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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 The total energy of an interacting inhomogeneous electron gas in the presence of an external potential Vext(r ) is a functional of the density E Vext ( r ) ( r )dr F [ ] 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 Kohn Sham equations Total energy LDA, GGA 1 ( r ) ( r ) E To [ ] Vext ( r )d r dr dr E xc [ ] 2 | r r | Ekinetic Ene Ecoulomb Eee Exc exchange-correlation vary non interacting 1-electron equation (Kohn Sham) 1 { 2 Vext ( r ) VC ( ( r )) Vxc ( ( r ))} i ( r ) i i ( r ) 2 (r ) | i |2 i EF Walter Kohn, Nobel Prize 1998 Chemistry A simple picture of LDA Look at the “LDA” from a different angle Slater, X Gunnarsson-Lundqvist ………… Exc = -∫ dx n(x) e 2/ 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) Role of „Gradient corrected functionals“ Becke, Perdew, Wang, Lee, Perdew ,Burke, Ernzerhof Yang, Parr …… ‟87 – „92 PBE …… „96 Use n and ∂n/∂x to correct LDA in regions of low density Substantial improvement in energy differences DFT ground state of iron LSDA NM fcc in contrast to experiment GGA LSDA FM GGA bcc GGA Correct lattice constant Experiment FM LSDA bcc 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 CoO why is GGA better than LSDA Central Co atom distinguishes Vxc VxcGGA VxcLSDA between Co and Co Angular correlation DFT thanks to Claudia Ambrosch (Graz) GGA follows LDA Overview of DFT concepts Form of Full potential : FP potential “Muffin-tin” MT atomic sphere approximation (ASA) pseudopotential (PP) Relativistic treatment of the electrons exchange and correlation potential fully-relativistic Local density approximation (LDA) semi-relativistic Generalized gradient approximation (GGA) non relativistic Beyond LDA: e.g. LDA+U 1 2 k V ( r )i ikik 2 Kohn-Sham equations Representation Basis functions non periodic (cluster) of solid plane waves : PW periodic augmented plane waves : APW (unit cell) Treatment of linearized “APWs” spin analytic functions (e.g. Hankel) Spin polarized atomic orbitals. e.g. Slater (STO), Gaussians (GTO) non spin polarized numerical How to solve the Kohn Sham equations Total energy LDA, GGA 1 ( r ) ( r ) E To [ ] Vext ( r )d r dr dr E xc [ ] 2 | r r | Ekinetic Ene Ecoulomb Eee Exc exchange-correlation vary non interacting 1-electron equation (Kohn Sham) 1 { 2 Vext ( r ) VC ( ( r )) Vxc ( ( r ))} i ( r ) i i ( r ) 2 (r ) | i |2 i EF 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) APW Augmented Plane Wave method The unit cell is partitioned into: Bloch wave function: atomic spheres atomic partial waves Interstitial region Plane Waves (PWs) unit cell Rmt rI Full potential VLM YLM (r ) ˆ r R PW: e Atomic partial wave LM i ( k K ). r VK e iK . r rI K join aKmu ( r , )Ym ( r ) ˆ m Slater‘s APW (1937) Atomic partial waves m aK u (r , )Ym (r ) m ˆ Energy dependent basis functions lead to H Hamiltonian Non-linear eigenvalue problem S overlap matrix Computationally very demanding One had to numerically search for the energy, for which the det(H-ES) vanishes. Linearization of energy dependence LAPW suggested by O.K.Andersen, Phys.Rev. B 12, 3060 (1975) kn [ A m m ( k n )u ( E , r ) Bm ( k n )u ( E , r )]Ym ( r ) ˆ join PWs in value and slope Atomic sphere LAPW Plane Waves (PWs) PW i ( k K n ). r e Full-potential in LAPW The potential (and charge density) can be of general form SrTiO3 (no shape approximation) VLM (r )YLM (r ) ˆ r R Full V (r ) { LM K VK e iK . r rI potential Inside each atomic sphere a local coordinate system is used (defining LM) Muffin tin approximation Ti TiO2 rutile O 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 Problems of the LAPW method: EFG Calculation for Rutile TiO2 as a function of the Ti-p linearization energy Ep exp. EFG Electronic Structure E Ti 3d / O 2p EF O 2p Hybridized w. „ghostband“ Ti 4p, Ti 3d P. Blaha, D.J. Singh, P.I. Sorantin and K. Schwarz, Phys. Rev. B 46, 1321 (1992). Ti- 3p ONE SOLUTION Treat all the states in a single energy Electronic Structure window: • Automatically orthogonal. E Ti 3d / O 2p • Need to add variational freedom. EF • Could invent quadratic or cubic APW methods. O 2p Hybridized w. -1/2 cG ei(G+k)r (r) = { (A G Ti 4p, Ti 3d lmul(r)+Blmůl(r)+Clmül(r)) Ylm(r) lm Problem: This requires an extra matching condition, e.g. second derivatives Ti- 3p continuous method will be impractical due to the high planewave cut-off needed. 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) THE LAPW+LO METHOD Key Points: 1.The local orbitals should only be used for those atoms and Shape of H and S angular momenta where they are needed. 2.The local orbitals are just another way to handle the <G|G> augmentation. They look very different from atomic functions. 3.We are trading a large number of extra planewave coefficients for some clm. 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) kn Am ( kn )u ( E , r )Ym ( r ) ˆ m lo [ AmuE1 BmuE1 ]Ym ( r ) ˆ 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 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) 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 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) (L)APW methods spin polarization APW + local orbital method shift of d-bands (linearized) augmented plane wave methodband Lower Hubbard k = C k n k n (spin up) Kn Upper Hubbard band Total wave function k = C k n k(spin down) n n…50-100 PWs /atom Kn <|H | > < E > Variational method: < E >= <| > Ck =0 n Generalized eigenvalue problem H C= ESC Flow Chart of WIEN2k (SCF) Input n-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 Structure: a,b,c,,,, R , ... Structure optimization k ε IBZ (irred.Brillouin zone) iteration i [ 2 V ( )] k E k k S C DFT Kohn-Sham Kohn Sham F k C kn kn V() = VC+Vxc Poisson, DFT k kn no Ei+1-Ei < Variational E Ckn 0 method yes Generalized eigenvalue HC ESC Etot, force problem Minimize E, force0 k * k properties Ek E F 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 ) 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 The WIEN2k authors 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 …. 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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