finite-temperature second-order green’s function approach...
TRANSCRIPT
Finite-temperature second-order Green’s function approach to
electronic correlations in solidsAlexander A. Rusakov1, Tran Nguyen Lan1,2, Sergei Iskakov2, and Dominika Zgid1
University of Michigan, 1 Department of Chemistry, 2 Department of Physics
Washington, D.C.August, 2017
Quantum chemistry of solids: constellation of challenges
Quantum chemistry of solids: constellation of challenges
• Large system size => computational complexity
Quantum chemistry of solids: constellation of challenges
• Large system size => computational complexity
• Small or disappearing band gaps => finite T is essential
Quantum chemistry of solids: constellation of challenges
• Large system size => computational complexity
• Small or disappearing band gaps => finite T is essential
• Metals: divergence of perturbative schemes (MP2)
Quantum chemistry of solids: constellation of challenges
• Large system size => computational complexity
• Small or disappearing band gaps => finite T is essential
• Metals: divergence of perturbative schemes (MP2)
• Electronic correlations govern electronic structure: need to go beyond mean-field methods
Quantum chemistry of solids: constellation of challenges
• Large system size => computational complexity
• Small or disappearing band gaps => finite T is essential
• Metals: divergence of perturbative schemes (MP2)
• Electronic correlations govern electronic structure: need to go beyond mean-field methods
• Prohibitive cost of wavefunction methods
Simpler than Ψ, richer than ρ(1,1’)
Simpler than Ψ, richer than ρ(1,1’)
Built-in temperature dependence
Simpler than Ψ, richer than ρ(1,1’)
Built-in temperature dependence
Electronic correlations: addressed in a
systematically improvable fashion
Simpler than Ψ, richer than ρ(1,1’)
Built-in temperature dependence
Electronic correlations: addressed in a
systematically improvable fashion
Computational tractability
Simpler than Ψ, richer than ρ(1,1’)
Built-in temperature dependence
Electronic correlations: addressed in a
systematically improvable fashion
Computational tractability
Simpler than Ψ, richer than ρ(1,1’)
Built-in temperature dependence
Electronic correlations: addressed in a
systematically improvable fashion
Computational tractability
Grand canonical one-electron Green’s function G(1,1’,ω=iωn), ωn=(2n+1)π/β, β=1/kBT
Simpler than Ψ, richer than ρ(1,1’)
Built-in temperature dependence
Electronic correlations: addressed in a
systematically improvable fashion
Computational tractability
Grand canonical one-electron Green’s function G(1,1’,ω=iωn), ωn=(2n+1)π/β, β=1/kBT
1-electron properties
Simpler than Ψ, richer than ρ(1,1’)
Built-in temperature dependence
Electronic correlations: addressed in a
systematically improvable fashion
Computational tractability
Grand canonical one-electron Green’s function G(1,1’,ω=iωn), ωn=(2n+1)π/β, β=1/kBT
1-electron properties
spectral properties
Simpler than Ψ, richer than ρ(1,1’)
Built-in temperature dependence
Electronic correlations: addressed in a
systematically improvable fashion
Computational tractability
Grand canonical one-electron Green’s function G(1,1’,ω=iωn), ωn=(2n+1)π/β, β=1/kBT
1-electron properties
spectral properties
energy
Simpler than Ψ, richer than ρ(1,1’)
Built-in temperature dependence
Electronic correlations: addressed in a
systematically improvable fashion
Computational tractability
Grand canonical one-electron Green’s function G(1,1’,ω=iωn), ωn=(2n+1)π/β, β=1/kBT
1-electron properties
spectral properties
energy thermodynamics
One-particle many-body Green’s function
One-particle many-body Green’s function
Gij(!) =D N
0
���ai⇥! �H+ EN
0
⇤�1a†j
��� N0
E+ c.c.
One-particle many-body Green’s function
Gij(!) =D N
0
���ai⇥! �H+ EN
0
⇤�1a†j
��� N0
E+ c.c. —difficult
One-particle many-body Green’s function
Gij(!) =D N
0
���ai⇥! �H+ EN
0
⇤�1a†j
��� N0
E+ c.c.
G0(!) = [! � F]�1
—difficult
One-particle many-body Green’s function
Gij(!) =D N
0
���ai⇥! �H+ EN
0
⇤�1a†j
��� N0
E+ c.c.
G0(!) = [! � F]�1
—difficult
— from 1-body operator, easy
One-particle many-body Green’s function
Gij(!) =D N
0
���ai⇥! �H+ EN
0
⇤�1a†j
��� N0
E+ c.c.
G0(!) = [! � F]�1
⌃(!) = G�10 (!)�G�1(!)
—difficult
— from 1-body operator, easy
One-particle many-body Green’s function
Gij(!) =D N
0
���ai⇥! �H+ EN
0
⇤�1a†j
��� N0
E+ c.c.
G0(!) = [! � F]�1
⌃(!) = G�10 (!)�G�1(!) — self-energy
—difficult
— from 1-body operator, easy
One-particle many-body Green’s function
Gij(!) =D N
0
���ai⇥! �H+ EN
0
⇤�1a†j
��� N0
E+ c.c.
G0(!) = [! � F]�1
⌃(!) = G�10 (!)�G�1(!) — self-energy
⌃(2)ij (⌧) = �
X
klmnpq
Gkl(⌧)Gmn(⌧)Gpq(�⌧)Vimqk(2Vjnpl � Vjlpn)
—difficult
— from 1-body operator, easy
One-particle many-body Green’s function
Gij(!) =D N
0
���ai⇥! �H+ EN
0
⇤�1a†j
��� N0
E+ c.c.
G0(!) = [! � F]�1
⌃(!) = G�10 (!)�G�1(!) — self-energy
⌃(2)ij (⌧) = �
X
klmnpq
Gkl(⌧)Gmn(⌧)Gpq(�⌧)Vimqk(2Vjnpl � Vjlpn)
—difficult
— from 1-body operator, easy
— 2nd-order perturbative approximation
One-particle many-body Green’s function
Gij(!) =D N
0
���ai⇥! �H+ EN
0
⇤�1a†j
��� N0
E+ c.c.
G0(!) = [! � F]�1
⌃(!) = G�10 (!)�G�1(!) — self-energy
⌃(2)ij (⌧) = �
X
klmnpq
Gkl(⌧)Gmn(⌧)Gpq(�⌧)Vimqk(2Vjnpl � Vjlpn)
—difficult
— from 1-body operator, easy
— 2nd-order perturbative approximation
cf. MP2:
One-particle many-body Green’s function
Gij(!) =D N
0
���ai⇥! �H+ EN
0
⇤�1a†j
��� N0
E+ c.c.
G0(!) = [! � F]�1
⌃(!) = G�10 (!)�G�1(!) — self-energy
⌃(2)ij (⌧) = �
X
klmnpq
Gkl(⌧)Gmn(⌧)Gpq(�⌧)Vimqk(2Vjnpl � Vjlpn)
—difficult
— from 1-body operator, easy
— 2nd-order perturbative approximation
cf. MP2:
⌃(2)ij (!) =
1
2
X
ars
V̄irasV̄jras
! � ✏i � ✏a � ✏b+
1
2
X
abr
V̄aibrV̄jarb
! + ✏j � ✏a � ✏b
Self-energy in atomic orbitals
⌃0g11 (⌧)
Self-energy in atomic orbitals⌃(2)
ij (⌧) = �X
klmnpq
Gkl(⌧)Gmn(⌧)Gpq(�⌧)Vimqk(2Vjnpl � Vjlpn)
⌃0g11 (⌧)
Self-energy in atomic orbitals⌃(2)
ij (⌧) = �X
klmnpq
Gkl(⌧)Gmn(⌧)Gpq(�⌧)Vimqk(2Vjnpl � Vjlpn)
cost: O(No5Nc4Nτ)
⌃0g11 (⌧)
Self-energy in atomic orbitals⌃(2)
ij (⌧) = �X
klmnpq
Gkl(⌧)Gmn(⌧)Gpq(�⌧)Vimqk(2Vjnpl � Vjlpn)
· · ·|G00
ij (τ)| > |G01ij (τ)| > · · · · · · > |G0g
ij (τ)|
0 1 g2cost: O(No5Nc4Nτ)
⌃0g11 (⌧)
Self-energy in atomic orbitals⌃(2)
ij (⌧) = �X
klmnpq
Gkl(⌧)Gmn(⌧)Gpq(�⌧)Vimqk(2Vjnpl � Vjlpn)
· · ·|G00
ij (τ)| > |G01ij (τ)| > · · · · · · > |G0g
ij (τ)|
0 1 g2
Rusakov, Zgid, J. Chem. Phys. 144, 054106 (2016)
cost: O(No5Nc4Nτ)
⌃0g11 (⌧)
⌃0g(⌧)
Self-energy in atomic orbitals⌃(2)
ij (⌧) = �X
klmnpq
Gkl(⌧)Gmn(⌧)Gpq(�⌧)Vimqk(2Vjnpl � Vjlpn)
· · ·|G00
ij (τ)| > |G01ij (τ)| > · · · · · · > |G0g
ij (τ)|
0 1 g2
Σ decays rapidly in AO basis
Rusakov, Zgid, J. Chem. Phys. 144, 054106 (2016)
cost: O(No5Nc4Nτ)
⌃0g11 (⌧)
⌃0g(⌧)
Self-energy in atomic orbitals⌃(2)
ij (⌧) = �X
klmnpq
Gkl(⌧)Gmn(⌧)Gpq(�⌧)Vimqk(2Vjnpl � Vjlpn)
· · ·|G00
ij (τ)| > |G01ij (τ)| > · · · · · · > |G0g
ij (τ)|
0 1 g2
Σ decays rapidly in AO basis
Rusakov, Zgid, J. Chem. Phys. 144, 054106 (2016)
cost: O(No5Nc4Nτ)
⌃0g11 (⌧)
computational cost can be dramatically reduced
⌃0g(⌧)
Self-consistent GF2
Self-consistent GF2 G(ω) = G0(ω) = [ω - F]-1
Self-consistent GF2 G(ω) = G0(ω) = [ω - F]-1 Hartree-Fock, DFT, …
Self-consistent GF2 G(ω) = G0(ω) = [ω - F]-1 Hartree-Fock, DFT, …
Self-consistent GF2 G(ω) = G0(ω) = [ω - F]-1
Σ(τ) = ΣGF2[G(τ)]
Hartree-Fock, DFT, …
Self-consistent GF2 G(ω) = G0(ω) = [ω - F]-1
Σ(τ) = ΣGF2[G(τ)]real space
atomic orbitals benefits of Σ locality
Hartree-Fock, DFT, …
Self-consistent GF2 G(ω) = G0(ω) = [ω - F]-1
Σ(τ) = ΣGF2[G(τ)]real space
atomic orbitals benefits of Σ locality
Hartree-Fock, DFT, …
Self-consistent GF2 G(ω) = G0(ω) = [ω - F]-1
Σ(τ) = ΣGF2[G(τ)]
Gcorr(ω,k) = [ω - Fk - Σ(ω,k)]-1
real space atomic orbitals
benefits of Σ locality
Hartree-Fock, DFT, …
Self-consistent GF2 G(ω) = G0(ω) = [ω - F]-1
Σ(τ) = ΣGF2[G(τ)]
Gcorr(ω,k) = [ω - Fk - Σ(ω,k)]-1
real space atomic orbitals
benefits of Σ locality
inversion in k-space
Hartree-Fock, DFT, …
Self-consistent GF2 G(ω) = G0(ω) = [ω - F]-1
Σ(τ) = ΣGF2[G(τ)]
Gcorr(ω,k) = [ω - Fk - Σ(ω,k)]-1
real space atomic orbitals
benefits of Σ locality
inversion in k-space
Hartree-Fock, DFT, …
Self-consistent GF2 G(ω) = G0(ω) = [ω - F]-1
Σ(τ) = ΣGF2[G(τ)]
Gcorr(ω,k) = [ω - Fk - Σ(ω,k)]-1
ρcorr = -2Gcorr(τ=β), F[ρcorr]
real space atomic orbitals
benefits of Σ locality
inversion in k-space
Hartree-Fock, DFT, …
Self-consistent GF2 G(ω) = G0(ω) = [ω - F]-1
Σ(τ) = ΣGF2[G(τ)]
Gcorr(ω,k) = [ω - Fk - Σ(ω,k)]-1
ρcorr = -2Gcorr(τ=β), F[ρcorr]
real space atomic orbitals
benefits of Σ locality
inversion in k-space
Hartree-Fock, DFT, …
updated mean-field term
Self-consistent GF2 G(ω) = G0(ω) = [ω - F]-1
Σ(τ) = ΣGF2[G(τ)]
Gcorr(ω,k) = [ω - Fk - Σ(ω,k)]-1
ρcorr = -2Gcorr(τ=β), F[ρcorr]
real space atomic orbitals
benefits of Σ locality
inversion in k-space
Hartree-Fock, DFT, …
updated mean-field term
Self-consistent GF2 G(ω) = G0(ω) = [ω - F]-1
Σ(τ) = ΣGF2[G(τ)]
Gcorr(ω,k) = [ω - Fk - Σ(ω,k)]-1
ρcorr = -2Gcorr(τ=β), F[ρcorr]
G(ω,k) = [ω - Fk[ρcorr] - Σ(ω,k)]-1
real space atomic orbitals
benefits of Σ locality
inversion in k-space
Hartree-Fock, DFT, …
updated mean-field term
Self-consistent GF2 G(ω) = G0(ω) = [ω - F]-1
Σ(τ) = ΣGF2[G(τ)]
Gcorr(ω,k) = [ω - Fk - Σ(ω,k)]-1
ρcorr = -2Gcorr(τ=β), F[ρcorr]
G(ω,k) = [ω - Fk[ρcorr] - Σ(ω,k)]-1
real space atomic orbitals
benefits of Σ locality
inversion in k-space
Hartree-Fock, DFT, …
updated mean-field term
Self-consistent GF2 G(ω) = G0(ω) = [ω - F]-1
Σ(τ) = ΣGF2[G(τ)]
Gcorr(ω,k) = [ω - Fk - Σ(ω,k)]-1
ρcorr = -2Gcorr(τ=β), F[ρcorr]
G(ω,k) = [ω - Fk[ρcorr] - Σ(ω,k)]-1
new Green’s function enters Σ evaluation until self-consistency is reached
Self-consistent GF2 G(ω) = G0(ω) = [ω - F]-1
Σ(τ) = ΣGF2[G(τ)]
Gcorr(ω,k) = [ω - Fk - Σ(ω,k)]-1
ρcorr = -2Gcorr(τ=β), F[ρcorr]
G(ω,k) = [ω - Fk[ρcorr] - Σ(ω,k)]-1
Self-consistent GF2 G(ω) = G0(ω) = [ω - F]-1
Σ(τ) = ΣGF2[G(τ)]
Gcorr(ω,k) = [ω - Fk - Σ(ω,k)]-1
ρcorr = -2Gcorr(τ=β), F[ρcorr]
G(ω,k) = [ω - Fk[ρcorr] - Σ(ω,k)]-1
• Self-consistent G does not depend on the initial G0
Self-consistent GF2 G(ω) = G0(ω) = [ω - F]-1
Σ(τ) = ΣGF2[G(τ)]
Gcorr(ω,k) = [ω - Fk - Σ(ω,k)]-1
ρcorr = -2Gcorr(τ=β), F[ρcorr]
G(ω,k) = [ω - Fk[ρcorr] - Σ(ω,k)]-1
• Self-consistent G does not depend on the initial G0
• Energy is uniquely defined
Self-consistent GF2 G(ω) = G0(ω) = [ω - F]-1
Σ(τ) = ΣGF2[G(τ)]
Gcorr(ω,k) = [ω - Fk - Σ(ω,k)]-1
ρcorr = -2Gcorr(τ=β), F[ρcorr]
G(ω,k) = [ω - Fk[ρcorr] - Σ(ω,k)]-1
• Self-consistent G does not depend on the initial G0
• Energy is uniquely defined
• Thermodynamic properties within a conserving approximation
Phase diagram of 1D hydrogen solid
Phase diagram of 1D hydrogen solid…… H H H H H H
Phase diagram of 1D hydrogen solid
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2
…… H H H H H H
Phase diagram of 1D hydrogen solid
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2GF2, β = 100
…… H H H H H H
Phase diagram of 1D hydrogen solid
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2GF2, β = 100
…… H H H H H H
Phase diagram of 1D hydrogen solid
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2GF2, β = 100
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__________________________________________________________________________________________________________________ ______________________________________________________________________________________ ____________________________________________________________________ __________________________________________________________ __________________________________________________ ____________________________________________
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
0.0
0.5
1.0
1.5
2.0
natu
ral o
ccup
atio
n
0.0
0.5
1.0
1.5
2.0
…… H H H H H H
Phase diagram of 1D hydrogen solid
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2GF2, β = 100
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__________________________________________________________________________________________________________________ ______________________________________________________________________________________ ____________________________________________________________________ __________________________________________________________ __________________________________________________ ____________________________________________
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
0.0
0.5
1.0
1.5
2.0
natu
ral o
ccup
atio
n
0.0
0.5
1.0
1.5
2.0
…… H H H H H H
Phase diagram of 1D hydrogen solid
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2GF2, β = 100
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__________________________________________________________________________________________________________________ ______________________________________________________________________________________ ____________________________________________________________________ __________________________________________________________ __________________________________________________ ____________________________________________
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
0.0
0.5
1.0
1.5
2.0
natu
ral o
ccup
atio
n
0.0
0.5
1.0
1.5
2.0
…… H H H H H H
Phase diagram of 1D hydrogen solid
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2GF2, β = 100
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__________________________________________________________________________________________________________________ ______________________________________________________________________________________ ____________________________________________________________________ __________________________________________________________ __________________________________________________ ____________________________________________
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
0.0
0.5
1.0
1.5
2.0
natu
ral o
ccup
atio
n
0.0
0.5
1.0
1.5
2.0
metal
…… H H H H H H
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2GF2, β = 100
Phase diagram of 1D hydrogen solid…… H H H H H H
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2GF2, β = 100
Phase diagram of 1D hydrogen solid…… H H H H H H
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2GF2, β = 100
Phase diagram of 1D hydrogen solid…… H H H H H H
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__________________________________________________________________________________________________________________ ______________________________________________________________________________________ ____________________________________________________________________ __________________________________________________________ __________________________________________________ ____________________________________________
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
0.0
0.5
1.0
1.5
2.0
natu
ral o
ccup
atio
n
0.0
0.5
1.0
1.5
2.0
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2GF2, β = 100
Phase diagram of 1D hydrogen solid…… H H H H H H
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__________________________________________________________________________________________________________________ ______________________________________________________________________________________ ____________________________________________________________________ __________________________________________________________ __________________________________________________ ____________________________________________
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
0.0
0.5
1.0
1.5
2.0
natu
ral o
ccup
atio
n
0.0
0.5
1.0
1.5
2.0
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2GF2, β = 100
Phase diagram of 1D hydrogen solid…… H H H H H H
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__________________________________________________________________________________________________________________ ______________________________________________________________________________________ ____________________________________________________________________ __________________________________________________________ __________________________________________________ ____________________________________________
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
0.0
0.5
1.0
1.5
2.0
natu
ral o
ccup
atio
n
0.0
0.5
1.0
1.5
2.0
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2GF2, β = 100
Phase diagram of 1D hydrogen solid…… H H H H H H
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__________________________________________________________________________________________________________________ ______________________________________________________________________________________ ____________________________________________________________________ __________________________________________________________ __________________________________________________ ____________________________________________
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
0.0
0.5
1.0
1.5
2.0
natu
ral o
ccup
atio
n
0.0
0.5
1.0
1.5
2.0
band insulator
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2GF2, β = 100
Phase diagram of 1D hydrogen solid…… H H H H H H
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2GF2, β = 100
Phase diagram of 1D hydrogen solid…… H H H H H H
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2GF2, β = 100
Phase diagram of 1D hydrogen solid…… H H H H H H
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__________________________________________________________________________________________________________________ ______________________________________________________________________________________ ____________________________________________________________________ __________________________________________________________ __________________________________________________ ____________________________________________
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
0.0
0.5
1.0
1.5
2.0
natu
ral o
ccup
atio
n
0.0
0.5
1.0
1.5
2.0
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2GF2, β = 100
Phase diagram of 1D hydrogen solid…… H H H H H H
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__________________________________________________________________________________________________________________ ______________________________________________________________________________________ ____________________________________________________________________ __________________________________________________________ __________________________________________________ ____________________________________________
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
0.0
0.5
1.0
1.5
2.0
natu
ral o
ccup
atio
n
0.0
0.5
1.0
1.5
2.0
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2GF2, β = 100
Phase diagram of 1D hydrogen solid…… H H H H H H
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__________________________________________________________________________________________________________________ ______________________________________________________________________________________ ____________________________________________________________________ __________________________________________________________ __________________________________________________ ____________________________________________
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
0.0
0.5
1.0
1.5
2.0
natu
ral o
ccup
atio
n
0.0
0.5
1.0
1.5
2.0 open-shell singletstrong correlation
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2GF2, β = 100
Phase diagram of 1D hydrogen solid…… H H H H H H
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__________________________________________________________________________________________________________________ ______________________________________________________________________________________ ____________________________________________________________________ __________________________________________________________ __________________________________________________ ____________________________________________
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
0.0
0.5
1.0
1.5
2.0
natu
ral o
ccup
atio
n
0.0
0.5
1.0
1.5
2.0 open-shell singletstrong correlation
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
-1.30
-1.20
-1.10
-1.00
-0.90
-0.80
-0.70
E, a
.u.
HFMP2GF2, β = 100
Phase diagram of 1D hydrogen solid…… H H H H H H
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__________________________________________________________________________________________________________________ ______________________________________________________________________________________ ____________________________________________________________________ __________________________________________________________ __________________________________________________ ____________________________________________
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00R, Å
0.0
0.5
1.0
1.5
2.0
natu
ral o
ccup
atio
n
0.0
0.5
1.0
1.5
2.0
Mott insulator
open-shell singletstrong correlation
Phase diagram of 1D hydrogen solid
T, K
R, Å1.0 2.0 3.0 4.03000
12000
metalband
insulatorGF2
cannot converge
Mottinsulator
In collaboration with Alicia Welden Welden, Rusakov, Zgid, J. Chem. Phys. 145, 204106 (2016)
Phase diagram of 1D hydrogen solid
T, K
R, Å1.0 2.0 3.0 4.03000
12000
metalband
insulatorGF2
cannot converge
Mottinsulator
In collaboration with Alicia Welden Welden, Rusakov, Zgid, J. Chem. Phys. 145, 204106 (2016)
Phase diagram of 1D hydrogen solid
T, K
R, Å1.0 2.0 3.0 4.03000
12000
metalband
insulatorGF2
cannot converge
Mottinsulator
In collaboration with Alicia Welden Welden, Rusakov, Zgid, J. Chem. Phys. 145, 204106 (2016)
Phase diagram of 1D hydrogen solid
T, K
R, Å1.0 2.0 3.0 4.03000
12000
metalband
insulatorGF2
cannot converge
Mottinsulator
In collaboration with Alicia Welden Welden, Rusakov, Zgid, J. Chem. Phys. 145, 204106 (2016)
Phase diagram of 1D hydrogen solid
GF2 band gap in 1D BN
B BN N ……R=1.45 Å
ANO-pVDZ basis
β = 100
GF2 band gap in 1D BN
In collaboration with Alicia Welden Welden, Rusakov, Zgid, J. Chem. Phys. 145, 204106 (2016)
B BN N ……R=1.45 Å
ANO-pVDZ basis
β = 100
GF2 band gap in 1D BN
In collaboration with Alicia Welden Welden, Rusakov, Zgid, J. Chem. Phys. 145, 204106 (2016)
B BN N ……R=1.45 Å
ANO-pVDZ basis
β = 100
GF2 band gap in 1D BN
Band gap decreases as T rises
In collaboration with Alicia Welden Welden, Rusakov, Zgid, J. Chem. Phys. 145, 204106 (2016)
B BN N ……R=1.45 Å
ANO-pVDZ basis
β = 100
Realistic solids: embedded GF2
Realistic solids: embedded GF2
Realistic solids: embedded GF2
Realistic solids: embedded GF2
0central cell
Realistic solids: embedded GF2
0central cell
Realistic solids: embedded GF2
0central cell
g
Realistic solids: embedded GF2
Σ0g(τ) decreases with intercell distance
0central cell
g
Realistic solids: embedded GF2
GF2 region: Σ(τ)GF2
Realistic solids: embedded GF2
SCF
GF2 region: Σ(τ)GF2
Realistic solids: embedded GF2
SCF
GF2 region: Σ(τ)GF2
GF2-in-SCF embedding
3D lithium hydride
3D lithium hydrideGF2-in-HF embedding:
3D lithium hydrideGF2-in-HF embedding:GF2: 25 primitive cells
3D lithium hydrideGF2-in-HF embedding:GF2: 25 primitive cells
in HF: 881 primitive cells
3D lithium hydride
-10 -5 0 5 10 15 20 25E, eV
0
0.2
0.4
0.6
0.8
DO
S
GF2: Li (2s1p), H(3s1p)GF2: Li (3s2p1d), H (3s1p)HF: Li (3s2p1d), H(3s1p)
GF2-in-HF embedding:GF2: 25 primitive cells
in HF: 881 primitive cells
3D lithium hydride
Band gaps:
-10 -5 0 5 10 15 20 25E, eV
0
0.2
0.4
0.6
0.8
DO
S
GF2: Li (2s1p), H(3s1p)GF2: Li (3s2p1d), H (3s1p)HF: Li (3s2p1d), H(3s1p)
GF2-in-HF embedding:GF2: 25 primitive cells
in HF: 881 primitive cells
3D lithium hydride
Band gaps:
-10 -5 0 5 10 15 20 25E, eV
0
0.2
0.4
0.6
0.8
DO
S
GF2: Li (2s1p), H(3s1p)GF2: Li (3s2p1d), H (3s1p)HF: Li (3s2p1d), H(3s1p)
GF2-in-HF embedding:GF2: 25 primitive cells
in HF: 881 primitive cells
HF: 15 eV
3D lithium hydride
Band gaps:
-10 -5 0 5 10 15 20 25E, eV
0
0.2
0.4
0.6
0.8
DO
S
GF2: Li (2s1p), H(3s1p)GF2: Li (3s2p1d), H (3s1p)HF: Li (3s2p1d), H(3s1p)
GF2-in-HF embedding:GF2: 25 primitive cells
in HF: 881 primitive cells
HF: 15 eVGF2, small basis: 8 eV
3D lithium hydride
Band gaps:
-10 -5 0 5 10 15 20 25E, eV
0
0.2
0.4
0.6
0.8
DO
S
GF2: Li (2s1p), H(3s1p)GF2: Li (3s2p1d), H (3s1p)HF: Li (3s2p1d), H(3s1p)
GF2-in-HF embedding:GF2: 25 primitive cells
in HF: 881 primitive cells
HF: 15 eVGF2, small basis: 8 eVGF2, larger basis: 5 eV
3D lithium hydride
Band gaps:
-10 -5 0 5 10 15 20 25E, eV
0
0.2
0.4
0.6
0.8
DO
S
GF2: Li (2s1p), H(3s1p)GF2: Li (3s2p1d), H (3s1p)HF: Li (3s2p1d), H(3s1p)
GF2-in-HF embedding:GF2: 25 primitive cells
in HF: 881 primitive cells
HF: 15 eVGF2, small basis: 8 eVGF2, larger basis: 5 eVExperiment: 4.99 eV
Beyond GF2: self-energy embedding in 1D hydrogen solid, STO-6G
Beyond GF2: self-energy embedding in 1D hydrogen solid, STO-6G
…… H H H H H H
Beyond GF2: self-energy embedding in 1D hydrogen solid, STO-6G
…… H H H H H H
Beyond GF2: self-energy embedding in 1D hydrogen solid, STO-6G
FCI FCI FCI…… H H H H H H
Beyond GF2: self-energy embedding in 1D hydrogen solid, STO-6G
FCI FCI FCI
GF2 GF2…… H H H H H H
Beyond GF2: self-energy embedding in 1D hydrogen solid, STO-6G
FCI FCI FCI
GF2 GF2…… H H H H H H
0.80 1.00 1.20 1.40 1.60 1.80 2.00R(H-H), Å
-1.10
-1.05
-1.00
-0.95
-0.90
-0.85
E pe
r -(H
-H)-
unit,
a.u
.
HFGF2SEET (FCI/GF2)AFQMC, TDE AFQMC data from Simons
Collaboration benchmark for H chains, by Shiwei Zhang
Beyond GF2: self-energy embedding in 1D hydrogen solid, STO-6G
FCI FCI FCI
GF2 GF2
T.N. Lan, Rusakov, and D. Zgid, in preparation
SEET vs. QMC extrapolated to the thermodynamic limit: mH energy deviations
…… H H H H H H
0.80 1.00 1.20 1.40 1.60 1.80 2.00R(H-H), Å
-1.10
-1.05
-1.00
-0.95
-0.90
-0.85
E pe
r -(H
-H)-
unit,
a.u
.
HFGF2SEET (FCI/GF2)AFQMC, TDE AFQMC data from Simons
Collaboration benchmark for H chains, by Shiwei Zhang
Beyond GF2: self-energy embedding in 1D hydrogen solid, STO-6G
Beyond GF2: self-energy embedding in 1D hydrogen solid, STO-6G
Self-consistent GF2
R=1.75 A
T=31K
Beyond GF2: self-energy embedding in 1D hydrogen solid, STO-6G
Self-consistent GF2
R=1.75 A
T=31K
SEET (FCI/GF2) for 2 orthogonalized AOs
R=1.75 A
T=31K
Beyond GF2: self-energy embedding in 1D hydrogen solid, STO-6G
Self-consistent GF2
R=1.75 A
T=31K
SEET (FCI/GF2) for 2 orthogonalized AOs
R=1.75 A
T=31K_
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0
0.5
1
1.5
2
natu
ral o
ccup
atio
ns
GF2
SEET
Conclusions and future work
Conclusions and future work• Self-consistent temperature-dependent GF2 in atomic
basis: a relatively simple, but generally adequate electronic correlation method for solids with realistic Hamiltonians
Conclusions and future work• Self-consistent temperature-dependent GF2 in atomic
basis: a relatively simple, but generally adequate electronic correlation method for solids with realistic Hamiltonians
• Small gaps: GF2 can perform better than MP2
Conclusions and future work• Self-consistent temperature-dependent GF2 in atomic
basis: a relatively simple, but generally adequate electronic correlation method for solids with realistic Hamiltonians
• Small gaps: GF2 can perform better than MP2
• GF2 can partially capture strong correlations
Conclusions and future work• Self-consistent temperature-dependent GF2 in atomic
basis: a relatively simple, but generally adequate electronic correlation method for solids with realistic Hamiltonians
• Small gaps: GF2 can perform better than MP2
• GF2 can partially capture strong correlations
• Local self-energy at the level beyond GF2 can lead to substantially more accurate methods — self-energy embedding (SEET)
Conclusions and future work• Self-consistent temperature-dependent GF2 in atomic
basis: a relatively simple, but generally adequate electronic correlation method for solids with realistic Hamiltonians
• Small gaps: GF2 can perform better than MP2
• GF2 can partially capture strong correlations
• Local self-energy at the level beyond GF2 can lead to substantially more accurate methods — self-energy embedding (SEET)
• Work on Green’s function methods with a multi-level self-energy approximation in realistic solids is underway
• Prof. Dominika Zgid and the Zgid group at the University of Michigan
• Prof. Edward Valeev (Virginia Tech.)
• Dr. Ireneusz Bulik (Yale U., Gaussian)
• Funding from DOE, Grant No. ER16391
• Organizers of the section
• Audience for attention
Acknowledgements