Beyond standard DFT

(1)

EPJ Web of Conferences 14, 01004 (2011) DOI: 10.1051/epjconf/20111401004

“ Fundamentals of Thermodynamic Modelling

of Materials ”

November 15-19, 2010

INSTN – CEA Saclay, France

Organized by

Bo SUNDMAN

[email protected]

Constantin MEIS [email protected]

PROFESSOR & TOPIC

Georg KRESSE

Vienna University, Austria

Beyond standart

DFT

(2)

Beyond standard DFT

And a guide to the many DFT’s

Georg Kresse

Faculty of Physics

Universität

Wien

Funded by the Austrian FWF

2

3/1/2011 Beyond standard DFT

Ab initio: What does it mean?

Ab initio

↔ free of parameters

Parameter free equivalent to using

Schrödinger Equations

Basics 80 years old (1926)

Exceedingly active research area

Virtual Matter Laboratory

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One electron:

3D-grid

128 Kbyte (60 Textpages)

Two electrons:

6D-grid

1 Gbyte (PC)

Three electrons: 9D-grid

8200 Terrabyte (Supercmp.)

W. Kohn

five electrons

five 3D sets

1923 Vienna

1940 Canada (Kindert.)

1950 Carnegien Mellon

1984 Santa Barbara

1998 Nobel Prize

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3 3/1/2011

One electron

Beyond standard DFT

Electronic structure methods and

one-electron theories

Density functional theory, DFT not improved in last 15 years

Hartree Fock theory → hybrid functionals

RPA:

Nozières, Phys. Rev.

111 , 442; Hedin, Phys. Rev.

139 , A796

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Hartree Fock

theory → hybrid functionals

RPA:

Nozières, Phys. Rev. 111, 442; Hedin, Phys. Rev. 139, A796

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Gradient corrected functionals

6

Optimized for

GGA

What does it do

Solids

PBEsol

AM05, WC

Less gradient corr.

Stronger binding

General purpose

PBE

Balanced gradient corr.

Molecules

rPBE

BLYP

Stronger gradient corr.

Weaker binding

References for functionals

GGA:

PBEsol:

J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E.

Scuseria, L. A. Constantin, X. Zhou, and K. Burke, Phys. Rev. Lett. 100,

136406 (2008).

AM05:

R. Armiento and A. E. Mattsson, Phys. Rev. B 72, 085108 (2005).

WC:

Z. Wu and R. E. Cohen, Phys. rev. B 73, 235116 (2006).

PBE:

J. P. Perdew, K. Burke, M. Ernzerhof, Phys.Rev.Lett. 77, 3865 (1996).

RPBE:

B. Hammer, L. B. Hansen, J. K. Norskov, Phys. Rev. B 59, 7413

(1999).

Hybrids:

PHEh/PBE0:

C. Adamo and V. Barone, J.Chem.Phys. 110, 6158(1999). J. P.

Perdew, M. Ernzerhof, and K. Burke, J. Chem. Phys. 105, 9982 (1996).

B3LYP:

Becke A. D. J.Chem.Phys. 98, 5648 (1993) Gaussian NEWS Vol.

5͑2͒

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Gaussian Inc., Pittsburgh, PA,

1994͒

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7

(5)

8

Electronic structure methods and

one-electron theories

Density functional theory, GGA not improved in last 10 years

Hartree Fock theory → hybrid functionals

GW RPA:

Nozières, Phys. Rev.

111 , 442; Hedin, Phys. Rev.

139 , A796

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GW RPA:

Nozières, Phys. Rev. 111, 442; Hedin, Phys. Rev. 139, A796

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Hartree-Fock theory

Lacks correlation and much too large band gaps

Expectation value

Hartree or electrostatic interaction between electrons

Exchange interaction between electrons (anti-symmetry)

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10

Bare exchange in HF

The exchange interaction between two particles

11

GW and RPA: The correlation

-1

The electrons move in the exchange potential screened by

all other electrons as a result of Coulomb-correlation

(7)

12

Hybrids: screened exchange

Screened exchange:

Screening system dependent

Direct Coulomb correlation

For bulk materials dielectric

matrix is approximately

diagonal in reciprocal space

Ɛ

-1

_(G)

Strong screening for small G

(static screening properties)

No screening for large G

Hybrids: ¼ is a compromise

Perdew, Ernzerhof, and Burke,

J. Chem. Phys. 105, 9982 (1996).

Vacuum no

screening

Insulators

weak screening

Semiconductors/ metals

strong screening

hybrids

S

st

Electronic structure methods and

one-electron theories

Density functional theory, GGA not improved in last 10 years

Hartree Fock

theory → hybrid functionals

RPA & GW:

Nozières, Phys. Rev.

111 , 442; Hedin, Phys. Rev.

139 , A796

(8)

14

Overview

Ab intio DFT: Local and Hybrid functionals

Evaluate performance and accuracy of different local

functionals

Or rather give a feeling how they perform

When hybrid functionals are better than local functionals

Prototypical solids: lattice constants, bulk moduli and

phonons

Band gaps

Localized electrons: ceria, transition metal oxides, ...

Choosing the right functional is still very difficult

There is no “best” general purpose functional

15

Take home messages

In many cases, hybrid functionals

are a step

forward compared to local functionals

But not a universal improvement

¼ exact exchange is a good compromise for

semiconductors and some insulators

Structural properties

Energies

Band gaps

Going further is difficult

(9)

16

VASP: PAW methods

P. Blöchl, PRB 50, 17953 (1994), G. Kresse, et. al. PRB 59, 1758 (1998).

PAW method:

full potential (all-electron method)

Core-valence interaction is described at the appropriate

level; core electrons are

frozen at DFT level

Plane waves everywhere in space (pseudo)

LCAO corrections in the spheres (one center terms)

PW

PS-LCAO

AE-LCAO

PBEh and HSE functional

The PBEh (PBE0) exchange-correlation functional

1 The HSE03 (HSE06) functional

2

(10)

HSE versus PBEh: convergence with k points

1

18

1

_{J. Paier, M. Marsman, K. Hummer, G. Kresse, I.C. Gerber, and J.G. Angyan,}

J. Chem. Phys.

124 , 154709 (2006).

Example: fcc aluminum

PBEh

HSE

Flavors of Semi-local and Hybrid functionals

Hybrids increase binding strength (most of the time)

Reduce self-interaction and increase band gap

Increase spin polarization energy

19

Optimized

for

GGA

Hybrids

Performance of DFT

functional

Solids

PBEsol

AM05, WC

WC1

Stronger binding

Smaller lattice const.

General

purpose

PBE

PBEh

HSE06

Balanced gradient

Molecules

rPBE

BLYP

B3LYP

(11)

Gradient corrected functionals

20

Optimized for

GGA

What does it do

Solids

PBEsol

AM05, WC

Less gradient corr.

Stronger binding

General purpose

PBE

Balanced gradient corr.

Molecules

rPBE

BLYP

Stronger gradient corr.

Weaker binding

PBE: Lattice constants and bulk moduli

Lattice constants

Bulk moduli

Paier, M. Marsman,

K. Hummer, G. Kresse,…, J. Chem. Phys. 122

, 154709 (2006)

PBE: MRE 0.8 %, MARE 1.0 %

(12)

22

HSE: MRE 0.2 %, MARE 0.5 %

HSE: MRE -3.2 %, MARE 6.4 %

PBE: MRE 0.8 %, MARE 1.0 %

PBE: MRE -9.8 %, MARE 9.4 %

PBE: Lattice constants and bulk moduli

Paier, M. Marsman,

K. Hummer, G. Kresse,…, J. Chem. Phys. 122

, 154709 (2006)

23

HSE: MRE 0.2 %, MARE 0.5 %

PBEsol, AM05: MARE 0.6 %

HSE: MRE -3.2 %, MARE 6.4 %

PBEsol, AM05:

MARE 7.1 %

PBE: MRE 0.8 %, MARE 1.0 %

PBE: MRE -9.8 %, MARE 9.4 %

PBE: Lattice constants and bulk moduli

(13)

24

B3LYP: Lattice constants and bulk moduli

PBE: MRE 0.8 %, MARE 1.0 %

B3LYP: MRE 1.0 %, MARE 1.2 %

PBE: MRE -9.8 %, MARE 9.4 %

B3LYP: MRE -10.2 %, MARE 11.4 %

J. Paier, et al., J. Chem. Phys.

127 , 24103 (2007).

Bond distances: Hybrid functionals

PBE becomes worse with increasing mass

PBEsol slightly more balanced

Hybrids decrease lattice constants (except for metals)

Optimized

for

Hybrids

Performance of DFT

functional

Solid

PBEsol

AM05

WC1

Quite good bond

length

General

purpose

PBE

PBEh

HSE06

Too large bond length

Molecules

rPBE

BLYP

B3LYP

(14)

Exception: Multivalent oxides e.g. ceria

26

CB

_VB

f

Usual from

DFT to hybrid

J.L.F. Silva, …, G. Kresse,

Phys. Rev. B

75 , 045121 (2007).

semilocal DFT

Hybrid HSE

unsual

Hybrids reduced metallicity and increase

lattice constants in „metals“

3d transition metal oxides

Hybrids

substantially

improve upon

PBE

HSE latt. const.

and local spin

mag. moments

are excellent

27

PBE

HSE

EXPT.

MnO

a

_o

E

_g

4.44

0.93

4.44

2.8

4.45

3.9 FeO

a

_o

E

_g

4.30 metal

4.33

2.2

4.33

2.4 CoO

a

_o

E

_g

4.22 metal

4.26

3.4

4.25

2.5 NiO

a

_o

E

_g

4.19

0.81

4.18

4.2

4.17

4.0 M. Marsman

et al., J. Phys.: Condens. Matter

20 , 64201 (2008).

(15)

Bond distances: Hybrid functionals

Hybrids generally reduce the lattice constants

Reduced self-interaction can lead to localization of

electrons in metals with concomitant significant increase in

lattice constant

If PBE predicts too small lattice constants, something wrong

28

Optimized

for

Hybrids

Performance of DFT

functional

Solid

PBEsol

AM05

WC1

Quite good bond

length

General

purpose

PBE

PBEh

HSE06

Too large bond length

Molecules

rPBE

BLYP

B3LYP

Much too large bond

length

Hybrids: atomization energies

J. Paier, R. Hirsch, M. Marsmann, G. Kresse, J. Chem. Phys.

122 , 234102 (2005).

Promising, but why weaker binding energies,

despite smaller lattice constants and bond length

INCREASED SPIN POLARIZATION IN ATOMS

overbound

(16)

Results for solids: atomization energies

1

30

Paier, M. Marsman

, K. Hummer, G. Kresse,…, J. Chem. Phys. 122

, 154709 (2006)

J. Paier, M. Marsman, and G. Kresse, J. Chem. Phys.

127 , 24103 (2007)

HSE yields slightly decreased atomization energies

despite smaller lattice constants (stronger bonding)

INCREASED SPIN POLARIZATION IN ATOMS

larger errors for

d-metals

underbound

overbound

B3LYP Solids: atomization energies

1

31 Beyond standard DFT

1

_{J. Paier, M. Marsman, and G. Kresse, J. Chem. Phys.}

₁₂₇

_{, 24103 (2007).}

Huge errors for

metals

underbound

overbound

(17)

Atomization energies

Hybrids increase bond strength

But also increase spin polarization energy (atom)

In total: slightly smaller atomization energies, despite

smaller bond length

32

Optimized

for

Hybrids

What does it do

Solids

PBEsol

AM05

WC1

1-2nd row: overbound

3rd- row: overbound

General

purpose

PBE

PBEh

HSE06

1-2nd row: overbound

3rd- row: ok

Molecules

rPBE

BLYP

B3LYP

1-2nd row: ok

3rd- row: way underbound

Heats of formation

Heats of formation w.r.t normal state at ambient conditions

in kJ/mol

PBE

PBEsol

HSE

EXP

LiF

570

573

591 619 (614)

NaF

522

540 577 (573)

NaCl

355

371 413 (411)

MgO

516

533

541 604 (597)

MgH

₂

52

60

64 78 (68)

AlN

262

280

286 321 (313)

SiC

52

53

64 78 (68)

Mg(bulk metal) + H

₂

→ MgH

₂

(18)

34

DFT band gaps

DFT band gaps much

too small

One needs to be

carefull if DFT is

applied to determine

Band gaps

Effective masses

Impurity levels

Band alignments

Electron localization

35

One of the great lies: The band issue

DFT is only accurate for ground state properties

hence the error in the band gap does not matter

The band gap is a well defined ground state property

wrong using local and semi-local functionals

Fundamental gap

Large errors in LDA/GGA/HF

Lack of Integer-discontinuity

in the LDA/GGA/HF

LDA/GGA

in

]

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(

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[

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1 [

(

VBMAX

CBMIN

N

E

N

E

N

E

N

E

I

A

E

_g

(19)

Hybrid functionals for solids: Band gaps

Band gaps improved

But fairly larger errors

prevail for materials

with weak screening

(ε<4)

for these materials

half-half functionals

are quite accurate but

these will be worse for

the rest !

36

<4

Vibrational properties: Phonons

Kresse, Furthmüller, Hafner, EPL

32 , 729 (1995).

Hummer, Harl, Kresse, Phys. Rev. B

80 , 115205 (2009).

C

Si

(20)

38

C

Si

Sn

Ge

Vibrational properties: Phonons

Kresse, Furthmüller, Hafner, EPL

32 , 729 (1995).

Hummer, Harl, Kresse, Phys. Rev. B

80 , 115205 (2009).

The random phase approximation

Perform DFT calculation

Calculate the response function:

Gonze and Fuchs, Phys. Rev. B

65 , 235109 (2002).

Correlation energy:

39 3/1/2011 The random phase approximation

basis

auxilary

an

in

response

particle

t

independen

the

is

0

0 (

)

(

)

]

1 Tr[ln(

i

v

i

v

d

3 basis

occupied

unocc

occ

0 (

,

)

n

occupied

N

basis

i

a

i

a

i

a

)

(21)

40

RPA: screened exchange

Screened exchange:

Screening system dependent

Direct Coulomb correlation

For bulk materials dielectric

matrix is approximately

diagonal in reciprocal space

Ɛ

-1

_(G)

Strong screening for small G

(static screening properties)

No screening for large G

Hybrids: ¼ is a compromise

Perdew, Ernzerhof, and Burke,

J. Chem. Phys. 105, 9982 (1996).

Vacuum no

screening

Insulators

weak screening

Semiconductors/ metals

strong screening

hybrids

S

st

RPA: Lattice constants for semic. and insulators

(22)

RPA: Heats of formation

Heats of formation w.r.t normal state at ambient conditions

in kJ/mol

42

PBE

Hartree-Fock

RPA

EXP

LiF

570

664

609 619 (614)

NaF

522

607

567 577 (573)

NaCl

355

433

405 413 (411)

MgO

516

587

577 604 (597)

MgH

₂

52

113

72 78 (68)

AlN

262

350

291 321 (313)

SiC

51

69

64 78 (68)

Mg(bulk metal) + H

₂

→ MgH

₂

J. Harl, G. Kresse, PRL 103, 056401 (2009)

609 619 (614)

567 577 (573)

405 413 (411)

577 604 (597)

72 78 (68)

291 321 (313)

64 78 (68)

43

RPA: for rare-gas solids: Ne, Ar and Kr

(23)

44

Covalent versus Van der Waals

Carbon

Graphite versus

Diamond

1/d

4 _behavior

J. Harl, G. Kresse,

PRL 103, 056401 (2009).

S. Lebeque, et al.

PRL in print.

Surface energies

CO adsorption energy

No semi-local

function works

RPA

increases surface

energy

but

decreases adsorption

energy

Good agreement

with experiment

L. Schimka

, …, G. Kresse,

(24)

Acknowledgement

FWF for financial support

The group for their

great work...

RPA: J. Harl, L. Schimka,

SOSEX:

A. Grüneis, M. Marsman

You for listening

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