Random-phase approximation and beyond for materials: concepts, practice, and future perspectives. Xinguo Ren
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1 Random-phase approximation and beyond for materials: concepts, practice, and future perspectives Xinguo Ren University of Science and Technology of China, Hefei USTC-FHI workshop on frontiers of Advanced Electronic Structure Methods Hefei, June 14-18, 216
2 Success and limitations of DFT in computational materials science Density functional theory (DFT) in its semi-local approximations is enormously successful in simulating materials properties and in materials designs, but has certain (intrinsic) limitations. Typical situations: Systems with mixed bonding characteristics Systems with near degeneracy ( strong correlation ) Systems where both ground-state energies and charge excitation energy gaps are important. Heterogeneous systems (different dielectric property in different subsystems)
3
4 A zoo of electronic-structure methods Workhorse LDA/GGAs Deficiencies Lack of van der Waals; Self-interaction errors Lack of functional derivative discontinuity; Insufficiency of KS spectrum Consequences Inaccuracy for groundstate energies Inaccuracy for quasi-particle excitation energies Possible fixes MP2 RPA CCSD(T) DFT+U SIC-LDA GW vdw-df Hybrid functionals Double hybrids LDA+DMFT DFT+vdW DFT-D DFT+G
5 Brief history of RPA Correlation energy of homogeneous electron gas (HES): -- Divergence problem of order-by-order perturbation theory Separation of collective modes and quasiparticle modes: -- First appearance of the RPA concept (Bohm & Pines, 195) Self-consistent field approach to HES -- Lindhard function (Lindhard, 1954) RPA = sum of ring diagrams to infinite order -- (Brueckner & Gell-Mann, 1957) The concept of screened Coulomb interaction W -- (Hubbard, 1957) The GW approximation to the self-energy Ʃ=iGW -- (Hedin, 1965)
6 RPA as a first-principles method With the framework of adiabatic-connection fluctuation-dissipation (ACFD) theorem, RPA can be formulated as an approximate, but fully nonlocal exchange-correlation (XC) energy functional. The XC energy E xc d d The coupling constant The density response function drdr' Langreth & Perdew, Phys. Rev. B 15, 2884 (1977). Gunnarsson & Lundqvist, Phys. Rev. B 13, 4274 (1976). ( r, r', i) n( r) ( r r') v( r, r' ) Coulomb interaction
7 RPA as a first-principles method With the framework of adiabatic-connection fluctuation-dissipation (ACFD) theorem, RPA can be formulated as an approximate, but fully nonlocal exchange-correlation (XC) energy functional. The XC energy E RPA xc d d drdr' RPA The coupling constant Langreth & Perdew, Phys. Rev. B 15, 2884 (1977). Gunnarsson & Lundqvist, Phys. Rev. B 13, 4274 (1976). First application to real molecules. The RPA density response function Furche, Phys. Rev. B 64, (21). Fuchs & Gonze, Phys. Rev. B (22). ( r, r', i) n( r) ( r r') v( r, r' ) Applications to molecules, solids, surfaces, and layered materials (26-present). Coulomb interaction For a review, see XR, P. Rinke, C. Joas, and M. Scheffler, J. Mater. Sci. 47, 7447 (212)
8 Why is RPA interesting? Automatic and seamless inclusion of van der Waals (vdw) interactions; non-additive, anisotropic, and many-body screening effects are properly accounted for. Self-interaction (delocalization) errors are reduced, by including exact-exchange. Different bonding (ionic, covalent, metallic, vdw situations are treated on equal footings. Static correlation (partly) captured => excellent chemical reaction barrier heights
9 Theoretical description of vdw interactions A B vdw (dispersion) stems from the coupling between spontaneous quantum charge fluctuations at a distance. First principles description of vdw interactions is (computationally) a challenge! CCSD(T) treats vdw accurately, but way too expensive. Within the DFT context: DFT+ pairwise C 6 /R 6 summation; DFT+ many-body dispersion Explicitly non-local density functionals: vdw-df Orbital-dependent functionals, via the ACFD frameworks, e.g., RPA
10 RPA description of vdw interactions A B v v RPA= summation of ring diagrams RPA captures the non-local coupling between spontaneous quantum charge fluctuations separated in space. It can be shown analytically Dispersion consistent J. Dobson, in Topics in Condensed Matter Physics, Ed. M. P. Das (Nova, New York, 1994)
11 RPA at the long range: quality of C 6 at the RPA level
12 Basic formalism behind RPA 1. Kohn-Sham eigenvalues and eigenfunctions ) ( ) ( ˆ n KS r r n n h 2. Independent-particle (Kohn-Sham) response function n m n m m n n m n m i f f i, * * ) ' ( ) ' ( ) ( ) ( ) ( 2 ), ', ( r r r r r r ] ) ( χ ) ) ( Tr[ln(1-2 1 RPA iω v iω v χ d E c 4. RPA correlation energy calculation 3. Basis representation ') ( ) ( ) ( ) ',, (, r r r r P i i P
13 RPA calculations in practice (so far )
![First-Principles code package The Fritz Haber Institute ab-initio molecular simulations (FHI-aims) package Numeric atom-centered basis functions ( i[ lm] r) uil ( r) Ylm( rˆ) Conventional DFT (LDA,](/docs-images/93/112821583/images/14-0.jpg "GGAs) All-electron, full potential Periodic and finite systems on equal footing Favorable scaling (w.r.t system size and CPU cores) V. Blum et al., Comp. Phys. Comm.")
14 First-Principles code package The Fritz Haber Institute ab-initio molecular simulations (FHI-aims) package Numeric atom-centered basis functions ( i[ lm] r) uil ( r) Ylm( rˆ) Conventional DFT (LDA, GGAs) All-electron, full potential Periodic and finite systems on equal footing Favorable scaling (w.r.t system size and CPU cores) V. Blum et al., Comp. Phys. Comm. 18, 2175 (29) Beyond-DFT methods Hybrid functionals, MP2, RPA and beyond Quasiparticle excitation energies: G W, scgw, MP2, and beyond X. Ren et al., New J. Phys. 14, 532 (212)
15 RPA for Ar2
16 RPA for S22 test set
17 RPA applied to the CO adsorption puzzle XR, P. Rinke, and M. Scheffler, Phys. Rev. B 8, 4542 (29) LDA/GGA => hollow site Exp/RPA => on-top site See also L. Schimka et al., Nature Materials 9, 741 (21).
18 RPA for Chemical reaction barrier heights 38 hydrogen transfer chemical reaction barrier Height (HTBH38) and 38 non-hydrogen transferchemical reaction barrier height (NHTBH38) Ref. Data: Zhao,Gonzalez-Garcia, Truhlar, J. Phys. Chem. A 19, 212 (25) XR, P. Rinke, G.E. Scuseria, M. Scheffler, Phys Rev. B 88, 3512 (213).
19 Beyond RPA: insights from the Diagrammatic representation RPA E c Compare to MP2 E c
20 Screened second-order exchange MP2
21 Screened second-order exchange MP2 The SOSEX improves the atomization energy of covalently bound molecules, T but does very little for vdw complexes.
22 Decomposing the RPA binding energy for Ar 2 Lennard-Jones Exact-exchange with Kohn-Sham orbitals has too strong Pauli repulsion!
23 There is yet another contribution: the singles 1, KS ˆ ˆ ˆ H H H e n n E n H () KS ˆ, KS KS () ˆ H E (1) () EXX KS E E E Now b a j i b a j i b a j i a i a i a i H H E,,, 2,,, 2 KS (2) ' ' i j b a b i j a, KS ' KS (1) ˆ H E i a a v v i KS xc HF x, ) ( ) ( 2 ˆ 1 Hxc ext 2 KS N k k k k v v H r r
24 There is yet another contribution: the singles ˆ H e Hˆ KS Hˆ 1, Fock operator Hˆ KS () n E n n () KS KS (1) KS 2 ' KS E ˆ H, E Hˆ, occ unocc i fˆ SE a Ec EXX () (1) EKS E E i a i a Kohn-Sham orbitals Now 2 2 KS a a, b H ' I Originally i H (2) derived within the framework of ' Rayleigh-Schrödinger i, j E perturbation theory i, a i a i, j, a, b XR, A. Tkatchenko, P. Rinke, M. Scheffler, PRL 16, 1533 (211) Rederived within the framework of Adiabatic-connection fluctuation dissipation a theorem a without freezing the density i i b j i a b j J. Klimes, M. Kaltak, E. Maggio, and G. Kresse, JCP 143, (215) i j a b
25 RPA+SE for Ar 2 S Singles contributions correct the too strong Pauli repulsion arising from the too extended GGA charge density. XR, A. Tkatchenko, P. Rinke, and M. Scheffler, Phys. Rev. Lett. 16, 1533 (211).
26 Renormalized Singles Excitation (rse) Contributions XR, P. Rinke, G.E. Scuseria, M. Scheffler, Phys Rev. B 88, 3512 (213).
27 The concept of renormalized second-order perturbation theory (rpt2) (PT2) rpt2 = RPA+SOSEX+rSE XR, P. Rinke, G.E. Scuseria, M. Scheffler, Phys Rev. B 88, 3512 (213).
28 rpt2 for Ar 2 XR, P. Rinke, G.E. Scuseria, M. Scheffler, Phys. Rev. B 88, 3512 (213).
29 The performance of rpt2 Van der Waals Cohesive energy Bond length Barrier heights rpt2: most balanced approach for atomization energy, van der Waals interaction, and chemical reaction barrier heights!
30 RPA + Singles for Solids rse fixed the problem of RPA!
31 What prevents RPA-based methods from widespread use? Computationally RPA is 2-3 orders of magnitude more expensive that semi-local GGA DFT, for a system with O(1) atoms. severely limited in the systems that can be tackled Need to mind about unoccupied states tough basis set convergence problem RPA Gradient is not readily available Production RPA calculations are only done with fixed geometry (I d love to have the RPA geometry of molecules on metal surfaces!) No uniquely defined self-consistency scheme No well-defined RPA orbitals or RPA densities We will work on these issues in the Partner Group!
32 Computational times FHI-aims tier 2 basis set. The situation gets more severe since RPA needs bigger basis set to converge.
33
34 Self-consistency issue A self-consistent Green function theory can be formulated and implemented at the GW level, that gives both ground-state energy and quasiparticle spectrum. (see, e.g., F. Caruso, P. Rinke, XR, A. Rubio, M. Scheffler, Phys. Rev. B 88, 7515 (212)) However, too expensive for materials binding energies are not great very difficult to go beyond GW Practical way to go: Formulating self-consistent theory in terms of the density matrix.
35 A dreamed theory (?) A self-adapted vdw-inclusive hybrid functional Computationally similar to the HSE-type hybrid functional. The mixing and screening parameters are adjusted on-the-fly based on the dielectric property of the system. Including seamlessly a vdw part (a non-local vdw functional based on density matrix should be possible, as Langreth-Lundquist vdw-df is only based density) Applicable to most materials of mixed bonding characters?
36 Patrick Rinke Matthias Scheffler ( Aalto U.) (FHI-Berlin) Volker Blum (Duke U.) Gustavo Scuseria (Rice U.)
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