The SCC-DFTB method applied to organic and biological systems: successes, extensions and problems.

1
The SCC-DFTB method applied to organic and
biological systems successes, extensions and
problems.

Marcus Elstner Physical and Theoretical
Chemistry Technical University of Braunschweig
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DFTB non-self-consistent scheme
Consider a case, where you know the DFT ground
state density ?G already (exactly or in good
approximation ?in )
Then the energy can given by (Foulkes Haydock
PRB 1989)
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DFTB non-self-consistent scheme
DFTB consider input density ?0 as superposition
of neutral atomic densities
LCAO basis
TB energy
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DFTB non-self-consistent scheme

• No charge transfer between atoms ? very good
  results for homonuclear systems (Si, C),
  hydrocarbons etc.
• Complete transfer of one charge between atoms?
  Also does not fail for ionic systems (e.g. NaCl)
• - Harrison
• - Slater, (Theory of atoms and molecules)
• Problematic case everything in between

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DFTB non-self-consistent scheme

• Problems
• HCOOH CO and C-O bond lengths equalized
• H2N-CHO and peptides N-C and CO bond lengths
  equalized
• non-CT systems
• CO2 vibrational frequencies
• CCCCC.. chains, dimerization, end effects

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DFTB non-self-consistent scheme
Problem charge transfer between atoms
overestimated due to electronegativity
differences between atoms? need balancing force
onsite e-e interaction of excess charge is
missing!
C O
C 0 O 0 C 1 O -1 C 0.5
O -0.5

• Non scf scheme ok
• no charge transfer
• transfer of one electron

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DFTB non-self-consistent scheme
?0 ?
Try to keep H0?? since it works well for many
systems
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DFT total energy
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Second order expansion of DFT total energy
Expand E? at ?0, which is the reference
density used to calculate the H0??
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Second order expansion of DFT total energy
Write density fluctuations as a sum of atomic
contributions
(I) (II) (III)
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(I) Hamiton matrix elements
Introduce LCAO basis
(I)
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Second order expansion of DFT total energy
Write density fluctuations as a sum of atomic
contributions
(I) (II) (III)
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(II) Repulsive energy contribution

• pair potentials
• exponentially decaying

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Second order expansion of DFT total energy
Write density fluctuations as a sum of atomic
contributions
(I) (II) (III)
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(III) Second order term
Monopolapproximation
Two limits
New parameter U? calculated for every element
from DFT
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(III) Second order term
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Combine the two limits
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(III) Second order term Klopman-Ohno
approximation
???
R??
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Determination of Gamma in DFTB

• - Consider atomic charge densities
• Rcov
• Calculate coulomb integrals (? ) for 2 spherical
  charge densities
• -deviation from 1/R for small R
• R0 1/? 3.2 UHubbard

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Klopman-Ohno vs DFTB Gamma
1/r
DFTB-?
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Approximate density-functional theory Elstner et
al. Phys. Rev. B 58 (1998) 7260
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Determination of Erep
HC-CH H2C-CH2 H3C-CH3
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Hamilton-Matrixelements

• non-scc neglect of red contributions

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Comparison to SE models Matrix elements

• Extended Hueckel (can be derived from DFT)

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Comparison to SE models Matrix elements

• Fenske Hall

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Comparison to SE models Matrix elements

• formal similarity in Hamiltonmatrixelements
• Very different in determination of matrixelements
• DFTB incorporate strengths, but also
• fundamental weaknesses of DFT

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Differences w.r. to SE models e.g. J
e.g. MNDO
Approx. by multipole-multipole interaction
Coulomb part J
e.g. CNDO
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Differences to SE models e.g. J
CNDO
Coulomb part accounts for e-e interaction due to
interaction of atomic charges looks similar to
2nd order term in DFTB.
MNDO simple charge-charge? higher multipoles
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Differences to SE models e.g. J
CNDO
Coulomb part accounts for e-e interaction due to
interaction of atomic charges similar to 2nd
order term in DFTB.
MNDO simple charge-charge? higher multipoles
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Differences to SE models e.g. J
DFTB how is e-e interaction treated? consider J
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Extensions of DFTB

• FAQS
• better basis sets (e.g. double zeta)
• higher order expansion
• monopole ? multipole
• other reference density
• why Mulliken charges?
• better fitting of Erep

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Approximate density-functional theory
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Extensions

• FAQS
• better basis sets ? much higher cost

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Extensions

• FAQS
• higher order expansion
• monopole ? multipole
• inspection of gamma?
• No additional cost!

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Determination of Gamma
deviation from 1/R for small R R0
1/? 3.2 Uhubbard Is this valid throughout
the periodic table? What is the relation
between atomic size and chemical hardness?
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Gamma Rcov 1/U ?
Si-Cl
R covalent
B-F
H
U-Hubbard
N
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Gamma requires 3.2Rcov 1/U?
H
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U vs Rcov Hydrogen atom
U-Hubbard
O
N
H
C
Si
R covalent
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U vs Rcov H not in line!
U-Hubbard
In DFTB, H is 0.73A instead of 0.33A!
N
H

• Gamma requires 3.2Rcov 1/U
• size of H overestimated based on hardness value
  H has same size like N!

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On-site interaction and coulomb scaling H

• UH for the on-site interaction of H should not
  be changed!
• However, UH is a bad measure for the size of H!
• Leads to too large H-atoms! I.e. coulomb
  interaction is damped too fast due to
  artificial overlap effect!
• modify coulomb-scaling for H!

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Modified Gamma for H-bondingchange only X-H
interaction!
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Modified Gamma for H-bonding

• Water dimer 3.3 kcal
• 4.6 kcal
• standard DFTB H-bonds 1-2 kcal too low
• mod Gamma 0.3-0.5 kcal too low

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H-bonds water clusterMP2 from KS Kim et al 2000
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Expansion to higher order?
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Charged systems with localized charge
E.g. H2O ? OH- H
Description of OH- O is very negative, is
the approximation of a constant Hubbard value
(chemical hardness) appropriate? Deprotonation
energy B3LYP/6-311G(2d2p) 397
kcal/mole SCC-DFTB 424 kcal/mole
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Problems with charged systems inclusion of third
order correction into DFTB

• charge dependent Hubbard
• U(q) U(q0) dU/dq (q-q0)
• Calculate dU/dq through U(q) consider atoms
  for different charge states.

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Deprotonation energies

• B3LYP vs SCC-DFTB and 3rd order correction Uq
• basis set dependence
• large charges on anions
• U(q) changes size of atom Rcov 1/U

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SCC-DFTB

• organic set available for H C N O S P Zn
  
• solids Ga,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

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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-lenghtsb) 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 3 Sattelmeyer Jorgensen, (to be
published)
Mean Absolute Errors in Calculated Heats of
Formation for Neutral Molecules Containing
the Elements C, H, N and O (kcal/mol). N
AM1 PM3 PDDG/PM3 SCC-DFTB Hydrocarbons 254
5.6 3.6 2.6 4.8 All Molecules 622
6.7 4.4 3.2 5.9 Training Set 134
6.1 4.3 2.7 7.0 Test Set 488
6.8 4.4 3.3 5.6

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SCC-DFTB Tests 3 Sattelmeyer Jorgensen, (to be
published)
Absolute Errors for Additional Molecular
Properties of CHNO-containing Species. N
AM1 PM3 PDDG/PM3 SCC-DFTB Bond lengths
(Å) 218 0.017 0.012 0.013
0.012 Bond angles (deg.) 126 1.5 1.7
1.9 1.0 Dihedral angles (deg.) 30
2.8 3.2 3.7 2.9 Dipole moments
(D) 47 0.23 0.25 0.23 0.39

• ok H-bonds, ions
• quite bad S

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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-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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Performance of DFTB

• Small molecules dont tell the whole story
• Test for large ones
• - peptides
• - DNA, sugar
• - other extended structures

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Secondary-structure elements for Glycine und
Alanine-based polypeptides
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 not convincing
• 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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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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TD-DFTB and excited states
Problems of TD-DFT Combination of DFTB and OM2!

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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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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

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DFT and VdW interactions

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DFT and VdW interactions

• 2 Problems
• Pauli repulsion exchange effect
• exp(R??) or 1/R12??
• - attraction due to correlation
• -1/R6??

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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)
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DFTB dispersion
Sponer et al. J.Phys.Chem. 100 (1996) 5590 Hobza
et al. J.Comp.Chem. 18 (1997) 1136stacking
energies in 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
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With help from
QM/MM DFTB Q. Cui, Madison H. Hu, J.
Herrmans UNC D. York, Minnesota A. Roitberg,
Florida
Morokuma, Witek Zheng, Irle IR, RAMAN, metals
DFTB Frauenheim, Seifert Suhai groups
Dispersion, DNA P. Hobza,
Nat. Academie, Prague
H. Liu, W. Yang, Duke O(N), COSMO, GB
DFG, Univ. Paderborn