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Approximate methods for large molecular systems

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  1. Approximate methods for large molecular systems Marcus Elstner Physical and TheoreticalChemistry, Technical Universityof Braunschweig

  2. Motivation Structure-formation, atomic-scale related properties and processes Si1600 MoS2 a-SiCN-ceramics Si21 C60-trimer defects, doping GaN-devices 4H-SiC-surfaces

  3. Reactions in biological Systems Alcohol DeHydrogenase Aquaporin Photosynthetic Reaction Center Catalysis Proton Transfer Photochemistry Electron/Energy Transfer bR Need QM description Photochemistry

  4. Computational challange • ~ 1.000-10.000 atoms • ~ ns molecular dynamics simulation • (MD, umbrella sampling) • weak bonding forces • chemical reactions • treatment of excited states

  5. Continuum electrostatics Molecular Mechanics SE-QM approx-DFT HF, DFT ‚multiscale business‘ fs ps ns time CI, MP CASPT2 Length scale nm predictivity

  6. Size problem: number of structures MD, MC, GA time scale of process MD, MC -- RP, TST ab initio, SE MM size of system: number of atoms

  7. Size problem: QM-Methods Hybride methods: QM/MM, QM/QM Linear scaling: O(N) SE/approx. Methods

  8. Semi-empirical /approximate methods • approximation, neglect and parametrization of interaction integrals from ab-initio and DFT methods • HF-based: • CNDO, INDO, MNDO, AM1, PM3, MNDO/d, OM1,OM2 • DFT-based: • SCC-DFTB, DFT- 3center- tight binding (Sankey) • Fireballs --- > Siesta DFT code • ~ 1000 atoms, ~ 100 ps MD

  9. Approximate density-functional theory:SCC-DFTBSelf consistent - charge density functional tight-binding • Seifert (1980-86): Int. J. Quant Chem., 58, 185 (1996). • O-LCAO; 2-center approximation: approximate DFT • http://theory.chm.tu-dresden.de • Frauenheim et al. (1995): Phys. Rev. B 51, 12947 (1995). • efficient parametrization scheme: DFTB • www.bccms.uni-bremen.de • Elstner et al. (1998): Phys. Rev. B 58, 7260 (1998). • charge self-consistency: SCC-DFTB • www.tu-bs.de/pci approximate DFT

  10. Extensions and Combinations: TD-DFTB-LR O(N)-QM/MM divide+conquer H. Liu W. Yang Duke Univ QM/MM AMBER: Han, Suhai DKFZ CHARMM: Cui, Karplus; Harvard TINKER: Liu, Yang; Duke CEDAR: Hu, Hermans; NC Univ SCC-DFTB Solvent Cosmo: W. Yang GB: H. Liu DISPERSION P. Hobza, Prague Electron Transport A. Di Carlo TD-DFTB R. Allen Texas A&M

  11. SCC-DFTB: • available for H C N O S P Zn (Si, ...) • all parameters calculated from DFT • computational efficiency as NDO-type methods (solution of gen. eigenvalue problem for valence electrons in minimal basis)

  12. SCC-DFTB: Tests 1) Small molecules: covalent bond • reaction energies for organic molecules • geometries of large set of molecules • vibrational frequencies, 2) non-covalent interactions • H bonding • VdW 3) Large molecules (this makes a difference!) • Peptides • DNA bases

  13. SCC-DFTB: Tests 4) Transition metal complexes 5) Properties • IR, Raman, NMR • excited states with TD-DFT • Transport calculations

  14. SCC-DFTB: Reviews • Application to biological molecules • M. Elstner, et al. ,A self-consistent carge density-functional based tight-binding scheme for large biomolecules, phys. stat. sol. (b) 217 (2000) 357. • Elstner, et al. An approximate DFT method for QM/MM simulations of biological structures and processes. J. Mol. Struc. (THEOCHEM), 632 (2003) 29. • M. Elstner, The SCC-DFTB method and its application to biological systems, Theoretical Chemistry Accounts, in print 2006. 2) Focus on solids and nanostructures • T. Frauenheim, et al., Atomistic Simulations of complex materials: ground and excited state properties, J. Phys. : Condens. Matter 14 (2002) 3015. • Th. Frauenheim et al. A self-consistent carge density-functional based tight-binding method for predictive materials simulations in physics, chemistry and biology, phys. stat. sol. (b) 217 (2000) 41. • G. Seifert, in: Encyclopedia of Computational Chemistry (Wiley&Sons 2004)

  15. SCC-DFTB Tests 1: Elstner et al., PRB 58 (1998) 7260 • Performance for small organic molecules • (mean absolut deviations) • Reaction energiesa): ~ 5 kcal/mole • Bond-lenghtsa) : ~ 0.014 A° • Bond anglesb): ~ 2° • Vib. Frequenciesc): ~6-7 % • a) J. Andzelm and E. Wimmer, J. Chem. Phys. 96, 1280 1992. • b) J. S. Dewar, E. Zoebisch, E. F. Healy, and J. J. P. Stewart, J. Am. • Chem. Soc. 107, 3902 1985. • c) J. A. Pople, et al., Int. J. Quantum Chem., Quantum Chem. Symp. 15, 269 • 1981.

  16. SCC-DFTB Tests 2: T. Krueger, et al., J.Chem. Phys. 122 (2005) 114110. With respect to G2: mean ave. dev.: 4.3 kcal/mole mean dev.: 1.5 kcal/mole

  17. SCC-DFTB Tests: Accuracy for vib. freq., problematic case e.g.: Special fit for vib. Frequencies: Mean av. Err.: 59 cm-1 33 cm-1 for CH Malolepsza, Witek & Morokuma: CPL 412 (2005) 237. Witek & Morokuma, J Comp Chem. 25 (2004) 1858.

  18. H-bonded systems: water CCSD(T): 5.0 kcal/mole (Klopper et al PCCP 2000 2, 2227) BLYP: 4.2 kcal/mole PBE: 5.1 kcal/mole B3LYP: 4.6 kcal/mole HF: 3.7 kcal/mole (from Xu&Goddard, JCPA 2004) For larger systems: DFTB: 3.3 kcal/mole HF: 5.7 kcal/mole @ 6-31G* B3LYP: 6.8 kcal/mole @ 6-31G* ~2 kcal/mole BSSE (BSIE)

  19. H2O-dimer complexes Cs, C2v NH3-NH3- and NH3-H2O-dimer H-bondsHan et al. Int. J. Quant. Chem.,78 (2000) 459. Elstner et al. phys. stat. sol. (b) 217 (2000) 357. Elstner et al. J. Chem. Phys. 114 (2001) 5149. Yang et al., to be published. Coulomb interaction • ~1-2kcal/mole too weak • relative energies reasonable • structures well reproduced Model peptides: N-Acetyl-(L-Ala)nN‘-Methylamide (AAMA) + 4 H2O

  20. N Secondary-structure elements for Glycine und Alanine-based polypeptidesElstner, et al.. Chem. Phys. 256 (2000) 15 aR-helix N = 1 (6 stable conformers) 310 - helix stabilization by internal H-bonds between i and i+4 between i and i+3 • main problem for DFT(B): dispersion! • AM1, PM3, MNDO quite bad • OM2 much improved (JCC 22 (2001) 509) • DFTB very good for: • relative energies • geometries • vib. freq. o.k.!

  21. N Glycine and Alanine based polypeptides in vacuoElstner et al., Chem. Phys. 256 (2000) 15 Elstner et al. Chem. Phys. 263 (2001) 203 Bohr et al., Chem. Phys. 246 (1999) 13 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

  22. Strength of SCC-DFTB Structure of large molecules - dynamics - relative energies DNA: A. V. Shiskin, et al., Int. J. Mol. Sci. 4 (2003) 537. O. V. Shishkin, et al., J. Mol. Struc. (THEOCHEM) 625 (2003) 295.

  23. Problems: • same Problems as DFT • additional Problems: - except for geometries, in general lower accuracy than DFT • - slight overbinding (probably too low reaction barriers?!) • - too weak Pauli repulsion • - H-bonding (will be improved) • - hypervalent species, e.g. HPO4 or sulfur compounds • - transition metals: probably good geometries, ... ? • - molecular polarizability (minimal basis method!)

  24. SCC-DFTB vs. NDDO (MNDO, AM1, PM3) DFTB: • energetics of ONCH ok, S, P problematic • very good for structures of larger Molecules • vibrational frequencies mostly sufficient (less accurate than DFT) NDDO: • very good for energetics of ONCH (and others, even better than DFT) • structures of larger Molecules often problematic !!! • do NOT suffer from DFT problems e.g. excited states  Mix of DFTB and NDDO to combine strengths of both worlds

  25. DFT Problems: Ex: Self interaction error. J- Ex = 0 !: Band gaps, barriers Ex: wrong asymptotic form; - HOMO << Ip: virtual KS orbitals Ex: ‚somehow too local‘; overpolarizability, CT excitations Ec: ‚too local‘: Dispersion forces missing Ec: even much more ‚too local‘: isomerization reactions Multi-reference problem (1) –(3) of course related, cure: exact exchange!

  26. DFT Problems: (very) selective publications • Ex: PRB 23 (1981) 5048, JCP 109 (1998) 2604 • Ex: JCP 113 (2000) 8918, Mol. Phys. 97 (1999) 859. • Ex: JPCA 104 (2000) 4755, JCP 119 (2003) 2943. • Ec: JCP 114 (2001) 5149 • Ec: Angew. Chem. Int. Ed. 2006, 45, 4460 –4464 • Koch, Wolfram / Holthausen, Max C.A Chemist's Guide to Density Functional Theory, Wiley

  27. Problems of DFT-GGA • - overbinding of small molecules: CO...  B3LYP, rev-PBE10 kcal • transition metals: B3LYP, PB86 ..., spin states, energetics10-20 kcal • - vib. Freqencies: • conjugate systems: GGAs overpolarize PA‘s of respective proton donors10 kcal • - H-bonds: ok with DFT, HF (cancellation of errors), water structure? • proton transfer (PT) barriers: GGA<B3LYP < MP2< CCSD 2-4 kcal with B3LYP! • Solution1: don‘t worry or don‘t care  different functionals VERY different accuracy • Solution2: use something else • VdW- problem (dispersion)complete failure • ‚Solution‘: empirical dispersion for GGAs • excited states within TD-DFT: ionic, CT states, double excitations, Rydberg states • Solution: exact exchange or CI-based methods