electronic excitations by time-dependent dft and bethe ......electronic excitations by...

Electronic excitations by time-dependent DFT

and Bethe-Salpeter equation

Hong Jiang (蒋鸿)

College of Chemistry, Peking University

Hands-On DFT and Beyond: Frontiers of Advanced Electronic Structure and Molecular Dynamics Methods

Peking University, Beijing, China, July 30th to August 10th, 2018

Recommended references General textbooks on optical properties of materials

◆ P. Y. Yu and M. Cardona, Fundamentals of Semiconductors: Physics

and Materials Properties, (4th ed), Springer (2010).

◆ G. Grosso and G. P. Parravicini, Solid State Physics, Elsevier (2006).

◆ M. Fox, Optical Properties of Solids, Oxford Uni. Press (2001).

GW+BSE

◆ G. Onida, L. Reining, A. Rubio, Electronic excitations: density-

functional versus many-body Green’s-function approaches, Rev. Mod.

Phys. 74, 601 (2002).

◆ G. Strinati, Application of the Green's Functions Method to the Study

of the Optical Properties of Semiconductors, Riv. Nuovo Cimento 11,

1 (1988).

Recommended references

TDDFT

➢ E.K.U. Gross, J.F. Dobson, and M. Petersilka, Density functional theory

of time-dependent phenomena (Topics in Current Chemistry, vol 181,

Springer, 1996).

➢ M. R. L. Marques, E. K. U. Gross, Time-dependent density-functional

theory, Annu. Rev. Phys. Chem. 55, 427(2004).

➢ S. Botti et al., Time-dependent density-functional theory for extended

systems, Rep. Prog. Phys. 70, 357 (2007).

➢ M.E. Casida and M. Huix-Rotllant, Progress in Time-Dependent Density-

Functional Theory, Annu. Rev. Phys. Chem. 63, 287 (2012).

➢ C. A. Ullrich, Time-Dependent Density-Functional Theory: Concepts and

Applications (Oxford Uni. Press, 2012).

➢ M. R. L. Marques et al. (ed.), Fundamentals of Time-Dependent Density

Functional Theory, Springer (2012).

Short introduction to research in my group

http://www.chem.pku.edu.cn/jianghgroup

Theoretical Materials Chemistry (TMC) group

Materials for solar energy conversion

Molecular magnetic materials

GW-based electronic band structure theory

First-principles approaches for strong correlation

Theory & Computation

of d/f-electron materials

Surface and catalysis of transition metals

and oxides

All electron GW: GW with Augmented Planewaves

GAP (GW with Augmented Planewaves)

◆ Based on LAPW (no pseudopotentials !)

◆ Interfaced with WIEN2k (P. Blaha et al. (2001))

H. Jiang，R. I. Gomez-Abal, et al., Computer Phys. Commun., 184, 348 (2013).

Numerically accurate GW with LAPW+HLOs

Numerically accurate GW: LAWP enhanced by high-energy local

orbitals (HLOs）

+

LAPW+HLOs

LAPW

H. Jiang*, P. Blaha, Phys. Rev. B 93, 115203 (2016).

ZnO, Nk=23

Numerically accurate GW with LAPW+HLOs

H. Jiang*, P. Blaha, Phys. Rev. B 93, 115203 (2016).

ZnO

M.-Y. Zhang & HJ, in preparation (2018).

Ln2O3 band gaps by GW0@LDA+U

H. Jiang et al. Phys. Rev. Lett. 102, 126403(2009);

Phys. Rev. B 86, 125115(2012).

GW@DFT+U with LAPW+HLOs for f-oxides

Ce2O3

H. Jiang, Phys. Rev. B 97, 245132(2018).

Doubly screened hybrid functional

Cui, Wang, Zhang, Xu, HJ, J. Phys. Chem. Lett. 9, 2338(2018)

ACFDT-RPA：relative stability of TiO2 phases

Z.-H. Cui, F. Wu, and H. Jiang*, Phys. Chem. Chem. Phys. 18, 29914 (2016).

ACFDT-RPA: FeS2 phase stability

M.-Y. Zhang, Z.-H. Cui, HJ, J. Mater.

Chem. A, 6, 6606 (2018)).

Optical absorption: basic physics

Light-matter interactions: physical processes

P. Y. Yu and M. Cardona, Fundamentals of Semiconductors: Physics and Materials Properties,

(4th ed), Springer (2010).

Optical coefficients

R : coefficient of

reflection (reflectivity)

: refractive index c

nv

Absorption coefficient

0( ) zI z I e −=Dispersion:

( )n n =

Optical absorption: the dielectric constant

n n i= +Complex refractive index

Electromagnetic wave in a medium

( )

0( , ) ei kz tE z t E −=( )n in

kv c c

+= = =

( / )

0( , ) e ez

i nz c t cE z t E

−

−=2

2

0

2( , ) e ( )

zcI E z t I I

c

−

= =

M 1 2n i = +=

( )

( )

1/22 2

1 1 2

1/22 2

1 1 2

1

2

1

2

n

= + +

= − + +

in a medium

n

M2 Im (( )

( ) )nc nc

=

Independent particle approximation (IPA)2

0 int

1ˆ ˆ ˆˆ ( , ) ( )2 e

eH t V H H

m c

= + + +

p A r r

( )2

(0) (0)

2

02( , ) ˆ

i f f

i

e

i f i

eAe

m cP

→

= − −q r

eq p

(0) (0) (0) (0)

',i v f c = =k k

Fermi’s golden rule

( )2

2(0) (0)0

' '

2ˆ( , ) i

i f c v c v

e

eAP e

m

→

= − −

q

k k k k

rq e p

(0

int

)

ˆ( , )

ˆ

( )

. .

ˆ

e

e

i t

et

m c

e

m c

H t

e cA

c −

=

= +q r

A r p

pe

,i i

,f f

Independent particle approximation (IPA)

( )2

2(0) (0)0

, ,

2ˆ( , ) 2 i

c v c v

v ce

eAW e

m c

+ +

= − −

k q k

r

k k

q

k

qq e p

( )2

2(0) (0)0

' '

2ˆ( , ) i

i f c v c v

e

eAP e

m

→

= − −

q

k k k k

rq e p

( )2 2

2(0) (0)

2 2 2, ,

8 1ˆ( , ) i

c v c v

v ce

ee

m V

+ += − − k q k k q k

q r

k

q e p

For optical absorption around visible light regime: q0. .

➔ Dipole approximation

1 ( )ie O q +q r

What is missing?

Electron-hole interaction

(from F. Sottile’s talk)

Microscopic description of optical absorption

reducible polarizability

ext

(1)(1,2)

(2)V

=

irreducible polarizability (1)

(1,2)(2)

PV

=

ext H ext

1 2

(2)(1) (1) (1) (1) d(2)V V V V

= + = +

− r r

1 H

ext ext

(1)(1) (3)(1,2) (1,2) (3)

(2) (3) (2)

(1,2) (3) (1,3) (3,2)

VVd

V V

d v

− = = +

= +

On the other hand

1

ext

(1) (3)(1,2) (3) (3) (1,3) (3,2)

(3) (2)

Vd d P

V V

−= =

(1,2) (1,2) (3) (1,3) (3,2)d v P = −

1 1 1 1 11 ( , , ) ( , )s t t r x

Inverse dielectric function

dielectric function

Macroscopic dielectric constant and local field effect

Optical radiation field: a spatially smooth function

ext

(

x

)

e t;0( ; ) e

(

( ; )

; ) \ ( , ; ) e

i

i

V

V V

V

+

=

=

q r

q G r

G

G

r

r

q

q G

Macroscopic averaging of microscopic quantity

0

1 1

, ' ext; ' ,0 ext;0

'

( ; ) e

( ; ) ( ; ) ( ; ) ( ;

)

) )

;

;

(

(

iV

V V V

V

−

=

−

=

= =

q r

G G

G

G G G

G

r

q q q q q

q

Macroscopic dielectric function

ext; 0

M 1

0 0, ' 0

( ; ) 1( , )

( ; ) ( , )

V

V

=

−

= = =

= =G

G G G

qq

q q

Neglecting local field effect (LFE) M 0, ' 0( , ) ( , ) = =G Gq q

Adler, Phys. Rev. 126, 413 (1962); N. Wiser, Phys. Rev. 129, 62 (1963).

Theoretical approaches to optical absorption➢ Independent particle approximation (IPA)

➢ Random phase approximation (RPA i.e. IPA with LFE)

➢ Time-dependent density functional theory (TDDFT)

➢ Bethe-Salpeter equation (BSE)

M 0 0; 0, ' 0( , ) 1 ( ) ( , )v P = = =− G G Gq q q

0(1,2) (1,2) (3) (1,3) (3,2)d v P = − M 1

0, ' 0

1( , )

( , )

−

= =

=G G

qq

KS KS xc(1,2) (1,2) ( ) (1,3) (3,4) (3,4) (4,2)d v f = + + 34

1(1,2) (1,2) (3) (1,3) (3,2)d v − = +

M 1

0, ' 0

1( , )

( , )

−

= =

=G G

qq

KS 0

KS

(1)(1,2) (1,2)

(2)P

V

( )( )

* *

0

,

( ) ( ) ( ') ( ')( , '; )

i j j i

i j

i j j i

P f fi

= −

− − +

x x x xx x

0 0(1,2;1', 2 ') (1,2;1', 2 ') d(3456) (1,4;1',3) (3,5;4,6) (6,2;5,2 ')L L L K L= +

Time-dependent density-functional theory

( ) ( ) ( )2

ex xt H c

1( , ) , , ;[ ( , [ ( ', ')])] , ;; ( , )

2i ii t V t V t n t t t

tV n t

= − + + +

r r r r r rr

Time-dependent many-body problems

M. R. L. Marques, E. K. U. Gross, Annu. Rev. Phys. Chem. 55, 427 (2004).

Time-dependent many-electron systems

( ) ( )

( )2

2

ext

1

ˆ, ( ) ,

1 1ˆ ( ) ,2 2

N N

i i

i i j i j

i t H t tt

H t v tm=

=

= + +

−

r r

rr r

Time-dependence of the external potential

• coupling with moving nuclei

• coupling with a time-dependent field

( )ext , =( )

MI

I I

Zv t

t

−

−r

r R

( )ext , = ( , )M

I

I I

Zv t t

−+

−r E r r

r R

Runge-Gross Theorem

E. Runge, E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984).M. R. L. Marques, E. K. U. Gross, Annu. Rev. Phys. Chem. 55, 427 (2004).

There is a one-to-one correspondence between the external (time-

dependent) potential vext(r, t) and the electronic density n(r, t), for many-

body systems evolving from a fixed initial state.

( ) ( )

( )2

2

ext

1

ˆ, ( ) ,

1 1ˆ ( ) ,2 2

N N

i i

i i j i j

i t H t tt

H t v tm=

=

= + +

−

r r

rr r

( , ) '( , ) ( ) ( , ) '( , )v t v t c t n t n t + r r r r

Time-dependent Kohn-Sham equation

Action functional 1

0

ˆ[ ( )] ( ) ( ) ( )

[ ( )] ˆ0 ( ) ( ) 0( )

t

tA t t i H t t

t

A ti H t t

t t

−

= − =

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1

0 0

1

00

1

ext

ex

e

t

e

H xc

, ( ) ( ) , ,

ˆ

,

ˆ

, , , , ,t

t t

t

Tt t

t

t

i Tt

A

V

n t V t

A n t t t dt n t V t dt

n t V t dt dt A n tn t

−

= −

−

− + −

r r r

r r r r rr

Using the Runge-Gross theorem

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 1

0 0xc

c

ext H

xcx

, = , , , , +

, , ,

,,

,s

s

t t

t tT

T T

AA n t n t V t dt n t V t dt

A n t A n

n t

A n t n

A

A

n

t t

t

−

+ −

−

r r r r r

r r

r

r

r

r

Ansatz: For a given time-dependent interacting N-electron system with

a given initial state , there exists a fictious non-interacting N-

electron system that has the same time-dependent electron density

n(r,t).

0( )t t =

E. Runge and E. K. U. Gross Phys. Rev. Lett. 52, 997(1984).

Time-dependent Kohn-Sham equation

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

1 1

0 0

xc

ext H, = , , , ,

,

,

, ,

s

s

t t

T T

tT

t

A n t A n t A

A n t n t V t dt n t V

n t

tt tn dA −

+

−

−

+

r r

r

r

r r r rr

( )1

0

1

01

2

, = ( ) ( )

(

ˆ

1) ( )

2

s

t

Tt

N t

i it

i

i Tt

it

A n t t t dt

t t dt =

=

−

+

r

0( )i

A

t

=

( ) ( ) ( )2

ex xt H c

1( , ) , , ;[ ( , [ ( ', ')])] , ;; ( , )

2i ii t V t V t n t t t

tV n t

= − + + +

r r r r r rr

( ) xxc

c

),

( ,V t

A

n t

=

rr

( )2

1

, ( , )N

i

i

n t t=

=r r

( )xc ,A n t r

E. Runge and E. K. U. Gross Phys. Rev. Lett. 52, 997(1984).

Three ways of using TDDFT

KS KS xc

1( , '; ) ( , '; ) '' ''' ( , ''; ) ( '', '''; ) ( ''', '; )

'' '''d d f

+ +

x x x x x x x x x x x x

r - r

➢ Linear response (LR) TDDFT

➢ Casida equation: converting the LR-TDDFT integral equation to an

eigen-value equation

➢ Real-time TDDFT

• as a way to calculate optical spectrum

• Electron-nuclear coupling, strong laser field, non-linear optical properties……

1(1,2) (1,2) (3) (1,3) (3,2)d v − = +

M 1

0, ' 0

1( , )

( , )

−

= =

=G G

qq

( ) ( )* *

KS

,

( ) ( ) ( ') ( ') , ';

i

i j j i

i j

i j j i

f f

= −− + +

x x x x

x x

Linear response TDDFT （1）( ) ( ) ( )

( ) ( ) ( ) ( )

( )

(0) (1)

ext ext ext

(0) (1) (2)

(1) 3 (1)

ext

, = ,

, = , ,

, ' ' ( , ' ') ( ' ')

V t V V t

n t n n t n t

n t dt d r t t V t+

−

+

+ + +

=

r r r

r r r r

r r r r

Weak external perturbation

( ) ( )

ext

KS

extKS

( , ' ') '' '( '' '')

( ' ')'

( '' '') )( ' '

n t

V

n V t

V t

td t

t tt t d

V

=

x x

xx

x

xx x

x

( )( ) ( )

* *

KS K

,KS

S

( ) ( ) ( ') ( ')( , ' ') ,

( ' ')';

i

i j j i

i j

i j j i

n t

V tt t f f

= −

− + +

x x x xx x x x

x

x

Linear density response function

( ) ( ) ( ) ( )

( )

( ) ( ) ( )

KS ext H xc

KS

e

Hxc

ext

ee xc

xt

, , , ;[ ] , ;[ ]

( ) ( '' ''), ' ' d ''d ''

( '' '') ( ' ')

, ' ' d ''d '' ( '') ( '') , ''

( )

( '

''

')

'' '', ' '

V t V t V t n V t n

V t n tt t t

n t V t

t t t v t t f t t t

V t

t

t

V

= + +

= +

= + − − +

x x r r

x xx x x

x x

x x x r r x x

x

x x

x

Linear response TDDFT （2）( ) ( ) ( )

( ) ( ) ( ) ( )

( )

(0) (1)

ext ext ext

(0) (1) (2)

(1) 3 (1)

ext

, = ,

, = , ,

, ' ' ( , ' ') ( ' ')

V t V V t

n t n n t n t

n t dt d r t t V t+

−

+

+ + +

=

r r r

r r r r

r r r r

Weak external perturbation

( ) ( )

ext

KS

extKS

( , ' ') '' '( '' '')

( ' ')'

( '' '') )( ' '

n t

V

n V t

V t

td t

t tt t d

V

=

x x

xx

x

xx x

x

( )( ) ( )

* *

KS K

,KS

S

( ) ( ) ( ') ( ')( , ' ') ,

( ' ')';

i

i j j i

i j

i j j i

n t

V tt t f f

= −

− + +

x x x xx x x x

x

x

Linear density response function

( ) ( ) ( ) ( )

( )

( ) ( ) ( )

KS ext H xc

KS

e

Hxc

ext

ee xc

xt

, , , ;[ ] , ;[ ]

( ) ( '' ''), ' ' d ''d ''

( '' '') ( ' ')

, ' ' d ''d '' ( '') ( '') , ''

( )

( '

''

')

'' '', ' '

V t V t V t n V t n

V t n tt t t

n t V t

t t t v t t f t t t

V t

t

t

V

= + +

= +

= + − − +

x x r r

x xx x x

x x

x x x r r x x

x

x x

x

KS KS xc

1( , '; ) ( , '; ) '' ''' ( , ''; ) ( '', '''; ) ( ''', '; )

'' '''d d f

+ +


r - r

M. Petersilka, U. J. Gossmann, E. K. U. Gross, Phys. Rev. Lett. 76, 1212(1996).

Exchange correlation kernel

( ) xcxc

( ), ' '

( ' ')

V tf t t

n t

xx x

x

Causality dilemma:

Causality principle ➔ fxc (xt,x’t’) = 0 if t<t’.

but

( )

( )2

xcxcxc xc

( )( ), ' ' = = ' ',

( ' ') ( ) ( ' ')

A n tV tf t t f t t

n t n t n t

=

xxx x x x

x x x

Solution：Keldysh’s time-contour formalism to non-equilibrium dynamics

R. van Leeuwen, Phys. Rev. Lett. 80, 1280 (1998).

Lehmann representation of (x, x’;)

* *

†

( ) ( ') ( ') ( )( , '; )

( , ) ( )

ˆ ˆˆ( ) ( ) , ( ) ( ) ,

s

s

s

s

s s

s

s

s

n n n n

E N s E N

n N n N s N N

i

s

i

= −

− + + +

−

x x x x

x x

x x x x

Exercise: Using time-dependent perturbation theory to derive the

equation above with the following perturbation:

( )(1)

extˆ '( )= ,

N

i

i

H t V t r

S. Botti et al. Rep. Prog. Phys. 70, 357 (2007).

Casida equation

* *( ) ( ') ( ') ( )ˆ( , '; ) ( ) ( ) ,s s s s

s

s s s

n n n nn N n N

is

i

= −

− + + +

x x x xx x x x

KS KS xc

1( , '; ) ( , '; ) '' ''' ( , ''; ) ( '', '''; ) ( ''', '; )

'' '''d d f

+ +


r - r

( ) ( )

( ) ( )* *

=

A B X 1 0 X

B A Y 0 -1 Y

( ) ( )

( )

* *

, ' ' ' ' ' '

* *

, ' ' ' '

' ( ) ( ) ( , '; ) ( ') ( ')

' ( ) ( ) ( , '; ) ( ') ( ')

ai a i a i aa ii a i a i

ai i a a i i a

A d d f

B d d f

− +

Hxc

Hxc

x x x x x x x x

x x x x x x x x

Tamm-Dancoff approximation (TDA): B 0

= AX XM. E. Casida, in Density Functional Methods, Part I (ed. DP Chong), p. 155. World Sci. (1995);

M. A. L. Marques, E. K. U. Gross, in A Primer in Density Functional Theory (2003).

Approximations in TDDFT

( ) xcxc

(, ' '

)

( ' ')f t

V t

n tt

x

xx x( ) x

xcc

),

( ,V t

A

n t

=

rr

( )1 1

0 0ee eexc , ( ) ( ) (ˆ ) (ˆ )ˆ ˆ

t t

t tA n t t t dti T i Tt dtV tV

t t

−

− − −

−

r

Adiabatic approximation:

( ) ( )(adiabatic (DF) A)

xc xc ( , ),V t V n t= rr

( ) xc

( , )

ALDA

(adiabatic)

x

(LDA

x

c

)

c

( )

( ( , )) (

, ' '

') ( ')

n n t

f tV n

n

f n t t t

t

=

⎯⎯⎯→ −

=

−

r

x x x

x x

Main limiting factors:

❑ Accuracy of Vxc(r,t): especially, asymptotic behaviors for finite systems.

❑ Spatial dependence of fxc(r,r’), especially for extended systems.

❑ time/energy dependence (memory effects) of the kernel fxc(r,r’;)

TDDFT performances: molecules

S. Botti et al., Rep. Prog. Phys. 70, 357 (2007).

Na8

TDDFT performances: molecules

D. Jacquemin, et al. Phys. Chem. Chem. Phys., 2011, 13, 16987–16998

Thiel’s databasesinglet triplet

TDDFT performances: solids

Botti S, et al. Phys. Rev. B 72 125203(2005)

LiF

Bethe-Salpeter Equation (BSE) for exciton effects

Two-particle correlation function

G. Strinati, Riv. Nuovo Cimento, 11, 1 (1988).

Two-particle correlation function

2 1 1(1,2;1', 2 ') (1,2;1', 2 ') (1,1') (2,2 ')L G G G − +

( ) ( ) ( ) ( ) ( )2 † †

2ˆ(1,2;1', 2 ') i 1 2 2 ' 1'G N NT = −

Two-particle Green’s function 1 1 1 1 11 ( , , ) ( , )s t t r x

For optical absorption

1 2 1 2 11 2 1 2 1 2 1 2 2( , ; ' , ' ) ( , ; ', '; )t t t tLt tL + + −x x x x x x x x

( ) ( )

* *

1 1 2 2 2 2 1 11 2 1 2

†

( , ') ( ', ) ( , ') ( ', )( , ; ', '; )

ˆ ˆ( , ') ' ,

( , ) ( )

s s s s

s

s

s s

s

X X X XL i

X N N s

E N

i

E

i

N s

− + +

= −

−

−

x x x x x x x x

x x x x

x x x x

Lehmann representation

Bethe-Salpeter equation (BSE)

G. Strinati, Riv. Nuovo Cimento, 11, 1 (1988).

0 0

0 1 1

(1,2;1', 2 ') (1,2;1', 2 ') d(3456) (1,4;1',3) (3,5;4,6) (6,2;5,2 ')

(1,2;1', 2 ') (1,2 ') (2,1')

L L L K L

L G G

= +

Hxc

1 1

(3,4) (3,4)(3,5;4,6) (3,4) (5,6) (3,6)

(6,5) (6,5)K i v

G G

= − +

12 2 2 2

2 2 2

(1,1')(1, ;1', ' )

( ', ; )

GL t t

U t

+ =x xx x

( )(1,2) 1,2;1 ,2iL + += −

M. Rohlfing, S. Louie, Phys. Rev. B 62, 4927 (2000)

( ) ( )

( )

( ) ( )

( )

( ) ( ) ( )

* *

1 1 2 2 2 2 1 1

0 1 2 1 2

,

*

1 1 1 1

, ' ', , ' ',( , ; ', '; ) i

i i

, ' '

vc vc vc vc

v c c v c v

ij j i

X X X XL

E E E E

X

= −

− − + + − −

x x x x x x x x

x x x x

x x x x

( ) ( ) ( ) ( )* *

1 1 2 2 2 2 1 1

1 2 1 2

, ' ', , ' ',( , ; ', '; )

s s s s

s s s

X X X XL i

i i

= −

− + + −

x x x x x x x xx x x x

( ) ( ) ( )occ unocc

*

; ,, ' , ' ',s vc s vc vc s vc

v c

X A X B X = + x x x x x x

BSE in the matrix form (1)

0 0

0 1 1

(1,2;1', 2 ') (1,2;1', 2 ') d(3456) (1,4;1',3) (3,5;4,6) (6,2;5,2 ')

(1,2;1', 2 ') (1,2 ') (2,1')

L L L K L

L G G

= +

1 2 1 2 11 2 1 2 1 2 1 2 2( , ; ' , ' ) ( , ; ', '; )t t t tLt tL + + −x x x x x x x x( )*

1

( ) ( '), ';

sgn( )

n n

n n n

GE i E

=

− − −

x xx x


( ) ( ) ( )occ unocc

*

, ,, ' , ' ',s vc s vc vc s vc

v c

X A X B X = + x x x x x x

( ) ( ) ( )

( ) ( ) ( )

AA AB' ', ,. ' ' , ' ' , ' '

BA BB' ', ,, ' ' . ' ' , ' '

v c s vc sc v vc v c vc v c s vc v c s

s

v c s vc svc v c s c v vc v c vc v c s

A AE E K K

B BK E E K

− + = − − − −

( ) ( )

( ) ( )

AA *

, ' ' 3 4 3 5 4 6 ' ' 6 5

AB *

, ' ' 3 4 3 5 4 6 ' ' 5 6

(3456) ( , ) , ; , ; ( , )

(3456) ( , ) , ; , ; ( , )

vc v c vc v c

vc v c vc v c

K i d X K X

K i d X K X

=

=

x x x x x x x x

x x x x x x x x

BSE in the matrix form (2)

KAB≈0


( ) ( )AA

. ' ' , ' ' ' ', ,

' '

c v vc v c vc v c s v c s s vc s

v c

E E K A A − + =

1

(3,4)(3,5;4,6) (3,4) (5,6) (3,6)

(6,5)

(3,4) (5,6) (3,6) (3,6) (4,5) (3 ,4)

(3,5;4,6) (3,5;4,6)x d

K i vG

i v i W

K K

+

= − +

= − +

+

x,AA *

, ' ' ' '

d,AA *

, ' ' ' '

' ( , ) ( , ') ( ', ')

' ( , ') ( , '; 0) ( , ')

vc v c vc v c

vc v c vc v c

K d d X v X

K d d X W X

=

= − =

x x x x r r x x

x x x x r r x x

1

0W

G

Tamm-Dancoff approximation (TDA)

(1,2) (1,2)iG W =

( ) ( ) ( )

( ) ( ) ( )

AA AB' ', ,. ' ' , ' ' , ' '

BA BB' ', ,, ' ' . ' ' , ' '

v c s vc sc v vc v c vc v c s vc v c s

s

v c s vc svc v c s c v vc v c vc v c s

A AE E K K

B BK E E K

− + = − − − −


BSE in the effective Hamiltonian formalism

( ) ( )AA

. ' ' , ' ' ' ', ,

' '


v c

E E K A A − + =

el hole el-hole

*

,

,

ˆ ˆ ˆ

( , ') ( ) ( ')

s s s

s vc s c v

v c

H H H

A

+ + =

=x x x x


( ) ( )AA

. ' ' , ' ' ' ', ,

' '


v c

E E K A A − + =

Spin structure of BSE-TDA

( )

( )

d ( =1) ( =1) ( =1)

d ( =0) ( =0) ( =0)2

S S S

s s s

x S S S

s s s

+ =

+ + =

D K A A

D K K A A

x *

, ' ' ' '

d *

, ' ' ' '

' ( , ) ( , ') ( ', ')

' ( , ') ( , '; 0) ( , ')

vc v c vc v c

vc v c vc v c

K d d X v X

K d d X W X

=

= − =

r r r r r r r r

r r r r r r r r


( ) ( )AA

. ' ' , ' ' ' ', ,

' '


v c

E E K A A − + =

BSE for periodic systems

Periodic systems

v v

c c

→

→ +

k

k q

AA AA

, ; ' ' ' ' ' , ; ' ', ' ' , '

AA

, ' ' ' , ' ( )

v c v c v c v c

vc v c

K K

K

+ + + +=

k k q k k q k k q k k q q q

k k q qq

( ) ( ) ( ) ( ) ( )AA

. ' ' ' , ' ' ' ' ' ', ,

' ' '

c v vc v c vc v c v c s s vc s

v c

E E K A A+ − + = k q k k k k k k k

k

q q q q

Dimension of the BSE in the matrix form: Nv Nc Nk

Onida, et al. Rev. Mod. Phys. 74, 601 (2002).

flow diagram for GW-BSE calculations

2

;

, ,

00

( )

( ) 1 lim ( )

i

c v vc s

v c

M

s s

e A

v qi

=→

= −− +

q r

k+q k

k

Gq

q

DFT (LDA/GGA)

KS

i

G0

0 0 0P iG G=

01 vP = −

1

0W v −=0 0iG W =

QP

i

Kvc,v’c’

KS

i

BSE ( , )

( )

s

s e h

M

r r

Example: Si

Albrecht et al., Phys. Rev. Lett. 80, 4510 (1998).

Im ( )M

Examples: Na4 cluster

Onida et al., Phys. Rev. Lett. 75, 818 (1995)

GW+BSE: carbon nanotubes

Spataru et al. Phys. Rev. Lett. 92, 077402 (2004)

CNT(3,3)

CNT(5,0)

CNT(8,0)

GW+BSE:2D materials

Z. Jiang et al. Phys. Rev. Lett. 118, 266401 (2017)

J.-H. Choi, et al. Phys. Rev. Lett. 115,

066403 (2015)

Concluding remarks

❑ Electronic excitations are currently one of the most challenging

frontiers of electronic structure theory.

❑ TDDFT: reasonably accurate for close-shell organic molecules, but

difficult to be systematically improved, and problematic for charge

transfer excitations, double excitations and extended systems.

❑ GW+BSE: accurate for moderately correlated insulating systems, but

computationally expensive, and problematic for strongly correlated

systems.

❑ Electronic excitations are always strongly coupled to nuclear dynamics

➔ electron-phonon (vibration) coupling are crucial.

electronic excitations by time-dependent dft and bethe ......electronic excitations by...

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