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 TheoreticalChemistry Technical Universityof Braunschweig

2. DFTB: non-self-consistent scheme Consider a case, where you know the DFT ground state density Galready (exactly or in good approximation in): Then the energy can given by (Foulkes& Haydock PRB 1989):

3. DFTB: non-self-consistent scheme DFTB: consider input density 0as superposition of neutral atomic densities LCAO basis: TB energy:

4. DFTB: non-self-consistent scheme • No charge transfer between atoms  very good results for homonuclear systems (Si, C), hydrocarbonsetc. • 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

5. DFTB: non-self-consistent scheme • Problems: • HCOOH: C=O and C-O bond lengths equalized • H2N-CH=O and peptides: N-C and C=O bond lengths equalized • non-CT systems: • CO2 vibrational frequencies • C=C=C=C=C.. chains, dimerization, end effects

6. 2pC 2pC 2pO 2pO 2pC 2pO 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

7. DFTB: non-self-consistent scheme 0  Try to keep H0 since it works well for many systems

8. DFT total energy

9. Second order expansion of DFT total energy Expand E[ ] at0, which is the reference density used to calculate the H0

10. Second order expansion of DFT total energy Write density fluctuations as a sum of atomic contributions (I) (II) (III)

11. (I) Hamiton matrix elements Introduce LCAO basis: (I)

12. Second order expansion of DFT total energy Write density fluctuations as a sum of atomic contributions (I) (II) (III)

13. (II) Repulsive energy contribution • pair potentials • exponentially decaying

14. Second order expansion of DFT total energy Write density fluctuations as a sum of atomic contributions (I) (II) (III)

15. (III) Second order term Monopolapproximation: Two limits: New parameter U: calculated for every element from DFT

16. 2pC 2pC 2pO 2pO 2pC 2pO (III) Second order term

17. Combine the two limits

18. (III) Second order term: Klopman-Ohno approximation  R

19. 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 • R=0: 1/ = 3.2 UHubbard

20. Klopman-Ohno vs DFTB Gamma 1/r DFTB-

21. Approximate density-functional theoryElstner et al. Phys. Rev. B 58 (1998) 7260

22. Hamilton-Matrixelements • non-scc: neglect of red contributions

23. Comparison to SE models: Matrix elements • Extended Hueckel (can be derived from DFT)

24. Comparison to SE models: Matrix elements • Fenske Hall

25. 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

26. Differences w.r. to SE models: e.g. J e.g. MNDO Approx. by multipole-multipole interaction Coulomb part J: e.g. CNDO

27. 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

28. Differences to SE models: e.g. J DFTB: how is e-e interaction treated? consider J

29. 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

30. Approximate density-functional theory

31. Extensions • FAQS: • better basis sets  much higher cost

32. Extensions • FAQS: • higher order expansion • monopole  multipole • inspection of gamma? • No additional cost!

33. Determination of Gamma deviation from 1/R for small R R=0: 1/ = 3.2 Uhubbard Is this valid throughout the periodic table? What is the relation between ‚atomic size‘ and chemical hardness?

34. Gamma: Rcov ~ 1/U ? Si-Cl R covalent B-F H U-Hubbard N

35. Gamma requires : 3.2*Rcov= 1/U? H

36. U vs Rcov: Hydrogen atom U-Hubbard O N H C Si R covalent

37. U vs Rcov: H not in line! U-Hubbard In DFTB, H is 0.73A instead of 0.33A! N H • Gamma requires: 3.2*Rcov= 1/U • size of H overestimatedbased on hardness value: H has same size like N!

38. 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!

39. Modified Gamma for H-bondingchange only X-H interaction!

40. 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

41. H-bonds: water clusterMP2 from KS Kim et al 2000

42. Expansion to higher order?

43. 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/6-311++G(2d2p): 397 kcal/mole SCC-DFTB: 424 kcal/mole

44. 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.

45. 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

46. 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)

47. 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

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

49. 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.

50. 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