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  1. DFT and VdW interactions Marcus Elstner Physical and Theoretical Chemistry, Technical Universityof Braunschweig

  2. E ~ 1/R6 DFT and VdW interactions • 2 Problems: • Pauli repulsion: exchange effect • ~ exp(R) or 1/R12 • - attraction due to correlation • ~ -1/R6

  3. E ~ 1/R6 DFT Problem • B88 exchange: too repulsive ? • PBEx/PW91x: too attractive • already at Ex only level • LDA finds often binding! Ex ?? • fix Ex • correlation Ec? Ec ??

  4. Ar2 with Ex only • B too repulsive, • PW91x too “attractive” • Complete mess with • DFT Wu et al. JCP 115 (2001) 8748

  5. Popular Functionals: role of Ex BPW91 BLYP B3LYP PW91 B3LYP contains only 20% HF exchange! Xu & Yang JCP 116 (2002) 515

  6. Popular Functionals: role of Ec Xu & Yang JCP 116 (2002) 515 BPW91 BLYP B3LYP PW91 • BPW91 vs PW91: attraction only due to exchange!!!!! • Correlation not significant for PW91 and LYP

  7. Popular Functionals: role of Ec Perez-Jorda et al. JCP 110 (1999) 1916 DFT HFx + Ec: some Ec lead to (over-) binding, some don’t!

  8. Does overlap matter? GGA DFTB Xu & Yang JCP 116 (2002) 515 Elstner et al. JCP 114 (2001) 5149

  9. E ~ 1/R6 DFT and VdW interactions: the problem Exc = ?? Ec = 0

  10. DFT and VdW interactions: solutions Adding empirical dispersion Elstneret al. JCP 114 (2001) 5149 Xu & Yang JCP 116 (2002) 515 Zimmerli et al. JCP 120 (2004) 2693 Grimme JCC 25 (2004) 1463 DFT model for empircal dispersion on top of HF Becke & Johnson JCP 124 (2006) 014104 Put it into the pseudopotential v. Lilienfeld et al. PRB 71 (2005) 195119 Find a new dispersion functional Dion, et al. Phys. Rev. Lett. 92 (2004) 246401; [JCP 124 (2006) 164106] Kamiya et al. JCP 117 (2002) 6010.

  11. Adding empirical dispersion Following the idea of HF+dis: Add f (R) C6 /R6to DFT total energy C6 empirical values Elstner, Hobza et al. JCP 114 (2001) 5149 To be successfull: Ex should be well-behaved (i.e. like HF) Ec: double counting

  12. E ~ 1/R6 Dispersion forces - Van der Waals interactionsElstner et al. JCP 114 (2001) 5149 Etot = ESCC-DFTB - f (R) C6 /R6  C6 via Slater-Kirckwood combination rules of atomic polarizibilities after Halgreen, JACS 114 (1992) 7827. damping f(R) = [1-exp(-3(R/R0)7)]3 R0 = 3.8Å (für O, N, C)

  13. How to get Dispersion coefficients?Halgren JACS 114 (1992) 7827 London, Phys. Chem. (Leipzig) B 11(1930) 222 Slater & Kirkwood. Phys. Rev. 37 (1931) 682. Kramer & Herschbach J. Chem. Phys. 53 (1970) 2792 effective electron number

  14. DFTB input Etot = ESCC-DFTB - f (R) C6 /R6 f(R) = [1-exp(-3(R/R0)7)]3 • R0: e.g. 3.8 for ONC • Atomic polarizabilities: • hybridisation dependent • Effective electron number (from Halgren)

  15. DFTB + dispersion Sponer et al. J.Phys.Chem. 100 (1996) 5590; Hobza et al. J.Comp.Chem. 18 (1997) 1136stacking energiesin MP2/6-31G* (0.25), BSSE-corrected ( + MP2-values) • Hartree-Fock, no stacking • AM1, PM3, repulsive interaction (2-10) kcal/mole • MM-force fields strongly scatter in results vertical dependence twist-dependence

  16. DFT + empirical dispersion: 1st generation 1) Problem of unbalanced Ex: 2) Problem of Ec?? Which one to choose?  Large variation in results when adding dispersion Wu and Wang 2002 Zimmerli et al 2004

  17. DFT and empirical dispersion Does not work for all Exc functionalsproperly Wu and Wang 2002 Zimmerli et al.2004 From Wu and Yang 2002

  18. DFT + empirical dispersion: 2nd generation 1) Problem of unbalanced Ex: 2) Problem of Ec?? Which one to choose?  Large variation when adding dispersion Grimme 2004: scale BLYP + dispersion with 1.4 scale PW91 + dispersion with 0.7

  19. f (R)  C6 /R6 • choice of C6 coefficients • Choice of damping function

  20. Choice of C6 coefficients • hybridisation dependence vs. atomic values • empirical values •  Very similar in various approaches

  21. Choice of damping function • various functional forms • - Fermi-function • - f(R) = [1-exp(-3(R/R0)7)]3 • choice of “cutoff” radius from Grimme 2004

  22. Choice of fdamp • fdamp balances several effects • - contribution from Ex/Ec in overlap region • - double counting of Ec • BSSE and BSIE • missing higher order terms 1/R**8 … • Determination completely empirical Choose, to reproduce interaction energies for large set of stacked compounds

  23. Choice of fdamp However, form of fdamp may be crucial Location of minimum For A-A stack From Wu and Yang 2002

  24. Grimme JCC 25 (2004) 1463 • s6: • PW91: 0.7 • BLYP: 1.4 • Scaling: • -Basis set dependent • functional dependent • hybridisation dependence • empirical vs. new fits •  Very similar in various approaches

  25. DFT + empirical dispersion: 3rd generation • 1) Problem of unbalanced Ex: • 2) Problem of Ec?? Which one to choose? •  Large variation in results when adding dispersion • mix PW91x and Bx • revPBE • meta GGA?? • + balanced damping function, no scaling

  26. DFT + empirical dispersion: 2nd generation Grimme JCC 25 (2004) 1463: scale BLYP + disp with 1.4 scale PW91 + disp with 0.7 DFT + empirical dispersion: 1st generation 1) Problem of unbalanced Ex: 2) Problem of Ec?? Which one to choose?  Large variation in results when adding dispersion Wu and Wang JCP 116 (2002) 515 Zimmerli et al. JCP 120 (2004) 2693 3rd generation: revPBE, XLYP and s6=1

  27. Applications of DFTB-D

  28. Benzene (from Irle/Morokuma, Emory)

  29. Benzene (from Irle/Morokuma, Emory) RHF, MP2 (both CP corrected) and DFTB DE on benzene dimers:

  30. Hybride materials

  31. O(N)-QM/MM-molecular-dynamics for DNA-dodecamer in H2OElstner et al. in preparation • DNA-Dodecamer 758 + 2722 H2O + 22 Na • periodic BC-Ewald-summation • dispersion in QM-region • MD-simulation at 300 K • parallel-16 processors SP2energy/forces: 1 – 2 sec.  10 ps/day 1-st stable QM/MM ns-scale dynamic simulation

  32. Intercalation: Ethidium – ATReha et al JACS 2003

  33. N Secondary-structure elements for Glycine und Alanine-based polypeptides: ß-sheets, helices and turnElstner, et a.. Chem. Phys. 256 (2000) 15 For increasing N: energetics of different conformers, geometries, vibrations N = 1 (6 stable conformers) aR-helix 310 - helix N-fold periodicity stabilization by internal H-bonds between i and i+4 between i and i+3

  34. N Glycine and Alanine based polypeptides in vacuoElstner et al., Chem. Phys. 256 (2000) 15 Relative energies, structures and vibrational properties: N=1-8 N = 1 (6 stable conformers) E relative energies (kcal/mole) B3LYP (6-31G*) MP2 MP4-BSSE SCC-DFTB Ace-Ala-Nme C7eq C5ext C7ax MP4-BSSE: Beachy et al, BSSE ‚corrected‘ at MP2 level

  35. Polypeptides in vacuoEffect of dispersion: favors more compact structures (6-31G*) N = 2 BLYP B3LYP HF MP2 SCC-DFTB Ace-Ala2-Nme C7eq C5ext BI BII BI` BII` DFT: relative stability of compact vs. extended structures?

  36. Secondary structure formationElstner et al., Chem. Phys. 256 (2000) 15 E DFT/DFTB ? 310 - helix aR-helix N peptide size DFT: crossover only for N~20 !!  solvation??

  37. Secondary structure:Influence of aqueous solutionCui et al, JPCB 105 (2001) 569 310 – helix: occurence for N<8 in database QM/MM MD of octa-Alanine: 310 - helix converts into aR-helix within 10 ps 310 - helix aR-helix Situation in Protein?

  38. energy and interatomic forces parallel (16-node SP2): 2 sec. MD simulation for 0.35 ns Molecular-dynamics for Crambin in H2O-solution O(N)-QM/MM simulationLiu et al. PROTEINS 44 (2001) 484 Crambin (639) + 2400 H2O

  39. Influence of DispersionLiu et al. PROTEINS 44 (2001) 484 QM/MM MD-Simulation Crambin in Solution HF DFT/DFTB ? MP2 SCC-DFTB + DIS 

  40. Enkephalin: ~30 local minima 3 cluster Jalkanen et al. to be published single bend double bend compact extended C5

  41. Enkephalin: MP2/6-31G* vs DFTB-dis//DFTB-dis compact  extended kcal c a b conformer Rel. energy (kcal) vs. conformer

  42. Enkephalin: MP2/6-31G* vs DFTB//DFTB-dis compact  extended kcal conformer

  43. Enkephalin: MP2 vs B3LYP//DFTB-dis compact  extended kcal conformer

  44. Enkephalin: MP2 vs B3LYP-dis//DFTB-dis compact  extended kcal conformer

  45. Enkephalin: MP2 vs PBE+dis//DFTB-dis compact  extended kcal conformer

  46. Enkephalin: MP2 vs PBE//DFTB-dis compact  extended kcal conformer

  47. Enkephalin: MP2 vs PBE+dis//DFTB-dis compact  extended kcal conformer

  48. CONCLUSIONS • Dispersion favors compact structures ~ 15 kcal/mole • MP2/6-31G*: • - internal BSSE • - higher level correlation contribution • -PBE and B3LYP differ in stability of extended (C5) confs • -B3LYP overestimates Pauli repulsion: N-H...

  49. DFT+large soft matter structures: don‘t do without dispersion! • large impact on relative energies • stabilizes more compact structures: • relevant secondary structures may • not be stable without!