Role of van der Waals Interactions in Physics, Chemistry, and Biology
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1 Role of van der Waals Interactions in Physics, Chemistry, and Biology How can we describe vdw forces in materials accurately? Failure of DFT Approximations for (Long-Range) Van der Waals Interactions 1
2 Failure of DFT Approximations for (Long-Range) Van der Waals Interactions Failure of DFT Approximations for (Long-Range) Van der Waals Interactions 2
3 Failure of DFT Approximations for (Long-Range) Van der Waals Interactions Failure of DFT Approximations for (Long-Range) Van der Waals Interactions EX+cRPA is O.K., though the C 6 coefficient is slightly too small. In general, using standard xc functionals : o Intermolecular equilibrium distances are overestimated by as much as an Angstrom or more o Many molecules and crystals are unbound o Proteins adopt wrong conformations o Large molecules do not adsorb on surfaces 3
4 accuracy accuracy Perdew s Dream: Jacob s Ladder in Density-Functional Theory The exchange-correlation functional our favorite 5 unoccupied ψ i (r), EX + crpa, as given by ACFD 4 occupied ψ i (r), hybrids (B3LYP, PBE0, HSE, ) 3 τ (r), meta-gga (e.g., TPSS) 2 n(r), Generalized Gradient Approximation 1 n(r), Local-Density Approximation τ(r) : Kohn-Sham kinetic-energy density EX: exact exchange: crpa : random-phase approximation for correlation ACFD : adiabatic connection fluctuation dissipation theorem Bohm, Pines (1953); Gell-Mann, Brueckner (1957); Gunnarsson, Lundqvist (1975, 1976); Langreth, Perdew (1977); X. Ren, P. Rinke, C. Joas, and M. S., Invited Review, Mater. Sci. 47, 21 (2012) Perdew s Dream: Jacob s Ladder in Density-Functional Theory The exchange-correlation functional our favorite 5 unoccupied ψ i (r), EX + crpa, as given by ACFD 4 occupied ψ i (r), hybrids (B3LYP, PBE0, HSE, ) 3 τ (r), meta-gga (e.g., TPSS) 2 n(r), Generalized Gradient Approximation 1 n(r), Local-Density Approximation τ(r) : Kohn-Sham kinetic-energy density EX: exact exchange: crpa : random-phase approximation for correlation ACFD : adiabatic connection fluctuation dissipation theorem Bohm, Pines (1953); Gell-Mann, Brueckner (1957); Gunnarsson, Lundqvist (1975, 1976); Langreth, Perdew (1977); X. Ren, P. Rinke, C. Joas, and M. S., Invited Review, Mater. Sci. 47, 21 (2012) 4
5 Exact Exchange plus Correlation in RPA E xc = E x EX + E c RPA E x EX The orbitals for evaluating E x EX are different in Hartree-Fock and Kohn-Sham DFT. --The numerical technique to evaluate E x EX 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) crpa Formulated within DFT Framework E xc = E x EX + E c RPA χ 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. 5
6 RPA in practice: In practical calculations, RPA is done perturbatively on top of a LDA/GGA or PBE0 reference. The RPA total energy = E = (T s + E ext + E Hartree + E x EX + E c RPA Pros and Cons of EX + crpa The good aspects: Exchange is treated at the exact exchange level (so far with PBE or PBE0 orbitals). Essentially no self-interaction error. EX+cRPA is fully non-local, and vdw interactions are included automatically and seamlessly (highly accurate). Screening is taken into account. Thus, EX + crpa works for molecules, insulators, metals with various bonding natures -- in contrast to MP2. And more Problems? An A is still there. For example, there is self-correlation due to the KS response function. Bond strengths of molecules are too weak. Let s improve on this. 6
7 Level 5 plus Viewed in the Many-Body Framework Perturbation theory: H = H 0 + H with H 0 ϕ n = E n ϕ n and ϕ n = Slater det. ϕ 0 = ground state, ϕ i, a = single excitations, ϕ ij, ab = double exci. = ϕ 0 H 0 ϕ 0, (1) = ϕ 0 H ϕ 0 ϕ 0 H ϕ n 2 ϕ 0 H ϕ i, a 2 ϕ 0 H ϕ ij,ab 2 + E + 0 E n E + 0 E + i, a E + 0 E n 0 i, a ij, ab ij,ab single excitations double excitations (2) = Σ = Σ + Σ Adding all ring diagrams from higher order perturbations: X. Ren, P. Rinke, C. Joas, and M. S., Invited Review, Mater. Sci. 47, 21 (2012) X. Ren, P. Rinke, G.E. Scuseria, and M. Scheffler, Phys. Rev. B, in print (2013) Level 5 plus Viewed in the Many-Body Framework Perturbation theory: H = H 0 + H with H 0 ϕ n = E n ϕ n and ϕ n = Slater det. ϕ 0 = ground state, ϕ i, a = single excitations, ϕ ij, ab = double exci. = ϕ 0 H 0 ϕ 0, (1) = ϕ 0 H ϕ 0 ϕ 0 H ϕ n E (2) 0 = Σ 2 ϕ 0 H ϕ i, a = Σ 2 ϕ 0 H ϕ ij,ab + Σ 2 + E + 0 E n E + 0 E + i, a E + 0 E n 0 i, a ij, ab ij,ab single excitations double excitations Using HF input, this is Møller-Plesset perturbation theory, MP2 Adding all ring diagrams from higher order perturbations: X. Ren, P. Rinke, C. Joas, and M. S., Invited Review, Mater. Sci. 47, 21 (2012) X. Ren, P. Rinke, G.E. Scuseria, and M. Scheffler, Phys. Rev. B, in print (2013) 7
8 ϕ 0 H ϕ n 2 ϕ 0 H ϕ i, a 2 ϕ 0 H ϕ ij,ab 2 E (2) Level 5 plus 0 = Σ = Σ + Σ n 0 Viewed in the Many-Body i, a Framework ij, ab E n E i, a E ij,ab single excitations double excitations Adding all ring diagrams from higher order perturbations: = single excitations crpa SOSEX X. Ren, A. Tkatchenko, P. Rinke, and M.S., PRL 106, (2011). J. Paier, X. Ren, P. Rinke, A. Grüneis, G. Kresse, G. E. Scuseria, M.S., NJP (2012). X. Ren, P. Rinke, C. Joas, and M. S., Invited Review, Mater. Sci. 47, 21 (2012) X. Ren, P. Rinke, G.E. Scuseria, and M. Scheffler, Phys. Rev. B, in print (2013) ϕ 0 H ϕ n 2 ϕ 0 H ϕ i, a 2 ϕ 0 H ϕ ij,ab 2 E (2) Level 5 plus 0 = Σ = Σ + Σ n 0 Viewed in the Many-Body i, a Framework ij, ab E n E i, a E ij,ab single excitations double excitations Adding all ring diagrams from higher order perturbations: = single excitations crpa SOSEX X. Ren, A. Tkatchenko, P. Rinke, and M.S., PRL 106, (2011). J. Paier, X. Ren, P. Rinke, A. Grüneis, G. Kresse, G. E. Scuseria, M.S., NJP (2012). X. Ren, P. Rinke, C. Joas, and M. S., Invited Review, Mater. Sci. 47, 21 (2012) X. Ren, P. Rinke, G.E. Scuseria, and M. Scheffler, Phys. Rev. B, in print (2013) 8
9 Mean absolute error to CCSD(T) (mev) Performance of rpt2 for Weak Intermolecular Interactions: S22 Test Set rpt2 achieves chemical accuracy (1 kcal/mol ~ 43 mev) -- same performance for the S66 test set -- Alexandre Tkatchenko Xinguo Ren PBE Patrick (TS et al.) 0 CCSD(T): Jurecka, Sponer, Cerny, Hobza, PCCP (2006). Langreth-Lundqvist : Gulans, Puska, Nieminen, PRB (2009); rpt2: X. Ren et al. PRL (2011) and to be published. TS: A. Tkatchenko and M.S., PRL (2009); A. Tkatchenko et al., JCP (2009). Atomization Energies with rpt2 rpt2 solids: C, Si, SiC, BN, BP, AlN, AlP, LiH, LiF, LiC, MgO. J. Paier, X. Ren, P. Rinke, G. Scuseria, A. Grueneis, G. Kresse, and M.S., New J. Phys. 14, (2012). 9
10 E vdw-df Langreth, Lundqvist et al. [Dion et al., PRL 92 (2004)] Exchange: x-revpbe E vdw DF x vdw DF c E E revpbe x Correlation: split into a local and non-local part LDA c 3 d rd 3 r' n( r) ( r, r') n( r ') Modifications X: J. Klimeš et al. J. Phys.: Condens. Matter 22 (2010) C: O. Vydrov and T. Van Voorhis, Phys. Rev. Lett. 103 (2009). New version: vdw-df2 [K. Lee, É. D. Murray, L. Kong, B. I. Lundqvist, and D C. Langreth, Phys Rev, B 82 (2011)] Changed X ( x-pw86) as well as C (modified Φ(r, r )). 10
11 TS Approach to vdw Description in DFT S. Grimme: add tails E = Σ C / R6 vdw I, J 6, I, J I, J Alexandre Tkatchenko For details, see: DFT+vdW: A. Tkatchenko and M.S., PRL 102 (2009). MP2+ΔvdW: A. Tkatchenko, R. DiStasio, M. Head-Gordon, and M.S., JCP (2009). TS Approach to vdw Description in DFT S. Grimme: add tails E = Σ C / R6 vdw I, J 6, I, J I, J Chu and Dalgarno, JCP 212 (2004) Alexandre Tkatchenko Calculated by DFT on the fly C 6 is a functional of the density, C 6 = C 6 [n]. For details, see: DFT+vdW: A. Tkatchenko and M.S., PRL 102 (2009). MP2+ΔvdW: A. Tkatchenko, R. DiStasio, M. Head-Gordon, and M.S., JCP (2009). Excellent description as long as the pair-wise model holds. 11
12 Mean absolute error to CCSD(T) (mev) Performance for Intermolecular Interactions: S22 Test Set Alex Tkatchenko Xinguo Ren Patrick Rinke (TS et al.) 0 S22: Jurecka, Sponer, Cerny, Hobza, PCCP (2006). Langreth-Lundqvist S22: Gulans, Puska, Nieminen, PRB (2009); EX+cRPA S22: X. Ren et al. PRL (2011); TS: A. Tkatchenko and M.S., PRL (2009); A. Tkatchenko et al., JCP (2009). 12
13 Metal Surfaces, Correct Long-Range Theory α TS (iω) The interaction energy is not additive; it is not a simple sum over pairwise interactions. Z For large Z: E(Z) = C 3 / Z 3 ε(iω) combined with the TS approach: PBE + vdw surf I. E. Dzyaloshinskii, E. M. Lifshitz, and L. P. Pitaevskii, Adv. Phys. (1961); C. Mavroyannis, Mol. Phys. 6 (1963); A. D. McLachlan, Mol. Phys. 7 (1964); E. Zaremba and W. Kohn, PRB (1976). A. Tkatchenko and M.S. PRL 102 (2009). V. G. Ruiz, W. Liu, E. Zojer, A. Tkatchenko, and M.S., PRL 108 (2012). Validation of The Theoretical Approach and Accuracy: Benzene on (111) Cu, Ag, Au, Pd, Pt, Rh, Ir Considered Geometries 13
14 Validation of The Theoretical Approach and Accuracy: Benzene on (111) Cu, Ag, Au, Pd, Pt, Rh, Ir Considered Geometries C 6 H 6 Metals with PBE + vdw surf Bz/Pt(111) ΔE vdw = 0.82 ev Bz/Au(111) ΔE vdw = 0.68 ev ΔE vdw = 1.15 ev (Å) Minimum, actuated by vdw interaction PBE+vdW surf PBE vdw (Å) Concerted Effect of Covalency and van der Waals Bonding Wei Liu, J. Carrasco, B. Santra, A. Michaelides, M.S., and Alexandre Tkatchenko Pt 1.48 Au
15 Benzene on (111) Cu, Ag, Au, Pd, Pt, Rh, Ir: Comparison of Different Theoretical Methods a) H. Ihm et al., J. Phys. Chem. B 108, (2004). b) A. Wander et al., Surf. Sci. 249, 21 (1991). c) D. Syomin et al., J. Phys. Chem. B, 105, 8387 (2001). 2. Example: Hybrid Inorganic/Organic Systems (HIOS) solid organic S. Blumstengel et al. HIOS HIOS: a novel type of material; new physics What is the atomic structure of and at the interface? What are the electronic properties? What is the nature of charge carriers? 15
16 Van der Waals Interactions at Interfaces The Example of Me(111) PTCDA perylene-tetracarboxylic acid dianhydride ( C 24 H 8 O 6 ) Which Theoretical Method? PTCDA Ag(111) Experiment: A. Hauschild et al., PRB 81 (2010) Theory review: L. Romaner, C. Draxl, et al. NJP (2009); A. Tkatchenko et al., MRS Bulletin (2010) PTCDA@Ag(111): PBE+vdW surf Alex Tkatchenko PBE+vdW surf PBE+vdW Victor G. Ruiz Experiment: A. Hauschild et al., PRB 81 (2010) EX+cRPA: M. Rohlfing and T. Bredow, PRL 101 (2008). V. G. Ruiz, W. Liu, E. Zojer, M. S., and A. Tkatchenko, PRL 108 (2012). Egbert Zojer 16
17 Many-Body Dispersion 17
18 18
19 vdw Effects Are Important for Bonding Between Atoms, But Also Beyond! What is next? o metals o elastic properties o vibrations o electron-phonon coupling o and more 19
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