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Density functional theory (DFT) and the concepts of the augmented-plane-wave plus local orbitals (APW+lo) method

Density functional theory (DFT) and the concepts of the augmented-plane-wave plus local orbitals (APW+lo) method. Karlheinz Schwarz Institute of Materials Chemistry TU Wien. Walter Kohn and DFT. DFT Density Functional Theory. Hohenberg-Kohn theorem.

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Density functional theory (DFT) and the concepts of the augmented-plane-wave plus local orbitals (APW+lo) method

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  1. Density functional theory (DFT) and the concepts of the augmented-plane-wave plus local orbitals (APW+lo) method Karlheinz Schwarz Institute of Materials Chemistry TU Wien

  2. Walter Kohn and DFT

  3. DFT Density Functional Theory Hohenberg-Kohn theorem The total energy of an interacting inhomogeneous electron gas in the presence of an external potential Vext(r ) is a functional of the density  In DFT 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. DFT treats both, exchange and correlation effects, but approximately

  4. Kohn Sham equations Total energy LDA, GGA Ekinetic non interacting Ene EcoulombEee Exc exchange-correlation vary  1-electron equation (Kohn Sham)

  5. Walter Kohn, Nobel Prize 1998 Chemistry

  6. A simple picture of LDA Slater, Gunnarsson-Lundqvist ………… Look at the “LDA” from a different angle Exc = -∫ dx n(x) e2/ R(x) R(x) interpreted as the radius of the ‘exchange-correlation hole’ surrounding an electron at the point x. R(x) is a length: What length could it be? Plausible assumption, the average distance between the electrons? R(x) ≈ γ-1 n-1/3(x) Exc = - γ e2 ∫ dx n4/3(x)

  7. Role of „Gradient corrected functionals“ Becke, Perdew, Wang, Lee, Yang, Parr …… ’87 – ‘92 Perdew ,Burke, Ernzerhof PBE …… ‘96 Use n and ∂n/∂x to correct LDA in regions of low density Substantial improvement in energy differences

  8. DFT ground state of iron • LSDA • NM • fcc • in contrast to experiment • GGA • FM • bcc • Correct lattice constant • Experiment • FM • bcc LSDA GGA GGA LSDA

  9. CoO AFM-II total energy, DOS • CoO • in NaCl structure • antiferromagnetic: AF II • insulator • t2g splits into a1g and eg‘ • GGA almost splits the bands

  10. CoO why is GGA better than LSDA • Central Co atom distinguishes • between • and • Angular correlation

  11. DFT thanks to Claudia Ambrosch (Graz) GGA follows LDA

  12. Overview of DFT concepts Form of potential Full potential : FP “Muffin-tin” MT atomic sphere approximation (ASA) pseudopotential (PP) Relativistic treatment of the electrons exchange and correlation potential fully-relativistic semi-relativistic non relativistic Local density approximation (LDA) Generalized gradient approximation (GGA) Beyond LDA: e.g. LDA+U Kohn-Sham equations Representation of solid Basis functions non periodic (cluster) periodic (unit cell) plane waves : PW augmented plane waves : APW linearized “APWs” analytic functions (e.g. Hankel) atomic orbitals. e.g. Slater (STO), Gaussians (GTO) numerical Treatment of spin Spin polarized non spin polarized

  13. How to solve the Kohn Sham equations Total energy LDA, GGA Ekinetic non interacting Ene EcoulombEee Exc exchange-correlation vary  1-electron equation (Kohn Sham)

  14. APW based schemes • APW (J.C.Slater 1937) • Non-linear eigenvalue problem • Computationally very demanding • LAPW (O.K.Anderssen 1975) • Generalized eigenvalue problem • Full-potential • Local orbitals (D.J.Singh 1991) • treatment of semi-core states (avoids ghostbands) • APW+lo (E.Sjöstedt, L.Nordstörm, D.J.Singh 2000) • Efficiency of APW + convenience of LAPW • Basis for K.Schwarz, P.Blaha, G.K.H.Madsen, Comp.Phys.Commun.147, 71-76 (2002)

  15. APW Augmented Plane Wave method • The unit cell is partitioned into: • atomic spheres • Interstitial region • Bloch wave function: • atomic partial waves • Plane Waves (PWs) unit cell Rmt Full potential PW: Atomic partial wave join

  16. Slater‘s APW (1937) Atomic partial waves Energy dependent basis functions lead to Non-linear eigenvalue problem H Hamiltonian S overlap matrix Computationally very demanding One had to numerically search for the energy, for which the det(H-ES) vanishes.

  17. Linearization of energy dependence LAPW suggested by O.K.Andersen, Phys.Rev. B 12, 3060 (1975) join PWs in value and slope Atomic sphere LAPW PW Plane Waves (PWs)

  18. Full-potential in LAPW • The potential (and charge density) can be of general form (no shape approximation) SrTiO3 Full potential • Inside each atomic sphere a local coordinate system is used (defining LM) Muffin tin approximation Ti TiO2 rutile O

  19. Core, semi-core and valence states For example: Ti • Valences states • High in energy • Delocalized wavefunctions • Semi-core states • Medium energy • Principal QN one less than valence (e.g. in Ti 3p and 4p) • not completely confined inside sphere • Core states • Low in energy • Reside inside sphere

  20. Electronic Structure E Ti 3d / O 2p EF O 2p Hybridized w. Ti 4p, Ti 3d Ti- 3p Problems of the LAPW method: EFG Calculation for Rutile TiO2 as a function of the Ti-p linearization energy Ep exp. EFG „ghostband“ P. Blaha, D.J. Singh, P.I. Sorantin and K. Schwarz, Phys. Rev. B 46, 1321 (1992).

  21. -1/2cG ei(G+k)r G  (Almul(r)+Blmůl(r)+Clmül(r)) Ylm(r) lm ONE SOLUTION • Treat all the states in a single energy window: • Automatically orthogonal. • Need to add variational freedom. • Could invent quadratic or cubic APW methods. Electronic Structure E Ti 3d / O 2p EF O 2p Hybridized w. Ti 4p, Ti 3d { (r) = Problem: This requires an extra matching condition, e.g. second derivatives continuous method will be impractical due to the high planewave cut-off needed. Ti- 3p

  22. Local orbitals (LO) • LOs are • confined to an atomic sphere • have zero value and slope at R • Can treat two principal QN n for each azimuthal QN  ( e.g. 3p and 4p) • Corresponding states are strictly orthogonal • (e.g.semi-core and valence) • Tail of semi-core states can be represented by plane waves • Only slightly increases the basis set (matrix size) D.J.Singh, Phys.Rev. B 43 6388 (1991)

  23. THE LAPW+LO METHOD • Key Points: • The local orbitals should only be used for those atoms and angular momenta where they are needed. • The local orbitals are just another way to handle the augmentation. They look very different from atomic functions. • We are trading a large number of extra planewave coefficients for some clm. Shape of H and S <G|G>

  24. New ideas from Uppsala and Washington E.Sjöststedt, L.Nordström, D.J.Singh, SSC 114, 15 (2000) • Use APW, but at fixed El (superior PW convergence) • Linearize with additional lo (add a few basis functions) LAPW PW APW • 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

  25. Improved convergence of APW+lo • force (Fy) on oxygen in SES vs. # plane waves • in LAPW changes sign and converges slowly • in APW+lo better convergence • to same value as in LAPW SES (sodium electro solodalite) K.Schwarz, P.Blaha, G.K.H.Madsen, Comp.Phys.Commun.147, 71-76 (2002)

  26. Relativistic effects For example: Ti • Valences states • Scalar relativistc • mass-velocity • Darwin s-shift • Spin orbit coupling on demand by second variational treatment • Semi-core states • Scalar relativistic • No spin orbit coupling • on demand • spin orbit coupling by second variational treatment • Additional local orbital (see Th-6p1/2) • Core states • Full relativistic • Dirac equation

  27. Relativistic semi-core states in fcc Th • additional local orbitals for 6p1/2 orbital in Th • Spin-orbit (2nd variational method) J.Kuneš, P.Novak, R.Schmid, P.Blaha, K.Schwarz, Phys.Rev.B. 64, 153102 (2001)

  28. (L)APW methods • spin polarization • shift of d-bands • Lower Hubbard band (spin up) • Upper Hubbard band (spin down) APW + local orbital method (linearized) augmented plane wave method Total wave function n…50-100 PWs /atom Variational method: Generalized eigenvalue problem

  29. Flow Chart of WIEN2k (SCF) Input rn-1(r) lapw0: calculates V(r) lapw1: sets up H and S and solves the generalized eigenvalue problem lapw2: computes the valence charge density lcore mixer yes no converged? done! • WIEN2k: P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Luitz

  30. Structure: a,b,c,,,, R , ... Structure optimization k ε IBZ (irred.Brillouin zone) iteration i SCF DFT Kohn-Sham Kohn Sham V() = VC+Vxc Poisson, DFT k Ei+1-Ei <  Variational method no yes Generalized eigenvalue problem Etot, force Minimize E, force0 properties

  31. Brillouin zone (BZ) • Irreducibel BZ (IBZ) • The irreducible wedge • Region, from which the whole BZ can be obtained by applying all symmetry operations • Bilbao Crystallographic Server: • www.cryst.ehu.es/cryst/ • The IBZ of all space groups can be obtained from this server • using the option KVEC and specifying the space group (e.g. No.225 for the fcc structure leading to bcc in reciprocal space, No.229 )

  32. WIEN2k software package An Augmented Plane Wave Plus Local Orbital Program for Calculating Crystal Properties Peter Blaha Karlheinz Schwarz Georg Madsen Dieter Kvasnicka Joachim Luitz November 2001 Vienna, AUSTRIA Vienna University of Technology

  33. The WIEN2k authors

  34. Development of WIEN2k • Authors of WIEN2k P. Blaha, K. Schwarz, D. Kvasnicka, G. Madsen and J. Luitz • Other contributions to WIEN2k • C. Ambrosch-Draxl (Univ. Graz, Austria), optics • U. Birkenheuer (Dresden), wave function plotting • R. Dohmen und J. Pichlmeier (RZG, Garching), parallelization • R. Laskowski (Vienna), non-collinear magnetism • P. Novák and J. Kunes (Prague), LDA+U, SO • B. Olejnik (Vienna), non-linear optics • C. Persson (Uppsala), irreducible representations • M. Scheffler (Fritz Haber Inst., Berlin), forces, optimization • D.J.Singh (NRL, Washington D.C.), local orbitals (LO), APW+lo • E. Sjöstedt and L Nordström (Uppsala, Sweden), APW+lo • J. Sofo (Penn State, USA), Bader analysis • B. Yanchitsky and A. Timoshevskii (Kiev), space group • and many others ….

  35. International co-operations • More than 500 user groups worldwide • 25 industries(Canon, Eastman, Exxon, Fuji, A.D.Little, Mitsubishi, Motorola, NEC, Norsk Hydro, Osram, Panasonic, Samsung, Sony, Sumitomo). • Europe: (ETH Zürich, MPI Stuttgart, Dresden, FHI Berlin, DESY, TH Aachen, ESRF, Prague, Paris, Chalmers, Cambridge, Oxford) • 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) • far east: AUS, China, India, JPN, Korea, Pakistan, Singapore,Taiwan (Beijing, Tokyo, Osaka, Sendai, Tsukuba, Hong Kong) • Registration at www.wien2k.at • 400/4000 Euro for Universites/Industries • code download via www (with password), updates, bug fixes, news • User’s Guide, faq-page, mailing-list with help-requests

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