Relativistic effects inRelativistic effects indensity functional theorydensity functional theory
Pina Romaniello and Paul L. de Pina Romaniello and Paul L. de BoeijBoeijTheoretical chemistry groupTheoretical chemistry group
Overview
Relativistic effects How to describe relativity within density
functional theory Summary
Part II Relativity in solids within TDCDFT Results
Part I
Relativistic effects
• The law of physics must be the same in all inertialsystems
• Absolute time
Galilean Transformation
Relativistic effects
• The law of physics must be the same in all inertialsystems
• Absolute time
• The law of physics are the same in all inertialsystems
• The velocity of light in free space c is an universalconstant and thus the same in all inertial systems
Galilean Transformation Lorentz Transformation
Relativistic effects
• The law of physics must be the same in all inertialsystems
• Absolute time
• The law of physics are the same in all inertialsystems
• The velocity of light in free space c is an universalconstant and thus the same in all inertial systems
Galilean Transformation Lorentz Transformation
• four-dimentional space
Relativistic effects
For an electron in a Coulomb-like potential !(r)
quantum mechanicsandRelativistic effects
For an electron in a Coulomb-like potential !(r)
Scrödinger equation
quantum mechanicsandRelativistic effects
For an electron in a Coulomb-like potential !(r)
Scrödinger equation Dirac equation
quantum mechanicsandRelativistic effects
For an electron in a Coulomb-like potential !(r)
mass-velocity Darwin spin-orbit coupling
Scrödinger equation Dirac equation
quantum mechanicsandRelativistic effects
mass-velocity:Variation of the mass with the velocity
Darwin:Correction to the non-local interactionbetween electron and Coulomb field
spin-orbit coupling:Interaction of the electron spin magneticmoment with magnetic field due to themotion of the electron in the electrostaticfield of the proton
Contraction and
stabilization of s- and p-
shells;
Expansion and
destabilization of the d-
and f-shells
Splitting of orbitals withangular momentum
quantum mechanicsandRelativistic effects
Some definitions
••• 4- component 4- component 4- component spinorsspinorsspinors• 4- component spinors
••• 4x4 4x4 4x4 Dirac Dirac Dirac matricesmatricesmatrices• 4x4 Dirac matrices
••• 2x2 Pauli matrices 2x2 Pauli matrices 2x2 Pauli matrices• 2x2 Pauli matrices
••• 4-component quantity 4-component quantity 4-component quantity AAAµµµ• 4-component vector Aµ
Density functional theory
Stationary system of N interacting electrons
Scrödingerequation
APPROXIMATE SOLUTIONS!
DFT
Density functional theory
" Hohenberg-Kohn theorems
(1984)
Density functional theory
" Hohenberg-Kohn theorems
(1984)
Density functional theory
" Hohenberg-Kohn theorems
(1984)
Universal functional…UNKNOWN
Density functional theory
"
Kohn-Sham noninteracting system
Hohenberg-Kohn theorems(1984)
Universal functional…UNKNOWN
Density functional theory
"
Kohn-Sham noninteracting system
Hohenberg-Kohn theorems(1984)
Universal functional…UNKNOWN
Exchange-correlation potentialUNKNOWN
Extension of DFT
Time dependent density functional theory Current density functional theory Time dependent current density functional
theory Spin density functional theory Spin current density functional theory….
Linear response in solids
RDFT (quantum electrodynamical approach)
• N interacting electron system in an external four potential
RDFT (quantum electrodynamical approach)
• N interacting electron system in an external four potential
"Relativistic
Hohenberg-Kohn theorem
RDFT (quantum electrodynamical approach)
• N interacting electron system in an external four potential
Dirac-Kohn-Shamequations
"Relativistic
Hohenberg-Kohn theorem
RDFT (quantum electrodynamical approach)
• N interacting electron system in an external four potential
"
• Effective potentials
Relativistic Hohenberg-Kohn theorem
Dirac-Kohn-Shamequations
RDFT (quantum electrodynamical approach)
• N interacting electron system in an external four potential
"
• Effective potentials
• Current (in the no-pair approximation)SCF
scheme
Relativistic Hohenberg-Kohn theorem
Dirac-Kohn-Shamequations
0th-component
The current
Gordon decomposition
0th-component
spatial-components
paramagnetic particlecurrent
diamagnetic particlecurrent
spin density
coupling 2-spinors
The current
Gordon decomposition
0th-component
spatial-components
The current
CDFT CSDFT
Gordon decomposition
0th-component
spatial-components
The current
CDFT
#
*DFT SDFT*H. MacDonald and S. H. Vosko,
J. Phys. C 12, 2977 (1979)
CSDFT
Dirac-Kohn-Shamequation
From the Dirac to the Zora equation
Dirac-Kohn-Shamequation
2-component zeroth order regular approximation equation
Unitarytransformation
Summary Part I
Relativistic effects are pure theoretical concept
Dirac equation combines quantum mechanicsand relativity
Relativistic density functional approach is basedon the four component current
Only the density and the total vector currenthave physical meaning
• Kohn-Sham effective one-electron scheme
Linear response in TDCDFT
Linear response in TDCDFT• Kohn-Sham effective one-electron scheme
microscopicCoulomb gauge
• Kohn-Sham effective one-electron scheme
microscopicCoulomb gauge
SCFscheme
Linear response in TDCDFT
• Kohn-Sham effective one-electron scheme
microscopicCoulomb gauge
Dielectric functions and EELS
Linear response in TDCDFT
• Kohn-Sham effective one-electron scheme
microscopicCoulomb gauge
Linear response in TDCDFT
q$0q≈0
For q≈0 only electrons closeto Fermi surface contribute
Inter-band contribution
Intra-band contribution
k k+q k
%1(k)
%2(k)
%1(k)
%F
k k+q k
Inter- and Intra-band contributions Transitions between occupied unoccupied bands
Transitions within the same partially-occupied band
nonmetallic systems
metallic systems
scalar effects spin-orbit coupling spin-magnetic field coupling
2-component zeroth order regular approximation equation
scalar effects spin-orbit coupling spin-magnetic field coupling
2-component zeroth order regular approximation equation
Scalar relativistic current operator (response)
1-component ZORA Kohn-Sham equation (ground state)
Fully relativistic current operator (response)
2-component ZORA Kohn-Sham equation (ground state)
?
… and some results
Computational information
ADF-BAND package
Slater-type TZ2P basis set
(Adiabatic) local density approximation
… and some results Band structure of W (BCC)
NR; R SR; R
… and some results Fermi cross-sections of W (BCC)
NR
SR
R
… and some results Interband Absorption of W (BCC)
40
30
20
10
076543210
! 2("
)
" (eV)
exp
… and some results
exp
NR
40
30
20
10
076543210
! 2("
)
" (eV)
Interband Absorption of W (BCC)
… and some results
exp
NR
SR
40
30
20
10
076543210
! 2("
)
" (eV)
Interband Absorption of W (BCC)
… and some results
exp
NR
SR
R
Interband Absorption of W (BCC)
40
30
20
10
076543210
! 2("
)
" (eV)
… and some results Dielectric functions of Au (FCC)
exp
-10
-5
0
5
10
76543210
! 1("
)! 2
(")
" (eV)
… and some results Dielectric functions of Au (FCC)
exp
-10
-5
0
5
10
76543210
! 1("
)! 2
(")
" (eV)
NR
~3.5 eV
… and some results Dielectric functions of Au (FCC)
exp
NR
-10
-5
0
5
10
76543210
! 1("
)! 2
(")
" (eV)
SR~1.9 eV
… and some results Dielectric functions of Au (FCC)
exp
NR
SR
-10
-5
0
5
10
76543210
! 1("
)! 2
(")
" (eV)
R
… and some results Dielectric functions of ZnTe (zincblende)
exp
20
10
0
-101086420
30
20
10
0
! 2 ("
)! 1
(")
" (eV)
… and some results Dielectric functions of ZnTe (zincblende)
exp
NR20
10
0
-101086420
30
20
10
0
! 2 ("
)! 1
(")
" (eV)
… and some results Dielectric functions of ZnTe (zincblende)
exp
NR
SR20
10
0
-101086420
30
20
10
0
! 2 ("
)! 1
(")
" (eV)
… and some results Dielectric functions of ZnTe (zincblende)
exp
NR
SR
R
20
10
0
-101086420
30
20
10
0
! 2 ("
)! 1
(")
" (eV)
… and ….
The end
