Theory and Practice of Density Functional Theoryorion.pt.tu-clausthal.de/atp/downloads/111004_dftlecture.pdf · Density functional theory maps interacting electrons onto non-interacting

http://www.pt.tu-clausthal.de/atp/

Peter E. BlöchlInstitute for Theoretical Physics

Clausthal University of Technology, Germany

Theory and Practice of Density Functional Theory

1

Tuesday, October 4, 11

Standard model of Solid State Physics

• particles: nuclei and electrons• interaction: electrostatics • equations of motion:

– Schrödinger equation: – Poisson equation:

2

i~@t| i = H| i

r2�(~r) =1

✏0⇢(~r)

but: exponential wall of many-particle physicsproblem: quantum mechanics + interaction


Density functional theorymaps interacting electrons onto non-interacting quasi-electrons in an effective potential

• one-particle Schrödinger equation

• limited to ground state properties• based on an exact theorem

– but: implementation requires approximations

3

✓� ~22m

~r2 + veff (~r)

◆ n(~r) = n(~r)✏n


4

McKee et al.,Science 300,1726 (2003)

Ojima,Yoshimura, Ueda PRB65, 75408 (2004)

Hu et al. PRB65,75408(2002)

Norga et al. MRS Proc. 2004

gate

n!doped n!doped

p!doped

Metal!Oxide Field!Effect Transistor (MOSFET)

silicon waver

source drain

channel

gate oxide

p!doped

Appetizer: high-K oxides for new transistorsFörst, Ashman, Schwarz, Blöchl, Nature 53, 427 (2004)

Ashman, Först, Schwarz, Blöchl, PRB 69, 75309 (2004)Först, Ashman, Blöchl, German patent DE10303875


• Minimize total energy functional:

• Solve Kohn-Sham equations:

Total energy functional

5

� ~22m

~r2 + veff

(~r)

� n

(~r) = n

(~r)✏n

and veff

(~r) =

Zd3r0

e2[n(~r0) + Z(~r0)]

4⇡✏0|~r � ~r0|| {z }vHartree(~r)

+�E

xc

�n(~r)| {z }µxc(~r)

E[| n

i, fn

] =X

n

fn

h n

| ~p2

2m|

n

i| {z }

Ekin

+1

2

Zd3r

Zd3r0

e2[n(~r) + Z(~r)][n(~r0) + Z(~r0)]

4⇡✏0

|~r � ~r0|| {z }EHartree

+ Exc

[n(~r)]| {z }Exc

�X

n,m

⇤n,m

(h n

| m

i � �n,m

)| {z }

constraint

with n(~r) =X

n

fn

| (~r)|2

| {z }electron density

Z(~r) = �X

j

Zj

�(~r � ~Rj

)

| {z }charge density of nuclei in electron charges




6

� ~22m

~r2 + veff

(~r)

� n

(~r) = n

(~r)✏n

and veff

(~r) =

Zd3r0

e2[n(~r0) + Z(~r0)]

4⇡✏0|~r � ~r0|| {z }vHartree(~r)

+�E

xc

�n(~r)| {z }µxc(~r)

E[| n

i, fn

] =X

n

fn

h n

| ~p2

2m|

n

i| {z }

Ekin

+1

2

Zd3r

Zd3r0

e2[n(~r) + Z(~r)][n(~r0) + Z(~r0)]

4⇡✏0


+ Exc

[n(~r)]| {z }Exc

�X

n,m

⇤n,m

(h n

| m

i � �n,m

)| {z }

constraint

with n(~r) =X

n

fn

| (~r)|2


Z(~r) = �X

j

Zj

�(~r � ~Rj

)






7

� ~22m

~r2 + veff

(~r)

� n

(~r) = n

(~r)✏n

and veff

(~r) =

Zd3r0

e2[n(~r0) + Z(~r0)]

4⇡✏0|~r � ~r0|| {z }vHartree(~r)

+�E

xc

�n(~r)| {z }µxc(~r)

E[| n

i, fn

] =X

n

fn

h n

| ~p2

2m|

n

i| {z }

Ekin

+1

2

Zd3r

Zd3r0

e2[n(~r) + Z(~r)][n(~r0) + Z(~r0)]

4⇡✏0


+ Exc

[n(~r)]| {z }Exc

�X

n,m

⇤n,m

(h n

| m

i � �n,m

)| {z }

constraint

with n(~r) =X

n

fn

| (~r)|2


Z(~r) = �X

j

Zj

�(~r � ~Rj

)




Self-consistency loop

8

� ~22m

~r2 + veff

(~r)

� n

(~r) = n

(~r)✏n

and veff

(~r) =

Zd3r0

e2[n(~r0) + Z(~r0)]

4⇡✏0|~r � ~r0|| {z }vHartree(~r)

+�E

xc

�n(~r)| {z }µxc(~r)

E[| n

i, fn

] =X

n

fn

h n

| ~p2

2m|

n

i| {z }

Ekin

+1

2

Zd3r

Zd3r0

e2[n(~r) + Z(~r)][n(~r0) + Z(~r0)]

4⇡✏0


+ Exc

[n(~r)]| {z }Exc

�X

n,m

⇤n,m

(h n

| m

i � �n,m

)| {z }

constraint

with n(~r) =X

n

fn

| (~r)|2


Z(~r) = �X

j

Zj

�(~r � ~Rj

)


veff

(⌅r) = vext

(⌅r) +

Zd3r0

e2n(1)(⌅r0)

4⇤⇥0|⌅r � ⌅r0|+

�Exc

[n(1)]

�n(1)(⌅r)

�~22me

⇤r2 + veff (⇤r)� �n

�⇥n(⇤x) = 0

n(1)(⇥r) =X

n

fnX

�

�⇤n(⇥x)�n(⇥x)


• Paradigm shift: – eigenvalue problems minimize energy

• Access to the molecular dynamics

Car-Parrinello method

9

Car, Parrinello, PRL 55, 2471(1985)


E0

=µ

minn(~r)

⇢FW [n] +

Zd3r v

ext

(~r)n(~r)� µ

✓Zd3r n(~r)�N

◆

| {z }charge constraint

�

=µ

minn(~r)

⇢Ts

[n]| {z }F

0[n]

+

Zd3r v

ext

(~r)n(~r) + FW [n]� F 0[n]| {z }EH+Exc

�µ

✓Zd3r n(~r)�N

◆

| {z }charge constraint

�

Levy’s constrained searchIs there a density functional?constrained search: proof of existence by presenting a construction principle1. sort all fermionic many-particle wave functions

according to their density

2. For each density find

a. the wave function with lowest kinetic energy and

b. the wave function with lowest kinetic and interaction energy

This defines two universal density functionals, F0[n] and FW[n]

3. determine energy and density of the ground state for a given external potential by minimization over all densities

- potential energy depends only on the density

10

W

00 0

0

W W W

>

|!">

|!#>|!$>

|!%>

F [n4(r)]F [n1(r)]

F [n1(r)]

F [n2(r)]

F [n2(r)] F [n3(r)]

F 3(r)]

F [n4(r)]

[n

n1(r) n2(r) n3(r) n4(r)

|!&>

|!'> |!(>

|!)

FW [n] =�W

(~r),EW

min| i

h |T + W | i| {z }kinetic + interaction

+

Zd3r �W (~r)

✓h |n(~r)| i � n(~r)

◆

| {z }density constraint

�EW

✓h | i � 1

◆

| {z }norm constraint

F 0[n] =�0

(~r),E0

min| i

h |T | i| {z }kinetic

+

Zd3r �0(~r)

✓h |n(~r)| i � n(~r)

◆

| {z }density constraint

�E0

✓h | i � 1

◆

| {z }norm constraint


n(r1)h(r1,r2)

|r2−r1|0

n2n(2)(r2,r1)

n(2)(r2,r1)

0

Exchange-correlation hole• interaction energy

• two-particle density– n(2)(r,r’)= probability density of finding one electron

at r and another at r’ times N(N-1) (number of pairs)

• exchange correlation hole hxc(r,r’)

• properties of the exchange correlation hole– each electron interacts with N-1 other electrons:

hole integrates to -1 electron (Charge sum rule)

– two electrons with the same spin cannot be at the same position (Pauli principle)

– the hole vanishes at large distances (short-sightedness)

11

Ee�e =1

2

Zd3r

Zd3r0

e2n(2)(~r, ~r0)

4⇡✏0|~r � ~r0|

n(2)(~r, ~r0) = n(1)(~r)n(1)(~r0) + n(1)(~r)hxc

(~r0,~r)

Ee�e

=1

2

Zd3r

Zd3r0

e2n(1)(~r)n(1)(~r0)

4⇡✏0|~r � ~r0|| {z }Hartree energy

+

Zd3r n(1)(~r) ·

"1

2

Zd3r0

e2hxc

(~r, ~r0)

4⇡✏0|~r � ~r0|

#

| {z }potential energy of exchange and correlation


Model for Exc

• properties of the exchange correlation hole

– each electron interacts with N-1 other electrons: hole integrates to -1 electron (Charge sum rule)

– two electrons with the same spin cannot be at the same position (Pauli principle)

– the hole vanishes at large distances (short-sightedness)

• Model:

– homogeneously charged sphere

For a free electron gas not much worse than a Hartree Fock calculation

12

(r0)=φ−e

φ(r)−e

h(r,r0)r0

4 0πεe2 n(r0)2π

g (r,r0)n(r)

r

r

r

3334

exchange-correlation energy per electron


Exc

=

Z 1

0d�

Zd3r n(~r)

1

2

Zd3r0

e2h�

(~r, ~r0)

4⇡✏0|~r � ~r0|| {z }dE/d�

=

Zd3r n(~r)

1

2

Zd3r0

e2R 10 d�h

�

(~r, ~r0)

4⇡✏0|~r � ~r0|

H(�) = H0 + �W

Exc

[n(1)] = FW [n(1)]� F 0[n(1)]� EHartree

Correlation energy

• Exchange energy uses hole of non-interacting electrons

• Coulomb repulsion pushes electrons away

• deformation of wave function raises kinetic energy

•adiabatic connection:– switch the interaction slowly on and

integrate opposing “force”

– never evaluate kinetic energy explicitly

13

Coulomb hole

Correlation energy is due to the balance between energy gain by lowering the

Coulomb repulsion and kinetic energy cost

h (r

r0

r

0,r)h (r

h (r0,r)

!1

0

0

!1

r

h (r


Jacobs ladder to heaven

14

5. Exact: Constrained search

4. Hybrid functionals: Include exact (nonlocal) exchange: left-right correlations

3. Meta-GGA: use kinetic energy density to estimate the flexibility of the electron gas

2. GGA: asymmetry of the xc-hole favors surfaces and thus weakens bonds

1. LDA: xc-hole from free electron gas: strong overbinding


Local density approximation (LDA)

• Local density aproximation (LDA) imports the “exact” exchange correlation hole of a free electron gas

15

0 0

r−r00−0.3a 0−0.2a 00.1a0−0.1a r0

r−r000.3a00.2a00.1a

r−r0−1.5a0 −1.0a0 −0.5a0 r0 00.5a

00.5a 01.0a 01.5a 02.0a

Exact

LSD

R

R

Exact

LSD

LSD

ExactLSD

Exact

R

Exc

=

Zd3r n(~r)✏

xc

(n(~r))

from free electron gas

local

Why does it work?Sum rules!

• charge sum rule exact• only spherical average matters• only first moment of radial hole matters

✏xc

=1

2

4⇡e2

4⇡✏0

Z 1

0dr r

⌧h(~r0,~r0 + ~r)

�

angle

1 = 4⇡

Z 1

0dr r2

⌧h(~r0,~r0 + ~r)

�

angle


Generalized gradient approximation (GGA)

• an electron far away from an atom sees a positive ion

• the exchange hole is entirely on the atom.

• the LDA hole is centered on the electron and smeared out over a wide region.

• LDA overestimates the exchange energy at the surface

• GGA stabilizes the surfaces (overbinding reduced)

16

✏xc

(r ! 1) = �1

2

e2

4⇡✏r�1

| {z }GGA

<< A exp(��r)| {z }LDA

reduced gradient is largest in the tails

GGA correction stabilizes tails

binding energies are dramatically improved

DFT became of interest to chemists

rs| n|Δ

rsrs r0

n

−h(r0,r)

r

n reduced gradient x=|�n|/n4/3


Hybrid functionals• adiabatic connection

• approximate integral by weighted sum of the values at the end-points of the interval

– Hartree-Fock exchange introduces nonlocality

– finite band gaps for Mott insulators

– correct dissociation limit of bonds

– related to GW, LDA+U, etc.

17

Exc

=

Z 1

0d�

Zd3r n(~r)

1

2

Zd3r0

e2h�

(~r, ~r0)

4⇡✏0|~r � ~r0|| {z }dE/d�

=

Zd3r n(~r)

1

2

Zd3r0

e2R 10 d�h

�

(~r, ~r0)

4⇡✏0|~r � ~r0|

H(�) = H0 + �W

d3r 4 0|r−r’|πεe2h (r,r’)λ

21

Hartree−Fock(hole of non−interacting electrons)

xcε

0 1 λ

λ=0 no interaction:exchange

λ=1 full interaction

λ-average (area or dashed line):exchange-correlation

Exc =

Zd� Uxc(�)

⇡ 1

4Uxc(� = 0) +

3

4Uxc(� = 1)

=1

4UHFxc| {z }

EHFx

�1

4UGGAxc| {z }

EGGAx

+1

4UGGAxc (� = 0) +

3

4UGGAxc (� = 1)

| {z }EGGA

xc

= EGGAxc +

1

4

✓EHF

x � EGGAx

◆


Band-gap problem

• GGA’s underestimate band gaps– in silicon 0.7 eV instead

of 1.17 eV

• Many transition metal oxides are metals instead of insulators!

• Problem: energy of average density instead of average of energies

18

e=

g

LDA

Eg

correct

dN

dE

E

N

E

10!1 2


Electronic structure methods:how to solve the Kohn-Sham equations

19

pseudopotentialsHamann-Bachelet-Schlueter, Troulier-Martins, Kleinman-Bylander, Vanderbilt ultrasoft

linear methodslinear augmented muffin tin-orbital (LMTO) method linear augmented plane wave (LAPW) method

projector augmented wave method

augmented wave methodsKorringa Kohn Rostocker (KKR), augmented plane wave (APW) method


atomic valence wave functions (s and d) of Pt

r

What is the problem?

1. wave function oscillates strongly in the atomic region (Coulomb singularity)

2. wave function needs to be very flexible in the bonding region and the tails (Chemistry)

3. most electrons (core) are “irrelevant”

4. relativistic effects

5. tiny but finite nucleus

20

K.Sc

hwar

z Ta

lk/W

WW

Fro

m A

PW t

o LA

PW t

o (L

)APW

+lo


Strategies

Pseudopotentials

21

• Chop off singular potential• node-less wave functions• no core states• no information on inner electrons• transferability problems

Augmented waves

• start with envelope function• replace incorrect shape with

atomic partial waves• complex basisset• retain full information on wave

functions and potential

LMTO, ASW, LAPW, APW, KKR, PAW, (all-electron)

Hamann-Bachelet-Schlüter, Kerker, Kleinman-Bylander, Troullier Martins, Ultrasoft,etc


PAW augmentation

22

| i|{z}all-electron

=

pseudoz}|{| i +

P↵ |�↵ihp↵| ˜ iz}|{| 1i|{z}

1-center, all-el.

�

1-center, pseudoz}|{| 1i|{z}

P↵ |˜�↵ihp↵| ˜ i

+ -=

= -+


PAW transformation theory

23

| i = | i+X

↵

⇣|�↵i � |�↵i

⌘hp↵| i

plane waves atomic partial wave

auxiliarypartial wave

projector function

plane waves

T = 1 +X

R

SR SR =X

↵

⇣|�↵i � |�↵i

⌘hp↵|

all-electronz}|{| i|{z}true

= T

pseudoz}|{| i|{z}

auxiliary

with

0 1 2 3r [a0]

-1

0

1

2

ps

d

Fe


PAW transformation theory

24

| i = | i+X

↵

⇣|�↵i � |�↵i

⌘hp↵| i

plane waves atomic partial wave

auxiliarypartial wave

projector function

plane waves

T = 1 +X

R

SR SR =X

↵

⇣|�↵i � |�↵i

⌘hp↵|

all-electronz}|{| i|{z}true

= T

pseudoz}|{| i|{z}

auxiliary

with


=

+

PAW augmentation

25

| i|{z}all-electron

=

pseudoz}|{| i +

P↵ |�↵ihp↵| ˜ iz}|{| 1i|{z}

1-center, all-el.

�

1-center, pseudoz}|{| 1i|{z}

P↵ |˜�↵ihp↵| ˜ i

= + -


E =X

n

h n

|�~22m

e

~r2| n

i+Z

d3r v(~r)n(~r)

+1

2

Zd3r

Zd3r0

e2[n(~r) + Z(~r)][n(~r0) + Z(~r0)]

4⇡✏0|~r � ~r0| + Exc

[n(~r)]

E1R

=X

i,j2R

Di,j

h�j

|�~22m

e

~r2|�i

i+Nc,RX

n2R

h�c

n

|�~22m

e

~r2|�c

n

i

+1

2

Zd3r

Zd3r0

e2[n1(~r) + Z(~r)][n1(~r0) + Z(~r0)]

4⇡✏0|~r � ~r0| + Exc

[n1(~r)]

E1R

=X

i,j2R

Di,j

h�j

|�~22m

e

~r2|�i

i+Z

d3r v(~r)n1(~r)

+1

2

Zd3r

Zd3r0

e2[n1(~r) + Z(~r)][n1(~r0) + Z(~r0)]

4⇡✏0|~r � ~r0| + Exc

[n1(~r)]

PAW total energy

26

E0[{| ni, Rj}] = E +X

R

⇣E1

R � E1R

⌘Also, the total energy is written as sum of• an extended plane wave part• two one-center expansions for each site

“smart zero”

contains pseudized core(non-linear core correction)

Compensation density:


Accuracy

• VASP: [Paier,Hirschl, Marsman, Kresse JCP122,234102 (1005)]

• GPAW: [Carsten Rostgaard]

27

All-electron calculations with PAW (continued)

PBE atomization energies relative to Gaussian:


The End

28Tuesday, October 4, 11

Documents

Theory and Practice of Density Functional Theoryorion.pt.tu-clausthal.de/atp/downloads/111004_dftlecture.pdf · Density functional theory maps interacting electrons onto non-interacting