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Simulation of processes on nano scales using the DFTB method

Simulation of processes on nano scales using the DFTB method. Off-topic: DFTxTB: A quantum mechanical hybrid method. Joint LCAO ansatz:. T. D. D. MO. AO or cGTO. N D : Number if DFT basis functions N T : Number of TB basis functions. AO. Theor. Chem. Acc. 2005, 114, 68.

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Simulation of processes on nano scales using the DFTB method

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  1. Simulation of processes on nano scales using the DFTB method

  2. Off-topic: DFTxTB: A quantum mechanical hybrid method • Joint LCAO ansatz: T D D MO AO or cGTO ND: Number if DFT basis functions NT: Number of TB basis functions AO Theor. Chem. Acc. 2005, 114, 68

  3. Off-topic: DFTxTB: A quantum mechanical hybrid method Kohn-Sham matrix: L(l) and K(k) mean that l and k run over the basis functions that belong to the L and K atomic centres. Theor. Chem. Acc. 2005, 114, 68

  4. DFTB implementation in deMon Experimental version of deMon http://www.demon-software.com parallelised using OpenMP (80% speedup), becomes sparse Calculate matrix elements For ca. 5000 basis functions 85% of CPU time, Order-3 LAPACK+BLAS (MKL, ACML, ATLAS…) Solve secular equations Calculate density and energy weighted density matrix BLAS (DSYRK) and Fortran90 intrinsics Calculate gradients parallelised using OpenMP (100% speedup)

  5. Calculation of matrix elements • All Overlap (S) and Kohn-Sham (F) integrals can be computed independently  simple massive parallelisation possible • If Slater-Koster tables are employed, we • can interpolate matrix elements quickly • know the interaction range of each pair of atoms and can screen efficiently • For interatomic distances of ~5 Å matrix elements start to vanish • sparse matrix algebra (sub Order-3) • linear scaling for memory usage • For the calculation of matrix elements there are no real limits for the applicability of the DFTB method.

  6. Representation of Slater-Koster tables Fitting to Chebycheff-polynomials by Porezag et al. (Phys. Rev. B 1995, 51, 12947) – idea abandoned due to numerical instabilities. In deMon: local fitting, analytical derivatives are in principle available

  7. DFTB implementation in deMon Experimental version of deMon http://www.demon-software.com parallelised using OpenMP (80% speedup), becomes sparse Calculate matrix elements For ca. 5000 basis functions 85% of CPU time, Order-3 LAPACK+BLAS (MKL, ACML, ATLAS…) Solve secular equations Calculate density and energy weighted density matrix BLAS (DSYRK) and Fortran90 intrinsics Calculate gradients parallelised using OpenMP (100% speedup)

  8. Solving the secular equations • This is the most time-consuming part of DFTB • Standard technique: Orthogonalisation of F (e.g. Cholesky decomposition) followed by diagonalisation • Popular algorithms: LAPACK 3 • Divide&Conquer (DQ) or Relatively Robust Representations (RRR) • claimed to be sub-Order-3 (sub Order-2 for RRR) • became much more stable in the past • no significant memory overhead required for RRR • give roughly a factor of 10 in performance compared to traditional diagonalisation methods • parallelisation possible (ScaLAPACK), but • message passing is significant  overall bad scalability • parallel versions are less stable than serial ones

  9. DFTB implementation in deMon Experimental version of deMon http://www.demon-software.com parallelised using OpenMP (80% speedup), becomes sparse Calculate matrix elements For ca. 5000 basis functions 85% of CPU time, Order-3 LAPACK+BLAS (MKL, ACML, ATLAS…) Solve secular equations Calculate density and energy weighted density matrix BLAS (DSYRK) and Fortran90 intrinsics Calculate gradients parallelised using OpenMP (100% speedup)

  10. Calculation of density matrix P, energy weighted density matrix W and gradients • Calculation of P and W involve essentially squaring a matrix: simple massive parallelisation possible • For the calculation of gradients, all arguments given before for the calculation of matrix elements apply: • fast calculation of derivatives • screening

  11. For large-scale simulations: Avoid diagonalisation! • Our approach: Car-Parrinello DFTB • Theory and standard implementation: M. Rapacioli, R. Barthel, T. Heine, G. Seifert, to be submitted to JCP • Parallelisation, sparsity, large scale behaviour, tricks of the trade: M. Rapacioli, T. Heine, G. Seifert, in preparation (JPCA special section DFTB)

  12. Car-Parrinello DFTB • Propagation of MO coefficients • S-1 is solved iteratively (conjugate gradient) • Only matrix-matrix operations are ^formally Order-3. These are computationally unproblematic (vectorisation and parallelisation) and become sparse “quickly”

  13. Illustrative applications of the DFTB method as implemented in deMon • Optimisation of many (~500,000) isomers • Long-time MD trajectories (ns region) • Doing nasty things with nano-scale systems • Explore complicated potential energy surfaces

  14. Local minima of many isomers • C36 has two isoenergetic isomers (36:14 and 36:15) • C36Hx, x=4,6, have been found in mass spectrometer. • But which isomer(s)? • Number of isomers to be calculated: J. Chem. Soc., Perkin Trans. 2, 2001, 487–490

  15. Which basis cage? dark: 36:14 based light: 36:15 based J. Chem. Soc., Perkin Trans. 2, 2001, 487–490

  16. Which are the stable isomers? side view top view side view top view (1,4) positions at equatorial hexagons! point group relative energy [kJ/mol] J. Chem. Soc., Perkin Trans. 2, 1999, 707–711

  17. Sc3N@C68: The first fullerene with adjacent pentagons • mass spectrum: Sc3N@C68 • graph theory: C68 must have adjacent pentagons • earlier calculations: adjacent pentagons energetically unfavoured • assumption: stabilisation by endohedral Sc3N molecule Nature 408 (2000) 427-428

  18. 13C and 45Sc NMR gives information on symmetry Graph theory: 11 isomers (point groups D3 and S6) out of 6332 are compatible with one 45Sc and 11+1 13C signals Nature 408 (2000) 427-428

  19. Which Sc3N@C68 isomer has been found? • minimum number of pentagon adjacencies: • 6140 and 6275. • 6140 is 120 kJ/mol more stable than all other isomers. • Added excess electrons (2, 4, 6) to simulate charge transfer increase the energy gap Nature 408 (2000) 427-428

  20. Simple explanation using Hückel and MO theory aromatic (4N+2 rule) 6e- not aromatic (hole in p system) antiaromatic (8 membered ring) • Sc3N@C68: 3 adjacent pentagons connected to Sc • ~2 electrons per adjacent pentagon  • isoelectronic with 10 membered ring (aromatic)

  21. Confirmation by 13C NMR fingerprint Nature 408 (2000) 427-428 J. Phys. Chem. A 2005, 109, 7068-7072

  22. 13C NMR in Sc3N@C80 dTMS [ppm] Magn. Res. Chem. 2004, 42,199

  23. IR spectrum of Sc3N@C80 unpublished

  24. Electromechanical properties of single-walled carbon nanotubes Rupture of CNT’s at different temperatures: DFTB-based Born-Oppenheimer MD with successive iterations of pulling the tubes until rupture Small 2005, 1, 399

  25. Elastic properties of SWCNT’s • Independent on temperature • Rupture at DL/L≈0.15 • Hooke-like behaviour up to DL/L≈0.1 300K: full circles 600K: squares 1000K: empty circles Small 2005, 1, 399

  26. Mechanical properties of inorganic nanotubes Golden Gate bridge, San Francisco, steel cables Golden Gate bridge, San Francisco, after reconstruction with nanotubes Thanks to Sibylle Gemming

  27. Electromechanical properties of CNTs Electronic transmission probability T(E) depends strongly on DL/L! Small 2005, 1, 399

  28. Axial tension of WS2 and MoS2 nanotubes • In standard materials: mechanical properties are affected, if not even determined, by defects • Nanotubes: almost defect free  mechanical properties of almost ideal structure can be studied, and superior mechanical properties can be achieved • Special structure of WS2/MoS2 particularly interesting regarding the axial tension

  29. Mechanical properties of MoS2 nanotubes - experiment Breaking a WS2 nanotube with an AFM, in-situ SEM Proc. Natl. Acad. Sci. USA 2006, 103, 523.

  30. Mechanical properties of MoS2 nanotubes - simulation Breaking a MoS2 nanotube with an AFM Almost harmonic behaviour until rupture! Proc. Natl. Acad. Sci. USA 2006, 103, 523.

  31. Speeding up the exploration of reaction mechanisms • Standard technique: • Get an idea of the transition state(s) (TS) • optimise each TS • Compute internal reaction coordinates • If no TS structure can be guessed, or if generality is required: • Scan potential energy surface • Nudged Elastic Band (NEB) method • Both are computationally very expensive • Our approach: • Get an idea of the PES with NEB/DFTB • Optimise TS with GGA-DFT • Compute IRC with GGA-DFT • Compute entropy corrections using GGA-DFT and harmonic approximation • Refine computations with higher level theory (MP2, CCSD(T), MR methods

  32. Ring formation in interstellar space NEB calculations (DFTB and DFT, deMon) IRC calculations, theory refinement, entropy corrections are still to be done Robert Barthel, TU Dresden, to be published

  33. Conclusions • DFTB is a very fast QM method, and problems to go to large-scale systems can be overcome relatively easily • DFTB is a very robust method and hence allows to • study many (~10n, n>5) systems in an automatised way • study rough processes, involving bond breaking and bond formation • study very long MD trajectories using the NVE ensemble with a numerical accuracy (energy conservation) comparable to MM methods • study finite (cluster, molecules) and infinite (solids, liquids, surfaces…) systems employing one method with identical approximations • predict stable subsystems without solving the complete problem • The accuracy of DFTB can be improved by SCC, but for the sake of losing the robustness of the method

  34. Acknowledgements • Theoretical Chemistry group at TU Dresden • Mathias Rapacioli • Knut Vietze • Robert Barthel • Viktoria Ivanovskaya • Helio A. Duarte • Gotthard Seifert • ZIH Dresden for computational facilities • Alexander v. Humboldt foundation • Gesellschaft Deutscher Chemiker • Deutsche Forschungsgemeinschaft • J. McKelvey, M. Elstner, T. Frauenheim for invitation

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