Spring College on Computational Nanoscience
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1 Spring College on Computational Nanoscience May 2010 At the Fifth Rung of Jacob's Ladder: A Discussion of Exact Exchange plus Local- and Nonlocal-density Approximations to the Correlation Functional Matthias SCHEFFLER FHI der Max-Planck Gesellschaft Theory Department Berlin Germany
2 At the fifth rung of Jacob s ladder: A Discussion of Exact Exchange plus Local- and Nonlocal- Density Approximations to the Correlation Functional f spd Biophysics Surface Energies and Adsorption Df Defects in Mtl Metals, Semiconductors, and Insulators Lanthanide Oxides m mm m nm space densityfunctional theory and beyond ab initio Molecular ElectronicDynamics Structure Theory Predictive modeling and simulations must address all time and space scales ab initio kinetic Monte Carlo Continuum Equations, Rate Equations, and Finite Element Modeling Last Friday time f s p s n s s m s s hours years the base Approximate xc functionals (e.g. LDA and GGAs) have been very successful but there are problems for certain bonding situations (vdw, hydrogen bonding, certain covalent bonds) for highly correlated situations, and for excited states. 1
3 The zoo of approximate xc functional is getting hardly comprehensible. -- Patchwork or sound science? HL VWN BP86 PKZB PW91 PBE r-pbe RPBE revpbe B88 PBE0 How to clean up this mess? The challenges: Find practical ways to correct the xc approximation and/or to control the errors. Efficient all-electron code; DFT also with hybrid functionals and TDDFT, Hartree- Fock +MP2, EX+cRPA, GW selfenergies, etc. Perdew s Dream: Jacob s Ladder in DFT accuracy? computationa al cost 5 unoccupied i (r), e.g., ACFD-RPA 4 occupied i (r), hybrid functional (e.g., B3LYP, PBE0) 3 (r), meta-gga (e.g., TPSS) 2 n(r), GGA (e.g., PBE) 1 n(r), LDA (r) ( ) : KS kinetic energy density ACFD : adiabatic connection fluctuation dissipation theorem (Bohm, Pines (1953); Gell-Mann, Brueckner (1957); Langreth, Perdew (1977); Gunnarsson, Lunqvist (1975, 1976) RPA : random phase approximation 2
4 Exact Exchange plus Correlation in RPA The orbitals for evaluating E x are different in Hartree-Fock and Kohn-Sham DFT. --The numerical technique to evaluate E x is the same. Adding correlation: on top of Hartree-Fock exchange: Møller-Plesset perturbation theory (MP2) on top of DFT exact exchange : random phase approximation (RPA) 3
5 RPA Formulated within DFT Framework E xc RPA = E x + E c RPA 0 == dynamical-response function of the Kohn-Sham system The approach gives total energies but no information on how the Kohn-Sham energies,, i.e. spectroscopy, will change. This (corresponding) change is given by the GW self-energy. Pros and Cons of EX+cRPA The good aspects: Exchange is treated at the exact exchange level (so far with PBE or PBE0 orbitals) vdw interactions are included (automatically; seamlessly). Screening is taken into account. Thus EX+cRPA works for metals/small gap systems -- in contrast to MP2. Right decay behavior at metal surfaces (Rohlfing, Bredos, PRL 101 (2008). An A is still there! In contrast to LDA and GGA, EX+cRPA does not benefit from (fortuitous) error cancellation between the exchange and correlation terms. The A clearly shows up in atomization energies of small molecules: A non-selfconsistent (PBE-based) RPA does not always improve over hybrid functionals, e.g. B3LYP or PBE0. 4
6 The RPA and vdw interactions in Comparison with Other Methods The RPA and vdw interactions in Comparison with Other Methods 5
7 The RPA and vdw interactions in Comparison with Other Methods EX + crpa is doing well, but the C 6 coefficient is slightly too small Another Example Where Present-Day xc Functionals Reveal (Severe) Limitations: CO adsorption at transition metal surfaces: LDA and GGA xc functionals dramatically fail to predict the correct adsorption site. For low coverage the theory gives the hollow site, but experimentally CO adsorbs on top. E.g.: For CO/Cu (111) the LDA error is 0.4 ev, and the GGA error is 02eV 0.2 ev. Feibelman, Hammer, Norskov, Wagner, Scheffler, Stumpf, Watwe, and Dumesic, The CO/Pt(111) puzzle. J. Phys. Chem. B 105, (2001). 6
8 EX+cRPA, MP2, and Others for the CO Adsorption Puzzle ) E fcc E top (ev LDA AM05 PBE0 PBE B3LYP & HSE exp. (1) (1) MP2 X. Ren, P. Rinke, M.S., PRB 80 (2009). RPA (2) hollow top RPA was evaluated with LDA, PBE and PBE0 orbitals. The difference is only 0.02 ev. Xinguo Ren (1) Difference to Stroppa et al. PRB 76(2007) is less than 0.04 ev. (2) Harl and Kresse, PRL 103 (2009) only 0.12 ev Patrick Rinke Wrong preference for the hollow site in LDA and GGA (PBE, AM05) possibly corrected with hybrid XC s (B3LYP, PBE0, HSE03) definitely corrected by RPA (and also by MP2). The CO Adsorption Puzzle gas phase 2 * lumo 5 homo O C O C adsorbed 2 * 5 2 * free CO clean Cu surface E F s-band d-band GW results: For details (RPA and GW) see X. Ren, P. Rinke, M.S., PRB 80 (2009). 7
9 Surface Energies from Y to Ag Theory: DFT-LDA M. Methfessel, D. Hennig, and M. S., Phys. Rev. B 46,, 4816 (1992) The trend in cohesive energies and surface energies is well understood (since long). Uncertainties in experimental estimates are significant (extrapolation from surface tension of liquids). For example, for Pd (111) the experimental value is 0.8 ± 0.15 ev/surface atom. 8
![(xc-better) E N cluster (LDA)] (1) Q.-M. Hu, K.](/docs-images/93/111716759/images/10-5.jpg "Reuter, and M. S.")
10 Surface Energies from Y to Ag We evaluate the total energy from a difference (1) E better-than-lda = E LDA + E xc. The total energy of a poly-atomic system can be written as Aloysius Soon E = N C N E N. C N is the number of atoms that are N-fold coordinated and E N is the energy contribution of each N-fold coordinated atom. We are not using this equation in full but only for the xc correction: E xc = N C N [ E N cluster (xc-better) E N cluster (LDA)] (1) Q.-M. Hu, K. Reuter, and M. S., PRL 98, (2007); and 99, (E) (2007). Some of the considered clusters: Pd 4 Pd 6 Pd 10 E = 4*E 3 E = 6*E 4 E = 6*E 4 + 3*E 5 + E 9 Pd 13 Pd 19 E = 12*E 5 + E 12 E = 6*E *E 7 + 1*E 12 9
11 Cohesive Energies and Surface Energies from Y to Ag differences with respect to LDA A. Soon et al., in preparation - xc(relative) 12 is PBE the difference of the PBEsol cohesive energy of AM05 xc-better btt and dthe BLYP LDA cohesive RPA: rpbe Harl&Kresse energy. (2009) RPA: ours B3LYP HSE03 experiment HSE06 RPA@PBE0 RPA@HSE06 For example: xc(relative) 12 - xc(relative) 9 is the difference of the fcc(111) surface energy of xc-better and the LDA. Cohesive Energies and Surface Energies from Y to Ag differences with respect to LDA A. Soon et al., in preparation PBE PBEsol AM05 BLYP rpbe B3LYP HSE03 experiment HSE06 RPA@PBE0 RPA@HSE06 Hybrids (HSE03, PBE0, B3LYP, M06 s) underestimate the cohesive energy severely. RPA is doing fine : For the cohesive energy it is worse than PBE; for the surface energies it is better than PBE (as good as LDA). 10
12 how about molecules? Hydrogen bonds and dispersion: S22 Jurecka, Sponer, Cerny, Hobza PCCP 8, 1985 (2006) MP2/ RPA (ev) Covalent bonds: small molecules CCSD(T) binding energies (ev) There is opportunity beyond the RPA F. Furche, Phys. Rev. B 64, (2001) The Role of vdw Forces in Organic Materials and Biophysics Alexandre Tkatchenko 11
13 EX + crpa is doing well, but the C 6 coefficient is slightly too small 12
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20 However, there are problems when applying the TS approach to bulk metals, as there are problems for all the other schemes, except EX+cRPA. Stability of the -helix structure of proteins (peptide chains): R O H N C C OH amino peptide carboxyl group bond group R 1 O O R 2 O O R 3 O R 4 O R n O H N C NC CN C NC CN C NC CN C... C N C C OH H H H H H H H H H H H H H H H -Faltblatt sheet Carboxylgruppe -Helix helix secondary structure 19
21 The Critical Role of vdw Intereactions: Structure and Stability of Polypeptides Fully extended structure (FES) Mariana Rossi Carvalho Volker Blum R O peptide: H - N - C - C - OH Alanine: R = CH 3 H H Alexandre Tkatchenko Structure and Stability of Polypeptides Fully extended structure (FES) Mariana Rossi Carvalho Volker Blum peptide: C, N, O Alanine: R = CH 3 Alexandre Tkatchenko 20
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23 Role of vdw Interaction on (Un)folding; Comparing, 3 10, and Helices for Ac-Ala 15 LysH + s (*) No of H-bomds PBE+vdW 3 10 PBE-pure Time (pico seconds) PBE+vdW 1000K Mariana Rossi Carvalho Volker Blum (*) H-bond = 2.5 between CO and NH groups PBE-pure Alexandre Tkatchenko Summary Developed accurate and efficient first-principle method for the long-range van der Waal interaction. The method can be coupled with DFT (e.g., PBE+vdW) and MP2 (MP2+ vdw). Application for example to (un)folding dynamics of polyalanine helices: o vdw forces increase the stability of polyalanine helices by ~ 100%. o Qualitative i differences in unfolding dynamics between PBE-pure and PBE+vdW. o Remarkable stability of polyalanine helices in vacuo is attributed to synergy between H-bonds and vdw forces. 22
24 Summary EX+cRPA is promising. Though the improvement over lower-level xc functionals is (in principle) systematic, agreement with experiment is not always better. There is room (need) for improvements; we have to go to self-consistent RPA (in some cases) and to higher order in the expansion. An estimate of errors due to the xc approximation is becoming possible. 23
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