Transcript The FP-LAPW and APW+lo methods Peter Blaha TU Wien Institute of Materials Chemistry
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 “Muffin-tin” MT atomic sphere approximation (ASA) pseudopotential (PP)
Full potential : FP
Relativistic treatment of the electrons non relativistic semi-relativistic
fully-relativistic
1 2
2
V
(
r
)
i k
i k
i k
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 “Muffin-tin” MT atomic sphere approximation (ASA)
Full potential : FP
pseudopotential (PP) Relativistic treatment of the electrons non relativistic semi-relativistic
fully-relativistic
1 2
2 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
V
(
r
)
i k
i k
i k
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 V ext (r ) is a functional of the density
E
V ext
(
r
) (
r
)
d r
F
[ ] Kohn-Sham: (still exact!)
E
T o
[ ]
V ext
(
r
)
d r
1 2 | (
r r
)
r
(
r
| )
d r
d r
E xc
[ ] E kinetic non interacting E ne E coulomb E ee E xc 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
E
T o
[ ]
V ext
(
r
)
d
r
1 2 | (
r
r
) (
r r
| )
d
r
d r
LDA, GGA
E xc
[ ] 1-electron equations (Kohn Sham) vary { 1 2 2
V ext
(
r
)
V C
( (
r
)) -Z/r |
r
(
r
)
r
|
d r
V xc
( (
r
))}
i
(
r
)
E xc
( )
i
i
(
r
) (
r
)
i
E F
|
i
| 2
E LDA xc E GGA xc
(
r
) (
r
) hom .
[
xc
(
r
)]
dr F
[ (
r
), (
r
)]
LDA
treats both, exchange and
dr
GGA
correlation effects approximately
Lagrange multiplier !
Walter Kohn, Nobel Prize 1998 Chemistry
Success and failure of “standard” DFT in solids
Standard LDA (GGA) gives insulator, magnetism,…) good description of structural and electronic properties of most solids (lattice parameters within 1-2%, at least qualitatively correct bandstructure, metal Problems: “localized” (correlated) electrons late 3d transition metal oxides/halides metals instead of insulators (FeO, FeF 2 , cuprates, …) nonmagnetic instead of anti-ferromagnetic (La 2 CuO 4 , YBa 2 Cu 3 O 6 ) 4f, 5f electrons all f-states pinned at the Fermi energy, “always” metallic orbital moments much too small “weakly” correlated metals FeAl is ferromagnetic in theory, but nonmagnetic experimentally 3d-band position, exchange splitting,…
Is LDA repairable ?
ab initio methods GGA: usually improvement, but often too small. Hartree-Fock: completely neglects correlation, very poor in solids Exact exchange: imbalance between exact X and approximate (no) C Hybrid-Functionals : mix of HF + LDA; good for insulators, poor for metals GW: gaps in semiconductors, but groundstate? expensive!
Quantum Monte-Carlo: very expensive not fully ab initio Self-interaction-correction: vanishes for Bloch states Orbital polarization: Hund’s 2 nd rule by atomic Slater-parameter LDA+U : strong Coulomb repulsion via external Hubbard U parameter DMFT: extension of LDA+U for weakly correlated systems
Concepts when solving Schrödingers-equation
Form of potential “Muffin-tin” MT atomic sphere approximation (ASA)
Full potential : FP
pseudopotential (PP) Relativistic treatment of the electrons non relativistic semi-relativistic
fully-relativistic
1 2
2 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
V
(
r
)
i k
i k
i k
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 R mt
r
I
PW: Basisset:
i
(
k
e
K
).
r
Atomic partial waves
m A K
m u
(
r
, )
Y
m
(
r
ˆ ) join u l (r, in a ) are the numerical solutions of the radial Schrödinger equation given spherical potential particular energy for a A lm K coefficients for matching the PW
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) Efficience of APW + convenience of LAPW Basis for K.Schwarz, P.Blaha, G.K.H.Madsen, Comp.Phys.Commun.147, 71-76 (2002)
Slater‘s APW (1937)
Atomic partial waves
m A K
m u
(
r
,
)
Y
m
(
r
ˆ
)
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
LAPW
suggested by O.K.Andersen, Phys.Rev. B 12, 3060 (1975)
k n
[
m A
m
(
k n
)
u
(
E
,
r
)
B
m
(
k n
)
u
(
E
,
r
)]
Y
m
(
r
) expand u l add
u
l
at fixed energy
u l
/
E l and A lm k , B lm k : join PWs in value and slope additional constraint requires more PWs than APW basis flexible enough for single diagonalization Atomic sphere LAPW APW PW
Full-potential in LAPW
(A.Freeman etal.) SrTiO 3 Full potential The potential (and charge density) can be of general form (no shape approximation)
V
(
r
) {
LM V LM
K V K
(
r
)
Y LM e
i K
.
r
(
r
ˆ )
r
R
r
I
Inside each atomic sphere a local coordinate system is used (defining LM) Muffin tin approximation
Ti TiO
2
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 TiO Ti p 2 as a function of the linearization energy E p 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)
LO
[
A
m u
E
1
B
m u
E
1
C
m u
E
2 ]
Y
m
(
r
ˆ ) LO is confined to an atomic sphere has zero value and slope at R can treat two principal QN n for each azimuthal QN (3p and 4p) 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 QAPW
La R
MT
= 3.3 a
0 LAPW+LO converges 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 like band.
Suggested settings: Two “energy” parameters, one for other for valence.
u u and (2) ů and the . Choose one at the semi-core position and the other at the
RK max
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 •Linearize , but at fixed E l (superior PW convergence) with additional lo (add a few basis functions)
k n
m A
m
(
k n
)
u
(
E
,
r
)
Y
m
(
r
ˆ )
lo
[
A
m u
E
1
B
m u
E
1 ]
Y
m
(
r
ˆ ) 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
better convergence in
APW+lo
For example: Ti
Relativistic treatment
Valence states Scalar relativistic 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-6p 1/2 ) Core states Fully relativistic Dirac equation
Relativistic semi-core states in fcc Th
additional local orbitals for 6p 1/2 orbital in Th Spin-orbit (2 nd variational method) J.Kuneš, P.Novak, R.Schmid, P.Blaha, K.Schwarz, Phys.Rev.B. 64, 153102 (2001)
Atomic forces (Yu et al.; Kohler et al.)
Total Energy: Electrostatic energy Kinetic energy XC-energy Force on atom :
U
[ ]
T
[ ]
E xc
[ ] 1 2
d
3
r
i n i
d
3
r
i
(
r
)
V es
(
r
)
d
3
r
(
r
)
V eff
1 2 (
r
) (
r
)
xc
(
r
)
F
tot
F HF
F core
F val d R
Z
V es
(
r
) Hellmann-Feynman-force Pulay corrections Core Valence expensive, contains a summation of matrix elements over all occupied states
F HF
F core
Z
1
m
lim 1
r
0
es V
1
m
(
r
r
)
r
Y
1
m
(
r
ˆ )
core
(
r
)
V eff
(
r
)
d r
F val
(
K V eff
2 (
r
)
i
)
K
*
val
(
r
) (
r
)
K d r
(
r
)
k
,
i dS
n i
K
c i
* ,
K
(
K
)
c i i
(
K
K
) (
K
K
)
H
i
K
Quantum mechanics at work
WIEN2k software package
WIEN97: ~500 users WIEN2k: ~1030 users mailinglist: 1500 users
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
http://www.wien2k.at
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 D.J.Singh (NRL, Washington D.C.), local oribtals (LO), APW+lo U. Birkenheuer (Dresden), wave function plotting T. Charpin (Paris), elastic constants R. Dohmen und J. Pichlmeier (RZG, Garching), parallelization P. Novák and J. Kunes (Prague), LDA+U, SO C. Persson (Uppsala), irreducible representations M. Scheffler (Fritz Haber Inst., Berlin), forces E. Sjöstedt and L Nordström (Uppsala, Sweden), APW+lo J. Sofo and J. Fuhr (Barriloche), Bader analysis B. Yanchitsky and A. Timoshevskii (Kiev), spacegroup R. Laskowski (Vienna), non-collinear magnetism B. Olejnik (Vienna), non-linear optics and many others ….
International co-operations
More than 500 user groups worldwide Industries (Canon, Eastman, Exxon, Fuji, A.D.Little, Mitsubishi, Motorola, NEC, Norsk Hydro, Osram, Panasonic, Samsung, Sony, Sumitomo).
Europe: (EHT 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 usersguide, faq-page, mailing-list with help-requests
WIEN2k- hardware/software
WIEN2k runs on any Unix/Linux clusters to supercomputers Fortran90 (dynamical allocation) platform from PCs, workstations, many individual modules , linked together with C-shell or perl-scripts f90 compiler, BLAS-library , perl5, ghostview, gnuplot, Tcl/Tk (Xcrysden), pdf-reader, www-browser •web-based GUI – w2web •real/complex version (inversion) •10 atom cells on 128Mb PC •100 atom cells require 1-2 Gb RAM •k-point parallel on clusters with common NFS ( slow •MPI/Scalapack network) parallelization for big cases (>50 atoms) and fast network •installation support for most platforms
How to run WIEN2k
WIEN2k consists of many independent F90 programs , which are linked together via C-shell scripts.
Each „ case “ runs in his own directory The „ master input “ is called Initialize a calculation: Run scf-cycle : ./case case.struct
init_lapw run_lapw (runsp_lapw) You can run WIEN2k using any www-browser and the w2web interface , but also at the command line of an xterm.
Input/output/scf files have endings programs: as the corresponding case.output1…lapw1; case.in2…lapw2; case.scf0…lapw0 Inputs are generated using STRUCTGEN (w2web) and init_lapw
w2web: the web-based GUI of WIEN2k
Based on www WIEN2k can be managed remotely via w2web Important steps: start w2web on all your hosts login to the desired host (ssh) w2web (at first startup you will be asked for username/password, port-number, (master-)hostname. creates ~/.w2web directory) use your browser and connect to the (master) host:portnumber mozilla http://fp98.zserv:10000 create a new session on the desired host (or select an old one)
w2web GUI (graphical user interface)
Structure generator spacegroup selection import cif file step by step initialization symmetry detection automatic input generation SCF calculations Magnetism (spin-polarization) Spin-orbit coupling Forces (automatic geometry optimization) Guided Tasks Energy band structure DOS Electron density X-ray spectra Optics
Spacegroup P4 2 /mnm
Structure given by:
spacegroup lattice parameter positions of atoms (basis)
Rutile TiO 2 :
P4 2 /mnm (136) a=8.68, c=5.59 bohr Ti: (0,0,0) O: (0.304,0.304,0)
Structure generator
Specify: Number of nonequivalent atoms lattice type (P, F, B, H, CXY, CXZ, CYZ) or spacegroup symbol if existing, you must use a SG-setting with inversion symmetry: Si: ± (1/8,1/8,1/8), not (0,0,0)+(1/4,1/4,1/4) !
lattice parameters a,b,c (in Å or bohr) name of atoms (Si) and fractional coordinates (position) as numbers (0.123); fractions (1/3); simple expressions (x-1/2,…) in fcc (bcc) specify just one atom, not the others in (1/2,1/2,0; …) „save structure “ updates automatically Z , r0, equivalent positions and generates case.inst
„set RMT and continue“: (specify proper “reduction” of NN-distances) non-overlapping „ as large as possible “ (saves time), but not larger than 3 bohr RMT for sp -elements 10-20 % smaller than for d Do not change RMT in a „ series “ of calculations ( f ) elements largest spheres not more than 50 % Exception: H in C-H or O-H bonds: larger than RMT~0.6
smallest bohr ( sphere RKMAX~3-4 ) „save structure – save+cleanup“
Program structure of WIEN2k
init_lapw initialization symmetry detection (F, I, C centering, inversion) input generation with recommended defaults quality (and computing time) depends on k-mesh and R.Kmax (determines #PW) run_lapw scf-cycle optional with SO and/or LDA+U different convergence criteria (energy, charge, forces) save_lapw tic_gga_100k_rk7_vol0 cp case.struct and clmsum files, mv case.scf file rm case.broyd* files
Task for electron density plot
A task consists of a series of steps that must be executed to generate a plot For electron density plot select states (e.g. valence e ) select plane for plot generate 3D or contour plot with gnuplot or Xcrysden
TiC electron density
Valence electrons NaCl structure (100) plane plot in 2 dimensions Shows charge distribution covalent bonding between the Ti-3d and C-2p electrons e g /t 2g symmetry
Properties with WIEN2k - I
Energy bands classification of irreducible representations ´character-plot´ (emphasize a certain band-character) Density of states including partial DOS with l and m- character (eg. p x Electron density, potential total-, valence-, difference-, spin-densities, 1-D, 2D- and 3D-plots (Xcrysden) X-ray structure factors , p y , p z ) of selected states Bader´s atom-in-molecule analysis, critical-points, atomic basins and charges .
n
0 spin+orbital magnetic moments (spin-orbit / LDA+U ) Hyperfine parameters hyperfine fields (contact + dipolar + orbital contribution) Isomer shift Electric field gradients
Properties with WIEN2k - II
Total energy and forces optimization of internal coordinates, (MD, BROYDEN) cell parameter only via E tot (no stress tensor) elastic constants for cubic cells Phonons via a direct method (based on forces from supercells) interface to PHONON (K.Parlinski) – phonon bandstructure, phonon DOS, thermodynamics, neutrons Spectroscopy core levels (with core holes) X-ray emission, absorption, electron-energy-loss (core valence/conduction bands including matrix elements and angular dep.) optical properties (dielectric function, JDOS including momentum matrix elements and Kramers-Kronig) fermi surface (2D, 3D)
Properties with WIEN2k - III
New developments (in progress) non-linear optics (B.Olejnik) non-collinear magnetism (R.Laskowski) transport properties (Fermi velocities, Seebeck, conductivity, thermoelectrics, ..) (G.Madsen) Compton profiles linear response (phonons, E-field) (C.Ambrosch-Draxl) stress tensor (C.Ambrosch-Draxl) exact exchange, GW, … ??
grid-computing
Advantage/disadvantage of WIEN2k
+ 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 + w2web (for novice users) - ? speed + memory requirements + very efficient basis for large spheres (2 bohr) (Fe: 12Ry, O: 9Ry ) - 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 APW+lo method (as implemented in WIEN2k) is one of them all electron full-potential highly accurate - benchmark for other methods many properties user friendly widely used development by several groups large user community used by many experimental groups