MBPT vs (TD)DFT - a fight or a wedding

Transcription

MBPT vs (TD)DFT
a fight or a wedding ?
Francesco Sottile
Laboratoire des Solides Irradiés
Ecole Polytechnique, Palaiseau - France
European Theoretical Spectroscopy Facility (ETSF)
Lyon, 14 December 2007
MBPT vs (TD)DFT
Francesco Sottile
Introduction
BSE and TDDFT up to 2002
The Mapping Theory Kernel
Conclusions and Perspectives
Outline
1
Introduction
2
3
Theory
Results
4
MBPT vs (TD)DFT
Francesco Sottile
Introduction
Outline
1
Introduction
2
3
Theory
Results
4
MBPT vs (TD)DFT
Francesco Sottile
Introduction
A rough Summary
DFT - TDDFT
√
fast (one-particle eqs)
× lack of functionals
MBPT (GW-BSE)
√
it works!
(physical ingredients)
× Cumbersome
Fast, efficient and reliable
MBPT vs (TD)DFT
Francesco Sottile
Introduction
A rough Summary
DFT - TDDFT
√
MBPT (GW-BSE)
√
it works!
× Cumbersome
MBPT vs (TD)DFT
Francesco Sottile
Introduction
A rough Summary
DFT - TDDFT
√
MBPT (GW-BSE)
√
it works!
× Cumbersome
MBPT vs (TD)DFT
Francesco Sottile
Introduction
A possible strategy
Combine the two approaches
MBPT vs (TD)DFT
Francesco Sottile
Introduction
A possible strategy
Dyson eq. for G
d
h
G = G0 + G0 (Σ − Vxc ) G
i
0 = G0 (Σ − Vxc ) G
Z
d(23)G0 (12) Σ(23) − Vxc (2)δ(23) G (31) = 0
MBPT vs (TD)DFT
Francesco Sottile
Introduction
A possible strategy
Dyson eq. for G
d
h
G = G0 + G0 (Σ − Vxc ) G
i
0 = G0 (Σ − Vxc ) G
Z
d(23)G0 (12) Σ(23) − Vxc (2)δ(23) G (31) = 0
MBPT vs (TD)DFT
Francesco Sottile
Introduction
A possible strategy
Dyson eq. for G
d
h
G = G0 + G0 (Σ − Vxc ) G
i
0 = G0 (Σ − Vxc ) G
Z
d(23)G0 (12) Σ(23) − Vxc (2)δ(23) G (31) = 0
MBPT vs (TD)DFT
Francesco Sottile
Introduction
A possible strategy
Dyson eq. for G
Z
d(23)G0 (12) Σ(23) − Vxc (2)δ(23) G (31) = 0
given a Σ (non-local and dynamic),
we obtain a Vxc (local and static) which provides the
same density
Sham-Schlüter equation
L.J.Sham and M.Schlter, Phys. Rev. Lett. 51, 1888 (1983)
MBPT vs (TD)DFT
Francesco Sottile
Introduction
A possible strategy
Dyson eq. for G
Z
d(23)G0 (12) Σ(23) − Vxc (2)δ(23) G (31) = 0
same density
MBPT vs (TD)DFT
Francesco Sottile
Introduction
A possible strategy
Dyson eq. for G
Z
d(23)G0 (12) Σ(23) − Vxc (2)δ(23) G (31) = 0
same density
MBPT vs (TD)DFT
Francesco Sottile
Introduction
A possible strategy
Dyson eq. for G
Z
d(23)G0 (12) Σ(23) − Vxc (2)δ(23) G (31) = 0
same density
MBPT vs (TD)DFT
Francesco Sottile
Introduction
A possible strategy
Z
Z
d(23)G0 (12) Σ(23) − Vxc (2)δ(23) G (31) = 0
d(23)G0 (12) Σx (23) − Vxc (2)δ(23) G0 (31) = 0
we obtain VxcEXX = G0 G0 vG0 (χ0 )−1
Exact-Exchange Approximation
MBPT vs (TD)DFT
Francesco Sottile
Introduction
A possible strategy
Z
Z
d(23)G0 (12) Σ(23) − Vxc (2)δ(23) G (31) = 0
d(23)G0 (12) Σx (23) − Vxc (2)δ(23) G0 (31) = 0
MBPT vs (TD)DFT
Francesco Sottile
Introduction
A possible strategy
Z
Z
d(23)G0 (12) Σ(23) − Vxc (2)δ(23) G (31) = 0
d(23)G0 (12) Σx (23) − Vxc (2)δ(23) G0 (31) = 0
MBPT vs (TD)DFT
Francesco Sottile
Introduction
A possible strategy
Z
Z
d(23)G0 (12) Σ(23) − Vxc (2)δ(23) G (31) = 0
d(23)G0 (12) Σx (23) − Vxc (2)δ(23) G0 (31) = 0
MBPT vs (TD)DFT
Francesco Sottile
Introduction
A possible strategy
Generalized SSE
p
h
G = G0 + G0 (Σ − Vxc ) G
i
M.Gatti et al, Phys. Rev. Lett. 99, 057401 (2007)
MBPT vs (TD)DFT
Francesco Sottile
Introduction
A possible strategy
Generalized SSE
p
h
G = G0 + G0 (Σ − Vxc ) G
i
M.Gatti et al, Phys. Rev. Lett. 99, 057401 (2007)
MBPT vs (TD)DFT
Francesco Sottile
Introduction
Outline
1
Introduction
2
3
Theory
Results
4
MBPT vs (TD)DFT
Francesco Sottile
Introduction
Optical Absorption Spectra of Solids
Semiconductors: Silicon
MBPT vs (TD)DFT
Francesco Sottile
Introduction
Insulators: Argon
MBPT vs (TD)DFT
Francesco Sottile
Introduction
ALDA bad for any solids!! though quick
BSE good but cumbersome
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The problem of Abs in solids. Towards a better understanding
Reining et al. Phys.Rev.Lett. 88, 66404 (2002)
Long-range kernel
de Boeij et al. J.Chem.Phys. 115, 1995 (2002)
Polarization density functional. Long-range.
Kim and Görling Phys.Rev.Lett. 89, 96402 (2002)
Exact-exchange
Sottile et al. Phys.Rev.B 68, 205112 (2003)
Long-range and contact exciton.
Botti et al. Phys. Rev. B 72, 125203 (2005)
Dynamic long-range component
Parameters to fit to experiments.
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The problem of Abs in solids. Towards a better understanding
Reining et al. Phys.Rev.Lett. 88, 66404 (2002)
Long-range kernel
de Boeij et al. J.Chem.Phys. 115, 1995 (2002)
Polarization density functional. Long-range.
Kim and Görling Phys.Rev.Lett. 89, 96402 (2002)
Exact-exchange
Sottile et al. Phys.Rev.B 68, 205112 (2003)
Long-range and contact exciton.
Botti et al. Phys. Rev. B 72, 125203 (2005)
Dynamic long-range component
Parameters to fit to experiments.
MBPT vs (TD)DFT
Francesco Sottile
Introduction
Beyond ALDA approximation
Abs in solids. Insights from MBPT
Parameter-free Ab initio kernels
Sottile et al. Phys.Rev.Lett. 91, 56402 (2003)
Full many-body kernel. Mapping Theory.
Marini et al. Phys.Rev.Lett. 91, 256402 (2003)
Full many-body kernel. Perturbation Theory.
−1
fxc = χ−1
0 GGWGG χ0
MBPT vs (TD)DFT
Francesco Sottile
Introduction
Outline
1
Introduction
2
3
Theory
Results
4
MBPT vs (TD)DFT
Francesco Sottile
Introduction
Outline
1
Introduction
2
3
Theory
Results
4
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The Mapping Theory
The idea
BSE works ⇒
MBPT vs (TD)DFT
we get the ingredients of the BSE
and we put them in TDDFT
Francesco Sottile
Introduction
The Mapping Theory
BSE: Excitonic Hamiltonian
h
4-point
0 0
0
BSE
v c
v c
H(vc)(v
(Ec − Ev ) δvv 0 δcc 0 + vvc
− Wvc
0c 0) =
MBPT vs (TD)DFT
i
0
Francesco Sottile
Introduction
The Mapping Theory
h
H BSE = (Ec − Ev ) + v − W MBPT vs (TD)DFT
4-point
i
Francesco Sottile
Introduction
The Mapping Theory
4-point
h
i
GW
H BSE =
c + ∆GW
−
−
∆
+
v
−
W
v
c
v
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The Mapping Theory
4-point
h
i
GW
H BSE =
c + ∆GW
−
−
∆
+
v
−
W
v
c
v
TDDFT: Polarizability equation
2-point
χ = χ0 + χ0 (v + fxc ) χ
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The Mapping Theory
4-point
h
i
GW
H BSE =
c + ∆GW
−
−
∆
+
v
−
W
v
c
v
TDDFT: written in transition space
h
H TDDFT = (c − v ) + v + fxc MBPT vs (TD)DFT
4-point
i
Francesco Sottile
Introduction
The Mapping Theory
4-point
h
i
GW
H BSE =
c + ∆GW
−
−
∆
+
v
−
W
v
c
v
h
H TDDFT = (c − v ) + v + fxc 4-point
i
The exchange-correlation kernel fxc has to take into account both
GW corrections and excitonic effects !!
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The Mapping Theory
h
H BSE = (Ec − Ev ) + v − W 4-point
i
h
H TDDFT = (Ec − Ev ) + v + fxc 4-point
i
Same starting point for both BSE and TDDFT:
the GW band-structure.
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The Mapping Theory
h
i
h
H TDDFT = (Ec − Ev ) + v + fxc 4-point
i
We concentrate, then, only on the excitonic effects.
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The Mapping Theory
h
i
h
H TDDFT = (Ec − Ev ) + v − W 4-point
i
We substitute the ‘unknown’ fxc with W .
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The Mapping Theory
The idea
We want to use W , but in a 2-point equation.
χ(12, ω) = χ0 (12, ω) + χ0 (13, ω) (v (34) + fxc (34, ω)) χ(42, ω)
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The Mapping Theory
The idea
We want to use W , but in a 2-point equation.
χ(12, ω) = χ0 (12, ω) + χ0 (13, ω) (v (34) + fxc (34, ω)) χ(42, ω)
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The Mapping Theory
χ = χ0 + χ0 (v + fxc ) χ
χ = (1 − χ0 v − χ0 fxc )−1 χ0
Let’s define an invertible matrix X (12, ω) =
P
vc
φv (1)φc (1)gvc (2, ω)
χ = XX −1 1 − χ0 v − χ0 X −1 X fxc
−1
χ0
−1
χ0
χ = X X − χ0 v X − χ0 X −1 X fxc X
χ = X X − χ0 vX − χ0 X −1 T
MBPT vs (TD)DFT
−1
χ0
Francesco Sottile
Introduction
The Mapping Theory
χ = χ0 + χ0 (v + fxc ) χ
χ = (1 − χ0 v − χ0 fxc )−1 χ0
P
vc
φv (1)φc (1)gvc (2, ω)
χ = XX −1 1 − χ0 v − χ0 X −1 X fxc
−1
χ0
−1
χ0
χ = X X − χ0 v X − χ0 X −1 X fxc X
χ = X X − χ0 vX − χ0 X −1 T
MBPT vs (TD)DFT
−1
χ0
Francesco Sottile
Introduction
The Mapping Theory
χ = χ0 + χ0 (v + fxc ) χ
χ = (1 − χ0 v − χ0 fxc )−1 χ0
P
vc
φv (1)φc (1)gvc (2, ω)
χ = XX −1 1 − χ0 v − χ0 X −1 X fxc
−1
χ0
−1
χ0
χ = X X − χ0 v X − χ0 X −1 X fxc X
χ = X X − χ0 vX − χ0 X −1 T
MBPT vs (TD)DFT
−1
χ0
Francesco Sottile
Introduction
The Mapping Theory
χ = χ0 + χ0 (v + fxc ) χ
χ = (1 − χ0 v − χ0 fxc )−1 χ0
P
vc
φv (1)φc (1)gvc (2, ω)
χ = XX −1 1 − χ0 v − χ0 X −1 X fxc
−1
χ0
−1
χ0
χ = X X − χ0 v X − χ0 X −1 X fxc X
χ = X X − χ0 vX − χ0 X −1 T
MBPT vs (TD)DFT
−1
χ0
Francesco Sottile
Introduction
The Mapping Theory
χ = χ0 + χ0 (v + fxc ) χ
χ = (1 − χ0 v − χ0 fxc )−1 χ0
P
vc
φv (1)φc (1)gvc (2, ω)
χ = XX −1 1 − χ0 v − χ0 X −1 X fxc
−1
χ0
−1
χ0
χ = X X − χ0 v X − χ0 X −1 X fxc X
χ = X X − χ0 vX − χ0 X −1 T
MBPT vs (TD)DFT
−1
χ0
Francesco Sottile
Introduction
The Mapping Theory
χ = χ0 + χ0 (v + fxc ) χ
χ = (1 − χ0 v − χ0 fxc )−1 χ0
P
vc
φv (1)φc (1)gvc (2, ω)
χ = XX −1 1 − χ0 v − χ0 X −1 X fxc
−1
χ0
−1
χ0
χ = X X − χ0 v X − χ0 X −1 X fxc X
χ = X X − χ0 vX − χ0 X −1 T
MBPT vs (TD)DFT
−1
χ0
Francesco Sottile
Introduction
The Mapping Theory
χ = X X − χ0 vX − χ0 X −1 T
−1
χ0
T (12, ω) = X (13, ω)fxc (34, ω)X (42, ω) =
XZ
d(34)gvc (1, ω)φv (3)φc (3)fxc (34, ω)φv 0 (4)φc 0 (4)gv 0 c 0 (2, ω)
vc
T (12, ω) =
X
gvc (1, ω) fxc gv 0 c 0 (2, ω)
vc
v 0c 0
TBSE (12, ω) =
X
gvc (1, ω) W gv 0 c 0 (2, ω)
vc
v 0c 0
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The Mapping Theory
χ = X X − χ0 vX − χ0 X −1 T
−1
χ0
T (12, ω) = X (13, ω)fxc (34, ω)X (42, ω) =
XZ
vc
T (12, ω) =
X
gvc (1, ω) fxc gv 0 c 0 (2, ω)
vc
v 0c 0
TBSE (12, ω) =
X
gvc (1, ω) W gv 0 c 0 (2, ω)
vc
v 0c 0
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The Mapping Theory
χ = X X − χ0 vX − χ0 X −1 T
−1
χ0
T (12, ω) = X (13, ω)fxc (34, ω)X (42, ω) =
XZ
vc
T (12, ω) =
X
gvc (1, ω) fxc gv 0 c 0 (2, ω)
vc
v 0c 0
TBSE (12, ω) =
X
gvc (1, ω) W gv 0 c 0 (2, ω)
vc
v 0c 0
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The Mapping Theory
χ = X X − χ0 vX − χ0 X −1 T
−1
χ0
T (12, ω) = X (13, ω)fxc (34, ω)X (42, ω) =
XZ
vc
T (12, ω) =
X
gvc (1, ω) fxc gv 0 c 0 (2, ω)
vc
v 0c 0
TBSE (12, ω) =
X
gvc (1, ω) W gv 0 c 0 (2, ω)
vc
v 0c 0
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The Mapping Theory
χ = X X − χ0 vX − χ0 X −1 T
−1
χ0
T (12, ω) = X (13, ω)fxc (34, ω)X (42, ω) =
X Z
vc
v 0c 0
T (12, ω) =
X
gvc (1, ω) fxc gv 0 c 0 (2, ω)
vc
v 0c 0
TBSE (12, ω) =
X
gvc (1, ω) W gv 0 c 0 (2, ω)
vc
v 0c 0
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The Mapping Theory
χ = X X − χ0 vX − χ0 X −1 T
−1
χ0
T (12, ω) = X (13, ω)fxc (34, ω)X (42, ω) =
X Z
vc
v 0c 0
T (12, ω) =
X
gvc (1, ω) fxc gv 0 c 0 (2, ω)
vc
v 0c 0
TBSE (12, ω) =
X
gvc (1, ω) W gv 0 c 0 (2, ω)
vc
v 0c 0
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The Mapping Theory
χ = X X − χ0 vX − χ0 X −1 T
−1
χ0
T (12, ω) = X (13, ω)fxc (34, ω)X (42, ω) =
X Z
vc
v 0c 0
T (12, ω) =
X
gvc (1, ω) fxc gv 0 c 0 (2, ω)
vc
v 0c 0
TBSE (12, ω) =
X
gvc (1, ω) W gv 0 c 0 (2, ω)
vc
v 0c 0
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The Mapping Theory
χ = X X − χ0 vX − χ0 X −1 T
−1
χ0
T (12, ω) = X (13, ω)fxc (34, ω)X (42, ω) =
X Z
vc
v 0c 0
T (12, ω) =
X
gvc (1, ω) fxc gv 0 c 0 (2, ω)
vc
v 0c 0
TBSE (12, ω) =
X
gvc (1, ω) W gv 0 c 0 (2, ω)
vc
v 0c 0
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The Mapping Theory
TDDFT 2-point equation containing W χ = X X − χ0 vX − χ0 X −1 T
TBSE (12, ω) =
X
−1
χ0
gvc (1, ω) W gv 0 c 0 (2, ω)
vc
v 0c 0
X (12, ω) =
X
φv (1)φc (1)gvc (2, ω)
vc
What about the application ??
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The Mapping Theory
TBSE (12, ω) =
X
−1
χ0
gvc (1, ω) W gv 0 c 0 (2, ω)
vc
v 0c 0
X (12, ω) =
X
φv (1)φc (1)gvc (2, ω)
vc
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The Mapping Theory
TBSE (12, ω) =
X
−1
χ0
gvc (1, ω) W gv 0 c 0 (2, ω)
vc
v 0c 0
X (12, ω) =
X
φv (1)φc (1)gvc (2, ω)
vc
MBPT vs (TD)DFT
Francesco Sottile
Introduction
Outline
1
Introduction
2
3
Theory
Results
4
MBPT vs (TD)DFT
Francesco Sottile
Introduction
The Mapping Theory: Results
Absorption of Silicon
60
Exp.
RPA
ALDA
50
40
30
20
10
0
MBPT vs (TD)DFT
3
4
ω (eV)
5
6
Francesco Sottile
Introduction
60
Exp.
RPA
ALDA
BSE
50
40
30
20
10
0
MBPT vs (TD)DFT
3
4
ω (eV)
5
6
Francesco Sottile
Introduction
60
Exp.
RPA
ALDA
BSE
MT kernel
50
40
30
20
10
0
3
4
ω (eV)
5
6
F.Sottile et al. Phys.Rev.Lett 91, 56402 (2003)
MBPT vs (TD)DFT
Francesco Sottile
Introduction
Absorption of Silicon Carbide and Diamond
20
20
BSE
MT
BSE
MT
15
15
10
10
5
5
0
10
5
eV
0
5
10
15
20
eV
last week preliminary results :-)
MBPT vs (TD)DFT
Francesco Sottile
Introduction
Absorption of Argon
exp
ALDA
BSE
-v0 Im {χ00}
15
10
5
0
10
MBPT vs (TD)DFT
12
ω (eV)
14
Francesco Sottile
Introduction
Absorption of Argon
exp
ALDA
BSE
Ab initio kernel
-v0 Im {χ00}
15
10
5
0
10
12
ω (eV)
14
F.Sottile, M.Marsili et al., PRB(R) 76, 161103 (2007)
MBPT vs (TD)DFT
Francesco Sottile
Introduction
Tested also on absorption of SiO2 , DNA bases, Ge-nanowires,
RAS of diamond surface, and EELS of LiF.
Marini et al. Phys.Rev.Lett. 91, 256402 (2003).
Bruno et al. Phys.Rev.B 72 153310, (2005).
Palummo et al. Phys.Rev.Lett. 94 087404 (2005).
Varsano et al. J.Phys.Chem.B 110 7129 (2006).
MBPT vs (TD)DFT
Francesco Sottile
Introduction
Outline
1
Introduction
2
3
Theory
Results
4
MBPT vs (TD)DFT
Francesco Sottile
Introduction
Conclusions
TDDFT is the method of choice
√
√
√
√
Absorption spectra of simple molecules
Electron energy loss spectra
Inelastic X-ray scattering spectroscopy
Absorption of Solids (BSE-like scaling)
MBPT vs (TD)DFT
Francesco Sottile
Introduction
Conclusions
DFT-MBPT
Functionals [ρ]
Extensions of TDDFT
⇒ Mapping Theory
⇒ Meta-GGA
⇒ TD-CDFT
⇒ OEP (EXX, etc.)
⇒ Orbital
dependency
⇒ Deformation
Theory
Today challenges
⇒ Open shells systems
⇒ Charge transfer excitations
⇒ Efficient calculations of Solids
MBPT vs (TD)DFT
Francesco Sottile
Introduction
Conclusions
DFT-MBPT
Functionals [ρ]
Extensions of TDDFT
⇒ Mapping Theory
⇒ Meta-GGA
⇒ TD-CDFT
⇒ OEP (EXX, etc.)
⇒ Orbital
dependency
⇒ Deformation
Theory
Today challenges
MBPT vs (TD)DFT
Francesco Sottile
Introduction
Conclusions
DFT-MBPT
Functionals [ρ]
Extensions of TDDFT
⇒ Mapping Theory
⇒ Meta-GGA
⇒ TD-CDFT
⇒ OEP (EXX, etc.)
⇒ Orbital
dependency
⇒ Deformation
Theory
Today challenges
MBPT vs (TD)DFT
Francesco Sottile
Introduction
Conclusions
DFT-MBPT
Functionals [ρ]
Extensions of TDDFT
⇒ Mapping Theory
⇒ Meta-GGA
⇒ TD-CDFT
⇒ OEP (EXX, etc.)
⇒ Orbital
dependency
⇒ Deformation
Theory
Today challenges
MBPT vs (TD)DFT
Francesco Sottile

MBPT vs (TD)DFT - a fight or a wedding

Transcription

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