Approximate methods for large molecular systems Marcus Elstner Physical and Theoretical Chemistry, Technical University of Braunschweig

C 60 -trimer Si 1600 MoS 2 4H-SiC-surfaces defects, doping GaN-devices Structure-formation, atomic-scale related properties and processes Motivation Si 21 a-SiCN-ceramics

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

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

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

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

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

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

Approximate density-functional theory: SCC-DFTB Self consistent - charge density functional tight-binding Seifert ( ): Int. J. Quant Chem., 58, 185 (1996). O-LCAO; 2-center approximation: approximate DFT Frauenheim et al. (1995): Phys. Rev. B 51, (1995). efficient parametrization scheme: DFTB Elstner et al. (1998): Phys. Rev. B 58, 7260 (1998). charge self-consistency: SCC-DFTB approximate DFT

Extensions and Combinations : 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 DISPERSION P. Hobza, Prague TD-DFTB-LR TD-DFTB R. Allen Texas A&M SCC-DFTB Solvent Cosmo: W. Yang GB: H. Liu Electron Transport A. Di Carlo

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)

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

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

SCC-DFTB: Reviews 1)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 ) 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)  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 )

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

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

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.

H-bonded systems: water CCSD(T): 5.0 kcal/mole (Klopper et al PCCP , 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: G* B3LYP: G* ~2 kcal/mole BSSE (BSIE)

H-bonds Han 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) Yang et al., to be published. -~1-2kcal/mole too weak - relative energies reasonable - structures well reproduced Model peptides: N-Acetyl-(L-Ala) n N‘-Methylamide (AAMA) + 4 H 2 O H 2 O-dimer complexes C s, C 2v NH 3 -NH 3 - and NH 3 -H 2 O-dimer Coulomb interaction

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

Glycine and Alanine based polypeptides in vacuo Elstner et al., Chem. Phys. 256 (2000) 15 Elstner et al. Chem. Phys. 263 (2001) 203 Bohr et al., Chem. Phys. 246 (1999) 13 N = 1 (6 stable conformers) N Relative energies, structures and vibrational properties: N=1-8 (6-31G*) C 7 eq C 5 ext C 7 ax MP4-BSSE MP2 B3LYP SCC-DFTB E relative energies (kcal/mole) MP4-BSSE: Beachy et al, BSSE corrected at MP2 level Ace-Ala-Nme

Strength of SCC-DFTB 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. Structure of large molecules - dynamics - relative energies

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. HPO 4 or sulfur compounds - transition metals: probably good geometries,... ? - molecular polarizability (minimal basis method!)

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

DFT Problems: (1) E x : Self interaction error. J- E x = 0 !: Band gaps, barriers (2) E x : wrong asymptotic form; - HOMO << I p : virtual KS orbitals (3) E x : ‚somehow too local‘; overpolarizability, CT excitations (4) E c : ‚too local‘: Dispersion forces missing (5) E c : even much more ‚too local‘: isomerization reactions (6)Multi-reference problem (1) –(3) of course related, cure: exact exchange!

DFT Problems: (very) selective publications (1)E x : PRB 23 (1981) 5048, JCP 109 (1998) 2604 (2)E x : JCP 113 (2000) 8918, Mol. Phys. 97 (1999) 859. (3)E x : JPCA 104 (2000) 4755, JCP 119 (2003) (4)E c : JCP 114 (2001) 5149 (5)E c : Angew. Chem. Int. Ed. 2006, 45, 4460 –4464 (6) Koch, Wolfram / Holthausen, Max C. A Chemist's Guide to Density Functional Theory, Wiley

Problems of DFT-GGA - overbinding of small molecules: CO...  B3LYP, rev-PBE 10 kcal - transition metals: B3LYP, PB86..., spin states, energetics kcal - vib. Freqencies: - conjugate systems: GGAs overpolarize  PA‘s of respective proton donors 10 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