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