Convergence behavior of RPA renormalized many-body perturbation...
Transcript of Convergence behavior of RPA renormalized many-body perturbation...
Convergence behavior of RPA renormalized many-bodyperturbation theory
Understanding why low-order, non-perturbative expansions work
Jefferson E. Bates, Jonathon Sensenig,Niladri Sengupta, & Adrienn Ruzsinszky
Department of Physics, Temple University
August 20, 2017
Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/2017 1 / 14
Introduction & Background
Electronic Instabilities
N2 dissociation with EXX kernel
Treating exchange to ∞-order causes instabilities even in simple systems.Renormalized perturbation theories offer robust solution.
Colonna, Hellgren, de Gironcoli Phys. Rev. B 90, 125150 (2014)
Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/2017 2 / 14
Introduction & Background
Outline
1 Introduction & BackgroundACFDT & RPA
2 Beyond-RPA CorrelationRPA Renormalization
3 ResultsConvergence behaviorBulk Phase Transitions
4 Conclusions & Acknowledgements
Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/2017 3 / 14
Introduction & Background ACFDT & RPA
Adiabatic Connection Fluctuation-Dissipation Theorem
E [ρ] = 〈Φ0[ρ]| Hα=1 |Φ0[ρ]〉+ EC[ρ]
Hα[ρ] = T + Ven + Vnn + αVee + Vα[ρ]
EC = −1∫
0
dα Re
∞∫0
du
2π〈V[χα(iu)− χ0(iu)]〉
χα: Density-density response function, V: bare Coulomb interaction
Density is constrained to physical (α = 1) ground state density.
Φ0 is a single-determinant of Kohn-Sham orbitals.
Zero-temperature fluctuation-dissipation theorem connects excited and groundstates
Langreth and Perdew, Phys. Rev. B 15, 2884 (1977)Eshuis, Bates, and Furche, Theor. Chem. Acc. 131, 1084 (2012)Ren et al., J. Mater. Sci. 47, 7447 (2012)
Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/2017 4 / 14
Introduction & Background ACFDT & RPA
Density-density Response Function
Dyson-like equation for TDDFT:
χ−1α (ω) =χ−1
0 (ω)− [Vα + f αxc (ω)]
χα =χ0 + χ0 [Vα + f αxc ]χα
Poles of χα(ω) at excitations of interacting system
Exact fxc: spatially non-local, complicated ω behavior
Electronic instabilities occur for some fxc
Random Phase Approximation : fxc = 0
χα =(1− χ0Vα)−1χ0
E RPAC =
∫ ∞0
du
2π〈ln[1− χ0(iu)V ] + χ0(iu)V 〉
Petersilka, Gossmann, and Gross, Phys. Rev. Lett. 76, 1212(1996)Lein, Gross, and Perdew, Phys. Rev. B 61, 13431 (2000)Colonna, Hellgren, de Gironcoli Phys. Rev. B 90, 125150 (2014)Erhard, Bleiziffer, Gorling Phys. Rev. Lett. 117, 143002 (2016)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4q (a.u.)
0.06
0.05
0.04
0.03
0.02
0.01
0.00
ε c(q
) (a.
u.)
RPAexact
HEG correlation energy per particle ; rs = 4
Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/2017 5 / 14
Introduction & Background ACFDT & RPA
Applications of RPA
Why RPA?
naturally incorporates dispersion
applicable to small-gap systems(metals)
EXX part is self-interaction free
less expensive than CCSD(T)
Shortcomings:
overestimates EC
tendency to underbind
self-correlation error
more expensive than semilocalDFT∗
Typically more accurate than semilocalfunctionals for:
basic properties of molecules & solids
adsorption of molecules on metal surfaces
adsorption of graphene on metal surfaces
binding energies & distances for weaklybound complexes
binding energies of layered materials
reaction energies & barriers, catalysisHarl, Schimka, Kresse, Phys. Rev. B 81, 115126 (2010)Lebegue et al. Phys. Rev. Lett. 105, 196401 (2010)Schimka et al. Nat. Mater. 9, 741 (2010)Bjorkman, Gulans, Krasheninnikov, Nieminen, Phys. Rev. Lett. 108,235502 (2012)Eshuis, Furche J. Phys. Chem. Lett. 2, 983 (2011)Olsen, Thygesen Phys. Rev. B 87, 075111 (2013)Schimka et al. Phys Rev. B 87, 214102 (2013)Burow, Bates, Furche, Eshuis J. Chem. Theory Comput. 10, 180(2014)Bao et al. ACS Catal. 5, 2070 (2015)Waitt, Ferrara, Eshuis J. Chem. Theory Comput. 12, 5350 (2016)
Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/2017 6 / 14
Beyond-RPA Correlation
Electron Gas Model KernelsNEO :
x-like (linear in α)
1- & 2-e self-correlation free
energy-optimized for HEG
CP07 :
xc kernel
compressability & 3rd -ω-moment sumrule, correct asymptotics
accurate for HEG correlation overwide rs range
Bates, Laricchia, and Ruzsinszky, Phys. Rev. B 93, 045119(2016)Constantin, Pitarke Phys. Rev. B 75, 245127 (2007)
χα = χα + χαf αxcχα
rADFT (LDA or PBE) :
renormalization eliminates divergenceof pair-density
can use any semilocal, adiabaticapprox. for fxc
x-only or xc forms possibleOlsen, Thygesen Phys. Rev. B 86, 081103(R) (2012)Olsen, Thygesen Phys. Rev. Lett. 112, 203001 (2014)Patrick, Thygesen J. Chem. Phys. 143, 102802 (2015)
Constraint satisfaction can be used to build model f αxc (ω)Many more than this, such as CDOP, RA, PGG, EXX, PSA, . . .
The choice of f αxc determines accuracy limit of bRPA methods
Heßelmann, Gorling Phys. Rev. Lett. 106, 093001 (2011)Colonna, Hellgren, de Gironcoli Phys. Rev. B 90, 125150 (2014)Erhard, Bleiziffer, Gorling Phys. Rev. Lett. 117, 143002 (2016)
Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/2017 7 / 14
Beyond-RPA Correlation RPA Renormalization
RPA Renormalization
Renormalization is refactorization
χ−1α =
[χ−1
0 − Vα]− f αxc −→ χ−1
α = χ−1α − f αxc
χα = χα + χαf αxcχα = (1− χαf αxc )−1 χα
Exact factorization of correlation energy : EC ∝ 〈V (χα − χ0)〉+ 〈V χαf αxcχα〉Beyond-RPA correlation is a functional of fxc
EC =E RPAC + ∆E bRPA
C [fxc]
∆E bRPAC [fxc] =− 1
2π
∫ ∞0
du
∫ 1
0
dα
× 〈V χα(iu)f αxc (iu)χα(iu)〉
Bates and Furche, J. Chem. Phys. 139, 171103 (2013)Bates, Laricchia, and Ruzsinszky, Phys. Rev. B 93, 045119 (2016)
0.0 0.5 1.0 1.5 2.0q (a.u.)
0.000
0.005
0.010
0.015
0.020
∆ε c
(q) (
a.u.
)
NEOrALDACP07exact
Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/2017 8 / 14
Beyond-RPA Correlation RPA Renormalization
Finite-order RPAr
Expanding χα in orders of χαf αxc . . .
χα = χα + χαf αxc χα + χαf αxc χαf αxc χα + χαf αxc χαf αxc χαf αxc χα + . . .
yields RPAr power series for ∆E bRPAC , with the n-th order term
∆E RPAr-nC [fxc ] = −
∫ 1
0
dα
∫ ∞0
du
2π〈V (χαf αxc )n χα〉
Both RPA and beyond-RPA correlation are obtained in a single calculation!
RPAr1 : χα ≈ χα + χαf αxc χα
eliminates electronic instabilities
preserves RPA’s static correlation
has analytic α integral for x-like f αxc
dominant (∼ 90%) part of ∆E bRPAC
RPAr-n : nth-order terms
do they converge?
relative contributions?
kernel dependent?
system dependent?Bates, Furche J. Chem. Phys. 139, 171103 (2013)Colonna, Hellgren, de Gironcoli Phys. Rev. B 90, 125150 (2014)Bates, Laricchia, and Ruzsinszky, Phys. Rev. B 93, 045119 (2016)Bates, Sensenig, Ruzsinszky Phys. Rev. B 95, 195158 (2017)
Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/2017 9 / 14
Results Convergence behavior
Spin-unpolarized Systems
RPAr convergence in Si-A4
(eV/Si2) rALDA
n E RPAr-nc ∆E RPAr-n
c
0 (RPA) –12.1975 0.00001 –8.6049 3.59262 –8.4141 0.19083 –8.3977 0.01644 –8.3960 0.00175 –8.3958 0.0002
∞ –8.3958 3.80170 1 2 3 4 5
n
1.0
0.0
-1.0
-2.0
-3.0
-4.0lo
g(-ΔEn c
) (eV
)
Spin-unpolarized RPAr ConvergenceCO molec ; rAPBEMg atom ; rAPBEMgO-B1 ; rAPBESi-A4 ; rAPBERh-A1 ; rAPBEFe-A1 ; rAPBEAl(111) ; rAPBEC-A4 ; CP07Al-A1 ; CP07
RPAr convergence is monotonic
RPAr shows no sensitivity to band gap or dimensionality
Speedup for RPAr1 : ∼ 2− 3x
Bates, Sensenig, Ruzsinszky Phys. Rev. B 95, 195158 (2017)
Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/2017 10 / 14
Results Convergence behavior
Spin-polarized systems
0 2 4 6 8 10 12n
1.0
0.0
-1.0
-2.0
-3.0
-4.0
log(
-En c
) (eV
)
RPAr@rAPBE ConvergenceBCNO
O2NiO-B1Co(0001)Fe-BCC
0 1 2 3 4 5 6n
1.0
0.0
-1.0
-2.0
-3.0
-4.0lo
g(-
En c) (
eV)
RPAr@rAPBEns ConvergenceBCNO
O2NiO-B1Co(0001)Fe-BCC
RPAr converges for FM, AFM, and spin-pol systems
Monotonic convergence a natural feature of RPA renormalization
Approximate spin-dependence in f αxc can hamper convergence rate
Spin-independent kernels behave more like spin-unpolarized systems
Bates, Sensenig, Ruzsinszky Phys. Rev. B 95, 195158 (2017)Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/2017 11 / 14
Results Bulk Phase Transitions
Pressure Induced Phase Transition
Pt is pressure where enthalpies oftwo phases are equivalent:
H = U + PV
H(Pt ,V1,U1) = H(Pt ,V2,U2)
band-gap and otherproperties change upontransition
useful applications in, e.g.,electronics and optics
thermal corrections importantfor nearly-degen. phases
High Pressure Phase
Low PressurePhase
RPAr1 captures nearly all of bRPA effects
f αxc tends to reduce energy gap & Pt vs RPA
Sengupta, Bates, Ruzsinszky submitted
Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/2017 12 / 14
Results Bulk Phase Transitions
Pressure Induced Phase Transition
Pt is pressure where enthalpies oftwo phases are equivalent:
H = U + PV
H(Pt ,V1,U1) = H(Pt ,V2,U2)
band-gap and otherproperties change upontransition
useful applications in, e.g.,electronics and optics
thermal corrections importantfor nearly-degen. phases
Zero-temperature Pt (GPa) :
Materials PBE SCAN RPA RPAr1 ∞-OSi 9.7 14.5 13.8 11.4 10.7Ge 8.1 11.3 11.2 10.4 10.1SiC 65.8 74.1 74.3 71.4 70.3GaAs 12.8 17.1 18.9 17.2 17.0SiO2 5.8 4.6 3.7 6.6 6.9Pb 12.2 17.5 18.9 17.1 16.9C 6.1 4.6 0.6 6.7 6.7BN 3.2 2.7 –1.4 0.9 1.1
RPAr1 captures nearly all of bRPA effects
f αxc tends to reduce energy gap & Pt vs RPA
Sengupta, Bates, Ruzsinszky submitted
Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/2017 12 / 14
Conclusions & Acknowledgements
Conclusions/Summary
RPA renormalization is a rapidly convergent MBPT based upon RPA
RPAr is not sensitive to band-gap or dimensionality
Choice of kernel, spin-dependence impacts convergence behavior
RPAr1 recovers 99% of bRPA correlation effects in pressure induced phasetransitions
Accuracy of RPA renormalization vs expt limited by choice of fxc
Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/2017 13 / 14
Conclusions & Acknowledgements
Acknowledgements
Thanks to . . .
Christopher Patrick & Kristian Thygesen(DTU)
Jon Sensenig & Niladri Sengupta
Adrienn Ruzsinszky
John Perdew
. . . and you for your attention!Funding/computational resources provided by:
NSF
DOE
Temple Owlsnest
Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/2017 14 / 14
Appendix
Origin of monotonic convergence
Using cyclic invariane of the trace
∆URPAr1c [fxc] = −〈V χfxcχ〉 = −〈V
12 χfxcχV
12 〉 ,
= 〈[
V12 χf
12
xc
] [V
12 χf
12
xc
]†〉 > 0
Can show this for any order of RPAr
∆URPAr-(2m+1)c [fxc] = 〈
[V
12 χ(fxcχ)m(fxc)
12
] [V
12 χ(fxcχ)m(fxc)
12
]†〉 ,
∆URPAr-(2m)c [fxc] = 〈
[V
12 (χfxc)m(χ)
12
] [V
12 (χfxc)m(χ)
12
]†〉 .
RPA renormalization specifically sums contributions beyond RPA that result in allcorrections having a fixed sign.
∆U(2)c,λ[fxc] ∝− λ〈
[χ
120 Vχ0f
12
xc,λ
] [χ
120 Vχ0f
12
xc,λ
]†〉 < 0
Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/2017 15 / 14
Appendix
N2 Dissociation
0 1 2 3 4 5 6 7 8n
1.0
0.0
-1.0
-2.0
-3.0
-4.0
log(
-En c
) (eV
)
N2 RPAr@rAPBE ConvergenceR=118 pmR=158 pmR=228 pmR=278 pm
R=118
R=158
R=228
R=278
5.2
5.4
5.6
5.8
6.0
ERPAr
nc
(eV)
bRPA Correlation CorrectionsRPAr1RPAr2RPAr3RPAr4
RPAr5RPAr6RPAr7
RPAr converges even upon dissociation for “stable” f αxc
Convergence slows as R increases
What happens for “unstable” f αxc (e.g. EXX)?
Colonna, Hellgren, de Gironcoli Phys. Rev. B 90, 125150 (2014)Bates, Sensenig, Ruzsinszky Phys. Rev. B 95, 195158 (2017)
Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/2017 16 / 14