Convergence behavior of RPA renormalized many-body perturbation theory
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1 Convergence behavior of RPA renormalized many-body perturbation 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/ / 14
2 Introduction & Background Electronic Instabilities N 2 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, (2014) Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/ / 14
3 Introduction & Background Outline 1 Introduction & Background ACFDT & RPA 2 Beyond-RPA Correlation RPA Renormalization 3 Results Convergence behavior Bulk Phase Transitions 4 Conclusions & Acknowledgements Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/ / 14
4 Introduction & Background ACFDT & RPA Adiabatic Connection Fluctuation-Dissipation Theorem E[ρ] = Φ 0[ρ] Ĥα=1 Φ0[ρ] + EC[ρ] Ĥ α[ρ] = ˆT + ˆV en + V nn + α ˆV ee + ˆV α[ρ] 1 E C = 0 dα Re 0 du 2π V[χ α(iu) χ 0 (iu)] χ α : Density-density response function, V: bare Coulomb interaction Density is constrained to physical (α = 1) ground state density. Φ 0 is a single-determinant of Kohn-Sham orbitals. Zero-temperature fluctuation-dissipation theorem connects excited and ground states 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/ / 14
5 Introduction & Background ACFDT & RPA Density-density Response Function Dyson-like equation for TDDFT: χ 1 α (ω) =χ 1 0 (ω) [V α + fxc α (ω)] χ α =χ 0 + χ 0 [V α + fxc α ] χ α Poles of χ α(ω) at excitations of interacting system Exact f xc: spatially non-local, complicated ω behavior Electronic instabilities occur for some f xc 0.00 Random Phase Approximation : f xc = 0 ˆχ α =(1 χ 0V α) 1 χ 0 E RPA C = 0 du ln[1 χ0(iu)v ] + χ0(iu)v 2π Petersilka, Gossmann, and Gross, Phys. Rev. Lett. 76, 1212 (1996) Lein, Gross, and Perdew, Phys. Rev. B 61, (2000) Colonna, Hellgren, de Gironcoli Phys. Rev. B 90, (2014) Erhard, Bleiziffer, Görling Phys. Rev. Lett. 117, (2016) ɛ c (q) (a.u.) RPA exact q (a.u.) HEG correlation energy per particle ; r s = 4 Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/ / 14
6 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 E C tendency to underbind self-correlation error more expensive than semilocal DFT Typically more accurate than semilocal functionals for: basic properties of molecules & solids adsorption of molecules on metal surfaces adsorption of graphene on metal surfaces binding energies & distances for weakly bound complexes binding energies of layered materials reaction energies & barriers, catalysis Harl, Schimka, Kresse, Phys. Rev. B 81, (2010) Lebègue et al. Phys. Rev. Lett. 105, (2010) Schimka et al. Nat. Mater. 9, 741 (2010) Björkman, Gulans, Krasheninnikov, Nieminen, Phys. Rev. Lett. 108, (2012) Eshuis, Furche J. Phys. Chem. Lett. 2, 983 (2011) Olsen, Thygesen Phys. Rev. B 87, (2013) Schimka et al. Phys Rev. B 87, (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/ / 14
7 Beyond-RPA Correlation Electron Gas Model Kernels NEO : CP07 : x-like (linear in α) 1- & 2-e self-correlation free energy-optimized for HEG xc kernel compressability & 3 rd -ω-moment sum rule, correct asymptotics accurate for HEG correlation over wide r s range Bates, Laricchia, and Ruzsinszky, Phys. Rev. B 93, (2016) Constantin, Pitarke Phys. Rev. B 75, (2007) χ α = ˆχ α + ˆχ αf α xc χ α radft (LDA or PBE) : renormalization eliminates divergence of pair-density can use any semilocal, adiabatic approx. for f xc x-only or xc forms possible Olsen, Thygesen Phys. Rev. B 86, (R) (2012) Olsen, Thygesen Phys. Rev. Lett. 112, (2014) Patrick, Thygesen J. Chem. Phys. 143, (2015) Constraint satisfaction can be used to build model fxc α (ω) Many more than this, such as CDOP, RA, PGG, EXX, PSA,... The choice of fxc α determines accuracy limit of brpa methods Heßelmann, Görling Phys. Rev. Lett. 106, (2011) Colonna, Hellgren, de Gironcoli Phys. Rev. B 90, (2014) Erhard, Bleiziffer, Görling Phys. Rev. Lett. 117, (2016) Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/ / 14
8 Beyond-RPA Correlation RPA Renormalization RPA Renormalization Renormalization is refactorization [ ] χ 1 α = χ 1 0 V α fxc α χ 1 α = ˆχ 1 α fxc α χ α = ˆχ α + ˆχ αf α xc χ α = (1 ˆχ αf α xc ) 1 ˆχ α Exact factorization of correlation energy : E C V (ˆχ α χ 0) + V ˆχ αf α xc χ α Beyond-RPA correlation is a functional of f xc E C =E RPA C E brpa C [f xc] = 1 2π + E brpa C [f xc] 1 0 du 0 dα V ˆχ α(iu)f α xc (iu)χ α(iu) ɛ c (q) (a.u.) NEO ralda CP07 exact Bates and Furche, J. Chem. Phys. 139, (2013) Bates, Laricchia, and Ruzsinszky, Phys. Rev. B 93, (2016) q (a.u.) Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/ / 14
9 Beyond-RPA Correlation RPA Renormalization Finite-order RPAr Expanding χ α in orders of ˆχ αf α xc... χ α = ˆχ α + ˆχ αf α xc ˆχ α + ˆχ αf α xc ˆχ αf α xc ˆχ α + ˆχ αf α xc ˆχ αf α xc ˆχ αf α xc ˆχ α +... yields RPAr power series for E brpa C EC RPAr-n [f xc] =, with the n-th order term 1 0 dα 0 du α V (ˆχαf xc ) n ˆχ α 2π Both RPA and beyond-rpa correlation are obtained in a single calculation! RPAr1 : χ α ˆχ α + ˆχ αfxc α ˆχ α RPAr-n : n th -order terms eliminates electronic instabilities preserves RPA s static correlation has analytic α integral for x-like f α xc dominant ( 90%) part of E brpa C Bates, Furche J. Chem. Phys. 139, (2013) Colonna, Hellgren, de Gironcoli Phys. Rev. B 90, (2014) Bates, Laricchia, and Ruzsinszky, Phys. Rev. B 93, (2016) Bates, Sensenig, Ruzsinszky Phys. Rev. B 95, (2017) do they converge? relative contributions? kernel dependent? system dependent? Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/ / 14
10 Results Convergence behavior Spin-unpolarized Systems (ev/si 2) RPAr convergence in Si-A4 E RPAr-n c ralda n 0 (RPA) E RPAr-n c log(-δe n c ) (ev) Spin-unpolarized RPAr Convergence CO molec ; rapbe Mg atom ; rapbe MgO-B1 ; rapbe Si-A4 ; rapbe Rh-A1 ; rapbe Fe-A1 ; rapbe Al(111) ; rapbe C-A4 ; CP07 Al-A1 ; CP n 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, (2017) Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/ / 14
11 Results Convergence behavior Spin-polarized systems log(- E n c) (ev) RPAr@rAPBE Convergence B C N O O 2 NiO-B1 Co(0001) Fe-BCC log(- E n c) (ev) RPAr@rAPBEns Convergence B O 2 C NiO-B1 N Co(0001) O Fe-BCC n n RPAr converges for FM, AFM, and spin-pol systems Monotonic convergence a natural feature of RPA renormalization Approximate spin-dependence in fxc α can hamper convergence rate Spin-independent kernels behave more like spin-unpolarized systems Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/ / 14
12 Results Bulk Phase Transitions Pressure Induced Phase Transition P t is pressure where enthalpies of two phases are equivalent: H = U + PV H(P t, V 1, U 1) = H(P t, V 2, U 2) band-gap and other properties change upon transition useful applications in, e.g., electronics and optics thermal corrections important for nearly-degen. phases High Pressure Phase Low Pressure Phase Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/ / 14
13 Results Bulk Phase Transitions Pressure Induced Phase Transition P t is pressure where enthalpies of two phases are equivalent: H = U + PV H(P t, V 1, U 1) = H(P t, V 2, U 2) band-gap and other properties change upon transition useful applications in, e.g., electronics and optics thermal corrections important for nearly-degen. phases Zero-temperature P t (GPa) : Materials PBE SCAN RPA RPAr1 -O Si Ge SiC GaAs SiO Pb C BN RPAr1 captures nearly all of brpa effects f α xc tends to reduce energy gap & P t vs RPA Sengupta, Bates, Ruzsinszky submitted Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/ / 14
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 phase transitions Accuracy of RPA renormalization vs expt limited by choice of f xc Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/ / 14
15 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/ / 14
16 Appendix Origin of monotonic convergence Using cyclic invariane of the trace U RPAr1 c [f xc] = V ˆχf xc ˆχ = V 2 1 ˆχfxc ˆχV 1 2, [ ] [ ] = V ˆχf 2 xc V ˆχf 2 xc > 0 Can show this for any order of RPAr [ ] [ ] Uc RPAr-(2m+1) [f xc] = V 1 2 ˆχ(fxc ˆχ) m (f xc) 2 1 V 2 1 ˆχ(fxc ˆχ) m (f xc) 1 2, [ ] [ ] Uc RPAr-(2m) [f xc] = V 1 2 (ˆχfxc) m (ˆχ) 2 1 V 1 2 (ˆχfxc) m (ˆχ) 1 2. RPA renormalization specifically sums contributions beyond RPA that result in all corrections having a fixed sign. [ ] [ U (2) c,λ [fxc] λ 2 χ 1 0 V χ 0f xc,λ χ 1 0 V χ 0f 2 xc,λ] 1 < 0 Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/ / 14
17 Appendix N 2 Dissociation log(- E n c) (ev) N 2 RPAr@rAPBE Convergence R=118 pm R=158 pm R=228 pm R=278 pm n E c RPAr n (ev) RPAr1 RPAr2 RPAr3 RPAr4 R=118 brpa Correlation Corrections R=158 RPAr5 RPAr6 RPAr7 R=228 R=278 RPAr converges even upon dissociation for stable f α xc Convergence slows as R increases What happens for unstable f α xc Colonna, Hellgren, de Gironcoli Phys. Rev. B 90, (2014) Bates, Sensenig, Ruzsinszky Phys. Rev. B 95, (2017) (e.g. EXX)? Jefferson E. Bates (Temple Univ.) RPAr Convergence 08/20/ / 14
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