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The Bethe-Salpeter EquationAn Introduction

Francesco Sottile

ETSF and Ecole Polytechnique, Palaiseau - France

Lyon, 14 December 2007

The Bethe-Salpeter Equation Francesco Sottile


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Generalities of Linear Response Approach The Bethe-Salpeter equation Results The EXC Code

Outline

1 Generalities of Linear Response Approach

2 The Bethe-Salpeter equationPolarizability in MBPTDefinition of BSEBSE in practice

3 Results

4 The EXC Code



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Outline



3 Results

4 The EXC Code



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Linear Response Approach

System subject to an external perturbation



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Vtot = 1Vext

Vtot = Vext + Vind


G S C C


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Vtot = 1Vext

Vtot = Vext + Vind

E = 1D


G li i f Li R A h Th B h S l i R l Th EXC C d


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Vtot = 1Vext

Vtot = Vext + Vind

E = 1D

Dielectric function

G liti f Li R A h Th B th S l t ti R lt Th EXC C d


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Vtot = 1Vext

Vtot = Vext + Vind

E = 1D

Dielectric function

Abs

Generalities of Linear Response Approach The Bethe Salpeter equation Results The EXC Code


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Vtot = 1Vext

Vtot = Vext + Vind

E = 1D

Dielectric function

Abs

EELS



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Vtot = 1Vext

Vtot = Vext + Vind

E = 1D

Dielectric function

Abs

EELS

X-ray



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Vtot = 1Vext

Vtot = Vext + Vind

E = 1D

Dielectric function

Abs

EELS

X-ray

R index




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Definition of polarizability

1 = 1 + v

is the polarizability of the system




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p pp p q


Polarizability

interacting system n = Vext

non-interacting system nni = 0

Vtot




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Polarizability



VtotSingle-particle polarizability

0 =

ij

i(r)

j(r)

i (r)j(r

)

(ij) + i

hartree, hartree-fock, dft, etc.

G.D. Mahan Many Particle Physics (Plenum, New York, 1990)




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Polarizability



Vtot

0 =ij

i(r)

j(r)

i (r)j(r

)

(i j) + i

i

unoccupied states

occupied states

j




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First approximation: IP-RPA

0 = ij

i(r)j(r)

i (r

)j(r)

(i

j

) + i

Abs = Im0

=

ij

|j|D|i|2 ( (j i))




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0 = ij

i(r)j(r)

i (r

)j(r)

(i

j

) + i

Abs = Im0

=

ij

|j|D|i|2 ( (j i))




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How to go beyond 0 ?




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Outline


2 The Bethe-Salpeter equation

Polarizability in MBPTDefinition of BSEBSE in practice

3

Results

4 The EXC Code




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Outline




3

Results

4 The EXC Code




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Many Body Perturbation Theory

Hedins equations

(1, 2) = i

d(34)G(1, 3)(3, 2, 4)W(4, 1+)

G(1, 2) = G0

(1, 2) +

d(34)G0

(1, 3)(3, 4)G(4, 2)

(1, 2, 3) = (1, 2)(1, 3)+

d(4567)

(1, 2)

G(4, 5)G(4, 6)G(7, 5)(6, 7, 3)

P(1, 2) = id(34)G(1, 3)G(4, 1+)(3, 4, 2)W(1, 2) = v(1, 2) +

d(34)v(1, 3)P(3, 4)W(4, 2)




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Hedins pentagon

G

P

W

G=G0

+G0G

=1

+(/

G)GG

P = GG

W=v+

vPW

=G

W

~

~

~

~ ~

~

~ ~




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Polarizability P is irreducible

P=n

Vtot; Vtot = Vext + VH

= G

1

Vtot= 1 +

Vtot

IrreducibleP

and Reducible

P=n

Vtot; =

n

Vext

= P+ Pv

Different quantities

P, ,G = time-ordered0, = retarded




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P=n


= G

1

Vtot= 1 +

Vtot

IrreducibleP

and Reducible

P=n

Vtot; =

n

Vext

= P+ Pv






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P=n


= G

1

Vtot= 1 +

Vtot

Irreducible P and Reducible

P=n

Vtot; =

n

Vext

= P+ Pv






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P=n


= G1

Vtot= 1 +

Vtot

Irreducible P and Reducible

P=n

Vtot; =

n

Vext

= P+ Pv






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Spectra in MBPT

Spectra in IP picture

IP-RPA

Abs = Im 0

i

unoccupied states

occupied states

j




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Spectra in MBPT

Spectra in GW approximation

GW-RPA

Abs = Im 0GW

0GW = P = iGG

i

occupied states

j

unoccupied (GW corrected) states




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Spectra in MBPT

GW pentagon

G

P

W

G=G0

+G0G

=1+

(

/G)G

G

P = GG

W=v+

vPW

=G

W

P=GG

~

~

~ ~

~ ~

~

~

~




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Spectra in MBPT

Spectra in GW-RPA

0 = ij

i(r)j(r)

i (r

)j(r)

(i j)

0GW =

ij

i(r)j(r)

i (r

)j(r)

(i + GWi ) j + GWj




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Spectra in MBPT

Spectra in GW-RPA

0 = ij

i(r)j(r)

i (r

)j(r)

(i j)

0GW =

ij

i(r)j(r)

i (r

)j(r)

(i + GWi ) j + GWj




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Spectra in MBPT

Spectra in GW-RPA




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Spectra in MBPT

GG Polarizability

P(1, 2) = i G(1, 2)G(2, 1+)




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Spectra in MBPT

GG Polarizability

P(1, 2) = i

d(34)G(1, 3)G(4, 1+)(3, 4, 2)




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Outline




3 Results

4 The EXC Code




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Bethe-Salpeter Equation

(1, 2, 3) = (1, 2)(1, 3)+

+

d(4567)

(1, 2)

G(4, 5)G(4, 6)G(7, 5)(6, 7, 3)




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Towards the Bethe-Salpeter

= 1 +

GGG

GG = GG+ GGG

GG

L = L0 + L0

GL

L(1234) = L0(1234) + L0(1256)(56)

G(78)L(7834)




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= 1 +

GGG

GG = GG+ GGG

GG

L = L0 + L0

GL

L(1234) = L0(1234) + L0(1256)(56)

G(78)L(7834)




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= 1 +

GGG

GG = GG+ GGG

GG

L = L0 + L0

GL

L(1234) = L0(1234) + L0(1256)(56)

G(78)L(7834)




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= 1 +

GGG

GG = GG+ GGG

GG

L = L0 + L0

GL

L(1234) = L0(1234) + L0(1256)(56)

G(78)L(7834)




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Towards the Bethe-Salpeter EquationFrom electron and hole propagation ... ..

L0(1234) = G(13)G(42) ...




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Towards the Bethe-Salpeter EquationFrom electron and hole propagation to the electron-hole interaction

L(1234) = L0(1234) + L0(1256)(56)

G(78)L(7834)




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Irreducible form of the Bethe-Salpeter equation

L(1234) = L0(1234) + L0(1256)(56)

G(78)

L(7834)

Reducible quantity

L = L + LvL




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L(1234)=L0(1234) + L0(1256)

v(57)(56)(78)+

(56)

G(78)

L(7834)




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Comparison with Linear Response quantities

(12) = n

(1)Vext(2)




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(12) = n

(1)Vext(2)

L(1234) =




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(12) = n

(1)Vext(2)

L(1234) = G(12)Vext(34)




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(12) = n

(1)Vext(2)

L(1133) = G(11)Vext(33)




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(12) = n

(1)Vext(2)

L(1133) = n(1)Vext(3)




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We have the (4-point)Bethe-Salpeter equation.

And now ?




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First point: Choosing

L(1234)=L0(1234) + L0(1256)

v(57)(56)(78)+

(56)

G(78)

L(7834)

BSE



B h S l E


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L(1234)=L0(1234) + L0(1256)

v(57)(56)(78)+

(56)

G(78)

L(7834)

Coulomb term

x(1, 2) = iG(12)v(21)

Time-Dependent Hartree-FockBSE



B h S l E i


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L(1234)=L0(1234) + L0(1256)

v(57)(56)(78)+

(56)

G(78)

L(7834)

Screened Coulomb term

GW(1, 2) = iG(12)W(21)

Standard Bethe-Salpeter equation(Time-Dependent Screened Hartree-Fock)

BSE



B h S l E i


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Choice of = GW

Everything should be coherently chosen

ground state calculation

i,

i G0KS ; 1RPA(r, r, ) ; W(r, r, ) = 1RPAv

Ei = i + GWi ; i i GGW(r, r, )



B th S l t E ti


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Choice of = GW



i,

i G0KS ; 1RPA(r, r, ) ; W(r, r, ) = 1RPAv




B th S l t E ti


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Choice of = GW



i, i

G0KS ; 1RPA(r, r, ) ; W(r, r, ) = 1RPAv




Bethe Salpeter Equation


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Choice of = GW



i, i


(r, r, ) = d

G0KS(r, r, )W(r, r, + )






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Choice of = GW



i, i


(r, r, ) = d

G0KS(r, r, )W(r, r, + )






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Choice of = GW



i, i


(r, r, ) = d

G0KS(r, r, )W(r, r, + )






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Choice of = GW



i, i


(r, r, ) = d

G0KS(r, r, )W(r, r, + )






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Choice of = GW



i, i


(r, r, ) = d

GGW(r, r, )W(r, r, + )






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L

=L

0

+L

0 v+

G L

Approx. WG

= 0





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L

=GG

+GGv

[GW]

G L

Approx. WG

= 0





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L = GG + GG [vW] L

Approx. WG

= 0





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L = L0 + L0 [vW] L





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L(1234) = L0(1234)+

+ L0(1256) [v(57)(56)(78)

W(56)(57)(68)]L(7834)





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p q


L(1234) = L0(1234)+

+ L0(1256) [v(57)(56)(78)

W(56)(57)(68)] L(7834)

Intrinsic 4-point equation

Correct!It describes the (coupled) progation of

two particles, the electron and the hole !





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p q


L(1234) = L0(1234)+

+ L0(1256) [v(57)(56)(78)

W(56)(57)(68)] L(7834)

Intrinsic 4-point equation

Correct!It describes the (coupled) progation of

two particles, the electron and the hole !

Exercise

Show that, ifW = 0, the equationfor L(1133) is a two-point

equation!





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p q

Bethe-Salpeter equation (4-points - space and time)

L(1234) = L0(1234)+

+ L0(1256) [v(57)(56)(78)

W(56)(57)(68)]L(7834)

W(12) = W(r1, r

2)(t

1t

2)

L(1234) = L(r1, r2, r3, r4; t t) = L(1234, )The Bethe-Salpeter Equation Francesco Sottile




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p q


L(1234) = L0(1234)+

+ L0(1256) [v(57)(56)(78)

W(56)(57)(68)]L(7834)

W(12) = W(r1, r

2)(t

1t

2)





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L(1234) = L0(1234)+

+ L0(1256) [v(57)(56)(78)

W(56)(57)(68)]L(7834)

W(12) = W(r1, r

2)(t

1t

2)





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Macroscopic Quantity from the contracted L

1 L(1234, ) =

L(r, r, r, r, )

2 M() = 1 limq0 v(q)drdr eiq(rr

)L(r, r, r, r, )

M() = 1

limq0 v(q)L(G = 0, G = 0, )





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1 L(1234, ) =

L(r, r, r, r, )


)L(r, r, r, r, )

M() = 1

limq0 v(q)L(G = 0, G = 0, )





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1 L(1234, ) =

L(r, r, r, r, )


)L(r, r, r, r, )

M() = 1

limq0 v(q)L(G = 0, G = 0, )





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BSE (4 space points - 1 frequency)

L(1234, ) = L0(1234, )+

+ L0

(1256, ) [v(57)(56)(78) W(56)(57)(68)]L(7834, )

How to solve it ?Really invert 4-point function for every frequency?





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BSE (4 space points - 1 frequency)

L(1234, ) = L0(1234, )+

+ L0

(1256, ) [v(57)(56)(78) W(56)(57)(68)]L(7834, )

How to solve it ?Really invert 4-point function for every frequency?



Outline


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3 Results

4 The EXC Code





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L(1234, ) = L0(1234, )+

+ L0(1256, ) [v(57)(56)(78) W(56)(57)(68)]L(7834, )





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L(1234, ) = L0(1234, ) + L0(1256, )K(5678)L(7834, )




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L(1234, ) = L0(1234, ) + L0(1256, )K(5678)L(7834, )

We work in transition space...

L(1234, ) L(n3n4)(n1n2)() =

=

d(1234)L(1234, )n1 (1)

n2 (2)n3 (3)

n4 (4) = L

The Bethe-Salpeter Equation Francesco Sottile Generalities of Linear Response Approach The Bethe-Salpeter equation Results The EXC Code



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L(1234, ) = L0(1234, ) + L0(1256, )K(5678)L(7834, )

L(n3n4)(n1n2)

() = L0 (n3n4)(n1n2)

() + L0 (n5n6)(n1n2)

()K(n7n8)(n5n6)

L(n3n4)(n7n8)

()


L(1234, ) L(n3n4)(n1n2)() =

=

d(1234)L(1234, )n1 (1)

n2 (2)n3 (3)

n4 (4) = L




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L(1234, ) = L0(1234, ) + L0(1256, )K(5678)L(7834, )

L(n3n4)(n1n2)

() = L0 (n3n4)(n1n2)

() + L0 (n5n6)(n1n2)

()K(n7n8)(n5n6)

L(n3n4)(n7n8)

()


L(1234, ) L(n3n4)(n1n2)() =

=

d(1234)L(1234, )n1 (1)

n2 (2)n3 (3)

n4 (4) = L

Clever choice of the basis n


Bethe-Salpeter Equation in transition space


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L(n3n4)(n1n2)

() = L0 (n3n4)(n1n2)

() + L0 (n5n6)(n1n2)

()K(n7n8)(n5n6)

L(n3n4)(n7n8)

()

... some trivial mathematical arzigogoli ...

L(n3n4)(n1n2)

() =

(En2 En1 )n1n3n2n4 + K(n3n4)(n1n2)1

L(n3n4)(n1n2)

() = [(En2 En1 )n1n3n2n4 + v W ]1




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L(n3n4)(n1n2)

() = L0 (n3n4)(n1n2)

() + L0 (n5n6)(n1n2)

()K(n7n8)(n5n6)

L(n3n4)(n7n8)

()


L(n3n4)(n1n2)

() =

(En2 En1 )n1n3n2n4 + K(n3n4)(n1n2)1

L(n3n4)(n1n2)

() = [(En2 En1 )n1n3n2n4 + v W ]1




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L(n3n4)(n1n2)

() = L0 (n3n4)(n1n2)

() + L0 (n5n6)(n1n2)

()K(n7n8)(n5n6)

L(n3n4)(n7n8)

()


L(n3n4)(n1n2)

() =

(En2 En1 )n1n3n2n4 + K(n3n4)(n1n2)1

L(n3n4)(n1n2)

() = [(En2 En1 )n1n3n2n4 + v W ]1




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L(n3n4)(n1n2)

() = L0 (n3n4)(n1n2)

() + L0 (n5n6)(n1n2)

()K(n7n8)(n5n6)

L(n3n4)(n7n8)

()


L(n3n4)(n1n2)

() =

(En2 En1 )n1n3n2n4 + K(n3n4)(n1n2)1

L(n3n4)(n1n2)

() = [(En2 En1 )n1n3n2n4 + v W ]1




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The Excitonic Hamiltonian

L(n3n4)(n1n2)

() = [(En2 En1 )n1n3n2n4 + v W ]1

L

(n3n4)

(n1n2)() =

1

Hexc Hexc = (En2 En1 )n1n3n2n4 + v W

Resonant vs Coupling

n1, n2, n3, n4 = v, c (k)

Hreso = (Ec Ev)vvcc+ v W




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L(n3n4)(n1n2)

() = [(En2 En1 )n1n3n2n4 + v W ]1

L

(n3n4)

(n1n2)() =

1



n1, n2, n3, n4 = v, c (k)





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L(n3n4)(n1n2)

() = [(En2 En1 )n1n3n2n4 + v W ]1

L

(n3n4)

(n1n2)() =

1



n1, n2, n3, n4 = v, c (k)


The Bethe Salpeter Equation Francesco Sottile Generalities of Linear Response Approach The Bethe-Salpeter equation Results The EXC Code



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L(n3n4)(n1n2)

() = L(r, r, ) =

(n1n

2)

(n3n4)

L(n3n4)(n1n2)

()n1 (r)

n2(r)n3 (r

)n4 (r)

L(n3n4)(n1n2)

() =L(G,G, ) =

(n1n2)(n3n4)

L(n3n4)(n1n2)

()

n1 (r)e

i(q+G)rn2 (r)n3 (r)ei(q+G

)rn4 (r)




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L(n3n4)(n1n2)

() = L(r, r, ) =

(n1n

2)

(n3n4)

L(n3n4)(n1n2)

()n1 (r)

n2(r)n3 (r

)n4 (r)

L(n3n4)(n1n2)

() =L(G,G, ) =

(n1n2)(n3n4)

L(n3n4)(n1n2)

()

n1 (r)e


)rn4 (r)




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L(n3n4)(n1n2)

() = L(r, r, ) =

(n1n

2)

(n3n4)

L(n3n4)(n1n2)

()n1 (r)

n2(r)n3 (r

)n4 (r)

L(n3n4)(n1n2)

() =L(G,G, ) =

(n1n2)(n3n4)

L(n3n4)(n1n2)

()

n1 (r)e


)rn4 (r)




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L(n3n4)(n1n2)

() = [(En2 En1 )n1n3n2n4 + v W ]1

L(n3n4)

(n1n2)() = [Hexc

]1

Hexc = [(En2 En1 )n1n3n2n4 + v W ]

Diagonalization Iterative inversion




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L(n3n4)(n1n2)

() = [(En2 En1 )n1n3n2n4 + v W ]1

L(n3n4)

(n1n2)() = [Hexc

]1



Th B th S l t E ti F s S ttil Generalities of Linear Response Approach The Bethe-Salpeter equation Results The EXC Code



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L(n3n4)(n1n2)

() = [(En2 En1 )n1n3n2n4 + v W ]1

L(n3n4)

(n1n2)() = [Hexc

]1



Th B th S l t E ti F S ttil Generalities of Linear Response Approach The Bethe-Salpeter equation Results The EXC Code



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Diagonalization case (only resonant approx)

Lvc

vc = [(Ec Ev) vvcc + v W ]1

1

H I =

|A >< A

|E

Spectrum within BSE

L(r, r, ) =

vc A

(vc) v(r)

c(r)

v(r

)c(r

)A(vc)

Eexc

+ i




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Lvc


1

H I =

|A >< A

|E

Spectrum within BSE

L(r, r, ) =

vc A

(vc) v(r)

c(r)

v(r

)c(r

)A(vc)

Eexc

+ i




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Lvc


1

H I =

|A >< A

|E

Spectrum within BSE

L(r, r, ) =

vc A

(vc) v(r)

c(r)

v(r

)c(r

)A(vc)

Eexc

+ i





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Spectrum in BSE (only resonant)

L(r, r, ) =

vcA(vc) v(r)c(r)v(r)c(r)A(vc) Eexc + i

0(r, r, ) = vc

v(r)c(r)

v(r

)c(r)

(c v) + i





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L(r, r, ) =

vcA(vc) v(r)c(r)v(r)c(r)A(vc) Eexc + i

0(r, r, ) = vc

v(r)c(r)

v(r

)c(r)

(c v) + i





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AbsBSE

() = Im L()=

vc

A

(vc)

c|D|v2

( Eexc

)

AbsIP-RPA() = Im 0()=vc |

c|D|v|

2(

(

c

v))





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BSE - creating and diagonalizing a (big) matrix

Hexc

(vc)(v

c

)

L

(vc)(vc) () =

A(vc) A

(vc)

Eexc

+ i





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Hexc

(vc)(v

c

)= (E

cE

v)

vv

cc + vv

c

vc Wv

c

vc

L

(vc)(vc) () =

A(vc) A

(vc)

Eexc

+ i





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Hexc

(vc)(v

c

)A

(vc)

= Eexc

A

(vc)

L

(vc)(vc) () =

A(vc) A

(vc)

Eexc

+ i





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Hexc

(vc)(v

c

)A

(vc)

= Eexc

A

(vc)

L

(vc)(vc) () =

A(vc) A

(vc)

Eexc

+ i





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Standard Approximations for BSEGround-state

pseudopotentialVxc local density approximation

Quasi-particle Many-Body Theory

GW approximation for W rpa, plasmon-pole modelGW = KS

Bethe-Salpeter equationW

G = 0W rpa, staticonly resonant term



The Bethe-Salpeter Soup


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Outline


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2 The Bethe-Salpeter equationPolarizability in MBPT

Definition of BSEBSE in practice

3 Results

4 The EXC Code



Bethe-Salpeter equation results: Semiconductors


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Albrecht et al., PRL 80, 4510 (1998)



Bethe-Salpeter equation results: Insulators


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Sottile, Marsili, et al., PRB (2007).



Bethe-Salpeter equation results: Molecule (Na4)


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Onida et al., PRL 75, 818 (1995)



Bethe-Salpeter equation results: Silicon Nanowires


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Bruno et al., PRL 98, 036807 (2007)



Bethe-Salpeter equation results: Hexagonal Ice


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Hahn et al., PRL 94, 37404 (2005)



Bethe-Salpeter equation results: Cu2O


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Bruneval et al., PRL 97, 267601 (2006)



Bethe-Salpeter equation results: EELS of Silicon


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Olevano and Reining, PRL 86, 5962 (2001)



Bethe-Salpeter equation results: Surface


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Rohlfing et al., PRL 85, 005440 (2000)



Bethe-Salpeter equation results: Surface


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Rohlfing et al., PRL 85, 005440 (2000)



Bethe-Salpeter equation results: liquid Water


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Garbuio et al., PRL 97, 137402 (2006)




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The EXC code - Sottile, Reining, Olevano, Onida, Albrechthttp://www.bethe-salpeter.org



Bethe-Salpeter equation: State-of-the-art


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DFT - ground stateGW - quasiparticle energies

BSE - optical and dielectric properties

several spectroscopies

variety of systems Cumbersome CalculationsThe Bethe-Salpeter Equation Francesco Sottile




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DFT - ground stateGW - quasiparticle energies

BSE - optical and dielectric properties

several spectroscopies

variety of systems Cumbersome CalculationsThe Bethe-Salpeter Equation Francesco Sottile




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Some references

Hanke and Sham, PRB 21, 4656 (1980)

Onida, Reining, Rubio, RMP 74, 601 (2002)

Strinati, Riv Nuovo Cimento 11, 1 (1988)



Outline


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2 The Bethe-Salpeter equationPolarizability in MBPT

Definition of BSEBSE in practice

3 Results

4 The EXC Code



What do we calculate ?

Dielectric function (only resonant case) 2


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M() = 1 limq0

v0(q)

(vc)

c|eiqr|vA(vc)

2Eexc

i

diagonalize excitonic Hamiltonian

H2p,exc(vc)(vc)A

(vc) = E

exc A

(vc)

H = (vc)

(vc)

266664

. . . . . .

. . .

. . .

. . .

. . .

377775

Hamiltonian:

H2p(vc)(vc)= (Ec Ev)vvcc + [v W]

(vc)(vc)

Ei = quasiparticle energies (GW)

W = 1v screened Coulomb interaction




Dielectric function (only resonant case) ( )

2


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M() = 1 limq0

v0(q)

(vc)

c|eiqr|vA(vc)

2Eexc

i


H2p,exc(vc)(vc)A

(vc) = E

exc A

(vc)

H = (vc)

(vc)

266664

. . . . . .

. . .

. . .

. . .

. . .

377775

Hamiltonian:


(vc)(vc)






Dielectric function (only resonant case) ( )

2


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M() = 1 limq0

v0(q)

(vc)

c|eiqr|vA(vc)

2Eexc

i


H2p,exc(vc)(vc)A

(vc) = E

exc A

(vc)

H = (vc)

(vc)

266664

. . . . . .

. . .

. . .

. . .

. . .

377775

Hamiltonian:


(vc)(vc)





What do we need ?


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structure, screening, quasiparticle files + input file

|v gs (LDA) wfs kss file

Ev Quasi-Particle energies gw file

1 for the screened interaction scr file



The excitonic Hamiltonian

H2p

E E + [ W ]


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H2p(vc)(vc) =

Ec Evvvcc + [v W](vc)(vc)

H2p(vc)(vc) = (c v) vvcc RPA-NLF

H2p(vc)(vc) = (c v) vvcc + v(vc)(vc) RPA

H2p(vc)(vc) = (Ec

Ev) vvcc + v(vc)(vc) GW

H2p(vc)(vc) = (Ec Ev) vvcc + v(vc)(vc) + W(vc)(vc) BSE





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H2p(vc)(vc) =

(c +

GWc ) (v + GWv )

vvcc + [v W](vc)(vc)



H2p(vc)(vc) = (Ec







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H2p(vc)(vc) =

(c +

GWc ) (v + GWv )

vvcc + [v W](vc)(vc)



H2p(vc)(vc) = (Ec







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H2p(vc)(vc) = ((c +

GWc ) (v + GWv )) vvcc + [v W](vc)(vc)



H2p(vc)(vc) = (Ec







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H2p(vc)(vc) = ((c +

GWc ) (v + GWv )) vvcc + [v W](vc)(vc)



H2p(vc)(vc) = (Ec






H2p =

E E + [v + W ]( )( )


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H p(vc)(vc) =

Ec Evvvcc + [v + W](vc)(vc)



H2p(vc)(vc) = (Ec





__ ____ ___ __

\ ___) \ \ / / / __)

| (__ \ \/ / | /

| __) > < | |

| (___ / /\ \ | \__

/ )_/ /__\ \__\ )_


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http://www.bethe-salpeter.org

program EXC version 2.3.4

built: 06 Dec 2007

calculate dielectric propertiesBethe-Salpeter equation code in frequency domain

reciprocal space on a transitions basis

Copyright (C) 1992-2007, Lucia Reining, Valerio Olevano,

Francesco Sottile, Stefan Albrecht, Giovanni Onida.

This program is free software; you can redistribute it

and/or modify it under the terms of the GNU General

Public License.

This program is distributed in the hope that it will be

useful, but WITHOUT ANY WARRANTY;The Bethe-Salpeter Equation Francesco Sottile


The input file

rpa, gw, exc (default) . . . . . . . . . . . . . . . . . . . . . . . type of calculationnlf, lf (default) . . . . . . . . . . . . . . . . . . . . . . . . . . with or w/o local fields

wdiag (default), wfull . . . . . . . . . . . . . . . . . . . . . . . . . . WGGGG or WGG(d f l ) li li i


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g ( ), . . . . . . . . . . . . . . . . . . . . . . . . . . orresonant (default), coupling . . . . . . . . coupling reso-antireso or notsoenergy (default=0.0) . . . . . . scissor operator energy [eV]matsh (default=all) ........number of G-shell for the GG

wfnsh (default=all) . . . . . plane waves shells for the nk(G)

nbands (default=all) last band included in the calculationlomo (default=1) . . . . . first band included in the calculationomegai (default=0.0) ......... frequency initial point [eV]omegae (default=10.0) . . . . . . . . . . frequency end point [eV]domega (default=0.01) . . . . . . . . . . . . . . . frequency step [eV]broad (default=domega) . . . . . . . lorentzian broadening [eV]haydock .............................. iterative diagonalization schemeniter (default=100) . . . . . . . . . iterations for haydock scheme



The input file


wdiag (default), wfull . . . . . . . . . . . . . . . . . . . . . . . . . . WGGGG or WGGt (d f lt) li li ti t


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g ,resonant (default), coupling . . . . . . . . coupling reso-antireso or notsoenergy (default=0.0) . . . . . . scissor operator energy [eV]matsh (default=all) ........number of G-shell for the GG





The input file


wdiag (default), wfull . . . . . . . . . . . . . . . . . . . . . . . . . . WGGGG or WGGt (d f lt) li li ti t


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g ,resonant (default), coupling . . . . . . . . coupling reso-antireso or notsoenergy (default=0.0) . . . . . . scissor operator energy [eV]matsh (default=all) ........number of G-shell for the GG





The input file


wdiag (default), wfull . . . . . . . . . . . . . . . . . . . . . . . . . . WGG

GG

or WGG

resonant (default) coupling coupling reso antireso or not


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gresonant (default), coupling . . . . . . . . coupling reso-antireso or notsoenergy (default=0.0) . . . . . . scissor operator energy [eV]matsh (default=all) ........number of G-shell for the GG





The input file



GG

or WGG



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gresonant (default), coupling . . . . . . . . coupling reso-antireso or notsoenergy (default=0.0) . . . . . . scissor operator energy [eV]matsh (default=all) ........number of G-shell for the GG





The input file



GG

or WGG



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resonant (default), coupling . . . . . . . . coupling reso-antireso or notsoenergy (default=0.0) . . . . . . scissor operator energy [eV]matsh (default=all) ........number of G-shell for the GG





The input file



GG

or WGG

resonant (default) coupling coupling reso-antireso or not


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resonant (default), coupling . . . . . . . . coupling reso-antireso or notsoenergy (default=0.0) . . . . . . scissor operator energy [eV]matsh (default=all) ........number of G-shell for the GG





The input file



GG

or WGG

resonant (default) coupling coupling reso-antireso or not


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resonant (default), coupling . . . . . . . . coupling reso antireso or notsoenergy (default=0.0) . . . . . . scissor operator energy [eV]matsh (default=all) ........number of G-shell for the GG





The input file



GG

or WGG

resonant (default), coupling coupling reso-antireso or not


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The input file



GG

or WGG

resonant (default), coupling . . . . . . . . coupling reso-antireso or not


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The input file



GG

or WGG



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( ), p g p gsoenergy (default=0.0) . . . . . . scissor operator energy [eV]matsh (default=all) ........number of G-shell for the GG





The input file



GG

or WGG



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( ), p g p gsoenergy (default=0.0) . . . . . . scissor operator energy [eV]matsh (default=all) ........number of G-shell for the GG


nbands (default=all) last band included in the calculationlomo (default=1) . . . . . first band included in the calculationomegai (default=0.0) ......... frequency initial point [eV]omegae (default=10.0) . . . . . . . . . . frequency end point [eV]domega (default=0.01) . . . . . . . . . . . . . . . frequency step [eV]

broad (default=domega) . . . . . . . lorentzian broadening [eV]haydock .............................. iterative diagonalization schemeniter (default=100) . . . . . . . . . iterations for haydock scheme



The input file



GG

or WGG



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p g p gsoenergy (default=0.0) . . . . . . scissor operator energy [eV]matsh (default=all) ........number of G-shell for the GG






The input file



GG

or WGG



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p g

soenergy (default=0.0) . . . . . . scissor operator energy [eV]matsh (default=all) ........number of G-shell for the GG






The input file



GG

or WGG



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soenergy (default=0.0) . . . . . . scissor operator energy [eV]matsh (default=all) ........number of G-shell for the GG





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