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MotivationCalculated Materials Properties
Computational ApproachesSummary
Computational Materials ScienceIntroduction
Volker Eyert
Institut für Physik, Universität Augsburg
Electronic Structure in a Nutshell
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Organization of the Course
Mon Tue Wed Thu
General Comp. Mat. DFT DFT LDAScience Basics Basics Introduct.
ASW Introduction Standard Full. Pot. Miscell.Method History ASW ASW
Background
ASW Installation Distribution CTRL File Miscell.Program Data Files
Cu, Be FeS2 CrO2
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Outline
1 Motivation
2 Calculated Materials Properties
3 Computational Approaches
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Outline
1 Motivation
2 Calculated Materials Properties
3 Computational Approaches
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Outline
1 Motivation
2 Calculated Materials Properties
3 Computational Approaches
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Outline
1 Motivation
2 Calculated Materials Properties
3 Computational Approaches
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
What Are We Aiming At?
Interpretation ofExperiments,
Understanding, and
Prediction ofMaterials Properties
-10
-8
-6
-4
-2
0
2
4
6
KWXΓLW
(E -
EF)
(eV
)
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
What Are We Aiming At?
Interpretation ofExperiments,
Understanding, and
Prediction ofMaterials Properties
-10
-8
-6
-4
-2
0
2
4
6
KWXΓLW
(E -
EF)
(eV
)
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
What Are We Aiming At?
0 10 20 30 40 50 0 10
20 30
40 50
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11
ρv / a.u.
Interpretation ofExperiments,
Understanding, and
Prediction ofMaterials Properties
-578.060000
-578.055000
-578.050000
-578.045000
-578.040000
-578.035000
-578.030000
-578.025000
-578.020000
-578.015000
-578.010000
80 90 100 110 120 130 140 150 160 170
Si phase stability
fccbcc
scβ-tin
dia
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Outline
1 Motivation
2 Calculated Materials Properties
3 Computational Approaches
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Structural and Mechanical Properties
Structural
geometries of molecules
crystal structures
electron densities
defect structures
interface structures
surface structures
adsorption
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Structural and Mechanical Properties
Structural
geometries of molecules
crystal structures
electron densities
defect structures
interface structures
surface structures
adsorption
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Structural and Mechanical Properties
Structural
geometries of molecules
crystal structures
electron densities
defect structures
interface structures
surface structures
adsorption
0 10 20 30 40 50 0 10
20 30
40 50
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11
ρv / a.u.
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Structural and Mechanical Properties
Structural
geometries of molecules
crystal structures
electron densities
defect structures
interface structures
surface structures
adsorption
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Structural and Mechanical Properties
Structural
geometries of molecules
crystal structures
electron densities
defect structures
interface structures
surface structures
adsorption
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Structural and Mechanical Properties
Structural
geometries of molecules
crystal structures
electron densities
defect structures
interface structures
surface structures
adsorption
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Structural and Mechanical Properties
Structural
geometries of molecules
crystal structures
electron densities
defect structures
interface structures
surface structures
adsorption
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Structural and Mechanical Properties
-16532.950000
-16532.900000
-16532.850000
-16532.800000
-16532.750000
-16532.700000
-16532.650000
-16532.600000
850 900 950 1000 1050 1100 1150 1200
E (
Ryd
)
V (aB3)
V0 = 1056.15 aB3
E0 = -16532.904060 Ryd
B0 = 170.94 GPa
Mechanicalcompressibility
elastic moduli
thermal expansion
vibrational properties
hardness
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Structural and Mechanical Properties
Mechanical
compressibility
elastic moduli
thermal expansion
vibrational properties
hardness
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Structural and Mechanical Properties
Mechanical
compressibility
elastic moduli
thermal expansion
vibrational properties
hardness
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Structural and Mechanical Properties
0.000000
0.000500
0.001000
0.001500
0.002000
0.002500
0.003000
0.003500
0.004000
0.004500
0.005000
0.12 0.122 0.124 0.126 0.128 0.13
Si fcc, TO(Γ) phonon
Mechanicalcompressibility
elastic moduli
thermal expansion
vibrational properties
hardness
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Structural and Mechanical Properties
Mechanical
compressibility
elastic moduli
thermal expansion
vibrational properties
hardness
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Electronic, Optical and Magnetic Properties
Electronic
band structuremetalsemiconductorinsulator
band gap
band offsets
density distributions
polarizabilities
ionization energies
electron affinities
electrostatic potential
work functions
-12
-10
-8
-6
-4
-2
0
2
4
6
KWXΓLW(E
- E
V)
(eV
)
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Electronic, Optical and Magnetic Properties
Electronicband structure
band gap
band offsets
density distributions
polarizabilities
ionization energies
electron affinities
electrostatic potential
work functions
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Electronic, Optical and Magnetic Properties
Electronic
band structure
band gap
band offsetsdensity distributions
electrical momentselectric field gradients
polarizabilities
ionization energies
electron affinities
electrostatic potential
work functions
0 10 20 30 40 50 0 10
20 30
40 50
0.02 0.025 0.03
0.035 0.04
0.045 0.05
0.055
ρv / a.u.
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Electronic, Optical and Magnetic Properties
Electronicband structure
band gap
band offsets
density distributions
polarizabilities
ionization energies
electron affinities
electrostatic potential
work functions
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Electronic, Optical and Magnetic Properties
Electronicband structure
band gap
band offsets
density distributions
polarizabilities
ionization energies
electron affinities
electrostatic potential
work functions
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Electronic, Optical and Magnetic Properties
Electronicband structure
band gap
band offsets
density distributions
polarizabilities
ionization energies
electron affinities
electrostatic potential
work functions
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Electronic, Optical and Magnetic Properties
Electronicband structure
band gap
band offsets
density distributions
polarizabilities
ionization energies
electron affinities
electrostatic potential
work functions
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Electronic, Optical and Magnetic Properties
Electronicband structure
band gap
band offsets
density distributions
polarizabilities
ionization energies
electron affinities
electrostatic potential
work functions
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Electronic, Optical and Magnetic Properties
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10 12 14
ε
E (eV)
Im εxxRe εxx
Optical
optical spectra
magneto-optical properties
dielectric response
luminescence
fluorescence
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Electronic, Optical and Magnetic Properties
Optical
optical spectra
magneto-optical properties
dielectric response
luminescence
fluorescence
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Electronic, Optical and Magnetic Properties
Optical
optical spectra
magneto-optical properties
dielectric response
luminescence
fluorescence
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Electronic, Optical and Magnetic Properties
Optical
optical spectra
magneto-optical properties
dielectric response
luminescence
fluorescence
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Electronic, Optical and Magnetic Properties
Optical
optical spectra
magneto-optical properties
dielectric response
luminescence
fluorescence
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Electronic, Optical and Magnetic Properties
0 10 20 30 40 50 0 10
20 30
40 50
-0.1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ρv / a.u.
Magnetic
spin-density distribution
magnetic moments
NMR chemical shifts
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Electronic, Optical and Magnetic Properties
Magnetic
spin-density distribution
magnetic moments
NMR chemical shifts
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Electronic, Optical and Magnetic Properties
Magnetic
spin-density distribution
magnetic moments
NMR chemical shifts
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Transport, Chemical, and Thermodynamic Properties
Transport
electrical conductivity
thermal conductivity
diffusion constants
permeability
Chemicalcatalytic properties
corrosion
surface reactivity
photochemical properties
Thermodynamic
binding energies
phase transitions
phase diagrams
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Transport, Chemical, and Thermodynamic Properties
Transport
electrical conductivity
thermal conductivity
diffusion constants
permeability
Chemicalcatalytic properties
corrosion
surface reactivity
photochemical properties
Thermodynamic
binding energies
phase transitions
phase diagrams
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Transport, Chemical, and Thermodynamic Properties
Transport
electrical conductivity
thermal conductivity
diffusion constants
permeability
Chemicalcatalytic properties
corrosion
surface reactivity
photochemical properties
Thermodynamic
binding energies
phase transitions
phase diagrams
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Transport, Chemical, and Thermodynamic Properties
Transport
electrical conductivity
thermal conductivity
diffusion constants
permeability
Chemicalcatalytic properties
corrosion
surface reactivity
photochemical properties
Thermodynamic
binding energies
phase transitions
phase diagrams
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Transport, Chemical, and Thermodynamic Properties
Transport
electrical conductivity
thermal conductivity
diffusion constants
permeability
Chemicalcatalytic properties
corrosion
surface reactivity
photochemical properties
Thermodynamic
binding energies
phase transitions
phase diagrams
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Transport, Chemical, and Thermodynamic Properties
Transport
electrical conductivity
thermal conductivity
diffusion constants
permeability
Chemicalcatalytic properties
corrosion
surface reactivity
photochemical properties
Thermodynamic
binding energies
phase transitions
phase diagrams
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Transport, Chemical, and Thermodynamic Properties
Transport
electrical conductivity
thermal conductivity
diffusion constants
permeability
Chemicalcatalytic properties
corrosion
surface reactivity
photochemical properties
Thermodynamic
binding energies
phase transitions
phase diagrams
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Transport, Chemical, and Thermodynamic Properties
Transport
electrical conductivity
thermal conductivity
diffusion constants
permeability
Chemicalcatalytic properties
corrosion
surface reactivity
photochemical properties
Thermodynamic
binding energies
phase transitions
phase diagrams
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Transport, Chemical, and Thermodynamic Properties
Transport
electrical conductivity
thermal conductivity
diffusion constants
permeability
Chemicalcatalytic properties
corrosion
surface reactivity
photochemical properties
Thermodynamic
binding energies
phase transitions
phase diagrams
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Transport, Chemical, and Thermodynamic Properties
Transport
electrical conductivity
thermal conductivity
diffusion constants
permeability
Chemicalcatalytic properties
corrosion
surface reactivity
photochemical properties
Thermodynamic
binding energies
phase transitions
phase diagrams
-578.060000
-578.055000
-578.050000
-578.045000
-578.040000
-578.035000
-578.030000
-578.025000
-578.020000
-578.015000
-578.010000
80 90 100 110 120 130 140 150 160 170
Si phase stability
fccbcc
scβ-tin
dia
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Transport, Chemical, and Thermodynamic Properties
Transport
electrical conductivity
thermal conductivity
diffusion constants
permeability
Chemicalcatalytic properties
corrosion
surface reactivity
photochemical properties
Thermodynamic
binding energies
phase transitions
phase diagrams
-578.060000
-578.055000
-578.050000
-578.045000
-578.040000
-578.035000
-578.030000
-578.025000
-578.020000
-578.015000
-578.010000
80 90 100 110 120 130 140 150 160 170
Si phase stability
fccbcc
scβ-tin
dia
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Calculated Electronic Properties of MetalsCohesive Energies
Moruzzi, Janak, Williams, 1978
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga
E/R
yd
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In
E/R
yd
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Calculated Electronic Properties of MetalsWigner-Seitz Radii
Moruzzi, Janak, Williams, 1978
2
2.5
3
3.5
4
4.5
5
5.5
K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga
WSR
/au
2
2.5
3
3.5
4
4.5
5
5.5
Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In
WSR
/au
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Calculated Electronic Properties of MetalsBulk Moduli
Moruzzi, Janak, Williams, 1978
0
500
1000
1500
2000
2500
3000
3500
4000
K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga
Bul
k m
odul
us/k
bar
0
500
1000
1500
2000
2500
3000
3500
4000
Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In
Bul
k m
odul
us/k
bar
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Calculated Electronic PropertiesARPES Data and Calculated Band Structure of Td-WTe2
Augustin et al., 2000
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Calculated Optical PropertiesDiagonal-Conductivity and Normal Incidence Reflectivity of CrO2
Brändle et al., 1993
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Calculated Electronic PropertiesExperimental and Calculated Fermi Surface of Cu and TiTe2
Straub et al., 1997
continuous penetration of a lifetime broadened peak intoa typically Gaussian energy window, given by the instru-mental response function.
This is illustrated on Cu~001! as an example. Figure 3shows the corresponding intensity map, which agrees withthat of other workers.4 The observed pattern approximatelyreflects a cut through the FS of the sp band, which is occu-pied in the center of the BZ. However, a closer inspection
shows that the position of the intensity maximum along theG-M direction of the surface BZ does not coincide with anindependent determination of the Fermi vector from the mea-sured dispersion of the sp band ~Fig. 4!, as obtained fromenergy distribution curves on the same surface. Rather, itappears slightly shifted towards the BZ center. Also shownin Fig. 3 is the gradient u¹kw(k)u of the intensity distribu-tion. In line with our above discussion, we now find two setsof maxima, the outer and more intense of which is inter-
FIG. 3. ~Color! Top: Fermi-level intensity map of Cu~001!. The surface Brillouin zone is indicated. Bottom: Modulus of the gradient(u¹kw(k)u) of the intensity map, together with the theoretical FS contour ~black line!.
13 476 55TH. STRAUB et al.
nent features are the ‘‘cigar-shaped’’ intensity maxima cen-tered at the M points, which originate from the Ti 3d band.The observed intensity pattern reflects the threefold symme-try of the crystal structure that requires a distinction betweenM and M 8 points.7 Though the electronic band structure it-self does not deviate much from sixfold symmetry in thehexagonal 1T structure,7 the optical transition matrix ele-ments are strongly affected by photoelectron diffraction offthe three Te atoms sitting atop each Ti site.13 Near the centerof the Brillouin zone ~BZ! three Te 5p bands cross the Fermilevel very close to each other, giving rise to a broad intensity
spot at the G point. Note that we can also access parts of thesecond BZ ~in the periodic zone scheme!. The identificationof the intensity maxima becomes possible by comparison toconventional band mapping results7 and to a correspondingFS cut derived from our ASW calculations @Fig. 1~b!#. Ap-parently, the overall shape of the measured intensity map isalready in good agreement with the calculated FS topology.However, the contours of the Ti 3d-derived electron pocketsat the M point~s! are clearly not resolved. This is a conse-quence of the very narrow ~occupied! width of the Ti 3dband. Its highest binding energy accessible by He-I radiation
FIG. 1. ~Color! ~a! Photoemission intensity map w(k) for excitations from the Fermi level of TiTe2 (hn521.2 eV!. The Brillouin zoneis indicated. ~b! Corresponding Fermi-surface contours obtained from an ASW band calculation. ~c! Modulus of the two-dimensionalgradient u¹kw(k)u of the map in ~a!. ~d! Modulus of the logarithmic gradient u¹klnw(k)u
13 474 55TH. STRAUB et al.
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Calculated Electronic PropertiesCalculated Electron Density of Si
0 10 20 30 40 50 0 10
20 30
40 50
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11
ρv / a.u.
0 10 20 30 40 50 0 10
20 30
40 50
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11
ρv / a.u.
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Calculated Electronic PropertiesCalculated Electron Density of Si
0 10 20 30 40 50 0 10
20 30
40 50
-0.1
-0.05
0
0.05
0.1
-∇ 2ρ / a.u.
0 10 20 30 40 50 0 10
20 30
40 50
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11
ρv / a.u.
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Calculated Optical PropertiesDielectric Function of Al2O3
Hosseini et al., 2005Ahuja et al., 2004
present work
0
0.1
0.2
0.3
0.4
0.5
0.6
0 5 10 15 20 25 30
ε
E (eV)
Im εxxIm εzz
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Calculated Optical PropertiesDielectric Function of Al2O3
Hosseini et al., 2005Ahuja et al., 2004
present work
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0 5 10 15 20 25 30
ε
E (eV)
Re εxxRe εzz
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Outline
1 Motivation
2 Calculated Materials Properties
3 Computational Approaches
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Theoretical Foundation
Newton’s 2nd Law
F = −∇E = ma
Schrödinger’s Equation
Hψ = Eψ
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Foundation of Density Functional Theory 1964/1965
Pierre C. Hohenberg Walter Kohn
Lu Jeu Sham
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Density Functional Theory I
Guidelines =⇒ to remember
theory for the ground stateno excitations (!)
electron density as the central variableno need for the many-body wave functionmany-body wave function and potential from the density
variational principle with respect to the electron densityminimization of total energy
many-body problem ⇐⇒ single-particle problemexact mapping
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Density Functional Theory I
Guidelines =⇒ to remember
theory for the ground stateno excitations (!)
electron density as the central variableno need for the many-body wave functionmany-body wave function and potential from the density
variational principle with respect to the electron densityminimization of total energy
many-body problem ⇐⇒ single-particle problemexact mapping
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Density Functional Theory I
Guidelines =⇒ to remember
theory for the ground stateno excitations (!)
electron density as the central variableno need for the many-body wave functionmany-body wave function and potential from the density
variational principle with respect to the electron densityminimization of total energy
many-body problem ⇐⇒ single-particle problemexact mapping
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Density Functional Theory I
Guidelines =⇒ to remember
theory for the ground stateno excitations (!)
electron density as the central variableno need for the many-body wave functionmany-body wave function and potential from the density
variational principle with respect to the electron densityminimization of total energy
many-body problem ⇐⇒ single-particle problemexact mapping
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Density Functional Theory IIMany-Body ⇐⇒ Single-Particle
Exact Mapping
many-body problemreal, unsolvable (!!!)density ρmb
single-particle problemfictitious, solvable (???)
density ρsp!= ρmb
effective single-particle potential
veff ,σ(r) = vext (r) + vH(r) + vxc,σ(r)
single-particle equations: Kohn-Sham equations[
−∇2 + veff ,σ(r) − εσ
]
ψσ(r) = 0
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Density Functional Theory IIMany-Body ⇐⇒ Single-Particle
Exact Mapping
many-body problemreal, unsolvable (!!!)density ρmb
single-particle problemfictitious, solvable (???)
density ρsp!= ρmb
effective single-particle potential
veff ,σ(r) = vext (r) + vH(r) + vxc,σ(r)
single-particle equations: Kohn-Sham equations[
−∇2 + veff ,σ(r) − εσ
]
ψσ(r) = 0
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Density Functional Theory IIMany-Body ⇐⇒ Single-Particle
Exact Mapping
many-body problemreal, unsolvable (!!!)density ρmb
single-particle problemfictitious, solvable (???)
density ρsp!= ρmb
effective single-particle potential
veff ,σ(r) = vext (r) + vH(r) + vxc,σ(r)
single-particle equations: Kohn-Sham equations[
−∇2 + veff ,σ(r) − εσ
]
ψσ(r) = 0
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations I
Kohn-Sham Equation
[
−∇2 + vext(r) + vH(r) + vxc,σ(r) − εσ(k)]
ψσ(k, r) = 0
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations I
Kohn-Sham Equation
[
−∇2 + vext(r) + vH(r) + vxc,σ(r) − εσ(k)]
ψσ(k, r) = 0
Kinetic Energy
non-relativistic
scalar-relativistic
fully relativistic
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations I
Kohn-Sham Equation
[
−∇2 + vext(r) + vH(r) + vxc,σ(r) − εσ(k)]
ψσ(k, r) = 0
External Potential (Ions)
non-periodic
periodic
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations I
Kohn-Sham Equation
[
−∇2 + vext(r) + vH(r) + vxc,σ(r) − εσ(k)]
ψσ(k, r) = 0
Effective Potential
all-electron muffin-tin
all-electron full potential
all-electron PAW
pseudopotential
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations I
Kohn-Sham Equation
[
−∇2 + vext(r) + vH(r) + vxc,σ(r) − εσ(k)]
ψσ(k, r) = 0
Exchange-Correlation Potential
non-spin-polarized
spin-polarized
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations I
Kohn-Sham Equation
[
−∇2 + vext(r) + vH(r) + vxc,σ(r) − εσ(k)]
ψσ(k, r) = 0
Exchange-Correlation Potential
local-density approximation (LDA)
generalized gradient approximation (GGA)
beyond LDA/GGA
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations I
Kohn-Sham Equation
[
−∇2 + vext(r) + vH(r) + vxc,σ(r) − εσ(k)]
ψσ(k, r) = 0
Exchange-Correlation Potential
local-density approximation (LDA)
generalized gradient approximation (GGA)beyond LDA/GGA
self-interaction correction (SIC)LDA+U, LDA+DMFT
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations I
Kohn-Sham Equation
[
−∇2 + vext(r) + vH(r) + vxc,σ(r) − εσ(k)]
ψσ(k, r) = 0
Exchange-Correlation Potential
local-density approximation (LDA)
generalized gradient approximation (GGA)beyond LDA/GGA
optimized effective potential (OEP)exact exchange (EXX)screened exchange (sX)
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations I
Kohn-Sham Equation
[
−∇2 + vext(r) + vH(r) + vxc,σ(r) − εσ(k)]
ψσ(k, r) = 0
Exchange-Correlation Potential
local-density approximation (LDA)
generalized gradient approximation (GGA)beyond LDA/GGA, beyond DFT
model GWGWtime-dependent DFT (TDDFT)
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations I
Kohn-Sham Equation
[
−∇2 + vext(r) + vH(r) + vxc,σ(r) − εσ(k)]
ψσ(k, r) = 0
Wave Function, Expanded in
atomic orbitals
plane waves
augmented plane waves
augmented spherical waves
fully numerical functions
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations I
Kohn-Sham Equation
[
−∇2 + vext(r) + vH(r) + vxc,σ(r) − εσ(k)]
ψσ(k, r) = 0
Wave Function, Expanded in
atomic orbitalsGaussians (GTO)Slater-type (STO)numerical (DMol, Siesta)
plane waves
augmented plane waves
augmented spherical waves
fully numerical functionsVolker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations I
Kohn-Sham Equation
[
−∇2 + vext(r) + vH(r) + vxc,σ(r) − εσ(k)]
ψσ(k, r) = 0
Wave Function, Expanded in
atomic orbitals
plane waves
augmented plane waves
augmented spherical waves
fully numerical functions
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations I
Kohn-Sham Equation
[
−∇2 + vext(r) + vH(r) + vxc,σ(r) − εσ(k)]
ψσ(k, r) = 0
Wave Function, Expanded in
atomic orbitals
plane wavesaugmented plane waves
augmented plane waves (APW)linear augmented plane waves (LAPW)full-potential linear augmented plane waves (FLAPW)
augmented spherical waves
fully numerical functionsVolker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations I
Kohn-Sham Equation
[
−∇2 + vext(r) + vH(r) + vxc,σ(r) − εσ(k)]
ψσ(k, r) = 0
Wave Function, Expanded in
atomic orbitals
plane waves
augmented plane wavesaugmented spherical waves
Korringa Kohn Rostoker (KKR)linear muffin-tin orbital (LMTO)augmented spherical wave (ASW)
fully numerical functionsVolker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations I
Kohn-Sham Equation
[
−∇2 + vext(r) + vH(r) + vxc,σ(r) − εσ(k)]
ψσ(k, r) = 0
Wave Function, Expanded in
atomic orbitals
plane waves
augmented plane waves
augmented spherical waves
fully numerical functions
courtesy of E. WimmerVolker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IIPotentials and Partial Waves
Full Potential
0 10 20 30 40 50 0 10
20 30
40 50
-4-3.5
-3-2.5
-2-1.5
-1-0.5
v / a.u.
Full Potentialspherical symmetric nearnuclei
flat outside the atomiccores
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IIPotentials and Partial Waves
John C. Slater Full Potential
spherical symmetric nearnuclei
flat outside the atomiccores
Muffin-Tin Approximation
approximate the potential
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IIPotentials and Partial Waves
John C. Slater Muffin-Tin Approximation
distinguish:atomic regions
muffin-tin spheresveff ,σ(r) = veff ,σ(|r|)
remainderinterstitial regionveff ,σ(r) = 0
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IIPotentials and Partial Waves
Muffin-Tin Potential Muffin-Tin Approximation
distinguish:atomic regions
muffin-tin spheresveff ,σ(r) = veff ,σ(|r|)
remainderinterstitial regionveff ,σ(r) = 0
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IIPotentials and Partial Waves
Muffin-Tin Potential Partial Waves
muffin-tin spheresveff ,σ(r) = veff ,σ(|r|)solve radial Schrödingerequation numerically
interstitial regionveff ,σ(r) = 0exact solutions
plane wavesspherical waves
match at sphere surface(„augment“)
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IIIPlane Waves ⇐⇒ Spherical Waves
Augmented Plane Waves
require periodic cell
good for solids and surfaces
good for metals, semiconductors, and insulators
not recommended for rare earth and actinides
inefficient for open structures
systematic but slow convergence of basis set
matrix elements easy to calculate
easy to implement
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IIIPlane Waves ⇐⇒ Spherical Waves
Augmented Plane Waves
require periodic cell
good for solids and surfaces
good for metals, semiconductors, and insulators
not recommended for rare earth and actinides
inefficient for open structures
systematic but slow convergence of basis set
matrix elements easy to calculate
easy to implement
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IIIPlane Waves ⇐⇒ Spherical Waves
Augmented Plane Waves
require periodic cell
good for solids and surfaces
good for metals, semiconductors, and insulators
not recommended for rare earth and actinides
inefficient for open structures
systematic but slow convergence of basis set
matrix elements easy to calculate
easy to implement
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IIIPlane Waves ⇐⇒ Spherical Waves
Augmented Plane Waves
require periodic cell
good for solids and surfaces
good for metals, semiconductors, and insulators
not recommended for rare earth and actinides
inefficient for open structures
systematic but slow convergence of basis set
matrix elements easy to calculate
easy to implement
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IIIPlane Waves ⇐⇒ Spherical Waves
Augmented Plane Waves
require periodic cell
good for solids and surfaces
good for metals, semiconductors, and insulators
not recommended for rare earth and actinides
inefficient for open structures
systematic but slow convergence of basis set
matrix elements easy to calculate
easy to implement
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IIIPlane Waves ⇐⇒ Spherical Waves
Augmented Plane Waves
require periodic cell
good for solids and surfaces
good for metals, semiconductors, and insulators
not recommended for rare earth and actinides
inefficient for open structures
systematic but slow convergence of basis set
matrix elements easy to calculate
easy to implement
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IIIPlane Waves ⇐⇒ Spherical Waves
Augmented Plane Waves
require periodic cell
good for solids and surfaces
good for metals, semiconductors, and insulators
not recommended for rare earth and actinides
inefficient for open structures
systematic but slow convergence of basis set
matrix elements easy to calculate
easy to implement
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IIIPlane Waves ⇐⇒ Spherical Waves
Augmented Plane Waves
require periodic cell
good for solids and surfaces
good for metals, semiconductors, and insulators
not recommended for rare earth and actinides
inefficient for open structures
systematic but slow convergence of basis set
matrix elements easy to calculate
easy to implement
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IVPlane Waves ⇐⇒ Spherical Waves
Augmented Spherical Waves
do not require periodic cell
good for solids and surfaces
good for metals, semiconductors, and insulators
good for all atoms including transition metals, rare earth,and actinides
still efficient for open structures
minimal basis set
computationally efficient
difficult to implement
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IVPlane Waves ⇐⇒ Spherical Waves
Augmented Spherical Waves
do not require periodic cell
good for solids and surfaces
good for metals, semiconductors, and insulators
good for all atoms including transition metals, rare earth,and actinides
still efficient for open structures
minimal basis set
computationally efficient
difficult to implement
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IVPlane Waves ⇐⇒ Spherical Waves
Augmented Spherical Waves
do not require periodic cell
good for solids and surfaces
good for metals, semiconductors, and insulators
good for all atoms including transition metals, rare earth,and actinides
still efficient for open structures
minimal basis set
computationally efficient
difficult to implement
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IVPlane Waves ⇐⇒ Spherical Waves
Augmented Spherical Waves
do not require periodic cell
good for solids and surfaces
good for metals, semiconductors, and insulators
good for all atoms including transition metals, rare earth,and actinides
still efficient for open structures
minimal basis set
computationally efficient
difficult to implement
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IVPlane Waves ⇐⇒ Spherical Waves
Augmented Spherical Waves
do not require periodic cell
good for solids and surfaces
good for metals, semiconductors, and insulators
good for all atoms including transition metals, rare earth,and actinides
still efficient for open structures
minimal basis set
computationally efficient
difficult to implement
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IVPlane Waves ⇐⇒ Spherical Waves
Augmented Spherical Waves
do not require periodic cell
good for solids and surfaces
good for metals, semiconductors, and insulators
good for all atoms including transition metals, rare earth,and actinides
still efficient for open structures
minimal basis set
computationally efficient
difficult to implement
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IVPlane Waves ⇐⇒ Spherical Waves
Augmented Spherical Waves
do not require periodic cell
good for solids and surfaces
good for metals, semiconductors, and insulators
good for all atoms including transition metals, rare earth,and actinides
still efficient for open structures
minimal basis set
computationally efficient
difficult to implement
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
DFT Implementations IVPlane Waves ⇐⇒ Spherical Waves
Augmented Spherical Waves
do not require periodic cell
good for solids and surfaces
good for metals, semiconductors, and insulators
good for all atoms including transition metals, rare earth,and actinides
still efficient for open structures
minimal basis set
computationally efficient
difficult to implement
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Summary
Density Functional Theory . . .
allows for an accurate determination of a still growingnumber of materials properties
is a ground state theory
is in principle exact
has given rise to a large variety of implementations
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Summary
Density Functional Theory . . .
allows for an accurate determination of a still growingnumber of materials properties
is a ground state theory
is in principle exact
has given rise to a large variety of implementations
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Summary
Density Functional Theory . . .
allows for an accurate determination of a still growingnumber of materials properties
is a ground state theory
is in principle exact
has given rise to a large variety of implementations
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Summary
Density Functional Theory . . .
allows for an accurate determination of a still growingnumber of materials properties
is a ground state theory
is in principle exact
has given rise to a large variety of implementations
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Further Reading I
P. Hohenberg and W. KohnInhomogeneous Electron GasPhys. Rev. 136, B864 (1964)
W. Kohn and L. J. ShamQuantum Density Oscillations in an InhomogeneousElectron GasPhys. Rev. 140, A1133 (1965)
W. KohnAn Essay on Condensed Matter Physics in the TwentiethCenturyRev. Mod. Phys. 71, S59 (1999)
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Further Reading II
R. M. Dreizler and E. K. U. GrossDensity Functional Theory (Springer, Berlin, 1990)
R. G. Parr and W. YangDensity-Functional Theory of Atoms and Molecules (OxfordUniversity Press, Oxford, 1989)
H. EschrigThe Fundamentals of Density-Functional Theory (Editionam Gutenbergplatz, Leipzig 2003)
Volker Eyert Computational Materials Science
MotivationCalculated Materials Properties
Computational ApproachesSummary
Further Reading III
J. Kübler and V. EyertElectronic structure calculationsin: Electronic and Magnetic Properties of Metals andCeramicsed by K. H. J. Buschow (VCH, Weinheim, 1992), pp. 1-145Volume 3A of Materials Science and Technology
E. WimmerPrediction of Materials Propertiesin: Encyclopedia of Computational Chemistryed by P. von Ragué-Schleyer (Wiley, New York, 1998)
Volker Eyert Computational Materials Science