HE ! = ! Modeling of Nanostructures and Materials

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Modeling of Nanostructures and Materials

Summer Semester 2013 Lecture

Jacek A. Majewski

Chair of Condensed Matter Physics Institute of Theoretical Physics

Faculty of Physics, Universityof Warsaw

E-mail: Jacek.Majewski@fuw.edu.pl

Jacek A. Majewski

Modeling of Nanostructures and Materials

Lecture 2

–

February 28, 2013

Density Functional Theory (DFT) – the key to the

Computational Materials Science T

Th hee B Ba assiiccss

Kohn-Sham realization of the DFT

Fundamental problem in materials science

A fundamental problem in materials science is

the prediction of condensed matter’s electronic structure

DNA - molecule

Crystal - diamond

C

₆₀

- molecule H ! = ! E

H = !

!

²

"

_!²

2M

_!

#

!

^! ^! _2m

²

^"

ⁱ²

#

i

^{+ 1} ₂ _|

_R^Z

^!

^!^Z^"^e²

!

! !

R_!

|

! ,!

# ^! _|

_R

^!

^Z^"^e²

!

! !

r_i

|

#

i,!

^{+ 1} ₂ _{| !r}

^e² i

! !

r_j

|

#

i, j

Materials Science:

Examples of Schrödinger Equation?

Ab-initio (first principles) Method –

ONLY Atomic Numbers {

Z

_i} as input parameters Materials are composed of nuclei and electrons the interactions are known ^{Z^!^,M^!^,

R!_!}

{!r

_i

}

Kinetic energy of nuclei

Kinetic energy of electrons

Nucleus-Nucleus interaction

Electron-Nucleus interaction

Electron-Electron interaction

(2)

Quantum Mechanics of Molecules and Crystals

Molecule or Crystal = a system of nuclei (Ions) and electrons

el Nucl

ˆ ˆ

H T U( x, X ) T == ++ ++

en ee NN

ˆ ˆ ˆ

U( x, X ) V ( x, X ) V ( x ) V ( X )== ++ ++

ˆV_en( x, X ) == !!Z_ae²

|! r_i!!!

R_a|

"

ia

"

_ˆV

ee( x ) == e²

|! r_i!!!

r_j|

i<< j

"

^ˆV^NN^{( X ) ==}

e²

|! R_a!!!

R_b|

a<<b

"

ˆT_el== 1

i==12m

!

N

!

^!^pⁱ²^{== !!} ^!_2m²

i==1

!

N

!

^!^!^!ⁱ² ˆT_Nucl== 1

a==12m

N_nucl

!

^P^!ⁱ²^{== !!} _2M^!²

a==1 a N_nucl

!

^!^!^!^a²

X !! {! R₁,!

R₂,…,! R_N

nucl} x !! {!

r₁,! r₂,…,!

r_N} ( M , X ,P )

Nuclei – mass M, coordinates X, and momenta P, Electrons – (m,x,p)

Kinetic energy of electrons Kinetic energy of the nuclei

Potential energy = The total Coulomb energy of nuclei and electrons

Electron-nucleus

Electron-Electron Nucleus-Nucleus

The Adiabatic Approximation (Born-Oppenheimer)

It is natural to consider the full Hamiltonian of the system to be the sum of an ionic and an electronic part

N el

ˆ ˆ ˆ H H == ++ H

N Nucl NN

ˆ ˆ ˆ

H == T ++ V ( X ) H ˆ

_el

== T V ( x, X ) V ( x ) ˆ

_el

++ ˆ

_en

++ ˆ

_ee

M. Born & J. R. Oppenheimer, Ann. Phys. 84, 457 (1927)

The Adiabatic Approximation (Born-Oppenheimer)

The Schrödinger equation for the electrons in the presence of fixed ions

el n n n

ˆH ! ( X,x ) E ( X )! ( X,x ) ==

Parametric dependence on ionic positions

The energy levels of the system of ions are determined by solving

ˆ

N

[ H ++ E( K', X )] !(Q,K', X ) "(Q )!(Q,K', X ) ==

The electronic energy contributes to the potential energy of the ion system.

This implies that the potential energy depends on the state of the electrons.

Adiabatic approximation – interacting electrons move in the ‘external’

potential of nuclei (ions) at fixed positions

en e e

ˆ ˆ ˆ ˆ

H T V == ++ ++ V

_!!

N 2 2 i 1 i

ˆT == 2m

==

# #

!!^" ""^!

ˆV_en== !!Z!e²

|"

r_i!!! R_a|

!

ia

!

^{== ˆ}^V^ext⁼⁼ ^v^ext⁽^r^!ⁱ

!

i

!

⁾

ˆV_e!!e== e²

|! r_i!!!

r_j|

i<< j

"

Quantum Mechanics:

System of N electrons in an external potential

E_nn== Z_aZ_be²

|! R_a!!!

R_b|

a<<b

"

##

$$%%

&&

''((

{! R₁,!

R₂,…}

H! E! ==

^!^({

"

R_!},"

r₁,"

r₂,…,"

r_N) !!!("

r₁,"

r₂,…,"

r_N)

N 10 !!

23 Many particle wave function

0

min | | ˆ min | ˆ ˆ

_{e e}

ˆ

_ext

|

N N

E

=

"#

"

H

" =

"#

"

T V

+

!

+

V

"

!(!r

₁

, !r

₂

,…,!r

_N

)

Ritz Variational Principle !! Ground State Energy of the system Many-particle wavefunction Schrödinger equation

Full minimization of the functional with respect to all allowed N-electron wave functions E[! ]

! | H |!ˆ E[! ] ! |!

<< >>

== << >>

E[! ] E!! 0

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Quantum Mechanics:

System of N electrons in an external potential H! E! ==

Schrödinger equation Exact analytical solutions are not known

even for two electrons !

Approximations are needed !

Concept of independent particles moving in an effective potential

Interacting particles Independent particles

!(!x1, !x2,…, !x₁₀23) !(!x1)!(!x2)…!(!x₁₀23)

Idea: consider electrons as independent particles moving in an effective potential

Hartree and Hartree-Fock Approximation Ansatz for the wave-function

!

_Hartree

(!x

1

, !x

2

,…, !x

N

) = !

1

(!x

1

)!

2

(!x

2

)....!

N

(!x

N

)

Hartree Method

Hartree-Fock Method

!

_H!F

(!x

1

, !x

2

,…, !x

N

) = 1

N!

!

₁

(!x

1

) !

2

(!x

1

) … !

N

(!x

1

)

!

₁

(!x

₂

) !

₂

(!x

₂

) … !

_N

(!x

₂

)

! ! !

!

₁

(!x

N

) !

2

(!x

N

) … !

N

(!x

N

)

!

i

- one-electron wavefunction of the ith level

Hartree-Fock Approximation

H F H F

H F

H F H F

! | H | ! ˆ E[! ]

!

^!!

| !

^!!

!!

!! !!

<< >>

== << >>

Variational Principle

H0

!

_i

(!x

i

) + ^! !

_j=1^N

! !

^*_j

(!x

j

)U(!x

i

, !x

j

) ^!

^j

^(!x

^j

^)d!x

^j

"

# $

% &!

i

(!x

i

) ' ^! )

_j=1^N

( !

^*_j

(!x

j

)U(!x

i

, !x

j

) ^!

ⁱ

^(!x

^j

^)d!x

^j

"

# $

% &!

j

(!x

i

) = !

i

!

_i

(!x

i

)

H == H₀++ 1

2 U(!!x_i,!!x_j)

i , j

!

H₀== H₀( i ) ==

!

i

!

"" 1

2!!_i²

!

i

!

^++Vext(!!r_i) U(!x_i,!x_j) == 1

|!r_i!!!r_j|

!

H F_!!

Spectrum of Electronic Hamiltonian:

What ab initio methods do we have?

Methods for computing the electronic structure Empirical Methods

Ab-initio Methods

Hartree-Fock

Method

+ Configuration Interaction

§!H-F - neglects completely

electron correlation

§!H-F+CI – is able to treat

ONLY few electrons

Density Functional Theory

Quantum Monte Carlo

Ø! Existing realizations of DFT allow accurate predictions for large systems Ø! Currently the method of choice in materials science

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P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964)

Density Functional Theory (DFT)

The DFT is based on two fundamental theorems for a functional of the one particle density.

!!( !r ) == !! ( !r₁, !r₂,…,!r_N)| !! ( !ˆr_i!! !r )|!! ( !r₁, !r₂,…,!r_N)

!

i

!

== N d!r

!!

2^,…,d!r^N^!^!^*^{( !r, !r}²^,…,!r^N^)!^{! ( !r,!r}²^,…,!r^N⁾

One particle density – Basic quantity of DFT

One particle density determines the ground state energy of the system

Modern formulation – constrained-search method of Mel Levy

Mel Levy, Proc. Natl. Acad. Sci. USA, vol. 76, No. 12, p.606 (1979).

Density Functional Theory – constrained search formulation

Mel Levy, Proc. Natl. Acad. Sci. USA, vol. 76, No. 12, p.606 (1979).

Functional of the one particle density F[!!]! min_"_"

!!!!!

" "

!!

| ˆT + ˆV

e!e

|" "

!!

The functional searches all many particle functions that yield the input density and then delivers the minimum of F [ !] !

!!(!r) T Vˆ ˆ++ _{e e}_!!

d!rvext(!r)!!(!r)+

!

^F[^!!^{]! E}⁰

d!rvext(!r)!!₀(!r)+

!

^F[^!!⁰^{]= E}⁰

Theorem I

Theorem II _E^!⁰₀ - ground state density

- ground state energy

Let us define function that minimizes "_min^! " |T V |"! ˆ ˆ++ e e_!! !

! !

min ˆ ˆe e min

F [ !]== " |T V |"++ _!! F [ ! ]₀ == " |T V |"_min^!⁰ ˆ ˆ++ _{e e}!! _min^!⁰

Proof of Theorem I:

d!rvext(!r)!!_(!r)+

!

^F[^!!^{]= d!rv}

!

^ext^(!r)^!!^(!r)+^!^!^min^"" ^{| ˆT + ˆV}^e!e^|^!^!^min^"" ⁼

= !!_min^"" | ˆVext+ ˆT + ˆV_e!e|!!_min^"" !E₀

Ritz variational principle

Density Functional Theory – constrained search formulation

Proof of Theorem II: E₀"" " |V_min^!⁰ ˆ_ext++ ++T V |"ˆ ˆ_{e e}_!! _min^!⁰

0 0

! !

0 ˆext ˆ ˆe e 0 min ˆext ˆ ˆe e min

" |V ++ ++T V |"_!! "" " |V ++ ++T V |"_!!

d!r!!ext(!r)!!₀(!r)+

!

^!^!⁰^{| ˆT + ˆV}^e!e^|^!^!⁰ ^!

!

^d!r^!!^ext⁽^!r)^!!⁰⁽^!r)+^!^!^min^""⁰ ^{| ˆT + ˆV}^e!e^|^!^!^min^""⁰

0 0

! !

0 ˆ ˆe e 0 min ˆ ˆe e min

" |T V |"++ _!! "" " |T V |"++ _!!

From variational principle

!0

"min

But, on the other hand, from the definition of

0 0

! !

0 ˆ ˆe e 0 min ˆ ˆe e min

" |T V |"++ !! "" " |T V |"++ !!

0 0

! !

0 ˆ ˆe e 0 min ˆ ˆe e min

" |T V |"_!! " |T V |"_!!

"

" ++ == ++

d!r!!ext(!r)!!₀(!r)+

!

^!^!⁰^{| ˆT + ˆV}^e"e^|^!^!⁰ ⁼^F[^!!⁰^{]+ d!r}

!

^!!^ext^(!r)^!!⁰^(!r)

!₀| ˆV_ext++ ˆT ++ ˆV_e!!e| !₀ ==F [ "₀] ++ d! r!_ext(!

r )!₀(!

""

r ) E₀==F [ !₀] ++ d!

r!_ext(! r )!₀(!

!!

r )

0 0

! !

0 min ˆ ˆe e min

F [ ! ]== " |T V |"++ !!

(A) (B) [(A) & (B) true]

Density Functional Theory – Constrained Search Formulation

The ground-state energy minimization procedure of can be divided into two steps

Relation to Ritz Variational Principle

!₂

!₁

!₄

!₃

!₅ !₆

!₇

! | H |!ˆ E[! ]

! |!

<< >>

== << >>

! !

0 " N ˆ ˆe e ˆext ! N " ! ˆ ˆe e ˆext

E [" ] min " |T V== "" ++ !! ++V |" ==min min " |T V"" ''%%## "" ++ !! ++V |" $$&&((

Percus-Levy partition of the N-electron Hilbert space

The inner minimization is constrained to all wave functions that give , while the outer minimization releases this constrain by searching all !!!!_(!r)_(!r)

Each shaded area is the set of that integrate to a particular . !!(!r) !

The minimization is over all such points. !!!N

The minimization for a particular is constrained to the shaded area associated with this , and is realized by one point (denoted by ) in this shaded area.

!!!" !

!

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Density Functional Theory – Constrained Search Formulation Relation to Ritz Variational Principle

E [ ! ] == F [ ! ] ++ d! r!_ext(!

r )!(!

!!

r )

In O N E function of 3 variables !!!

In 2

^N

wave functions of 3N variables

E₀[ ! ] == min_{" !}_!N " | ˆT ++ ˆVe""e++ ˆV_ext|" ==

==min_!!_!N##min_{" !}_!! "^!| ˆT ++ ˆVe""e++ ˆV_ext|"^!

$$%% &&

''((==

==min_!!_!N min_{" !}_!! "^!| ˆT ++ ˆVe""e|"^! ++ d! r#_ext(!

r )!(!

))

r )

##$$%% &&

''((==

==min

!!!N[ F [ ! ] ++ d_r#!

ext(_{r )!(}! !_{r )}

))

^{] ==}

==min

!!!NE[ ! ]

Density Functional Theory

PROBLEM: exact functional is unknown ! F [ !]

Thomas-Fermi-Method (probably the oldest approximation to DFT)

F[ !] T [ !] U[ !] E [ !] == ++ ++

xc

One needs a good approximation to F [ !]

! !

! e e ! min e e min

" !

! !

min e e min

ˆ ˆ ˆ ˆ

F [ !] min " |T V |" " |T V |"

T [ !] U [ !] " |V |"ˆ U [ !]

!! !!

"

!!

== ++ == ++

== ++ ++ !!

Kinetic energy

U[!!]= 12 d!rd!r'!!_(!r)!!_(!r')

|!r!!r'|

""

Classical Coulomb energy ^xc

E [ !]

Exchange & Correlation

The functional is universal in the sense that it is independent of the external potential (field) . F [ !] !!ext(!r)

T F

Vee^!! [ !] U [ !]""

T^T!F[!!]! 35(3!!²)^2/3!²

2m

!

d!r[^!!^(!r)]^5/3 and extensions

§! Thomas-Fermi-Dirac

§! Thomas-Fermi-Weizsacker

PROBLEM:

Very often these models give even qualitatively wrong results.

TT F^!! [ !]

Interacting particles Independent particles

!(!x1, !x2,…, !x₁₀23) !¹(_x!

1)!₂(_x!

2)…!₃(_x!

10²³)

Density Functional Theory (DFT) in Kohn-Sham realization

This reduction is rigorously possible !

DFT- The Kohn- Sham Method

W. Kohn & L. Sham (1965) invented an ingenious indirect approach to the kinetic- energy functional.

They turned density functional theory into a practical tool for rigorous calculations

W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965)

System of interacting

electrons with density

!( !r)

System of non-interacting electrons with the same density

!( !r)

The main idea:

“Real” system “Fictitious” or Kohn-Sham reference system

!!(!r)

T [ !] T [ !]

_S !!S(!r)=!!_(!r)

E [ !! ] == d!r!! !!

_ext

( !r )!!( !r ) ⁺⁺ ^T

^S

[ !! ] ++ U [ !! ] ++ E

_xc

[ !! ]

xc ee S

E [ !] V [ !] U[ !] T [ !] T [ !] == !! ++ !!

Exchange-correlation functional contains now the difference between kinetic energy functional of interacting and non-interacting electrons.

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The Kohn- Sham Method – Kinetic energy functional

Hamiltonian of the non-interacting reference system H

_S

= ! !

²

2m

i

!

N

^! ^!

ⁱ²

⁺ ^!!

^S

⁽ ^!r

ⁱ i

!

N

⁾

How the looks like ? T [ !]

_S

!!

_S

(!r) - local potential

!

! = 1

N! det[ !!

₁

, !!

₂

,…, !!

_N

] ˆh

_S

!!

_i

= ! !

²

2m

! !

²

+!!

_S

(!r)

!

"

# #

$

%

&

&!!

ⁱ

(!r)=!!

i

!!

_i

(!r) For this system there will be an

exact determinantal ground-state wave function

, where are the N lowest eigenstates of the one-electron Hamiltonian ⁱ

!

!! ( !r)= !! *

i

( !r)

i=1

!

N

^!!

ⁱ

⁽ ^!r)

The density

T

_S

[ !! _{]= Min}

!

!!""

! ! _{| ˆT |} ! ! = Min

!

!!""

##

_i

_|" ^!

²

2m

# !

²

| !!

_i

i=1

$

N

T [ !]

S

- can be defined by the constrained-search formula

The search is over all single-determinantal functions that yield the given density

.

!

The existence of the minimum has been proved by Lieb (1982).

!

is uniquely defined for any density.

T [ !]

S

T [ !] T [ !]

S

!!

! !

Crucial characteristics of the Kohn-Sham Method NOT

The Kohn- Sham Method – Kinetic energy functional

The Kohn-Sham Method: Variational Procedure

We cast the Hohenberg-Kohn variational problem in terms of the one-particle (Kohn-Sham) orbitals

E₀==min_!!"_"NE[ !! ] ==

==min_!!"_"N

{{

T_S[ !! ] ++U [ !! ] ++ E_xc[ !! ] ++ d!r##

$$

_ext(!r )!!( !r )

}}

==min_!!"_"N

{{

[ Min_%_{% "}_"!! % | ˆT |%% % ] ++U [ !! ] ++ E_xc[ !! ] ++ d!r##

$$

_ext(!r )!!( !r )

}}

==min

%

% ""N

{{

T_S[%% ] ++U [ !! [%% ]] ++ E_xc[ !! [%% ]] ++ d!r##

$$

_ext(!r )!![%% ]( !r )

}}

=={ &&mini}""N((T_S[{ &&_i}] ++U [ !! [{ &&_i}]] ++ E_xc[ !! [{ &&_i}]] ++

' '

_i==1^N

$$

d!r&&^*_i(!r )##_ext(!r )^&&i(!r ) ))**

++,, -- The dependence of the density on the orbitals is known ! { ! }i

!(!

r ) == !*_i(! r )

i==1

!

N

!

^!ⁱ⁽^{r )}^!

Variational search for the minimum of can be equivalently performed in the space of the orbitals .

E [ !]

{ ! }

i

Derivation of the Kohn-Sham Equations

Performing variational search for the minimum of one must actually constrain orbitals to be orthonormal E [ !]

Conservation of the number of particles Let us define the constrained functional of the N orbitals

d! r!^*_i(!

r )!_j(!

!!

r )^{== !}^ij

!

![{""_i}] == E [##] !! $$_ij

j==1

"

N

"

i==1

"

N

" ##

^d^r^!^!!ⁱ^*⁽^{r )}^!^!!^j⁽^!^{r )}

( )

where are Lagrange multipliers for the constrain ( ).

!

_ij

E [ !]

For to be minimum, it is necessary that

!"[{ # }] 0

_i

==

!!

!!""_i^*(!

r ) E [!!] !! ""_ij

j==1

"

N

"

i==1

"

N

" ##

^d^r'^!^!!ⁱ^*⁽^r')^! ^!!^j⁽^r')^!

$$%%

&&

''((

))==0 !!

!!""_i^*(!r)= !!##

!!""_i^*(!r)

!!

!!##

!!!² 2m

"

"²++!!_ext(! r ) ++!!_H(!

r ) ++!!_xc(!

## r )

$$%% &&

''((!!_i(!

r ) == !!_ij""_j(! r )

j==1

)

N

)

The variational procedure leads to equations:

Note:

!!_H(! r ) ==!!U

!!"" ⁼⁼ ^d r'!

!!

_|r ""^!!^!⁽^!^r')!

r' | !!_xc(!

r ) ==""E_xc[##]

""##

(7)

In Kohn-Sham method exchange-correlation functional can be split into separate exchange and correlation functional E [ !] E [ !] E [ !]_xc == _x ++ _c

E_x[!!] == !! 1

2 d!

""""

rd

#

i

#

^r'^!^!!i

*(!

r ) !!_j(! r )!!^*_j(!

r')

|! r !!!

r' |

#

j

$$

#

%%&& '' (())!!_i(!

r')

!!_xc(!

r ) ==""E_x[##]

""## ⁺⁺

""E_c[##]

""## == !!_x(!

r ) ++ !!_c(! r )

ˆH_KS== !!

!

² 2m

"

!

!²++

!!

_KS(

!

r ) is hermitian !!

!!

_ij is also hermitian Unitary transformation of diagonalizes ,

but the density and remain invariant. { ! }ⁱ

!

_ij

ˆHKS

ij i ˆKS j

( ! == " | H | " )

!!Kohn-Sham potential (local potential !) ^KS(

!r ) == !!

_ext(

!r )++!!

_H(

!r )++!!

_x(

!r )++!!

_c(

!r ) ( == !!

S(!r ) )

Derivation of the Kohn-Sham Equations

Exchange energy

functional Correlation energy

functional Exchange

potential Correlation

potential

!! !

²

2m

"

²

++ !!

_ext

( !r )++!!

_H

( !r )++!!

_x

( !r )++!!

_c

( !r )

##

$$ %% &&

'' ((!!

i

( !r ) == !!

_i

""

_i

( !r ) The Kohn- Sham Method –

The Kohn-Sham Equations

!! ( _{r ) ==} ! !! *

_i

( ! _{r )}

i==1

!

N

! ^!!

ⁱ

⁽ ^! ^{r )}

!E [ "]x

!" ^c

!E [ "]

!"

!!_H(! r ) ==!!U

!!"" ⁼⁼ ^d r'!

!!

_|^!r ""^!!⁽^!^r')! r' |

!!_ext(!

r ) == ""e² Z_s

|! r ""!

##_s""! R_n|

$

s,n

$

Schrödinger-like equations with local potential

These equation are nonlinear and must be solved iteratively (self-consistently)

The Kohn- Sham Method – ‚Aufbau‘ principle

HOMO LUMO

! ₁

! ₂

! N ^N

! ₊₊₁ unoccupied occupied How to calculate one particle density?

The Kohn- Sham Method – The Total Energy

E [ !! ] == !! !²

2m

# #

_i==1^N

""

d"r^!!ⁱ^*⁽^{!r )}^$^$^!²^!!ⁱ⁽!r )++U[ !! ] ++ E_x[ !! ] ++ E_c[ !! ] ++ d!r!!

""

_ext(!r )!!⁽^{!r )} E == !!_i

i==1

"

N

"

^{## 1}₂

%%%%

^d^{r d}^! ^!^r'^$$_|⁽_{r ##}^!^!^{r )}^$$_{r' |}^!⁽^r')^! ⁺⁺^E^x^[^$$^{] ++ E}^c^[^$$^{] ## d}

%%

^r(^! ^&&^x⁽^{r ) ++}^! ^&&^c⁽^!^{r ))}^$$⁽^{r )}^!

"$$$$$$$$$$$$$$$$$$$$$$$$$$$$#$$$$$$$$$$$$$$$$$$$$$$$$$$$$% so-called double counting correction

Sum of the one-particle Kohn-Sham energies

Energy of the reference system differs from the energy of ‘real’ system

!!_i

i==1

!

N

!

⁼⁼ ^""i| ""!² 2m

"

#

#²++!!_KS(! r )|!!_i

i==1

!

N

!

⁼⁼^TS[""] ++ d!

$$

r^!!^KS⁽^{r )}^! ^!!⁽^!^{r )}

(8)

Kohn-Sham energies may be considered as the zero order approximation to the energies of quasi-particles

in the many-particle theory.

Correlation energy functional (also ) is unknown for non-homogeneous systems

Physical meaning of the Kohn-Sham orbital energies ?

!

i

The Kohn- Sham Method – Problems

(Note, these energies were introduced as Lagrange multipliers)

E [ !]c

!max

== == !!

µ I

!!(! r ) == f_i

!

i

!

^!!ⁱ^*⁽^{r )}^!^!!ⁱ⁽^!^{r )} ⁱ

i

E !f

!! ==

!!

! ( r )c ! Strictly speaking there is none

The Kohn-Sham orbital energy of the highest occupied level is equal to the minus of the ionization energy,

Extension to non-integer occupation numbers 0 f 1!! i!!

Janak theorem (1978)

E [ !]c -! is known for homogeneous electron gas (constant density)

Interacting particles Independent particles

!(!x1, !x2,…, !x₁₀23) !¹(! x₁)!₂(!

x₂)…!₃(! x₁₀23)

Density Functional Theory (DFT) in Kohn-Sham realization

This reduction is rigorously possible !

DFT- The Kohn- Sham Method

W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965) System of interacting

electrons with density

!(!r)

System of non-interacting electrons with the same density

!( !r)

“Real” system “Fictitious” or Kohn-Sham reference system

!!

(_{r )}

!

T [ !] T [ !]

S

!!

_S(

!r )== !!( !r )

E[ !! ] == d!r!! !!

_ext

( !r )!!( !r ) ^++T

^S

[ !! ] ++U [ !! ] ++ E

_x

[ !! ] ++ E

_c

[ !! ]

!! ( !

r ) == !! *

_i

( ! r )

i==1

!

N

! ^!!

ⁱ

⁽ ^{r )} ^!

T

_s

[ !! ] == !! ! 2m

²

# #

_i==1^N

"" d"r ^!!

ⁱ^*

⁽ ^{!r )} ^$ ^$ ^!

²

^!!

ⁱ

⁽ ^{!r )}

unknown!!!

E_x[

!!

] == !! 1

2 d_rd

!

""""

#

i

#

^r'

^! ^!!

ⁱ^*⁽^{r )}

^! ^!!

^j_|⁽_{r !!}

^!

^{r )}

^! ^!! ^!

_{r' |}^*^j⁽^r')

^!

#

j

$$ #

%% && '' (( )) !!

_i(_r')

!

DFT: Implementations of the Kohn-Sham Method

Fully relativistic Semi-relativistic Non-relativistic

Non-periodic periodic

All-electron full potential All-electron muffin-tin All-electron PAW Pseudopotential

Non-spin-polarized Spin polarized

Beyond LDA

Generalized Gradient Approximation (GGA) Local Density Approximation (LDA)

GW (quasi-particles) EXX (exact exchange) sX-LDA

Time dependent DFT LDA+U

Atomic orbitals Plane Waves Augmentation Fully numerical (real space)

Gaussians(GTO) Slater type (STO) Numerical Plane waves (FPLAPW) Spherical waves (LMTO, ASW)

! 1 2 "

²

+ V

_ext

( !r) + V

_xc

( !r)

#

$ % &

' ( !

_"

= #

_"

!

_"

(9)

Exchange and Correlation Energy of Homogeneous Electron Gas

s B

r a

!! ""

== ##%% $$&&

1 3 1/ 3 4''((

x x

E == ## !! ""$$ %%&& '' e ==

1/ 3 2 4/ 3 hom

3 3

2 (( )) ((

** ^x == ## $$ %%!! ""&& '' e

hom 3 3 1/ 3 2 1/ 3

(( 2 ))

**

aB^!!

in ( ) ³

"" ^x s Ry

!! "" r

== ## $$&& %%''^{1/ 3} in

hom 3 92 1 [ ]

(( 2 4 ))

x^hom( )rs == !!0.91633/ [ ]r Rys

""

s s s s s

c s

s s s

A r B Cr r Dr r

r Ry

r r r

++ ++ ++ <<

!!""

== ##""%% ++ ++ $$

for for

hom

1 2

ln ln 1

( ) [ ]

/(1 ) 1

&&

'' (( ((

/

!! ==N ""

Exchange energy per unit volume Exchange energy per particle

!!(! k,!

r ) == 1

!

!eⁱk""^!!

Homogeneous electron gas (free electron gas or “jellium”) r

Wave functions: Constant electron density:

Dimensionless parameter characterizing density:

Quantum Monte-Carlo simulations for homogeneous electron gas

Correlation energy per particle

D. M. Ceperly & B. J. Alder, Phys. Rev. Lett. 45, 566 (1980) Parametrization: J. P. Perdew & A. Zunger, Phys. Rev. B 23, 5048 (1981)

A B C D, , , , , ,

!! "" ""

1 2- fitted parameters

Local Density Approximation (LDA)

E

_xc^LDA

[!!] == d!r !! !!(!r)""

_xc^hom

(!!( !r))

xc^hom

==

x^hom

++

c^hom

!! !! !!

In atoms, molecules, and solids the electron density is not homogeneous

!(r)

r

1

! = !

₁

( ) r

₁

r

!

₁ The main idea of the

Local Density Approximation

^: the density is treated locally as constant

GGA - Gradient Corrections to LDA

Gradient Expansion Approximation

E_xc^GEA

[ !! ] == E

_xc^LDA

[ !! ]++ d !

_r

!! ^!! ⁽

^{r )C}

^!

^xc

^[ ^!! ^{]| !!} _!! ^!! ₍

_{r )}

^! ⁽

^{r ) |}

^!

4/3²

] E

_xc^GGA

[!!] == d!r !! ^f

^xc

(!!(!r),!!!!(!r))

Generalized Gradient Approximation

J. P. Perdew & Y. Wang, Phys. Rev. B 33, 8800 (1986) D. C. Langreth & M. J. Mehl, Phys. Rev. B 28, 1809 (1983)

f

xc-!constructed to fulfill maximal number of “summation rules”

Exchange-correlation potential can be calculated very easily, since explicit dependence of E_xc on the density is known. !!

HE ! = ! Modeling of Nanostructures and Materials

Modeling of Nanostructures and Materials

Summer Semester 2013 Lecture

Jacek A. Majewski

Modeling of Nanostructures and Materials

Lecture 2

February 28, 2013

Density Functional Theory (DFT) – the key to the

Computational Materials Science T

Th hee B Ba assiiccss

Kohn-Sham realization of the DFT

Fundamental problem in materials science

DNA - molecule

Crystal - diamond

C

- molecule H ! = ! E

!

"

2M

#

! ! 2m

"

#

+ 1 2 |

!

! !

|

# ! |

!

! !

|

#

+ 1 2 | !r

! !

|

#

Materials Science:

Examples of Schrödinger Equation?

Z

{!r

}

Quantum Mechanics of Molecules and Crystals

ˆ ˆ

H T U( x, X ) T == ++ ++

"

"

"

"

"

"

!

!

!

!

!

!

!

!

The Adiabatic Approximation (Born-Oppenheimer)

It is natural to consider the full Hamiltonian of the system to be the sum of an ionic and an electronic part

ˆ ˆ ˆ H H == ++ H

ˆ ˆ ˆ

H == T ++ V ( X ) H ˆ

== T V ( x, X ) V ( x ) ˆ

++ ˆ

++ ˆ

The Adiabatic Approximation (Born-Oppenheimer)

ˆH ! ( X,x ) E ( X )! ( X,x ) ==

ˆ

[ H ++ E( K', X )] !(Q,K', X ) "(Q )!(Q,K', X ) ==

ˆ ˆ ˆ ˆ

H T V == ++ ++ V

# #

!

!

!

!

"

"

Quantum Mechanics:

^! ^! _2m

^"

^{+ 1} ₂ _|

^!

# ^! _|

^!

^{+ 1} ₂ _{| !r}