Density Functional Theory

Weitao Yang

Duke University

Peking University

South China Normal University

2H

Funding

NSF

NIH

DOE

Theory

Biological

Nano

Material

FHI-Beida

July 30 20181

Outline of Three Topics

1. Electronic Structure Theory Overview and

Introduction to DFT

HK Theorems, Kohn-Sham Equations and Density Functional

Approximations

2. Insight into DFT

Adiabatic Connections, Density Functional Approximations, Fractional

Perspectives of DFT, Challenges in DFT—Delocalization and Static

Correlation Error, Band Gaps, Chemical potentials, Derivative Discontinuity,

Physical Meaning of Frontier Orbital Energies (HOMO and LUMO).

3. Recent Developments of xc Functionals for Large

Systems

SCAN meta GGA

Localized Orbital Scaling Corrections

2

Key references

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

molecules”, Oxford Univ. Press, 1989

RM Dreizler, EKU Gross, “Density Functional Theory: An

Approximation to the Quantum Many-Body Problems”

Cohen, A. J., Mori-Sánchez, P. & Yang, W. “Challenges for

Density Functional Theory”. Chemical Reviews 112, 289–320,

2012.

SCAN: Jianwei Sun, Adrienn Ruzsinszky, and John P. Perdew

Phys. Rev. Lett. 115, 036402, 2015

LOSC: C. Li, X. Zheng, N. Q. Su, and W. Yang, National Science

Review, vol. 5, no. 2, pp. 203–215, 2018.3

Roles of Electrons

Matter

Molecules, Atoms

Nuclei, Electrons

C1

H

HH

H

Electrons are the glue that bind the

repulsive nuclei and determine the

structure of matter, from molecules

to macromolecules, to bulk materials. 6

Quantum Mechanics

The wave-like behavior of electrons

Electron Beam

7

Quantum Mechanics

Erwin Schrödinger

8

Adiabatic Approximation:

Born-Oppenheimer (BO) Approximation

9

The time-dependent Schrodinger equation for the entire system

(containing both electrons and nuclei)

The Quantum Adiabatic Theorem: A physical system remains in its

instantaneous eigenstate if a given perturbation is acting on it slowly

enough (and if there is a gap between the eigenvalue and the rest of the

Hamiltonian's spectrum).

The nuclei move slowly, compared to electrons, because of the difference in

mass.

Considering the nuclei motion as perturbation to the electrons, and

assuming that the nuclei move slowly enough, then the electrons are in its

instantaneous eigenstate of a given nuclei configuration. This is the BO

approximation.

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

10

Methods for Solving the Schrodinger Equation

for N electrons

Hª(x1;x2;:::xN) = Eª(x1;x2;:::xN)

11

1. Hartree-Fock mean-field theory

2. Perturbation Theory

3. Configuration interaction

4. Coupled-cluster theory

5. Quantum Monte Carlo

6. Green function theory (for excitation spectrum)

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

12

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.

13

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.

?

14

Electron density , 3-dimentional only

15

•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. Variational principle for ground state energy over all densities.

3.16

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

17

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

18

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)

19

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

20

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)

21

Introducing the electron density

hªjNX

i

vext(ri) jªi =

ZdxN jª(x1;x2;:::xN )j2

NX

i

vext(ri)

=

ZdxN jª(x1;x2;:::xN )j2

Zdrvext(r)

NX

i

±(r¡ ri)

=

Zdrvext(r)

ZdxN jª(x1;x2;:::xN )j2

NX

i

±(r¡ ri)

=

Zdrvext(r)½(r)

22

½(r) =

ZdxN jª(x1;x2;:::xN )j2

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

23

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

24

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 25

The variational principle for ground states

Most QM methods use the wavefunction as the

computational variable and work on its optimization.

26

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

27

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

h(Ki) is the height of kid Ki

H(J) = maxKi2class(J) fh(Ki)g, the max height of class J28

An alternative route

is the original variable

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

i

is the reduced variableJ

29

The constrained-search formulation of DFT

Levy, 197930

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)31

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 32

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).

33

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

The Thomas Fermi Approximation for T

34

The Thomas Fermi Approximation for T

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 35

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) = ¹

36

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

37

1 Question to the students

Question: What is the Thomas –Fermi kinetic energy functional in

2 dimensional space?

In 3dTTF [½] = CF

Zdr½(r)

53

CF =3

10(3¼2)

23

38

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

39

Functions and Functionals

I Ã f(x) I =R badxf(x)

E Ã ª(x) E[ª] = hªjH jªiE Ã ½(r) E[½] =?

• A function of one variable is a mapping from a number to a number

• A function of multi-variables is a mapping from muny numbers to a

number

• A functional is a mapping from a function to a number.

Functional is the continuous version of function of

infinitively many variables.

40

Functional derivatives

Functional Derivatives: Continuous extension of multivariable derivatives

41

Functional derivatives

dy(t) = lim²!0

[y(t+ ²dt)¡ y(t)]

²

= y0(t)dt

= lim²!0

dy(t+ ²t)

d²

±F [f(x)] = lim²!0

[F [f(x) + ²±f(x)]]¡ F [f(x)]

²

=

Zdx

±F

±f [x]±f [x]

= lim²!0

dF [f(x) + ²±f(x)]

d²

dy(t1; t2; :::) =X

i

@y(t1; t2:::)

@tidti

42

Functional derivatives

±F [f(x)] =

Zdx

±F

±f(x)±f(x)

The meaning of functional derivative: let ±f(x) = ±(x¡ x0), then

±F [f(x)] =

Zdx

±F

±f(x)±(x¡ x0) =

±F

±f(x0)

43

How to take functional derivatives

±F [f(x)] =

Zdx

±F

±f(x)±f(x)

F [f(x)] =

Zdxg(f(x))

±F [f(x)] = lim²!0

d

d²

Zdxg(f(x) + ²±f(x))

= lim²!0

d

d²

Zdx

½g(f(x)) +

dg(f(x))

df²±f(x)

¾

=

Zdx

½dg(f(x))

df±f(x)

¾

±F

±f(x)=

dg(f(x))

df 44

Functional derivatives

F [f(x)] =

Zdxg(f(x))

±F

±f(x)=dg(f(x))

df

F [f(x)] =

Zdxf(x) ±F

±f(x)= 1

TTF [½] = CF

Zdr½(r)

53

±TTF [½]

±½(r)=

5

3CF½(r)

23

45

Functional derivatives

ETF [½] = CF

Zdr½(r)

53 +

1

2

Zdrdr0

½(r)½(r0)

jr¡ r0j +

Zdrvext(r)½(r)

use ¹ for the constraint ofRdr½(r) =N

±

±½

½ETF [½]¡ ¹(

Zdr½(r)¡N)

¾= 0

The condition for the minimization of the energy functional is

5

3CF½(r)

23 +

Zdr0

½(r0)

jr¡ r0j + vext(r) = ¹

46

How to take functional derivatives

±F [f(x)] =

Zdx

±F

±f(x)±f(x)F [f(x)] =

Zdxg(f(x); f 0(x))

±F [f(x)] = lim²!0

d

d²

Zdxg(f(x) + ²±f(x); f 0(x) + ²±f 0(x))

= lim²!0

d

d²

Zdx

½g(f(x)) +

@g

@f²±f(x) +

@g

@f 0²±f 0(x)

¾

=

Zdx

½@g

@f±f(x)

¾+

Zdx

@g

@f 0±f 0(x)

=

Zdx

½@g

@f±f(x)

¾+

Z@g

@f 0d(±f(x))

=

Zdx

½@g

@f±f(x)

¾¡Z

dx

µd

dx

@g

@f 0

¶±f(x) +

@g

@f 0±f(x) jboundary

±F

±f(x)=dg(f(x))

df¡ d

dx

@g

@f 047

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

49

Kohn-Sham theory

½(r) =

NX

i

jÁi(r)j2 Ts[½] =

NX

i

hÁij ¡1

2r2 jÁii

² ½(r) =P

i 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

50

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

51

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 52

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 : 53

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

54

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. 55