6. • Electron density parameterized by KS single particle states
• Greens function in spectral representation, expressed in terms of quasi-
particle states leads to the quasi particle equation:
• The quasi-particle energies are the electron removal and addition
energies
KS v.s. Quasi-Particle equation
G(r, ʹr ;z) =
Ψr,n
qp
(r, z)Ψl,n
qp†
( ʹr, z)
z −εn
qp
(z)+iηsign(εn
qp
(z)-µ)n
∑
−
1
2
∇2
+VH (r)+Vext (r)
⎛
⎝
⎜
⎞
⎠
⎟Ψr,n
qp
(r, z)+ d ʹr Σ(r, ʹr ;εn
qp
(z))∫ Ψr,n
qp
( ʹr, z) =εn
qp
(z)Ψr,n
qp
(r, z)
)()()()()(
2
1
)()()(
KSKSKS
XCextH
2
occ
*KSKS
rrrrr
rrr
nnn
n
nn
VVV ψεψ
ψψρ
=⎟
⎠
⎞
⎜
⎝
⎛
+++∇−
= ∑
7. • Electron density parameterized by KS single particle states
• Greens function in spectral representation, expressed in terms of quasi-
particle states leads to the quasi particle equation:
• The quasi-particle energies are the electron removal and addition
energies
KS v.s. Quasi-Particle equation
Vxc: exchange correlation potential
HEDIN Phys Rev 139, A796, HYBERTSEN and LOUIE PRB 34, 5390
Σ: self-energy
)()()()()(
2
1
)()()(
KSKSKS
XCextH
2
occ
*KSKS
rrrrr
rrr
nnn
n
nn
VVV ψεψ
ψψρ
=⎟
⎠
⎞
⎜
⎝
⎛
+++∇−
= ∑
G(r, ʹr ;z) =
Ψr,n
qp
(r, z)Ψl,n
qp†
( ʹr, z)
z −εn
qp
(z)+iηsign(εn
qp
(z)-µ)n
∑
−
1
2
∇2
+VH (r)+Vext (r)
⎛
⎝
⎜
⎞
⎠
⎟Ψr,n
qp
(r, z)+ d ʹr Σ(r, ʹr ;εn
qp
(z))∫ Ψr,n
qp
( ʹr, z) =εn
qp
(z)Ψr,n
qp
(r, z)
8. Zeroth order quasi particle equation
• Full quasi particle equation
• 0th order corrections (diagonal elements only)
• Linearized 0th order corrections
),()(),())(;,(),()()(
2
1 qp
,
qpqp
,
qpqp
,extH
2
zzzzdzVV nrnnrnnr rrrrrrrr Ψ=ʹΨʹΣʹ+Ψ⎟
⎠
⎞
⎜
⎝
⎛
++∇− ∫ εε
KS
XC
KSKSKS
)(00
nnnnn
WG
n VZ ψεψεε −Σ+=
εn
G0W0
= εn
KS
+ ψn
KS
Σ(εn
G0W0
)−VXC ψn
KS
)()();,()()()(
2
1 qp
,
qpqp
,
qpqp
,extH
2
rrrrrrrr nrnnrnnr dVV Ψ=ʹΨʹΣʹ+Ψ⎟
⎠
⎞
⎜
⎝
⎛
++∇− ∫ εε
9. Advantages
• There is a closed set of equations for the self-energy
• Perturbative expansion in terms of the screened
interaction in stead of the bare Coulomb interaction
• The quasi-particle energies are the electron removal and
addition energies
• Extension to charge neutral excitations via the Bethe-
Salpeter equation (recently implemented in Turbomole)
• GW selfenergy -> RPA forces
Bare Coulomb interac-on Screened Coulomb interac-on
10. Hedin Equations
– Space time notation
(the numbers indicate a contracted space, time and spin index)
∫
∫
∫
∫
∫
+
+
Γ−=
+=
Γ
Σ
+−−=Γ
Σ+=
Γ=Σ
)1,4()2,4,3()3,1()34()2,1(
)2,4()4,3()3,1()34()2,1()2,1(
)3,7,6()5,7()6,4(
)5,4(
)2,1(
)4567()32()21()3,2,1(
)2,4()4,3()3,1()34()2,1()2,1(
)4,3,2()4,1()3,1()34()2,1(
00
GGdiP
WPvdvW
GG
G
d
GGdGG
WGdi
δδ
HEDIN Phys Rev 139, A796
11. Hedin Equations
– Space time notation
(the numbers indicate a contracted space, time and
spin index)
– Neglecting the second term in the vertex function
leads to the GW approximation for the self-energy
Fourier transformed to frequency domain:
ωωω
π
ω
dWEGe
i
E i
)()(
2
)( 0
−=Σ ∫
+
−
)1,2()2,1()2,1(
)2,4()4,3()3,1()34()2,1()2,1(
)2,4()4,3()3,1()34()2,1()2,1(
)2,1()2,1()2,1(
00
GGiP
WPvdvW
GGdGG
WGi
−=
+=
Σ+=
=Σ
∫
∫
+
HEDIN Phys Rev 139, A796
12. Full analy-c
– Calculate response in spectral representa-on
– Close connec-on to TDDFT (actually TDH)
– Analy-c expression of Sigma as a sum over
poles of G and W
– Calculate Sigma analy-cally
• Numerically exact except for finite basis
• Full analy-c structure of Sigma
• Expensive
Re n Σc
(εn ) n( )=
in ρm( )
2 εn −εi +Ωm
εn −εi +Ωm( )
2
+η2
i
occ
∑
+ an ρm( )
2
a
unocc
∑
εn −εa −Ωm
εn −εa −Ωm( )
2
+η2
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
m
∑
van Se*en et al. JCTC 9, 232 (2013)
18. Benchmark set for molecules: GW100
• Original Collaboration: KIT Karlsruhe, FHI Berlin, and Berkeley Lab US DoE
– TURBOMOLE: Gaussian basis sets, spectral representation via Casida
– FHI-Aims: numerical local orbitals, analytic continuation
– BerkeleyGW: plane waves, plasmon pole and real frequency integration
• 5 different ways to evaluate the self-energy
• well converged all electron reference values for IP and EA
• Follow ups
– CCSD(T) total energy reference (Klopper)
– Plane wave results by VASP (Kresse) and WEST (Galli)
– CP2k results (tes-ng their O(3) GW implementa-on)
– (par-al) Stochas-c GW results (tes-ng O(1) implementa-on)
– Evalua-on of 6 types of (par-al) self-consistency
– Dipole moments and Densi-es (new project in process)
– Core levels (new project in process)
gw100.wordpress.com
22. Problems when solving the QPE
n n
making it featureless and almost constant. Then the solution is unique in the re
t to us and eq. 32 may even be linearized so a single evaluation of ⌃ at the KS-
cient and there is no iteration process.2
x
εn
G0W0
= εn
KS
+ ψn
KS
Σ(εn
G0W0
)−VXC ψn
KS
23. Hard to converge
ms and TURBOMOLE and the plane wave code BerkeleyGW in the results
ection we will always used the extrapolated values. These will be referred to as
EXTRA’.
0,001 0,01 0,1
1/Nbasis
-2,00
-1,50
-1,00
-0,50
0,00
0,50ε
QP
H
(extra)-ε
QP
H
(basis)[eV]
SVP
TZVP
QZVP
gure 5: The deviation of the HOMO energies from the extrapolated complete basis set