All-Electron Full-Potential Calculations at O(ASA) Speed A Fata Morgana?

Transcription

1 All-Electron Full-Potential Calculations at O(ASA) Speed A Fata Morgana? Center for Electronic Correlations and Magnetism Institute for Physics, University of Augsburg February 4, 2008

2 Outline 1 2 3

3 Outline 1 2 3

4 Outline 1 2 3

5 Outline 1 2 3

6 Back in the 1930 s... John C. Slater Full Potential { spherical symmetric near nuclei v σ (r) : flat outside the atomic cores

7 Back in the 1930 s... John C. Slater Full Potential { spherical symmetric near nuclei v σ (r) : flat outside the atomic cores Muffin-Tin Approximation { vσ MT spherical symmetric in spheres (r) = constant in interstitial region

8 Back in the 1930 s... Full Potential (FeS 2 ) Muffin-Tin Potential v / a.u

9 Back in the 1930 s... Muffin-Tin Approximation distinguish: atomic regions Muffin-Tin Potential remainder

10 Back in the 1930 s... Muffin-Tin Approximation distinguish: atomic regions muffin-tin spheres v eff,σ (r) = v eff,σ ( r ) remainder Muffin-Tin Potential

11 Back in the 1930 s... Muffin-Tin Approximation distinguish: atomic regions muffin-tin spheres v eff,σ (r) = v eff,σ ( r ) remainder interstitial region v eff,σ (r) = 0 Muffin-Tin Potential

12 Back in the 1930 s... Partial Waves muffin-tin spheres v eff,σ (r) = v eff,σ ( r ) Muffin-Tin Potential interstitial region v eff,σ (r) = 0

13 Back in the 1930 s... Partial Waves muffin-tin spheres v eff,σ (r) = v eff,σ ( r ) solve radial Schrödinger equation numerically interstitial region v eff,σ (r) = 0 Muffin-Tin Potential

14 Back in the 1930 s... Partial Waves muffin-tin spheres v eff,σ (r) = v eff,σ ( r ) solve radial Schrödinger equation numerically interstitial region v eff,σ (r) = 0 exact solutions plane waves spherical waves Muffin-Tin Potential

15 Back in the 1930 s... Partial Waves muffin-tin spheres v eff,σ (r) = v eff,σ ( r ) solve radial Schrödinger equation numerically interstitial region v eff,σ (r) = 0 exact solutions plane waves spherical waves match at sphere surface ( augment ) Muffin-Tin Potential

16 Back in the 1930 s... Partial Waves muffin-tin spheres v eff,σ (r) = v eff,σ ( r ) solve radial Schrödinger equation numerically interstitial region v eff,σ (r) = 0 exact solutions plane waves spherical waves match at sphere surface ( augment ) Basis Functions matched partial waves augmented plane waves (APWs) muffin-tin orbitals (MTOs), augmented spherical waves (ASWs)

17 Back in the 1930 s... Wave Function expand in basis functions expansion coefficients from variational principle Basis Functions matched partial waves augmented plane waves (APWs) muffin-tin orbitals (MTOs), augmented spherical waves (ASWs)

18 Back in the 1930 s... Wave Function expand in basis functions expansion coefficients from variational principle Core States all-electron methods fully included orthogonal to partial waves Basis Functions matched partial waves augmented plane waves (APWs) muffin-tin orbitals (MTOs), augmented spherical waves (ASWs) used to describe valence states

19 Back in the 1930 s... Augmented Spherical Waves envelope functions augmented spherical waves A B A B r

20 Back in the 1970 s... Ole K. Andersen Linear Methods in Band Theory energy dependence of basis functions almost linear linearize (ϕ, ϕ) huge increase in computat. efficiency!

21 Back in the 1970 s... Ole K. Andersen Linear Methods in Band Theory energy dependence of basis functions almost linear linearize (ϕ, ϕ) huge increase in computat. efficiency! Linear Augmented Plane Wave (LAPW) muffin-tin approximation easy to implement good! full-potential at a low price basis functions from muffin-tin potential wave functions from full potential example: Wien2k large basis set ( 100 pw s/atom) bad!

22 Back in the 1970 s... Ole K. Andersen Linear Muffin-Tin Orbital (LMTO) based on spherical waves does not require crystalline periodicity natural interpretation of results difficult to implement bad! full-potential extension extremely difficult muffin-tin approximation (?) finite interstitial region large basis set: two functions per s-, p-, d-state still inefficient bad!

23 Back in the 1970 s... Ole K. Andersen Linear Muffin-Tin Orbital (LMTO) based on spherical waves does not require crystalline periodicity natural interpretation of results difficult to implement bad! full-potential extension extremely difficult muffin-tin approximation (?) finite interstitial region large basis set: two functions per s-, p-, d-state still inefficient bad!

24 Back in the 1970 s... Ole K. Andersen Linear Muffin-Tin Orbital (LMTO) based on spherical waves does not require crystalline periodicity natural interpretation of results atomic-sphere approximation (ASA) make spheres space-filling! interstitial region formally removed only numerical functions in spheres minimal basis set (s, p, d) very high computational efficiency O(ASA) speed!!! makes potential more realistic systematic error in total energy bad!

25 Back in the 1970 s... Ole K. Andersen Linear Muffin-Tin Orbital (LMTO) based on spherical waves does not require crystalline periodicity natural interpretation of results atomic-sphere approximation (ASA) make spheres space-filling! interstitial region formally removed only numerical functions in spheres minimal basis set (s, p, d) very high computational efficiency O(ASA) speed!!! makes potential more realistic systematic error in total energy bad!

26 Back in the 1970 s... Ole K. Andersen Linear Muffin-Tin Orbital (LMTO) based on spherical waves does not require crystalline periodicity natural interpretation of results atomic-sphere approximation (ASA) make spheres space-filling! interstitial region formally removed only numerical functions in spheres minimal basis set (s, p, d) very high computational efficiency O(ASA) speed!!! makes potential more realistic systematic error in total energy bad!

27 Iron Pyrite: FeS 2 Pyrite Pa 3 (T 6 h ) a = Å NaCl structure sublattices occupied by iron atoms sulfur pairs sulfur pairs 111 axes x S = rotated FeS 6 octahedra

28 FeS 2 : Structure Optimization ASW Version E (Ryd) x S

29 Towards a Full-Potential Spherical-Wave Method Conclusions ASA (space-filling atomic spheres) O(ASA) speed systematic error in total energy non-overlapping muffin-tin spheres prerequisite for accurate total energies larger basis set inefficient

30 Towards a Full-Potential Spherical-Wave Method Conclusions ASA (space-filling atomic spheres) O(ASA) speed systematic error in total energy non-overlapping muffin-tin spheres prerequisite for accurate total energies larger basis set inefficient Requirements restore interstitial region go to non-overlapping muffin-tin spheres go beyond constant-potential approximation inside muffin-tin spheres non-spherical contributions

31 Towards a Full-Potential Spherical-Wave Method Conclusions ASA (space-filling atomic spheres) O(ASA) speed systematic error in total energy non-overlapping muffin-tin spheres prerequisite for accurate total energies larger basis set inefficient Requirements restore interstitial region go to non-overlapping muffin-tin spheres go beyond constant-potential approximation inside muffin-tin spheres non-spherical contributions

32 Towards a Full-Potential Spherical-Wave Method Guidelines interstitial quantities expanded in plane waves straightforward to implement inefficient interstitial quantities expanded in spherical waves elegant, no periodicity required efficient difficult to implement

33 Towards a Full-Potential Spherical-Wave Method Guidelines interstitial quantities expanded in plane waves straightforward to implement inefficient interstitial quantities expanded in spherical waves elegant, no periodicity required efficient difficult to implement

34 ASW Method Characteristics similar to LMTO different linearization scheme different interstitial energy different implementations

35 ASW Method Characteristics similar to LMTO different linearization scheme different interstitial energy different implementations 0th Generation (Williams, Kübler, Gelatt, 1970s) PRB 19, 6094 (1979)

36 ASW Method Characteristics similar to LMTO different linearization scheme different interstitial energy different implementations 1st Generation (VE, 1990s) IJQC 77, 1007 (2000) completely new, monolithic implementation new algorithms improved accuracy, numerical stability much improved functionality, usability, and portability xanderson convergence acceleration scheme all LDA-parametrizations, most GGA-schemes still based on atomic-sphere approximation

37 ASW Method: Basic Formalism Wave Function Expanded in Basis Functions ψ σ (r) = Lκi c Lκiσ H Lκσ (r i) c Lκiσ determined variationally

38 ASW Method: Basic Formalism Wave Function Expanded in Basis Functions ψ σ (r) = Lκi c Lκiσ H Lκσ (r i) c Lκiσ determined variationally Augmented Spherical Wave H HLκσ Lκ I (r i) (r i) = H Lκσ (r i ) J L j L κσ(r j )B L Lκ interstitial region on-centre sphere i off-centre spheres j B L Lκ(R j R i ): structure constants ASW classified by atomic site R i, L = (l, m), decay κ, spin σ

39 ASW Method: Basic Formalism Augmented Spherical Waves envelope functions augmented spherical waves A B A B r

40 ASW Method: Basic Formalism Envelope Functions H I Lκ (r i) := iκ l+1 h (1) l (κr i )Y L (ˆr i ) h (1) l (κr i ): spherical Hankel function

41 ASW Method: Basic Formalism Envelope Functions H I Lκ (r i) := iκ l+1 h (1) l (κr i )Y L (ˆr i ) h (1) l (κr i ): spherical Hankel function Augmented Functions H Lκσ (r i ) := h lκσ (r i )Y L (ˆr i ) J L κσ(r j ) := j l κσ(r j )Y L (ˆr j ) h, j: numerical solutions of radial Kohn-Sham equation boundary conditions from envelope functions correspond to ϕ and ϕ of LMTO

42 ASW Method: Further Reading

43 Outline 1 2 3

44 Basic Principles Steps to be Taken remove total energy error due to overlap of atomic spheres reintroduce non-overlapping muffin-tin spheres restore interstitial region

45 Basic Principles Steps to be Taken remove total energy error due to overlap of atomic spheres reintroduce non-overlapping muffin-tin spheres restore interstitial region find representation of electron density and full potential inside muffin-tin spheres in the interstitial region

46 Basic Principles Steps to be Taken remove total energy error due to overlap of atomic spheres reintroduce non-overlapping muffin-tin spheres restore interstitial region find representation of electron density and full potential find representation of products of the wave function inside muffin-tin spheres in the interstitial region

47 Basic Principles Steps to be Taken remove total energy error due to overlap of atomic spheres reintroduce non-overlapping muffin-tin spheres restore interstitial region find representation of electron density and full potential find representation of products of the wave function find representation of products of the basis functions inside muffin-tin spheres in the interstitial region

48 Basic Principles Steps to be Taken remove total energy error due to overlap of atomic spheres reintroduce non-overlapping muffin-tin spheres restore interstitial region find representation of electron density and full potential find representation of products of the wave function find representation of products of the basis functions inside muffin-tin spheres use spherical-harmonics expansions in the interstitial region

49 Basic Principles Steps to be Taken remove total energy error due to overlap of atomic spheres reintroduce non-overlapping muffin-tin spheres restore interstitial region find representation of electron density and full potential find representation of products of the wave function find representation of products of the basis functions inside muffin-tin spheres use spherical-harmonics expansions in the interstitial region no exact spherical-wave representation available!

50 From Wave Functions to Electron Density Density inside MT-Spheres (Al) ρ v / a.u

51 From Wave Functions to Electron Density Products of Basis Functions in Interstitial Region p I (r) = n d n F n (r) d 3 r F n (r)pi (r) = n d n d 3 r F n (r)f n(r)

52 From Wave Functions to Electron Density Products of Basis Functions in Interstitial Region p I (r) = n d n F n (r) d 3 r F n (r)pi (r) = n d n d 3 r F n (r)f n(r) F n (r): plane waves integrals exact inefficient Weyrich 1988, Blöchl 1989, VE 1991, Savrasov 1992, Methfessel 2000

53 From Wave Functions to Electron Density Products of Basis Functions in Interstitial Region p I (r) = n d n F n (r) d 3 r F n (r)pi (r) = n d n d 3 r F n (r)f n(r) F n (r): spherical waves would be efficient integrals not known analytically Springborg/Andersen 1987, Methfessel 1988, VE 2002, VE 2006

54 From Wave Functions to Electron Density Interstitial Products of Basis Functions expand in atom-centered spherical waves (Hankel functions): p I (r) := ( H Lκ 1 σ (r i) ) H L κ 2 σ (r j)! = n H L κ 2 σ (r j): orbital basis set (obs) H Kη (r n ): product basis set (pbs) Kη d Lκ 1iL κ 2 j Kηnσ H Kη (r n ) coefficients from projection P: [ ] P p I (r) = P [( HLκ 1 σ (r i) ) H L κ 2 σ (r j) ] = d Lκ 1iL κ 2 j Kηnσ P [ H Kη (r n ) ] n Kη

55 From Wave Functions to Electron Density Interstitial Products of Basis Functions expand in atom-centered spherical waves (Hankel functions): p I (r) := ( H Lκ 1 σ (r i) ) H L κ 2 σ (r j)! = n H L κ 2 σ (r j): orbital basis set (obs) H Kη (r n ): product basis set (pbs) Kη d Lκ 1iL κ 2 j Kηnσ H Kη (r n ) coefficients from projection P: [ ] P p I (r) = P [( HLκ 1 σ (r i) ) H L κ 2 σ (r j) ] = d Lκ 1iL κ 2 j Kηnσ P [ H Kη (r n ) ] n Kη

56 From Wave Functions to Electron Density Interstitial Products of Basis Functions projection P: P [( H Lκ 1 σ (r i) ) H L κ 2 σ (r j) ] = n Kη d Lκ 1iL κ 2 j Kηnσ P [ H Kη (r n ) ]

57 From Wave Functions to Electron Density Interstitial Products of Basis Functions projection P: P [( H Lκ 1 σ (r i) ) H L κ 2 σ (r j) ] = n Kη d Lκ 1iL κ 2 j Kηnσ P [ H Kη (r n ) ] Springborg/Andersen 1987 integrate over interstitial region, use one-center expansions P K η n... ˆ= Ω I d 3 r H K η (r n )...

58 From Wave Functions to Electron Density Interstitial Products of Basis Functions projection P: P [( H Lκ 1 σ (r i) ) H L κ 2 σ (r j) ] = n Kη d Lκ 1iL κ 2 j Kηnσ P [ H Kη (r n ) ] Methfessel 1988 match values and slopes at MT-sphere surfaces P K η n... ˆ= ( ) d 2ˆr η 1 Y K (ˆr)... r rn =s n

59 From Wave Functions to Electron Density Interstitial Products of Basis Functions projection P: P [( H Lκ 1 σ (r i) ) H L κ 2 σ (r j) ] = n Kη d Lκ 1iL κ 2 j Kηnσ P [ H Kη (r n ) ] VE 2002 integrate over interstitial region, use one-center expansions (interstitial shells between AS- and MT-spheres) P K η n... ˆ= d 3 r m HK η (r n )... m Ω m

60 From Wave Functions to Electron Density Interstitial Products of Basis Functions projection P: P [( H Lκ 1 σ (r i) ) H L κ 2 σ (r j) ] = n experience Kη product basis set almost overcomplete d Lκ 1iL κ 2 j Kηnσ P [ H Kη (r n ) ] value/slope (Methfessel) matching most stable

61 From Wave Functions to Electron Density Density from Value/Slope Matching at MT-Radii (Al) ρ v / a.u. ρ v / a.u

62 From Electron Density to Full Potential Inside Muffin-Tin Spheres density, Hartree-potential and xc-potential numerically Interstitial Region density from value/slope matching Hartree-potential analytically xc-potential from value/slope matching

63 From Electron Density to Full Potential Inside Muffin-Tin Spheres density, Hartree-potential and xc-potential numerically Interstitial Region density from value/slope matching Hartree-potential analytically xc-potential from value/slope matching

64 From Full Potential to Basis Functions Previous Approaches project full potential to muffin-tin potential construct basis functions from muffin-tin potential no minimal basis set! (large basis set!) Present Approach project full potential to ASA potential construct basis functions from ASA potential minimal basis set! future use spheres larger than space-filling remove linearization error

65 From Full Potential to Basis Functions Previous Approaches project full potential to muffin-tin potential construct basis functions from muffin-tin potential no minimal basis set! (large basis set!) Present Approach project full potential to ASA potential construct basis functions from ASA potential minimal basis set! future use spheres larger than space-filling remove linearization error

66 Comparison of Approaches Ole K. Andersen ASA geometry used for basis functions minimal basis set ASA geometry used for density and potential error in total energy good! bad!

67 Comparison of Approaches Ole K. Andersen ASA geometry used for basis functions minimal basis set ASA geometry used for density and potential error in total energy Michael S. Methfessel MT geometry used for density and potential accurate total energy MT geometry used for basis functions large basis set good! bad! good! bad!

68 Comparison of Approaches Ole K. Andersen ASA geometry used for basis functions ASA geometry used for density and potential Michael S. Methfessel MT geometry used for density and potential MT geometry used for basis functions present approach ASA geometry used for basis functions minimal basis set O(ASA) speed MT geometry used for density and potential accurate total energy good! bad! good! bad! great! great!

69 Implementation 2nd Generation ASW (VE, 2000s) based on 1st generation code full-potential ASW method at O(ASA) speed! electron densities, spin densities electric field gradients elastic properties, phonon spectra optical properties based on linear-response theory direct calculation of Rσ and Iσ no Kramers-Kronig relations needed LDA+U method all flavours for double-counting terms (AMF, FLL, DFT)

70 Implementation 2nd Generation ASW (VE, 2000s) based on 1st generation code full-potential ASW method at O(ASA) speed! electron densities, spin densities electric field gradients elastic properties, phonon spectra optical properties based on linear-response theory direct calculation of Rσ and Iσ no Kramers-Kronig relations needed LDA+U method all flavours for double-counting terms (AMF, FLL, DFT)

71 Implementation 2nd Generation ASW (VE, 2000s) based on 1st generation code full-potential ASW method at O(ASA) speed! electron densities, spin densities electric field gradients elastic properties, phonon spectra optical properties based on linear-response theory direct calculation of Rσ and Iσ no Kramers-Kronig relations needed LDA+U method all flavours for double-counting terms (AMF, FLL, DFT)

72 Outline 1 2 3

73 Electronic Structure of BaTiO 3 Wien2k VASP 5 Energy (ev) ev 5 R X G M R G

74 Electronic Structure of BaTiO 3 Wien2k ASW Version (E - E V ) (ev) R X Γ M R Γ

75 Electronic Structure of BaTiO 3 Wien2k FPASW Version (E - E V ) (ev) R X Γ M R Γ New! much better agreement with other full-potential codes (valence-band width, valence states at M-point)

76 Fermi Surface of MoO 2 ASW Version 1.9 FPASW Version (E - E F ) (ev) 0 (E - E F ) (ev) Γ Y C Z Γ A E Z Γ B D Z Γ Y C Z Γ A E Z Γ B D Z New! no hole pocket near Z-point much better agreement with ARPES, de Haas-van Alphen

77 Iron Pyrite: FeS 2 Pyrite Pa 3 (T 6 h ) a = Å NaCl structure sublattices occupied by iron atoms sulfur pairs sulfur pairs 111 axes x S = rotated FeS 6 octahedra

78 FeS 2 : Partial Densities of States DOS (1/eV) Fe 3d S 3s S 3p Relevant states Fe 3d S 3s S 3p (E - E V ) (ev)

79 FeS 2 : Partial Densities of States 8 DOS (1/eV) Fe 3d t 2g Fe 3d e g Crystal field splitting Fe 3d in octahedron: t 2g and e g states without rotation 1 with rotation (E - E V ) (ev)

80 FeS 2 : Partial Densities of States 8 DOS (1/eV) Fe 3d t 2g Fe 3d e g Crystal field splitting Fe 3d in octahedron: t 2g and e g states without rotation 1 with rotation (E - E V ) (ev)

81 FeS 2 : Partial Densities of States 2 DOS (1/eV) S 3p x, 3p y S 3p z (E - E V ) (ev) Crystal field splitting S 3p: p x /p y vs. p z states without rotation with rotation: z axis 111

82 FeS 2 : Partial Densities of States 2 DOS (1/eV) S 3p x, 3p y S 3p z (E - E V ) (ev) Crystal field splitting S 3p: p x /p y vs. p z states without rotation with rotation: z axis 111

83 FeS 2 : Density and Laplacian Valence Electron Density ρ v / a.u

84 FeS 2 : Density and Laplacian Valence Electron Density Laplacian of Electron Density ρ v / a.u. - 2 ρ / a.u New! topological analysis (Bader analysis)

85 FeS 2 : Equilibrium Volume and Bulk Modulus V 0 = a B 3 E 0 = Ryd B 0 = GPa E (Ryd) V (a B )

86 FeS 2 : Equilibrium Volume and Bulk Modulus Lattice Constant NCPP Zeng and Holzwarth FPLO Opahle et al CRYSTAL98 Muscat et al CASTEP Muscat et al FPASW present work exp. Finklea et al exp. Will et al exp. Stevens et al. 91

87 FeS 2 : Equilibrium Volume and Bulk Modulus Bulk Modulus 187 LMTO Nguyen-Manh et al FPLO Opahle et al CRYSTAL98 Muscat et al CASTEP Muscat et al FPASW present work 148 exp. Drickamer et al exp. Will et al exp. Chattopadhyay and von Schnering exp. Fujii et al exp. Jephcoat and Olson exp. Ahrens and Jeanloz exp. Blachnik et al. 98

88 FeS 2 : From Atoms to the Solid V 0 = a B 3 E 0 = Ryd B 0 = GPa E (Ryd) V (a B )

89 FeS 2 : From Atoms to the Solid E (Ryd) V (a B 3 )

90 FeS 2 : Structure Optimization ASW Version 2.2 E (Ryd) x S

91 FeS 2 : Structure Optimization ASW Version 2.2 FPASW Version 2.3 E (Ryd) x S E (Ryd) x S

92 FeS 2 : Structure Optimization Sulfur Position NCPP Zeng and Holzwarth FPLO Opahle et al CRYSTAL98 Muscat et al CASTEP Muscat et al FPASW present work exp. Finklea et al exp. Will et al exp. Stevens et al. 91

93 Phase Stability in Silicon ASW Version E (Ryd) dia hex β-tin sc bcc fcc V (a B ) Bad β-tin structure most stable # nature (diamond structure)

94 Phase Stability in Silicon ASW Version 2.2 FPASW Version 2.4 E (Ryd) E (Ryd) dia hex β-tin sc bcc fcc dia hex β-tin sc bcc fcc V (a B ) V (a B ) New! diamond structure most stable pressure induced phase transition to β-tin structure

95 LTO(Γ)-Phonon in Silicon ASW Version Si fcc E (Ryd) x Si Bad no stable Si position # nature

96 LTO(Γ)-Phonon in Silicon ASW Version 2.2 FPASW Version Si fcc Si fcc E (Ryd) E (Ryd) x Si x Si New! phonon frequency: f calc = THz (f exp = THz)

97 Structure Optimization in PdCoO 2 Delafossite Structure

98 Structure Optimization in PdCoO 2 Delafossite Structure Total energy surface

99 Structure Optimization in PdCoO 2 Structural Data experiment theory a = 2.83 Å c = Å z O = a = Å c = Å z O = Total energy surface VE, R. Frésard, A. Maignan, Chem. Mat. (2008), in press

100 Dielectric Functions of Corundum Imaginary Part FLAPW, Hosseini et al., 2005 FPLMTO, Ahuja et al., 2004 FPASW Im ε xx Im ε zz 0.4 ε E (ev)

101 Dielectric Functions of Corundum Real Part FLAPW, Hosseini et al., 2005 FPLMTO, Ahuja et al., 2004 FPASW Re ε xx Re ε zz ε E (ev)

102 LDA+U-Calculations for Gadolinium DOS of ferromagnetic Gd Energetics DOS (1/eV) Gd total Gd 4f (LDA+U) Gd 4f (LDA) E AF E FE LDA LDA+U (in mryd/atom) (E - E F ) (ev)

103 Summary (Versions 2.3/2.4) ASA Geometry used for Basis Functions MT Geometry used for Density and Potential Accurate Total Energies O(ASA) Speed! Optical Properties implemented LDA+U-Method implemented What s Next? Forces? Automated Structure Optimization? Exact Exchange (EXX)? at O(ASA) speed?

104 Summary (Versions 2.3/2.4) ASA Geometry used for Basis Functions MT Geometry used for Density and Potential Accurate Total Energies O(ASA) Speed! Optical Properties implemented LDA+U-Method implemented What s Next? Forces? Automated Structure Optimization? Exact Exchange (EXX)? at O(ASA) speed?

105 Acknowledgments Darmstadt SFB 252 Stuttgart O. K. Andersen Augsburg T. Kopp, J. Mannhart, K.-H. Höck, W. Scherer Darmstadt J. Sticht Darmstadt P. C. Schmidt, M. Stephan

106 Acknowledgments Darmstadt SFB 252 Stuttgart O. K. Andersen Augsburg T. Kopp, J. Mannhart, K.-H. Höck, W. Scherer Darmstadt J. Sticht Darmstadt P. C. Schmidt, M. Stephan Cocoyoc Thank You for Your Attention!