n Run scf-cycle: run_lapw (runsp_lapw) n You can run WIEN 2 k using any www-browser and the w 2 web interface, but also at the command line of an xterm. n Input/output/scf files have endings as the corresponding programs: n n n case. output 1…lapw 1; case. in 2…lapw 2; case. scf 0…lapw 0 Inputs are generated using STRUCTGEN(w 2 web) and init_lapw
w 2 web: the web-based GUI of WIEN 2 k Based on www n WIEN 2 k can be managed remotely via w 2 web n Important steps: n start w 2 web on all your hosts n n use your browser and connect to the (master) host: portnumber n n login to the desired host (ssh) w 2 web (at first startup you will be asked for username/password, port -number, (master-)hostname. creates ~/. w 2 web directory) mozilla http: //fp 98. zserv: 10000 create a new session on the desired host (or select an old one)
w 2 web GUI (graphical user interface) n n Structure generator n spacegroup selection n import cif file step by step initialization n symmetry detection n automatic input generation SCF calculations n Magnetism (spin-polarization) n Spin-orbit coupling n Forces (automatic geometry optimization) Guided Tasks n Energy band structure n DOS n Electron density n X-ray spectra n Optics
Spacegroup P 42/mnm Structure given by: spacegroup lattice parameter positions of atoms (basis) Rutile Ti. O 2: P 42/mnm (136) a=8. 68, c=5. 59 bohr Ti: (0, 0, 0) O: (0. 304, 0)
Structure generator n Specify: Number of nonequivalent atoms n lattice type (P, F, B, H, CXY, CXZ, CYZ) or spacegroup symbol n n if existing, you must use a SG-setting with inversion symmetry: n Si: ±(1/8, 1/8), not (0, 0, 0)+(1/4, 1/4)! lattice parameters a, b, c (in Å or bohr) n name of atoms (Si) and fractional coordinates (position) n n n as numbers (0. 123); fractions (1/3); simple expressions (x-1/2, …) in fcc (bcc) specify just one atom, not the others in (1/2, 0; …) „save structure “ n updates automatically Z, r 0, equivalent positions and generates case. inst n „set RMT and continue“: (specify proper “reduction” of NN-distances) n non-overlapping „as large as possible“ (saves time), but not larger than 3 bohr n RMT for sp-elements 10 -20 % smaller than for d (f) elements n largest spheres not more than 50 % larger than smallest sphere n Exception: H in C-H or O-H bonds: RMT~0. 6 bohr (RKMAX~3 -4) n Do not change RMT in a „series“ of calculations n „save structure – save+cleanup“ n
Program structure of WIEN 2 k init_lapw n initialization n symmetry detection (F, I, Ccentering, inversion) n input generation with recommended defaults n quality (and computing time) depends on k-mesh and R. Kmax (determines #PW) n run_lapw n scf-cycle n optional with SO and/or LDA+U n different convergence criteria (energy, charge, forces) n save_lapw tic_gga_100 k_rk 7_vol 0 n cp case. struct and clmsum files, n mv case. scf file n rm case. broyd* files n
Task for electron density plot A task consists of n a series of steps n that must be executed n to generate a plot n For electron density plot n select states (e. g. valence e ) n select plane for plot n generate 3 D or contour plot with gnuplot or Xcrysden n
Ti. C electron density Valence electrons n Na. Cl structure n (100) plane n plot in 2 dimensions n Shows n charge distribution n covalent bonding n n n between the Ti-3 d and C-2 p electrons eg/t 2 g symmetry
Properties with WIEN 2 k - I Energy bands n classification of irreducible representations n ´character-plot´ (emphasize a certain band-character) n Density of states n including partial DOS with l and m- character (eg. px , py , pz ) n Electron density, potential n total-, valence-, difference-, spin-densities, r of selected states n 1 -D, 2 D- and 3 D-plots (Xcrysden) n X-ray structure factors n Bader´s atom-in-molecule analysis, critical-points, atomic basins and charges ( ) n spin+orbital magnetic moments (spin-orbit / LDA+U) n Hyperfine parameters n hyperfine fields (contact + dipolar + orbital contribution) n Isomer shift n Electric field gradients n
Properties with WIEN 2 k - II n Total energy and forces optimization of internal coordinates, (MD, BROYDEN) n cell parameter only via Etot (no stress tensor) n elastic constants for cubic cells n Phonons via a direct method (based on forces from supercells) n n n interface to PHONON (K. Parlinski) – phonon bandstructure, phonon DOS, thermodynamics, neutrons Spectroscopy core levels (with core holes) n X-ray emission, absorption, electron-energy-loss (core valence/conduction bands including matrix elements and angular dep. ) n optical properties (dielectric function, JDOS including momentum matrix elements and Kramers-Kronig) n fermi surface (2 D, 3 D) n
Properties with WIEN 2 k - III n New developments (in progress) non-linear optics (B. Olejnik) n non-collinear magnetism (R. Laskowski) n transport properties (Fermi velocities, Seebeck, conductivity, thermoelectrics, . . ) (G. Madsen) n Compton profiles n linear response (phonons, E-field) (C. Ambrosch-Draxl) n stress tensor (C. Ambrosch-Draxl) n exact exchange, GW, … ? ? n grid-computing n
Advantage/disadvantage of WIEN 2 k + robust all-electron full-potential method + unbiased basisset, one convergence parameter (LDA-limit) + all elements of periodic table (equal expensive), metals + LDA, GGA, meta-GGA, LDA+U, spin-orbit + many properties + w 2 web (for novice users) - ? speed + memory requirements + very efficient basis for large spheres (2 bohr) (Fe: 12 Ry, O: 9 Ry) - less efficient for small spheres (1 bohr) (O: 25 Ry) - large cells, many atoms (n 3, iterative diagonalization not perfect) - full H, S matrix stored large memory required + many k-points do not require more memory - no stress tensor - no linear response
Conclusion There are many ways to make efficient use of DFT calculations n APW+lo method (as implemented in WIEN 2 k) is one of them n all electron n full-potential n highly accurate - benchmark for other methods n many properties n user friendly n widely used n n development by several groups large user community used by many experimental groups
Thank you for your attention !
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