Density Functional Theory and Exchange-Correlation Functionals · 1. A functional of density; given a density, it gives a number. 2. A universal functional of density, independent

Density Functional Theory and

Exchange-Correlation Functionals

Weitao Yang, Duke University

Durham, North Carolina

2H

Funding

NSF

NIH

DOE

Theory

Biological

Nano

Material

Isfahan

May 2016 1

Outline

1. Introduction to the Fundamental Principles of DFT

2. Kohn-Sham Equations and Density Functional

Approximations

3. Fractional Perspectives of DFT

R. G. Parr and W. Yang, “Density functional theory of atoms

and molecules”, Oxford Univ. Press, 1989

A. J. Cohen, P Mori-Sanchez, and W. Yang, Chem. Rev, 2012

2

The Schrodinger Equation for N electrons

Hª(x1;x2;:::xN) = Eª(x1;x2;:::xN)

x= r; s

H = T + Vee +

NX

i

vext(ri)

T =

NX

i

¡1

2r2

Vee =

NX

i<j

1

rij

vext(r) = ¡X

A

¡ZA

rAi

3

Wavefunction--the curse of dimensionality

Hª=Eª# of electrons wavefunction amount of data

1

2

… … …

N

ª(r1)

ª(r1; r2) 106103

•The amount of data contained in wavefunction grows exponentially

with the number of electrons!

•The problem is caused by the exponential increase in volume

associated with adding extra dimensions to a (mathematical) space.

ª(r1;r2; :::rN) 103N

N » 1¡10000

4

The curse of dimensionality in QM

The problem is caused by the exponential increase in

volume associated with adding extra dimensions to a

(mathematical) space.

5

Exponential growth of information

N, positions and types of atoms

H

103N in ª(r1;r2; :::rN)

+

+, N, v(r)

v(r) is the electrostatic potential from the nuclei

m

v(r) =

AtomsX

A

¡ ZA

jr¡RAj

1.

2.

?

6

Electron density , 3-dimentional only

7

•Electron density is the measure of the probability of an

electron being present at a specific location.

•Experimental observables in X-ray diffraction for structural

determination of small and large molecules

½(r)

Electron density for anilineElectron density for the hydrogen atom

Distance from the nucleus

½(r) =1

¼exp(¡2r)

½(r)

Density Functional Theory

Density functional theory (DFT) (1964, 1965)

1

2.

8

½(r)()N;v(r)()ª

The Nobel Prize in Chemistry 1998

Walter Kohn Pierre C. Hohenberg Lu J. Sham

½(r) =X

i

jÁi(r)j2 Just the sum of molecular orbital densities

DFT: Exchange-correlation energy

9

E = E[½(r)]

= Kinetic energy + potential energy

+ Coulomb interaction energy

+ Exc[½(r)]

E

Exc[½(r)]

•Exchange-correlation energy

•The only unknown piece in the energy

•About 10% of E

Approximations in exchange-correlation energy

10

Exc[½(r)]

• 1965: Kohn and Sham, Local Density Approximation (LDA)

• 1980s-1990s: Axel Becke, Robert Parr, John Perdew

•Generalized Gradient Approximation (GGA)

•Hybrid Functionals (B3LYP, PBE0)

Density Functional Theory

• Structure of matter: atom, molecule, nano,

condensed matter

• Chemical and biological functions

• Electronic

• Vibrational

• Magnetic

• Optical (TD-DFT)

11

ISI Web of Science search for articles with topic “density functional theory”

12

0

10000

20000

30000

40000

50000

60000

103573 DFT records, March 2014

records

A deeper look at the Hamiltonian

PN

i vext(ri) =Rdrvext(r)

PN

i ±(r¡ ri)

H = T + Vee +

NX

i

vext(ri)

PN

i vext(ri)Only depends on atoms and molecules

Rdrf(r)±(r¡ r0) = f(r0)

13

Introducing the electron density

hªjNX

i

vext(ri) jªi =

ZdxN jª(x1;x2;:::xN )j2

NX

i

vext(ri)

=


Zdrvext(r)

NX

i

±(r¡ ri)

=

Zdrvext(r)


NX

i

±(r¡ ri)

=

Zdrvext(r)½(r)

14

½(r) =


NX

i

±(r¡ ri)

= hªjNX

i

±(r¡ ri) jªi

= N hªj ±(r¡ r1) jªi

= N

ZdxN jª(x1;x2;:::xN )j2 ±(r¡ r1)

= N

Zds1dx2:::dxN jª(rs1;x2;:::xN )j2

Introducing the electron density

15

½(r)the electron density

½(r)

1. The electron density at r in physical space

2. The probability of finding an electron at r

3. A function of three variables

½(r) ¸ 0

Zdr½(r) = N

Zdr jr½(r)j2 <1

16

For a determinant wave function

j©0i = jÂ1Â2:::ÂiÂj:::ÂNi½(r) = h©0j

NX

i

±(r¡ ri) j©0i

=

NX

i

h©0j ±(r¡ ri) j©0i

=

NX

i

Zdxi jÂi(xi)j2 ±(r¡ ri)

=

NX

i

Zdxi jÂi(xi)j2 ±(r¡ ri)

=

NX

i

ZdriX

si

jÁi(ri)¾(si)j2 ±(r¡ ri)

=

NX

i

jÁi(r)j2 17

The variational principle for ground states

Most QM methods use the wavefunction as the

computational variable and work on its optimization.

18

An alternative route

Goal: Looking for the max height, and the tallest kid in a school.

1) We can first look for the tallest kid in each class

2) then compare the result from each class

19


Goal: Looking for the max height, and the tallest kid in a school.

1) We can first look for the tallest kid in each class

2) then compare the result from each class

h(Ki) is the height of kid Ki

H(J) = maxKi2class(J) fh(Ki)g, the max height of class J20


is the original variable

H(J) = maxKi2class(J) fh(Ki)g is a constrained search

i

is the reduced variableJ

21

The constrained-search formulation of DFT

Levy, 197922

The constrained search

F [½(r)] = minª!½(r)

hªjT + Vee jªi1. A functional of density; given a density, it gives a

number.

2. A universal functional of density, independent of

atoms, or molecules.

B. The ground state energy is the minimum

E0 = min½(r)

Ev[½(r)]

= min½(r)

½F [½(r)] +

Zdrvext(r)½(r)

¾

A. We have a new

C. is the new reduced variable ½(r)23

The condition on the density

For what density the energy functional is defined?

-- It has to come from some wavefunciton, N-representable

-- the N-representability is insured if

F [½(r)] = minª!½(r)

hªjT + Vee jªi

½(r) ¸ 0

Zdr½(r) = N

Zdr jr½(r)j2 <1 24

The Thomas Fermi Approximation for T

1. Divide the space into many small cubes

2. Approximate the electrons in each cubes as independent

particles in a box (Fermions)

3. T= Sum of the contributions from all the cubes

² =h2

8ml2(n2x + n2y + n2z) =

h2

8ml2R2²

Eigenstate energies for particle in a cubic box

With quantum numbers nx; ny; nz = 1;2;3; ::::

Each state can have two electrons (spin up and spin down).

25

The number of states (with energy < ²) ©(²)

©(²) =1

8

4¼R3²

3=

¼

6(8ml2

h2²)

32

Density of states

g(²) =d©(²)

d²=

¼

4(8ml2

h2)32 ²

Two electrons occupy each state. The kinetic energy is

t = 2

Z ²F

0

²g(²)d² =8¼

5(2m

h2)32 l3²

52

F

N = 2

Z ²F

0

g(²)d² =8¼

3(2m

h2)32 l3²

32

F


26


To express t as a function of density

t =3

10(3¼2)

23 l3(

N

l3)53

N

l3! ½

l3 ! dr

TTF [½] =X

t = CF

Zdr½(r)

53

CF =3

10(3¼2)

23

---Local density approximation for T 27

The Thomas Fermi energy functional

J[½] = 12

Rdrdr0 ½(r)½(r

0)jr¡r0j

For the Vee functional, use the classical electron-electron interaction

energy

ETF [½] = CF

Zdr½(r)

53 +

1

2

Zdrdr0

½(r)½(r0)

jr¡ r0j +

Zdrvext(r)½(r)

The Thomas-Fermi total energy functional is

The minimization of ETF [½] leads to the Euler-Lagrange equation, with ¹

for the constraint ofRdr½(r) = N .

¹ =±Ev[½]

±½(r)=

±TTF [½]

±½(r)+ vJ(r) + vext(r)

or

5

3CF½(r)

23 +

Zdr0

½(r0)

jr¡ r0j + vext(r) = ¹28

The Thomas Fermi Dirac energy functional

Dirac exchange energy functional

ELDAx [½] = ¡CD

Z½(r)

43 dr

Slater, Gaspar, the X® approximation

EX®x [½] = ¡3

2®CD

Z½(r)

43 dr

CD = 34( 3¼)13

Adding the local approximation to the exchange energy

29

1 Question to the students

What is the Thomas –Fermi kinetic energy functional in

2 dimensional space?

In 3dTTF [½] = CF

Zdr½(r)

53

CF =3

10(3¼2)

23

30

Comments on TF theory

•Beautiful and simple theory, with density as

the basic variable

BUT,

•the energy is NOT a bound to the exact

energy

•no shell structure for atoms, no quantum

oscillation

•no bounding for molecules

•Main problem is in the local

approximation for T

31

2. Kohn-Sham Theory and Density Functional

Approximations

•Kohn-Sham equations

•Generalized Kohn-Sham equations

•Exchange and correlation energy functional

•Adiabatic Connections

•Density Functional Approximations

40

Kohn-Sham theory, going beyond the Thomas-Fermi

Approximation

Use a non-interacting electron system, —Kohn-Sham reference system,

to calculate the electron density and kinetic energy

j©si = jÂ1Â2:::ÂiÂj:::ÂNi

½(r) =

NX

i

jÁi(r)j2

Ts[½] = h©sjNX

i

¡1

2r2i j©si

=

NX

i

ZdriÁ

¤i (ri)

µ¡1

2r2i

¶Ái(ri)

=

NX

i

hÁij ¡1

2r2 jÁii

41

Kohn-Sham theory

½(r) =

NX

i

jÁi(r)j2 Ts[½] =

NX

i

hÁij ¡1

2r2 jÁii

² ½(r) =Pi jÁi(r)j2 is the true electron den-

sity, but Ts[½] is not T [½]: However, Ts[½] is

a very good approximation to Ts[½]

² The essence of KS theory is to solve Ts[½]

exactly in terms of orbitals

42

Kohn-Sham theory

½(r) =

NX

i

jÁi(r)j2 Ts[½] =

NX

i

hÁij ¡1

2r2 jÁii

Introduce the exchange correlation energy functional

F [½] = T [½] + Vee[½]

= Ts[½] + J [½] +Exc[½]

Exc[½] = T [½]¡ Ts[½] + Vee[½]¡ J [½]

J[½] = 12

Rdrdr0 ½(r)½(r

0)jr¡r0j

43

Kohn-Sham Equations

½(r) =

NX

i

jÁi(r)j2 Ts[½] =

NX

i

hÁij ¡1

2r2 jÁii

In terms of the orbitals, the KS total energy functional is now

Ev[½] =

NX

i

hÁij ¡1

2r2 jÁii+ J[½] +Exc[½] +

Zdrvext(r)½(r)

The g.s. energy is the minimum of the functional w.r.t. all

possible densities. The minimum can be attained by

searching all possible orbitals

W [Ái] = Ev[Ái]¡NX

i

"i f< ÁijÁi > ¡1g

±W [Ái]

±Ái(r)= 0 44

W [Ái] = Ev[Ái]¡NX

i

"i f< ÁijÁi > ¡1g±W [Ái]

±Ái(r)= 0

The orbitals fjÁiig are the eigenstates of an

one-electron local potential vs(r) if we have

explicit density functional for Exc[½] (KS equa-

tions)

µ¡12r2+ vs(r)

¶jÁii = "i jÁii ;

or a nonlocal potential vNLs (r,r0), if we have

orbital functionals for Exc[Ái] (generalized KS

equations)

µ¡12r2+ vNLs (r,r0)

¶jÁii = "GKSi jÁii : 45

vs(r) =±Exc[½]

±½(r)+ vJ(r) + vext(r)

vNLs (r,r0) =±Exc[±½s(r

0; r)]

±½s(r0; r)+ [vJ(r) + vext(r)] ±(r

0 ¡ r)

µ¡12r2+ vs(r)

¶jÁii= "i jÁii ;

µ¡12r2+ vNLs (r,r0)

¶jÁii= "GKSi jÁii :

Kohn-Sham (KS)

Generalized Kohn-Sham (GKS)

One electron Equations

46

Feastures of Kohn-Sham theory

1. With N orbitals, the kinetic energy is treated rigorously. In

comparison with the Thomas-Fermi theory in terms of

density, it is a trade of computational difficulty for

accuracy.

2. The KS or GKS equations are in the similar form as the

Hartree-Fock equations and can be solved with similar

efforts.

3. KS or GKS changes a N interacting electron problem into

an N non-interacting electrons in an effective potential.

4. The E_xc is not known, explicitly. It is about 10% of the

energy for atoms. 47

Adiabatic connection: from Kohn-Sham reference system to

the true physical system

½(r) =

NX

i

jÁi(r)j2

So far, the E_xc is expressed in a form that is not appealing, not

revealing, not inspiring…

What is the relation of the Kohn-Sham reference system to the true

physical system?

• The same electron density

• Anything else?48



½(r) =

NX

i

jÁi(r)j2

49



• Exc in terms of wavefunction!

• Compared with

• What is the integrand at ?

• Starting point to make approximations –

DFA (Density functional approximation)

52

The Local Density Approximation (Kohn-Sham, 1964)

ELDAxc [½] =

Zdr½(r)"xc(½(r))

"xc(½(r)) is the XC energy per particle of a

homogeneous electron gas of density ½: It is a

function of ½:

Dirac exchange energy functional

ELDAx [½] = ¡CD

Z½(r)

43dr

53

Beyond the Local Density Approximation

ELDAxc [½] =

Zdr½(r)"xc(½(r))

LDA Exc =Rdrf(½) VWN, PW,

GGA Exc =Rdrf(½;r½) BLYP, PW96, PBE

Hybrid Exc = c1EHFx + c2E

GGAxc B3LYP, PBE0

range-separated ....

54

55

Atomization energies for a few selected molecules, mH

57

Mean Absolute Errors (MAE) Thermochemistry(G3 set150), Barriers (HTBH42161 and

NHTB38151), Geometries (T96), Hydrogen Bonding and Polarizabilities

GGA and Meta-GGA

58



59



60

A large class of problems

• Wrong dissociation limit for molecules and ions

• Over-binding of charge transfer complex

• too low reaction barriers

• Overestimation of polarizabilities and hyperpolarizabilities

• Overestimation of molecular conductance in molecular electronics

• Incorrect long-range behavior of the exchange-correlation potential

• Charge-transfer excited states

• Band gaps too small

• Diels-Alder reactions, highly branched alkanes, dimerization of aluminum complexes

Fractional Charges

Delocalization Error61

Error Increases for systems with fractional number of electrons: Yingkai Zhang and WY, JCP 1998

too low energy for delocalized electrons

Savin, in Seminario, “Recent Developments and Applications of Modern DFT”, 1996

+

2H at the dissociation limit

62

DFT for fractional number of electronsfrom grand ensembles,

Perdew, Parr, Levy, and Balduz, PRL. 1982

63

Where can you find fractional charges?

WY, Yingkai Zhang and Paul Ayers, PRL, 2000 – pure states

+

2H

(0) (1)E E E

1

2

1 1 12 2 2

( ) ( ) 2 ( )E E E E

ProtonA

ProtonB

e

ProtonB

eProtonA

at the dissociation limit

e/2Proton

A

ProtonB

e/2

(1) (0)E E E

1 1 1 12 2 2 2

( ) : ( ) (0) (1) (1)E N E E E E 64

The linearity condition in fractional charges: The energy of e/2

ee/2

1 12 2

( ) (1)E E

0

0

-I

( )E N

65

E(N)

66

12 ( ) ( ) ( 1),

2For -convex, < delocalizedN E N E N E N

( )A dimer, with separation: each monomer has E N

12 ( ) ( ) ( 1),

2For -concave, > localizedN E N E N E N

2 x + e

67

Localization Error

Delocalization Error

Delocalization and Localization ErrorPaula Mori-Sanchez, Aron Cohen and WY, PRL 2008

Consequence of Delocalization Error1. predicts too low energy for delocalized distributions

2. gives too delocalized charge distributions 68

Delocalization Error

Define the Delocalization Error as the violation of the

linearity condition for fractional charges

Cohen, Mori-Sanchez and Yang, 2008 Science69

Delocalization Error vs. Self-Interaction Error (SIE)

Delocalization Error (Mori-Sanchez, Cohen and Yang, PRL 2008)• Two SIE-free functionals: Becke06, and MCY2 (2006) did not solve the problems (Self-

interaction-free exchange-correlation functional for thermochemistry and kinetics, Mori-Sanchez, Cohen and Yang, JCP 2006)

• Many-electron SIE were used in 2006 (Mori-Sanchez, Cohen and Yang, JCP 2006, A. Ruzsinszky, J. P. Perdew, G. I. Csonka, O. A. Vydrov, and G. E. Scuseria, JCP 2007)

• Delocalization Error agrees with SIE for one electron systems. For general systems, it reveals the true relevant mathematical error of approximate functionals, and captures the physical nature of the error—delocalization.

SIE (Perdew-Zunger 1982)• Exc error for one-electron systems• Self-Interaction Correction (SIC) forces the correction for

every one-electron orbital, improves atomic systems.• SIC does not improve molecular systems in general (kind

over corrections). • SIC fractional extension (0<N<1, Zhang and Yang, JCP 1998)

explained problem) +

2H

70

Delocalization Error in GGA, LDA, B3LYP

• Energy of dissociation of molecular ion: too low

• Charge transfer complex energy: too low

• Transition state energy: too low

• Charge transfer excitation energy: too low

• Band gap: too low

• Molecular conductance: too high

• (Hyper)polarizability for long molecules: too high

• Diels-Alder reaction products, highly branched alkanes, dimerization of aluminum complexes: too high

Too low energy for fractional charge systems

Cohen, Mori-Sanchez and Yang, 2008 Science71

Seeing the delocalization error 2 7( )Cl H O

Where is the negative charge ?

72

Another large class of problems

• huge error dissociation of chemical bonds

• transition metal dimmers

• some magnetic properties

• strongly correlated systems

• Mott insulators, high superconductors

• degeneracy and near degeneracyc

T

Fractional Spins

Static Correlation Error

Aron Cohen, Paula Mori-Sanchez and WY, JCP, 2008

73

The huge error in

breaking any bond

74

Where can you find fractional spins?

Aron Cohen, Paula Mori-Sanchez and Yang, 2008, JCP

Yang, Ayers and Zhang, PRL 2000

2H

( ) ( )E E E

1

2

2 ( ) 2 ( )E E E

Proton

A

Proton

B↑

Proton

B

Proton

A

at the dissociation limit

Proton

A

Proton

B

( ) ( )E E E

2( ) ( ) ( )E E E

↓

↑ ↓

↑↓↑↓

Fractional Spin H Atom: half spin up electron and half spin down electron75

The constancy condition: energy of fractional spins

( )E 2

( ) ( ) ( )E E E

↑

↑2

76

Static Correlation Error

Define the Static Correlation Error as the violation

of the constancy condition for fractional spins

Cohen, Mori-Sanchez and Yang, 2008, JCP; 2008 Science77

78

Why we have to deal with fractional number of electrons?

½(r)()ª(r1; r2; :::rN)

E = hªjH jªiE = E[½(r)]

Wavefunction TheoryDensity Functional Theory

Fractional charge can occur Integers, always!

Many-electron theories based on

•Green function

•Density matrix

Exact conditions on DFT

, ,[ ]i N i N i NiE c E E

1 1(1 ) (1 )N N N NE E E

, 1, 1(1 ) (1 )i N i j N j N Ni jE c d E E

•!! The exact XC functional cannot be an explicit and differentiable functional of the electron density/density matrix, either local or nonlocal.

•Valid for density functionals, and also for 1-body density matrix functionals, 2-RDM theory, and other many-body theories.

Fractional Charge: 1982: Perdew, Levy, Parr and Baldus

Fractional Spins: 2000, PRL, WY, Zhang and Ayers;2008, JCP, Cohen, Moris-Sanchez, and WY

Fractional Charges and Spins: 2009: PRL, Moris-Sanchez, Cohen and WY

79

: , ,Falt plan for Hydrogen H H and H

H+

H(↓)

H-

H(↑)

Fractional Charges and Spins: PRL 2009, Cohen, Moris-Sanchez, and WY

80

81

Band GapDefinition of fundamental gap

derivative

Only if is linear.

82

How can fundamental gap be predicted in DFT

• LUMO energy is the chemical potential for electron addition

• HOMO is the chemical potential for electron removal

• Fundamental gaps predicted from DFT with KS, or GKS

calculations, as the KS gap or the GKS gap

• For orbital functionals, the LUMO of the KS (OEP) eigenvalue

is NOT the chemical potential of electron addition.

Thus the KS gap is not the fundamental gap predicted by the

functional.

@Ev(N)

@N= hÁf jHe® jÁfi

WY, Mori-Sanchez and Cohen, PRB 2008, JCP 201283

Convex curve (LDA, GGA):derivative underestimates I, overestimates A, I-A is too small

Concave curve (HF):derivative overestimates I, underestimate A, I-A is too large

1,E

N EN

For Linear E(N)

84

How well can fundamental gap be predicted in DFT

• Fundamental gaps predicted from DFT with KS, or GKS

calculations, as the KS gap or the GKS gap

• Only works well if functionals have minimal

delocalization/localization error.

85

2012

86

Improving band gap prediction in density functional theory from

molecules to solidsPRL, 2011, Xiao Zheng, Aron J. Cohen, Paula Mori-

Sanchez, Xiangqian Hu, and Weitao Yang

Paula Mori-Sanchez(Univ. Autonoma Madrid)

Xiao Zheng(USTC)

Aron J. Cohen(Cambridge) Xiangqiang Hu

87

88

Nonempirical Scaling correction

To linearize an energy component, Ecomp, for a system of N + n (0 < n < 1) electrons, we construct

1 and 0 at E reproduces and , with linearly scales )(E compcomp nnnnN~

The SC to Ecomp is

It is then important to cast ΔEcomp into a functional form, so that it depends explicitly on ρ(r), or on Kohn-Sham first-order reduced density matrix ρs(r,r’)

88

89

Energy versus fractional electron number

The SC significantly restores linearity condition for energy89

90

Band gaps by various DFA and S-DFASummary of band-gap prediction for a variety of systems

The SC preserves the accuracy of I, A, and integer gaps, while it improves significantlyon HOMO and LUMO energies, and derivative gaps

S-MLDA predicts reasonable band gaps with consistent accuracy for systems of allsizes, ranging from atoms and molecules to solids

90

91

Band gaps of H-passivated Si nanocrystalsComputational details

Systems under study: H-passivated spherical Si nanocrystals. The largest system isSi191H148 with a diameter of 20 angstrom

Geometries optimized by B3LYP with Lanl2dz ECP basis set, except that the largestsystem Si191H148 is optimized with semiempirical PM3 method

Diffusive basis functions are important to obtain accurate HOMO-LUMO gaps: 6-31G for H atoms; an sp-shell with an exponent of 0.0237 au and a d-shell with anexponent of 0.296 au added to Lanl2dz basis for Si atoms

Si191H148

91

92

Band gaps of H-passivated Si nanocrystals

The S-MLDA HOMO-LUMO gaps agree well with the GW gaps

S-MLDA

LDA

92

Local Scaling Correction, PRL, 2015

Paula Mori-Sanchez

(Univ. Autonoma Madrid)

Xiao Zheng

(USTC)

Aron J. Cohen

(Cambridge))

Chen Li

(Duke)

93

Local fractional occupation numbers n(r)

Local Scaling Correction




• Polarizablity (longitudinal)

Hydrogen chain

2 n(H )


Polyethilene (PY)

2 nH(C ) H


Polythiophene (PT)

4 2 nH(C H S) H

• OH(H2O)n

Documents

Density Functional Theory and Exchange-Correlation Functionals · 1. A functional of density; given a density, it gives a number. 2. A universal functional of density, independent