Combining density-functional theory and many-body methods · 2016-12-14 · Combining...
Transcript of Combining density-functional theory and many-body methods · 2016-12-14 · Combining...
Combining density-functional theory andmany-body methods
Julien ToulouseUniversite Pierre & Marie Curie and CNRS, Paris, France
AlbuquerqueNew Mexico, USA
June 2016
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
Outline
1 A brief overview of DFT/many-body hybrids
2 Range-separated hybrids for the ground state
3 Range-separated hybrids for excited states
2/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
Outline
1 A brief overview of DFT/many-body hybrids
2 Range-separated hybrids for the ground state
3 Range-separated hybrids for excited states
3/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
Overview of DFT/many-body hybrids
1 Kohn-Sham DFT (1965):
E = minΦ
{
〈Φ|T + Vne|Φ〉+ EHxc[nΦ]}
where Φ is a single determinant
Semilocal density-functional approximations (DFAs) for Exc[n]:LDA, GGAs, meta-GGAs
=⇒ better accuracy still needed: self-interaction error, strong/staticcorrelation, van der Waals dispersion interactions
2 Hybrid approximations (Becke 1993):
Exc = axEHFx [Φ] + (1− ax)E
DFAx [n] + E
DFAc [n]
with Hartree-Fock (HF) exchange: EHFx [Φ] = 〈Φ|Wee|Φ〉 − EH[n]
one empirical parameter: ax ≈ 0.25
=⇒ reduces self-interaction error4/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
Overview of DFT/many-body hybrids
3 Double-hybrid approximations (Grimme 2006):
Exc = axEHFx [Φ] + (1− ax)E
DFAx [n] + (1− ac)E
DFAc [n] + acE
MP2c
with second-order Møller-Plesset (MP2) perturbative correlation:
EMP2c = −
occ∑
i<j
unocc∑
a<b
|〈Φabij |Wee|Φ〉|2
εa + εb − εi − εj
two empirical parameters: ax ≈ 0.5 and ac ≈ 0.25
=⇒ further reduces self-interaction error, partially account for van derWaals dispersion, but fails for strongly correlated systems
5/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
Overview of DFT/many-body hybrids
4 General DFT/many-body hybrid scheme (Sharkas, Toulouse, Savin 2011)
E = minΨ
{
〈Ψ|T + Vne + λWee|Ψ〉+ Eλ
Hxc[nΨ]}
with one empirical parameter λ and the complement density functional:
Eλ
Hx[n] = (1− λ)EHx[n] and Eλ
c [n] ≈ (1− λ2)Ec[n]
Single determinant: Ψ ≈ Φ =⇒ a hybrid approximation:
Exc = λEHFx [Φ] + (1− λ)EDFA
x [n] + (1− λ2)EDFA
c [n]
Second-order perturbation:one-parameter double-hybrid approximation:
Exc = λEHFx [Φ] + (1− λ)EDFA
x [n] + (1− λ2)EDFA
c [n] + λ2EMP2c
Non-perturbative approaches: Ψ =∑
n cnΦn
=⇒ for strong correlation (Sharkas, Savin, Jensen, Toulouse, JCP, 2012)
6/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
Overview of DFT/many-body hybrids
5 Range-separated DFT/many-body hybrid scheme (Savin 1996)
E = minΨ
{
〈Ψ|T + Vne + Wlree|Ψ〉+ E
srHxc[nΨ]
}
with a long-range interaction Wlree =
∑
i<j
erf(µrij )rij
a short-range density functional E srHxc[n]
and one empirical parameter µ controlling the range of the separation
Single determinant: Ψ ≈ Φ =⇒ a lrHF+srDFT hybrid:
Exc = Elr,HFx [Φ] + E
sr,DFAx [n] + E
sr,DFAc [n]
similar to the LC scheme (Hirao et al. 2001)
Many-body perturbation theory =⇒ lrMP2/lrRPA+srDFT hybrids:
Exc = Elr,HFx [Φ] + E
sr,DFAx [n] + E
sr,DFAc [n] + E
lr,MP2/RPAc
Angyan, Gerber, Savin, Toulouse, PRA, 2005; Toulouse, Gerber, Jansen, Savin,Angyan, PRL, 2009; Janesko, Henderson, Scuseria, JCP, 2009
Non-perturbative approaches: Ψ =∑
n cnΦn
Leininger, Stoll, Werner, Savin, CPL, 1997; Fromager, Toulouse, Jensen, JCP, 20077/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
Outline
1 A brief overview of DFT/many-body hybrids
2 Range-separated hybrids for the ground state
3 Range-separated hybrids for excited states
8/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
Range-separated hybrids: lrHF+srDFT
ElrHF+srDFTxc = E
lr,HFx [Φ] + E
sr,DFAx [n] + E
sr,DFAc [n]
Angyan, Gerber, Savin, Toulouse, PRA, 2005
Short-range exchange and correlation DFAs have been developed, e.g.:
short-range LDA:
EsrLDAx/c [n] =
∫
n(r) ǫsr,unifx/c (n(r)) dr
Toulouse, Savin, Flad, IJQC, 2004; Paziani, Moroni, Gori-Giorgi, Bachelet, PRB, 2006
short-range PBE:
EsrPBEx/c [n] =
∫
n(r) ǫsrPBEx/c (n(r),∇n(r)) dr
Toulouse, Colonna, Savin, JCP, 2005; Goll, Werner, Stoll, Leininger, Gori-Giorgi,
Savin, CP, 2006
9/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
lrHF+srDFT: reduction of the self-interaction error
Dissociation of H+2 molecule: H+
2 −→ H+0.5 + H+0.5
-70
-60
-50
-40
-30
-20
-10
0
10
0 2 4 6 8 10
Bin
din
g e
nerg
y (
kcal/
mo
l)
Internuclear distance (Angstrom)
exact
lrHF+srPBE
PBE0
PBE
=⇒ removal of the large self-interaction error comingfrom the long-range part of the PBE exchange
Mussard, Toulouse, submitted10/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
Adding long-range correlation: lrMP2/lrRPA+srDFT
Long-range MP2:
Elr,MP2c = −
occ∑
i<j
unocc∑
a<b
|〈Φabij |W
lree|Φ〉|2
εa + εb − εi − εj= +
Angyan, Gerber, Savin, Toulouse, PRA, 2005
Long-range direct RPA (dRPA) = sum of all direct ring diagrams
Elr,dRPAc = + + · · ·
Toulouse, Gerber, Jansen, Savin, Angyan, PRL, 2009; Janesko, Henderson, Scuseria, JCP, 2009
Long-range RPA with exchange (RPAx-SO2) = sum of all direct+ some exchange ring diagrams
Elr,RPAx-SO2c = + + +
+ + + · · ·Toulouse, Zhu, Savin, Jansen, Angyan, JCP, 2011
11/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
Fast basis convergence of long-range perturbation theory
Total energy of N2 calculated with Gaussian basis sets(µ = 0.5 bohr−1, srPBE functional, Dunning basis cc-pVXZ):
-109.45
-109.40
-109.35
-109.30
-109.25
VDZ VTZ VQZ V5Z V6Z
To
tal
en
erg
y (
ha
rtre
e)
Basis sets of increasing sizes
MP2lrMP2+srPBE
=⇒ Exponential basis convergence of lrMP2+srPBE
Franck, Mussard, Luppi, Toulouse, JCP, 2015 12/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
Test of lrMP2/lrRPA+srDFT on atomization energies
A G2 subset of 49 atomization energies of small molecules(µ = 0.5 bohr−1, srPBE functional, cc-pVQZ, spin unrestricted):
0
4
8
12
-20 -10 0 10
Sta
nd
ard
devia
tio
n (
kcal/
mo
l)
Mean error (kcal/mol)
MP2dRPARPAx-SO2lrMP2+srPBElrdRPA+srPBElrRPAx-SO2+srPBE
=⇒ Range separation decreases the standard deviation for all methods
=⇒ All range-separated methods give a mean error of ∼ 5 kcal/mol
Mussard, Reinhardt, Angyan, Toulouse, JCP, 201513/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
Test of lrMP2/lrRPA+srDFT on reaction barrier heights
DBH24/08 set: 24 barrier heights of reactions with small molecules(µ = 0.5 bohr−1, srPBE functional, aug-cc-pVQZ, spin unrestricted):
0
2
4
6
8
-1 0 1 2 3 4 5 6
Sta
nd
ard
devia
tio
n (
kcal/
mo
l)
Mean error (kcal/mol)
MP2dRPARPAx-SO2lrMP2+srPBElrdRPA+srPBElrRPAx-SO2+srPBE
=⇒ Range separation improves all methods
=⇒ All range-separated methods give a mean error . 1 kcal/mol
Mussard, Reinhardt, Angyan, Toulouse, JCP, 201514/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
Test of lrMP2/lrRPA+srDFT on weak interactions
S22 set: 22 equilibrium interaction energies of weakly-interacting molecularsystems from water dimer to DNA base pairs(µ = 0.5 bohr−1, srPBE functional, aug-cc-pVDZ):
−60
−40
−20
0
20
40
60
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22%
of
err
or
on
in
tera
cti
on
en
erg
y
system in S22 set
lrMP2+srPBElrdRPA+srPBElrRPAx−SO2+srPBE
H bonded dispersion dispersion+multipoles
=⇒ lrRPAx-SO2+srPBE/aVDZ gives a mean absolute error of ∼ 4%
Toulouse, Zhu, Savin, Jansen, Angyan, JCP, 2011 15/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
lrMP2+srDFT for periodic solid-state systems
periodic lrHF+srDFT, followed by local lrMP2 with localized orbitalsTest on cohesive energies (µ = 0.5 bohr−1, srPBE functional,p-aug-cc-pVDZ)
−20
−10
0
10
20
30
40
50
60
70
Ne
Ar
CO
2
NH
3
HC
N
LiH
LiF
Si
SiC
% o
f err
or
on
co
hesiv
e e
nerg
y
crystal
PBEMP2lrMP2
rare−gas molecular ionic semi−conductor
+srPBE
=⇒ lrMP2+srPBE improves over PBE but is similar to MP2
Sansone, Civalleri, Usvyat, Toulouse, Sharkas, Maschio, JCP, 201516/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
Outline
1 A brief overview of DFT/many-body hybrids
2 Range-separated hybrids for the ground state
3 Range-separated hybrids for excited states
17/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
Time-dependent range-separated hybrids
Linear-response TDDFT equation
χ−1(ω) = χ−10 (ω)− fHxc(ω)
=⇒ excitation energies, linear-response properties
Range separation for exchange kernel is now standard:
fxc = flr,HFx + f
sr,DFAx + f
DFAc
Tawada, Tsuneda, Yanagisawa, Yanai, Hirao, JCP, 2004
Here, range separation for both exchange and correlation kernels:
fxc = flr,HFx + f
sr,DFAx + f
sr,DFAc + f
lr,(2)c (ω)
Rebolini, Savin, Toulouse, MP, 2013; Rebolini, Toulouse, JCP, 2016
Other similar schemes: Pernal, JCP, 2012; Fromager, Knecht, Jensen, JCP, 2013;
Hedegard, Heiden, Knecht, Fromager, Jensen, JCP, 201318/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
Second-order correlation self-energy and kernel
EMP2c
= +
Σ(2)c (2, 1) =
1
2
+
1
2
δ
δG0(1, 2)
Ξ(2)c (2, 4; 1, 3)=
1, 4
2, 3
+
1, 4
2 3
δ
δG0(3, 4)
+
1
2
4
3
+
1
2, 3
4
+
1
2
3
4
+
1
2
3
4
19/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
From the Bethe-Salpeter equation to TDDFT
Bethe-Salpeter equation with a kernel depending on only one frequency:
χ(ω) = χ0(ω) +
∫
dω′dω′′χ0(ω
′, ω)Ξ(2)Hxc(ω
′ ± ω′′)χ(ω′′, ω)
Define an effective second-order kernel:
f(2)Hxc(ω) = χ−1
0 (ω)
[∫
dω′dω′′χ0(ω
′, ω)Ξ(2)Hxc(ω
′ ± ω′′)χ0(ω′′, ω)
]
χ−10 (ω)
which can be used in a TDDFT-like linear-response equation
χ(ω) = χ0(ω) + χ0(ω)f(2)Hxc(ω)χ(ω)
or, equivalently,
χ−1(ω) = χ−10 (ω)− f
(2)Hxc(ω)
Romaniello, Sangalli, Berger, Sottile, Molinari, Reining, Onida, JCP, 2009
Sangalli, Romaniello, Onida, Marini, JCP, 201120/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
In an orbital basis
Effective long-range second-order correlation kernel (lrBSE2):
flr,(2)c,ia,jb(ω) =
occ∑
k<l
unocc∑
c<d
〈Φai |W
lree|Φ
cdkl 〉〈Φ
cdkl |W
lree|Φ
bj 〉
ω − (εc + εd − εk − εl)
=⇒ The correlation kernel brings the effect of the double excitations
Calculation of excitation energies in two steps:
1 lrTDHF+srTDLDA calculation in the TDA: A X0 = ω0 X0
2 perturbative addition of lrBSE2 kernel: ω = ω0 + X†0 f
lr,(2)c (ω0) X0
Zhang, Steinmann, Yang, JCP, 2013
Rebolini, Toulouse, JCP, 2016
21/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
Test of lrBSE2+srTDDFT on excitation energies
56 singlet and triplet excitation energies of 4 small molecules N2, CO, H2CO,C2H4 (µ = 0.35 bohr−1, srLDA functional, TDA, Sadlej+ basis):
0
0.1
0.2
0.3
0.4
0.5
-1.2 -1 -0.8 -0.6 -0.4 -0.2 0
Sta
nd
ard
de
via
tio
n (
eV
)
Mean error (eV)
Valence TDLDAValence lrTDHF+srTDLDAValence lrBSE2+srTDLDARydberg TDLDARydberg lrTDHF+srTDLDARydberg lrBSE2+srTDLDA
=⇒ lrBSE2+srTDLDA provides a slight overall improvement overlrTDHF+srTDLDA
Rebolini, Toulouse, JCP, 2016 22/23
1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states
Summary and Acknowledgments
DFT/many-body hybrid methods based on a decompositionof the e-e interaction into long-range and short-range parts
lrHF+srDFT already helps to reduce the self-interaction error
lrMP2/lrRPA+srDFT has a fast basis convergence
lrMP2/lrRPA+srDFT accounts for van der Waals dispersioninteractions
lrMP2+srDFT has been applied to periodic solid-state systems
lrBSE2+srTDDFT for excitation energies: frequency-dependentsecond-order long-range correlation kernel bringing the effect of thedouble excitations
Acknowledgments
J. Angyan, B. Civalleri, F. Colonna, H.-J. Flad, O. Franck, E. Fromager, I.Gerber, G. Jansen, H. J. Aa. Jensen, E. Luppi, L. Maschio, B. Mussard, E.Rebolini, P. Reinhardt, G. Sansone, A. Savin, K. Sharkas, D. Usvyat, W. Zhu
23/23