Numerical Aspects of Many-Body Theory
• Choice of basis for crystalline solids• Local orbital versus Plane wave
• Plane waves ei(q+G).r
• Complete (in practice for valence space)• No all electron treatment (PAW?)• Large number of functions x.104
• Slow for HF exchange• Straightforward to code (abundance of Dirac delta’s)
• Local orbital (x - Ax)i(y - Ay)j e–(x - A)2
• Incomplete (needs care in choice of basis)• All electron possible and relatively inexpensive• Relatively small number of functions permits large unit cells to be treated• Relatively fast for HF exchange in gapped materials• Difficult to code (lattice sum convergence, exploitation of symmetry, ..)
G q
q+G
IBZ
Numerical Aspects of Many-Body Theory
• Coulomb Energy in real and reciprocal spaces
• Coulomb interaction
• Ewald form of Coulomb interaction
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Numerical Aspects of Many-Body Theory
• Density Matrix Representation of Charge Density
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tscoefficien expansion orbital local
tscoefficien expansion orbital waveplane
tionrepresenta waveplane
cell gth in site ith on orbital local
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function cellperiodic function Bloch
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i
r
Numerical Aspects of Many-Body Theory
• Coulomb Energy with real space representation of charge density
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12403
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iii
iii
Numerical Aspects of Many-Body Theory
• Coulomb Energy with reciprocal space representation of interaction
r r’
Numerical Aspects of Many-Body Theory
• Exchange Energy with real space representation of interaction
• No Ewald transformation possible since h sum is split• 3 lattice sums instead of 2• Absolute convergence neither guaranteed nor rapid
34g
12h4
g2
h3
01Exch PP) - - '() - '(
'-
1)()( E hrgr
rrh -r r r r’
34g
12h4
h3
g2
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'-
1)()( E hrhr
rrg-rr r r’
• Exchange Energy with reciprocal space representation of interaction
• q + G lattice sum instead of just G• Absolute convergence not guaranteed nor rapid
Gq,k,
Gq
rrGqrGq
rrGqrrrGqrGq
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n,
2
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Numerical Aspects of Many-Body Theory
r r’
Quasiparticle energies in solid Ne and Ar
)()(),,()()(ˆ
),(),(),(
)(),(),(),(),(
)(),()()(ˆ
)(),()(),()()(ˆ
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o
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H(1) Non-interacting Hamiltonianm
QP Quasiparticle amplitudem Quasiparticle energyQuasiparticle equation
Dyson equation
• Dyson and Quasiparticle equations F125
RPA Polarisability and Dielectric Function
tscoefficien Expansion
functions lorthonorma inof Expansion
)(),)F(( ddF
Fδ δ F
)()( d)()( dF)(),)F(( dd
F )()(F),F(
**kk
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ji,iij
221121
222111221121
2121
• Projection of functions onto orthogonal bases
notationDirac in Same FF
)()(FFFφ),F(
jiij
*jiijjijijjii
2121
RPA Polarisability and Dielectric Function
25 Eq Louie and Hybertsen Compare
BZ. first the to restricted is function. cellperiodic
a is as functions Bloch the
expandingby derived be can by differing points k to nrestrictio The
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221 121q
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.).(.
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iii
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• Projection of o onto plane wave basis
RPA Polarisability and Dielectric Function
function dielectric )(Hermitian dSymmetrise
space! in tionmultiplica becomes space in nConvolutio
function dielectric Hermitian-non
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),()(v~δ2),(ε~
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rGqrGq
rGqrGq
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ii
• Projection of o onto plane wave basis
• Dielectric bandstructure () expanded in eigenfunctions of static inverse dielectric function
Plasmon pole approximation for -1(q,)
ii
z
ii
iii
i
ii
i
ii
i
q
q
q
qqq
qqq
GGGG
GGGG'
11
21)(
)()(
1)(),(~
)()()0,(~
1
*1'
Pole strength zq and plasmon frequency q fitted at = 0 and several imaginary frequencies
Baldereschi and Tossatti, Sol. St. Commun. (1979)
Ar 15v Ar 1c
Energy dependence of self-energies in Ar
Nicastro, Galamic-Mulaomerovic and Patterson, J. Phys. Cond. Matt. (2001)
• Dielectric bandstructure and self energy
Self-energy operator matrix elements
Rohlfing, Kruger and Pollmann, Phys. Rev. B (1993)
HF exchange - looks like dynamically screened HFT
• Self-energy calculated from dielectric bandstructure
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n
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1
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1
fcc Ne DFT & GW bandstructures
Ne DFT
PP
GW
PP
DFT
AE
GW
AE
Expt.
15 -13.14 -19.37 -13.18 -19.10 -20.21
1c -1.35 0.86 -1.42 1.03 1.3
Wv 0.71 0.93 0.79 0.93 1.3
Eg 11.79 20.23 11.76 20.13 21.51
Expt. Runne and Zimmerer, Nucl. Instrum. Methods Phys. Res. B (1995)DFT/GW Galamic-Mulaomerovic and Patterson, Phys. Rev. B (2005)
Ne
DFTmxc
QPm
DFTm
DFTm
QPm VEEE
fcc Ar DFT & GW bandstructures
Ar DFT
PP
GW
PP
DFT
AE
GW
AE
Expt.
15 -9.74 -13.15 -10.27 -13.00 -13.75
1c -0.60 0.72 -0.76 0.81 0.4
Wv 1.35 1.73 1.32 1.85 1.7
Eg 9.14 13.87 9.51 13.81 14.15
Expt. Runne and Zimmerer, Nucl. Instrum. Methods Phys. Res. B (1995)DFT/GW Galamic-Mulaomerovic and Patterson, Phys. Rev. B (2005)
Ar
• Bethe-Salpeter Equation (F 558)
ioG(1,2)G(2,1) i.e. dressed Green’s function product
• K* proper part of electron/hole scattering kernel• o is a special case of the particle-hole Green’s function
• 4-index function
• (1,1,2,2) = o(1,1,2,2) + o(1,1,3,4) K*(3,4,5,6)(5,6,2,2)
Bethe-Salpeter Equation
K*
= +
oo
oo
2
)(ψ̂)(ψ̂)(ψ̂)(ψ̂),,,(G
12341234
Ti
K*
1
4 6
53
2
• Electron-hole scattering kernel K*
Bethe-Salpeter Equation
babaV kjiijkl
ℓ
k
j
i
bbaaV kjiijkl
ℓ
ik
j
bbabV kjiijkl
ℓ
i
k
j
bbabV kjiijkl
j
ik
ℓ
Time flows from left to right here
• Electron-hole scattering Lego
• Electron-hole pair scattering (summed in BSE)
• Electron-hole scattering (summed in screened electron-hole interaction)
Bethe-Salpeter Equation
Can’t have dangling ends
• Electron-hole scattering kernel K*
• K*(3,4,5,6) =
• Iteration of the Bethe-Salpeter equation leads to a series of the form
• = o + oK*o + oK*oK*o + oK*oK*oK*o + …
• Generates sums of ring and screened ladder diagrams
Bethe-Salpeter Equation
3 5
+ + + …
4 6
• Bethe-Salpeter Equation: Solution as an eigenvalue problem
• = o + o K* • (1 - o K* ) = o • = (1 - o K* ) -1 o • = (1 - o K* )-1 ( o
-1)-1
• = (o-1 - K* )-1
• -1= o-1 - K*
Bethe-Salpeter Equation
.(excitons) pairs hole-electron bound produces it ninteractio
hole-electron strong For strengths. oscillator and energies transition
particle single modifies energies. transition particle singlesimply are
seigenvalue of absence In . elements diagonal on/off and diagonal the
on energies transition particle single withequation eigenvalueMatrix
*
**
,*
,
K
KK
0c K
,
,
Look for zeros of -1 equivalent to poles of -1= o
-1 - K* = 0 an eigenvalue equation
• Bethe-Salpeter Equation: Expansion of functions of 2 or 4 variables
• Need all 4 arguments of o
Bethe-Salpeter Equation
lkjiijkl
i*k
*jiijkllkjilkji
lklkjiji
jiij
*jiijjijijjii
φφGφφG
)(φ)(φ)(φ)(φGφφφφφφGφφ
φφφφGφφφφ),,,G(
φFφF
)(φ)(φFφφφFφφφFφφ),F(
4321
4321
2121
1,t1 2,t2(1,2)
1,t1
2,t2
(1,2,3,4)
3,t3
4,t4
• Bethe-Salpeter Equation: Solution as an eigenvalue problem• o and o
-1 are diagonal in the basis of single particle states
Bethe-Salpeter Equation
)-(
...-)()ψ(ψ)-(
)()ψ()ψ()ψ(ψ)()ψ(ψdddd)(
...)-(
)()ψ()ψ()ψ(ψ ),,,,(
...)-(
)()ψ()ψ()ψ(ψ),,(
,
mn
mnmn
*
mn
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,o
mn
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unoccocc,
nm,
Ro
mn
*nn
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nm,
Ro
ii
i
i
i
411324
23 4321
13244321
122121
)()ψ(ψ * 23
1,t1 2,t2(1,2)
1,t1
2,t2
(1,2,3,4)
3,t3
4,t4
• Bethe-Salpeter Equation: Solution as an eigenvalue problem• K* in the basis of single particle states
Bethe-Salpeter Equation
)()ψ(ψ),()()ψ(ψdd )()ψ(ψ),,()()ψ(ψdd
)()ψ(ψ),(),(),()()ψ(ψdddd
)()ψ(ψ),,(),(),()()ψ(ψdddd
)()ψ()ψ,,,,()()ψ(ψdddd)(
),(),(),(),,(),(),( ),,,,(
****
**
**
***,
*
*
112122 21212121 21
4121324123 4321
4121423123 4321
41432123 4321
2132412142314321
vW
v
W
KK
vWK
)()ψ(ψ * 23
Direct term -W(1,2,)
1,t1 3,t3
2,t24,t4
1,t1 3,t3
2,t24,t4
Exchange term (singlet excitons only) v(1,2)
• Bethe-Salpeter Equation: Solution as an eigenvalue problem
Bethe-Salpeter Equation
stransition single to ingcorrespond
indices (compound) single are and
energy transition particle single a is
mn
,1-R
,o
,R,o
-
...)(
...- )(
i
Ne
)(ψ* 1
)(ψ 2
)(ψ* 2
)(ψ 1
v(q)
W(q)
v(q)
• Bethe-Salpeter Equation: numerical calculation of matrix elements
Bethe-Salpeter Equation
Direct term -W(1,2,)
1,t1 3,t3
2,t24,t4
1,t1 3,t3
2,t24,t4
Exchange term (singlet excitons only) v(1,2)
)()(),()()( c'cvv' 222111 k'kkk' W )()(),(v)()( c'v'vc 222111 k'k'kk
kkqr)G'qrG)q
GG,q,
GG,kk kkkk
GqGq
q',
.i(-.i(
'
1'
2
''' 'c'ecve'v''
)0,(e4
cvvcK
k'k'kk
TrGrGkk
kpkkpkqq kkkk
v'c'vc
.i-.i
0G2
2
'c'v',vc EE
'c''v
EE
vcˆˆ'v'e'c'vec
G
1e4 x 2
K
Excitons in solid Ne
Expt. Runne and Zimmerer Nucl. Instrum. Methods Phys. Res. B (1995). DFT/GW Galamic-Mulaomerovic and Patterson Phys. Rev. B (2005).
Singlet Ne energy levels, band gaps, binding energies (eV)
n En BSE En EXPT EB BSE EB EXPT
1 17.25 17.36 4.44 4.22
2 19.90 20.25 1.79 1.33
3 20.55 20.94 1.14 0.64
4 20.95 21.19 0.74 0.39
5 21.15 21.32 0.54 0.26
Eg
21.69
Eg
21.58
LT
0.30
LT
0.25
Excitons in solid Ar
Expt. Runne and Zimmerer Nucl. Instrum. Methods Phys. Res. B (1995). GW/BSE Galamic-Mulaomerovic and Patterson Phys. Rev. B (2005).
Singlet Ar energy levels, band gaps, binding energies (eV)
n En BSE En EXPT EB BSE EB EXPT
1 11.60 12.10 2.09 2.06
2 13.05 13.58 0.64 0.58
3 13.45 13.90 0.24 0.26
Eg
13.69
Eg
14.25
LT
0.36
LT
0.15