Relativistic effects in density functional theoryetsf.polytechnique.fr/system/files/users/promanie/Groningen06.pdf · Relativistic effects in density functional theory ... NR 20 10

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



• 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



• 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


For an electron in a Coulomb-like potential !(r)

quantum mechanicsandRelativistic effects


Scrödinger equation



Scrödinger equation Dirac equation



mass-velocity Darwin spin-orbit coupling

Scrödinger equation Dirac equation


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


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


" Hohenberg-Kohn theorems

(1984)



(1984)



(1984)

Universal functional…UNKNOWN


"

Kohn-Sham noninteracting system

Hohenberg-Kohn theorems(1984)



"

Kohn-Sham noninteracting system

Hohenberg-Kohn theorems(1984)


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



"Relativistic

Hohenberg-Kohn theorem



Dirac-Kohn-Shamequations

"Relativistic

Hohenberg-Kohn theorem



"

• Effective potentials

Relativistic Hohenberg-Kohn theorem




"

• Effective potentials

• Current (in the no-pair approximation)

SCFscheme

Relativistic Hohenberg-Kohn theorem


0th-component

The current

Gordon decomposition

0th-component

spatial-components

paramagnetic particlecurrent

diamagnetic particlecurrent

spin density

coupling 2-spinors

The current


0th-component

spatial-components

The current

CDFT CSDFT


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



SCFscheme




Dielectric functions and EELS





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


scalar effects spin-orbit coupling spin-magnetic field coupling


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


exp

NR

40

30

20

10

076543210

! 2(")

" (eV)

Interband Absorption of W (BCC)


exp

NR

SR

40

30

20

10

076543210

! 2(")

" (eV)



exp

NR

SR

R


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)


exp

-10

-5

0

5

10

76543210

! 1(")

! 2(")

" (eV)

NR

~3.5 eV


exp

NR

-10

-5

0

5

10

76543210

! 1(")

! 2(")

" (eV)

SR~1.9 eV


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)


exp

NR20

10

0

-101086420

30

20

10

0

! 2 (")

! 1 (")

" (eV)


exp

NR

SR20

10

0

-101086420

30

20

10

0

! 2 (")

! 1 (")

" (eV)


exp

NR

SR

R

20

10

0

-101086420

30

20

10

0

! 2 (")

! 1 (")

" (eV)

… and ….

The end

Relativistic effects in density functional theoryetsf.polytechnique.fr/system/files/users/promanie/Groningen06.pdf · Relativistic effects in density functional theory ... NR 20 10

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