DFTB Symposium

1
DFTB Symposium
Looking at DFTB from a Semiempirical
Perspective Walter Thiel Max-Planck-Institut für
Kohlenforschung, Mülheim, Germany ACS National
Meeting at San Francisco, 11 September 2006
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DFTB Original tight-binding approach

• LCAO-MOs from solution of secular equations
  with overlap. In usual matrix notation
  H0CSCE.
• Hamiltonian matrix elements calculated using
  the kinetic energy operator and an
  effective Kohn-Sham potential which is
  approximated as the sum of the Kohn-Sham
  potentials of associate neutral
  atoms (A, B).
• Basis orbitals and potentials V0 taken
  from DFT calculations on atoms.
• Only two-center terms computed
  
• Non-iterative tight-binding treatment.
• Total energy as sum of orbital energies and
  repulsive two-center correction terms
  determined by fitting the differences between
  reference DFT and tight-binding DFTB potential
  curves in suitable reference molecules.

G. Seifert, H. Eschrig, and W. Bieger, Z. Phys.
Chem. (Leipzig) 267, 529 (1986). G. Seifert, D.
Porezag, and T. Frauenheim, Int. J. Quantum Chem.
58, 185 (1996).
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SCC-DFTB Self-consistent-charge tight-binding
approach

• Improve DFTB by allowing for charge
  fluctuations.
• Second-order expansion of the DFT total energy
  with respect to the charge density variation
  relative to a chosen reference density.
• Charge density variation represented by sum of
  atomic contributions ?qA which are
  approximated by Mulliken charges
• Damped Coulomb interaction between
  Mulliken charges with correct asymptotic
  behavior for large distances (? 1/RAB) and for
  small distances (? one-center-term
  chemical hardness computed from PBE).
  
• Working equations
• Iterative SCF treatment.
• Parametrization of repulsive two-center terms
  Erep (part of E0).

M. Elstner, D. Porezag, G. Jungnickel, J. Elsner,
M. Haugk, T. Frauenheim, S. Suhai, and G.
Seifert, Phys. Rev. B 58, 7260 (1998).
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SCC-DFTB vs. MNDO-type methods Basic Features

Valence-electron SCF-MO treatment Minimal basis set of atomic orbitals Only one-center and two-center terms Type of integral approximation Overlap included in secular equations One-center integrals derived from Damped two-center two-electron integrals - with correct limits (R 0, R ?) Electrostatic balance (attraction/repulsion) Two-center one-electron integrals Repulsive atom-pair correction terms SCC-DFTB CNDO DFT DFT-TB MNDO-type NDDO - Exp empirical
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SCC-DFTB vs. MNDO-type methods Practical issues

Parametrization against Reference data in parametrization Number of parameters Computational scaling Estimated relative cpu time Analytic gradient Analytic Hessian SCC-DFTB DFT Energies 10 per atom pair N3 1.5 MNDO-type Exp Many 5-18 per atom N3 1.0
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SCC-DFTB validation by the Elstner group

• Reaction energies for 28 reactions involving
  22 small molecules (CHNO) Mean absolute
  deviation of 4.3 kcal/mol relative to G2
  reference data, compared with BLYP
  deviations of 5.1 (3.6) kcal/mol for
  cc-pVDZ(cc-pVTZ) basis.
• Harmonic frequencies for 196 normal modes of
  these 22 molecules (CHNO) Mean absolute
  deviation of 75 cm-1 from B3LYP/cc-pVTZ
  reference data.
• Bond lengths and bond angles of these
  molecules Excellent agreement with
  MP2/6-31G and BLYP/cc-pVTZ reference data, with
  mean absolute deviations of 0.017 (0.016) Å
  and 1.6 (1.4) vs. MP2 (BLYP).
• Occasional failures reported (H2O2 planar, CO
  energetics, N2H4 frequencies).

T. Krüger, M. Elstner, P. Schiffels, and T.
Frauenheim, J. Chem. Phys. 122, 114110 (2005).
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Heats of formation General considerations

• MNDO-type methods Evaluation from computed
  atomization energies and experimental heats
  of formations of the atoms.
• Implies that zero-point vibrational and
  thermal corrections are incorporated into
  through the parametrization.
• Not done in the SCC-DFTB parametrization.
• First option in SCC-DFTB Include
  zero-point vibrational and thermal corrections
  explicitly.
• Second option in SCC-DFTB Add empirical
  atomic increments for converting the computed
  total energies into heats of formations, or
  equivalently, treat as an adjustable
  parameter rather than computing it.

M. R. Ibrahim and P. v. R. Schleyer, J. Comput.
Chem. 6, 157 (1985). W. Thiel, Tetrahedron 44,
7393 (1988).
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SCC-DFTB heats of formation Explicit calculation

• Standard CHNO test set with 140 mostly organic
  molecules Strong overbinding in 139 cases
  (exception H2), mean absolute deviation of
  54.5 kcal/mol from experiment.
• Errors (kcal/mol) for selected molecules
  H2 28.0, methane -14.1, ethane -27.7, ethylene
  -18.8, acetylene -17.4, n-hexane -75.1,
  benzene -56.7, N2 -32.4, ammonia -18.2, HCN
  -34.2, water -13.3, dimethylether 38.1,
  CO -31.8, CO2 -37.5, formaldehyde -25.7,
  acetic acid -40.9, nitric acid -130.8.
• Errors increase with molecular size.
• Errors particularly large for triple bonds
  (N2, HCN, CO) and NO bonds.
• Typical overbinding per bond (kcal/mol) CH
  3-4, NH and OH 6-7, CC and CC 4-7, CC
  ca. 10, CN ca. 25.
• Direct calculation with explicit inclusion of
  zero-point vibrational and thermal
  corrrections leads to inaccurate heats of
  formation in SCC-DFTB.
• Potential problems with dissociation reactions.

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SCC-DFTB heats of formation Increment approach

• Applied by the Jorgensen group.
• Electronic energies of the atoms
  optimized by fitting against the
  experimental heats of formation of the PDDG/PM3
  training set with 134 reference molecules.
• Results (in eV)
• Use of the optimized values removes
  systematic errors from the SCC- DFTB
  atomization energies.
• Reaction energies are not affected.
• All subsequent results for SCC-DFTB heats of
  formation are based on these fitted
  values.

Element Optimized Computed H -7.7196 -6.4923 C -39.7799 -38.0530 N -59.9909 -56.1120 O -85.9785 -83.9753
K. W. Sattelmeyer, J. Tirado-Rives, and W. L.
Jorgensen, J. Phys. Chem. A 110, 13551 (2006).
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SCC-DFTB validation by the Jorgensen group
Energetics
Heats of formation - neutral hydrocarbons - neutral CHNO Molecules - ions and radicals Conformational energies Isomerization energies Hydrogen bond energies N 254 622 30 15 34 12 AM1 5.6 6.8 7.0 1.4 6.6 3.2 PM3 3.6 4.4 9.8 1.8 4.3 4.5 PDDG/PM3 2.6 3.2 10.0 1.8 2.4 4.1 SCC-DFTB 4.8 5.8 13.9 1.2 5.0 1.9 B3LYP/6-31G(d) (3.4) 0.4 3.1

• All values in kcal/mol, N comparisons.
• Experimental reference data, except for H-bond
  energies from CCSD(T).
• SCC-DFTB problems Molecules with NO bonds,
  three-membered rings, some small molecules
  (H2, H2CCH2).

K. W. Sattelmeyer, J. Tirado-Rives, and W. L.
Jorgensen, J. Phys. Chem. A 110, 13551 (2006).
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SCC-DFTB validation by the Jorgensen group Other
properties
Bond lengths (Å) Bond angles Dihedral angles (deg) Dipole moments (D) N 218 126 30 47 AM1 0.017 1.5 2.8 0.23 PM3 0.012 1.7 3.2 0.25 PDDG/PM3 0.013 1.9 3.7 0.23 SCC-DFTB 0.012 1.0 2.9 0.39

• N comparisons for neutral CHNO molecules
• Reference geometries from MP2/cc-pVTZ
• Reference dipole moments from gas-phase
  experiments

K. W. Sattelmeyer, J. Tirado-Rives, and W. L.
Jorgensen, J. Phys. Chem. A 110, 13551 (2006).
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Own validation Standard CHNO molecules
Mean absolute errors ( N comparisons) for a
standard validation set of mostly organic
compounds (C, H, N, O).  
Mean absolute errors ( N comparisons) for a
standard validation set of mostly organic
compounds (C, H, N, O).
Propertya N MNDO AM1 PM3 OM1 OM2 OM3 DFTB
?Hf (kcal/mol) 140 6.3 5.5 4.2 3.5 3.1 2.9 7.7
R (pm) 242 1.4 1.7 1.1 1.2 1.6 2.0 1.5
? (degree) 101 2.6 1.9 2.1 1.8 2.2 1.8 1.3
IP (eV) 52 0.46 0.35 0.42 0.32 0.26 0.45 3.82
µ (D) 53 0.35 0.26 0.27 0.25 0.28 0.27 0.37
? (cm-1) 112 241 172 151 189 155 120 90

1. Heats of formation ?Hf, bond lengths R, bond
   angles ?, vertical ionization potentials IP,
   dipole moments µ, vibrational wavenumbers ?.

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Own validation Heats of formation

• Mean absolute errors (kcal/mol) for N comparisons

Neutral CHNO molecules - Hydrocarbons - CHN compounds - CHO compounds - XNO compounds Anions Cations Radicals N 140 57 32 39 8 24 33 42 MNDO 6.3 5.9 6.2 4.8 16.3 14.4 11.5 11.9 AM1 5.5 4.9 4.6 5.5 11.4 11.3 9.8 10.6 OM2 3.1 1.7 3.9 4.5 2.9 8.4 7.2 5.0 DFTB 7.7 6.3 6.1 2.7 43.9 12.7 14.5 17.0
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Own validation G2 and G3 sets
Mean absolute errors (N comparisons) for heats of
formation (kcal/mol) Reference data from G2 and
G3 studies a,b
Compounds N G3 B3LYP MNDO AM1 PM3 OM1 OM2 OM3 DFTBe
G2 CHNO 81 0.69 2.35 7.72 7.37 6.77 4.39 3.36 3.82 9.19
G3 CHNO 47 0.94 7.12 7.13 6.27 4.43 4.36 3.15 3.62 4.50
G2 IPs 32 1.13 5.15 12.55 12.22 11.93 10.57 7.13 6.91 9.70
G2 EAs 29 0.97 3.57 15.44 11.80 9.22 13.70 8.60 8.39 12.30
Alkanes C1-C16 16 0.49c 15.44d 1.81 10.94d 2.24 1.54 2.03 0.44 5.99d

• a) L. A. Curtiss, K. Raghavachari, P. C.
  Redfern, and J. A. Pople, J. Chem. Phys. 112,
  7374 (2000).
• b) P. C. Redfern, P. Zapol, L. A. Curtiss,
  and K. Raghavachari, J. Phys. Chem. A 104, 5850
  (2000).
• c) G3 data only up to C8H18.
• Error increases with molecular size, e.g., up to
  30 kcal/mol for C16H34 in B3LYP.
• N78, 26, 22 in rows 1, 3, 4 (triplets excluded).

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Own validation Pericyclic reactions

• Reference data for barriers taken from
  experiment and published calculations at the
  CCSD(T), MP4, MP2, and B3LYP level.
• Reference data for transition structure mostly
  from B3LYP/6-31G, partly also from MP2 and
  CCSD(T) calculations.
• Reaction studied Diels-Alder reaction,
  electrocyclic ring opening, Cope and Claisen
  rearrangement, dipolar cycloaddition, ene
  reaction.
• Mean absolute deviations (N comparisons)

Barriers (kcal/mol) X Y bond lengths (Å) N 15 24 AM1 4.7 0.20 OM2 4.3 0.22 DFTB 10.4 0.11
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Own validation Peptides

• Reference data taken from RHF, B3LYP, and MP2
  calculations Geometries generally from
  RHF/6-31G or RHF/6-31G, relative energies
  generally from MP2/6-31G//RHF or
  LMP2/cc-pVTZ(-f)//RHF and sometimes from
  B3LYP/6-31G
• Reference systems for geometries
  N-methylacetamide complexes (3),
  Ac-Ala-NHMe dipeptides (7), Ac-(Gly)2-NHMe turns
  (4), Ac-(Gly)3-NHMe turns (5),
  Ac-(Ala)3-NHMe tetrapeptides (10), Ac-(Ala)n-NHMe
  (n2-6) helix and C7eq conformers (10).
• Reference systems for relative energies All
  except first and last group above.
• Mean absolute deviations (N comparisons)

Relative energies (kcal/mol) Backbone H-bond lengths (Å) Backbone dihedral angles (deg) N 22 67 190 AM1 2.0 0.22 17.0 OM2 1.7 0.34 12.0 DFTB 1.1 0.26 9.0
K. Möhle, H. J. Hofmann, and W. Thiel, J. Comput.
Chem. 22, 509 (2001).
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Own validation Hydrogen bond energies

• Reference geometries from B3LYP/aug-cc-pVTZ
  optimizations.
• Reference energies from single-point
  counterpoise-corrected MP2//B3LYP energies
  using the aug-cc-pVDZ and aug-cc-pVTZ basis sets
  and subsequent complete basis set
  extrapolation ??E (MP2/CBS).
• Higher-order correlation effects estimated
  from the difference between single- point
  CCSD(T) and MP2 calculations with the aug-cc-pVDZ
  basis ?Ecorr.
• Reference binding energy ?E0 ?E(MP2/CBS)
  ?Ecorr.
• Reference systems All 57 CHNO complexes from
  the MMFF94 data base, see T. A. Halgren, J.
  Comput. Chem. 17, 520 (1996).
• Mean absolute deviations of computed H-bond
  energies (kcal/mol)

Geometry Optimized N 57 AM1 2.8 OM2 1.5 DFTB 2.7
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Own validation Hydrogen bond geometries

• Reference geometries from B3LYP/aug-cc-pVTZ
  optimizations.
• Reference systems All 57 CHNO complexes from
  the MMFF94 data base, see T. A. Halgren, J.
  Comput. Chem. 17, 520 (1996).
• Mean deviations (N comparisons) N
  AM1 OM2 DFTB X H Y bond lengths
  (Å) 148 0.12 -0.14 -0.03 X H
  Y bond angles (deg) 74 -32.1 -10.3 0.1
  
• Mean absolute deviations (N comparisons)
  N AM1 OM2 DFTB X H Y bond
  lengths (Å) 148 0.25 0.20 0.08 X
  H Y bond angles (deg) 74 33.7
  12.1 6.2

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Treatment of excited states in large molecules

• Ref. 1 Excited state surfaces within TDDFT
  response theory
• Standard TDDFT and TD-DFTB share similar
  limitations in applicability and accuracy
  (for conjugated organic molecules).
• TDDFT with GGA/hybrid functionals should be
  applied to photochemical problems with
  great care.
• Issues Long-range charge transfer or
  polarization, multiconfigurational ground
  state.
• Ref. 2 Calculating absorption shifts for retinal
  proteins
• Comparison of different methods
• Recommendation Use SCC-DFTB for ground-state
  optimization or MD and OM2-GUGACI or ab
  initio SORCI for excitation energies.
• Ref. 3 Color tuning in rhodopsins
• Successful application of this approach to
  analyze spectral shifts between rhodopsins.

1 M. Wanko, M. Elstner et al, J. Chem. Phys.
120, 1674 (2004). 2 M. Wanko, W. Thiel, F.
Neese, M. Elstner et al, J. Phys. Chem. B 109,
3606 (2005).3 M. Hofmann, K. Schulten, W.
Thiel, M. Elstner et al, J. Am. Chem. Soc. 128,
10818 (2006).
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SCC-DFTB validation Assessment

• Viable alternative to established
  semiempirical methods
• Comparable overall accuracy
• Accuracy ranking dependent on systems and
  properties considered
• Present evidence suggest overall AM1 lt
  SCC-DFTB lt OM2, PDDG/PM3
• SCC-DFTB excellent for geometries
• SCC-DFTB performs well for biological systems
• SCC-DFTB may show large errors in unusual
  systems
• SCC-DFTB less suitable for excited states
• More elements need to be parametrized

N. Otte, M. Scholten, and W. Thiel, J. Phys.
Chem. A 111, 5751 (2007).
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Acknowledgements

• Marco Bocola Marcus Elstner
• Axel Koslowski Bill Jorgensen
• Nikolaj Otte Paul Strodel