5/27/2012. 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 1
2 Role of van der Waals Interactions in Physics, Chemistry, and Biology Karin Jacobs: Take van der Waals forces into account in theory, simulations, and experiment! The Kohn-Sham Ansatz of Density-Functional Theory Walter Kohn 2
3 The Kohn-Sham Ansatz of Density-Functional Theory Kohn-Sham (1965): Replace the original many-body problem by an independent electron problem that can be solved! E v [n] = T s [n] + v(r) n(r) d3 r + E artree [n] + E xc [n] With T s [n] the kinetic energy functional of independent electrons, and E xc [n] the unknown functional. The challenge is to find useful, approximate xc functionals. Walter Kohn The Kohn-Sham Ansatz of Density-Functional Theory Kohn-Sham (1965): Replace the original many-body problem by an independent Approximate xc functionals have been very electron problem that can be solved! successful E v [n] = T s [n] + v(r) n(r) d3 r + E artree [n] + E xc [n] With T s [n] the kinetic energy functional of independent electrons, and E xc [n] the unknown functional. The challenge is to find useful, approximate xc functionals. Walter Kohn 3
4 accuracy 5/27/2012 The Kohn-Sham Ansatz of Density-Functional Theory Kohn-Sham (1965): Replace the original many-body problem by an independent Approximate xc functionals have been very electron problem that can be solved! successful but there are problems E v [n] for = certain T s [n] + bonding v(r) n(r) situations d3 r + (vdw, E artree [n] + E xc [n] hydrogen bonding, certain covalent bonds) With for T s [n] highly the correlated kinetic energy situations, functional and of independent electrons, and E xc [n] the unknown functional. for excited states. The challenge is to find useful, approximate xc functionals. Walter Kohn 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, SE, ) 3 τ (r), meta-gga (e.g., TPSS) 2 n(r), Generalized Gradient Approx. 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); ; X. Ren, et al., Invited Review in J Mater Sci (2012) -- arxiv:
5 accuracy accuracy 5/27/2012 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, SE, ) 3 τ (r), meta-gga (e.g., TPSS) 2 1 n(r), n(r), Generalized Gradient Approx. Local-Density Approximation past and presence 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); ; X. Ren, et al., Invited Review in J Mater Sci (2012) -- arxiv: 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, SE, ) 3 τ (r), meta-gga (e.g., TPSS) 2 1 n(r), n(r), Generalized Gradient Approx. Local-Density Approximation past and presence 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); ; X. Ren, et al., Invited Review in J Mater Sci (2012) -- arxiv:
6 Level 5 plus Viewed in the Many-Body Framework Perturbation theory: = 0 + with 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 0 ϕ 0, (1) = ϕ 0 ϕ 0 ϕ 0 ϕ n E (2) 0 = Σ 2 ϕ 0 ϕ i, a = Σ 2 ϕ 0 ϕ ij,ab + Σ 2 E n E i, a E ij,ab n 0 i, a ij, ab single excitations double excitations Adding all ring diagrams from higher order perturbations: X. Ren, et al., Invited Review in J Mater Sci (2012) -- arxiv: Level 5 plus Viewed in the Many-Body Framework Perturbation theory: = 0 + with 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 0 ϕ 0, (1) = ϕ 0 ϕ 0 ϕ 0 ϕ n E (2) 0 = Σ 2 ϕ 0 ϕ i, a = Σ 2 ϕ 0 ϕ ij,ab + Σ 2 E n E i, a E ij,ab n 0 i, a ij, ab single excitations double excitations Using F input, this is Møller-Plesset perturbation theory, MP2 Adding all ring diagrams from higher order perturbations: X. Ren, et al., Invited Review in J Mater Sci (2012) -- arxiv:
7 Level 5 plus Viewed in the Many-Body Framework Perturbation theory: = 0 + with 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 0 ϕ 0, (1) = ϕ 0 ϕ 0 ϕ 0 ϕ n E (2) 0 = Σ 2 ϕ 0 ϕ i, a = Σ 2 ϕ 0 ϕ ij,ab + Σ 2 E n E i, a E ij,ab n 0 i, a ij, ab single excitations double excitations Adding all ring diagrams from higher order perturbations: X. Ren, et al., Invited Review in J Mater Sci (2012) -- arxiv: = ϕ 0 ϕ 0, = ϕ 0 ϕ 0 Level 5 plus Renormalized second order perturbation theory sets n 0 Viewed the reference. in the owever, Many-Body i, a it is quite Framework costly ij, ab and (at present) limited single to about excitations 100 atoms. Can we find a method that is efficient AND accurate? We need dispersion coefficients with an error 5%. ϕ 0 ϕ n 2 ϕ 0 ϕ i, a 2 ϕ 0 ϕ ij,ab 2 (2) = Σ = Σ + Σ E n E i, a E ij,ab 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, et al., Invited Review in J Mater Sci (2012) -- arxiv:
8 Mean absolute error to CCSD(T) (mev) 5/27/2012 C 6 Coefficients in the TS Scheme (schematic) S. Grimme: add tails E vdw = Σ C 6, / R 6 I, J 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. ead-gordon, and M.S., JCP (2009). Excellent description as long as the pair-wise model holds. Performance for Intermolecular Interactions: S22 Test Set Alex Tkatchenko Xinguo Ren Patrick Rinke (TS et al.) 0 S22: Jurecka, Sponer, Cerny, obza, 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). 8
9 Mean absolute error to CCSD(T) (mev) 5/27/2012 Performance for Intermolecular Interactions: S22 Test Set Alex Tkatchenko Xinguo Ren Patrick Rinke (TS et al.) 0 S22: Jurecka, Sponer, Cerny, obza, 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). amino group Stability of Secondary Structures structure of proteins (peptide chains): R 1 O N C N C C N C N C C N C N C C N C C N C O O R 2 O O O R 4 Secondary structure sheets R 3 peptide bond O... C R n carboxyl group O Tertiary structure R O N C C O Mariana Rossi Carvalho turns Volker Blum helices R = C 3 = alanine Alexandre Tkatchenko 9
10 Secondary Structure from First Principles Environment Solute-environment interactions (dieletric, -bonds, van der Waals,...) Intramolecular interactions (-bonds, electrostatics, van der Waals) Secondary Structure from First Principles Intramolecular interactions (-bonds, electrostatics, van der Waals) 10
11 Secondary Structure from First Principles First (necessary) step: Intramolecular interactions; Clean room conditions: gas phase Intramolecular interactions (-bonds, electrostatics, van der Waals) In the following: Ac-Ala n -Lys +, n = 1 15 Structure prediction: There are several hundred low energy structures Role of vdw Interaction on (Un)folding; Ac-Ala 15 Lys + Mariana Rossi Carvalho Volker Blum Alexandre Tkatchenko (*) -bond = 2.5 Å between CO and N groups PRL 106, (2011) 11
12 Role of vdw Interaction on (Un)folding; Ac-Ala 15 Lys + Mariana Rossi Carvalho Volker Blum (*) -bond = 2.5 Å between CO and N groups Alexandre Tkatchenko PRL 106, (2011) What Did We Learn from Studying Polypeptides? Density-functional theory (PBE+vdW) is able to: Predict geometry and properties, analyze stability and unfolding: - hydrogen bonding, - vdw interaction, and - vibrational entropy. Verification of structure predictions against experiment (vibrational spectroscopy). M. Rossi, V. Blum, et al., JPCL 1 (2010). A Tkatchenko, M Rossi, V Blum, J Ireta, M.S., PRL 106, (2011) 12
13 2. Example: ybrid Inorganic/Organic Systems (IOS) solid organic S. Blumstengel et al. IOS IOS: 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? Van der Waals Interactions at Interfaces The Example of Me(111) PTCDA perylene-tetracarboxylic acid dianhydride ( C 24 8 O 6 ) Which Theoretical Method? PTCDA Ag(111) Experiment: A. auschild et al., PRB 81 (2010) Theory review: L. Romaner, C. Draxl, et al. NJP (2009); A. Tkatchenko et al., MRS Bulletin (2010) 13
14 Metal Surfaces, Correct Long-Range Theory α TS (iω) The interaction energy is not additive; it is not a simple sum over pair-wise interactions. Z For large Z: E(Z) = C 3 / Z 3 ε(iω) To be 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). PTCDA@Ag(111): PBE+vdW surf Alex Tkatchenko PBE+vdW surf PBE+vdW Victor G. Ruiz Experiment: A. auschild 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 14
15 E (ev) 5/27/2012 Graphene on Ni(111) -- a novel bonding mechanism vdw-df vdw-df2 PBE EX+cRPA d (Å) Energy lowering due to change of orbital occupation which implies a reduction of EX energy: decrease in d z 2 and increase in Ni d xz and d yz. F. Mittendorfer, A. Garhofer, J. Redinger, J. Klimes, J. arl, G. Kresse, PRB 84 (2011). Minimum, actuated by vdw interaction. Summary Electronic-Structure Theory of Materials Were Are We? What Comes Next? Level 5 plus = second order renormalized perturbation theory provides a most balanced description of materials. For some issues (e.g. energy barriers) we need even better methods as well as self consistency. Need more efficient numerics (massive parallel, GPUs, hybrid-core, ) Building a Materials Library of hitherto Unknown Materials novel materials discovery with C. Draxl, K.-R. Müller, A. Rubio, N. Marzari, and others. emphasis on properties and functions. Transparencies of this talk and pre- and reprints: 15
16 Summary Electronic-Structure Theory of Materials Where Are We? What Comes Next? Level 5 plus = second order renormalized perturbation theory provides a most balanced description of materials. For some issues (e.g. energy barriers) we need even better methods as well as self consistency. Need more efficient numerics (massive parallel, GPUs, hybrid-core, ) Building a Materials Library of hitherto Unknown Materials novel materials discovery with C. Draxl, K.-R. Müller, A. Rubio, N. Marzari, and others. emphasis on properties and functions. Transparencies of this talk and pre- and reprints: Summary Electronic-Structure Theory of Materials Were Are We? What Comes Next? Level 5 plus = second order renormalized perturbation theory provides a most balanced description of materials. For some issues (e.g. energy barriers) we need even better methods as well as self consistency. Need more efficient numerics (massive parallel, GPUs, hybrid-core, ) Building a Materials Library of hitherto Unknown Materials novel materials discovery with C. Draxl, K.-R. Müller, A. Rubio, N. Marzari, and others. emphasis on properties and functions. Transparencies of this talk and pre- and reprints: 16
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