Approximate methods for large molecular systems

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Approximate methods for large molecular systems

Marcus Elstner Physical and Theoretical
Chemistry, Technical University of Braunschweig
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Motivation
Structure-formation, atomic-scale related
properties and processes
Si1600
MoS2
a-SiCN-ceramics
Si21
C60-trimer
defects, doping
GaN-devices
4H-SiC-surfaces
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Reactions in biological Systems
Alcohol DeHydrogenase
Aquaporin
Photosynthetic Reaction Center
Catalysis Proton Transfer Photochemistry Electron/
Energy Transfer
bR
Need QM description
Photochemistry
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Computational challange

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

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multiscale business
fs ps
ns time
CI, MP CASPT2
Length scale
nm
predictivity
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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
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Size problem QM-Methods
Hybride methods QM/MM, QM/QM

Linear scaling O(N)
SE/approx. Methods
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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 --- gt Siesta DFT
  code
• 1000
  atoms, 100 ps MD

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Approximate density-functional theorySCC-DFTB
Self 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
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Extensions and Combinations
TD-DFTB-LR
O(N)-QM/MM divideconquer 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 AM
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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)

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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

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SCC-DFTB Tests

• 4) Transition metal complexes
• 5) Properties
• IR, Raman, NMR
• excited states with TD-DFT
• Transport calculations

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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 (WileySons 2004)

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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.

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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

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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.
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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 XuGoddard,
JCPA 2004) For larger systems DFTB 3.3
kcal/mole HF 5.7 kcal/mole _at_
6-31G B3LYP 6.8 kcal/mole _at_ 6-31G 2
kcal/mole BSSE (BSIE)

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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) 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)n N-Methylamide
(AAMA) 4 H2O
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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 i4
between i and i3

• 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.!

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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
Relative energies, structures and vibrational
properties N1-8
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
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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.
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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!)

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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

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DFT Problems

• Ex Self interaction error. J- Ex 0 !
  Band gaps, barriers
• Ex wrong asymptotic form - HOMO ltlt 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!

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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

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Problems of DFT-GGA

• - overbinding of small molecules CO... ? B3LYP,
  rev-PBE 10 kcal
• transition metals B3LYP, PB86 ..., spin states,
  energetics 10-20 kcal
• - vib. Freqencies
• conjugate systems GGAs overpolarize? PAs of
  respective proton donors 10 kcal
• - H-bonds ok with DFT, HF (cancellation of
  errors), water structure?
• proton transfer (PT) barriers GGAlt B3LYP lt MP2lt
  CCSD 2-4 kcal with B3LYP!
• Solution1 dont worry or dont 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