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Thomas Heine Fakultät Mathematik und Naturwissenschaften, Institut für Physikalische Chemie und Elektrochemie Simulation of processes on nano scales using the DFTB method

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

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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. Off-topic: DFTxTB: A quantum mechanical hybrid method Theor. Chem. Acc. 2005, 114, 68

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

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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.

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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

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

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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

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

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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

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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)

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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”

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Illustrative applications of the DFTB method as implemented in deMon 1.Optimisation of many (~500,000) isomers 2.Long-time MD trajectories (ns region) 3.Doing nasty things with nano-scale systems 4.Explore complicated potential energy surfaces

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Local minima of many isomers 36:1436:15 xTotalDistinctNon- radical TotalDistinctNon- radical C 36 has two isoenergetic isomers (36:14 and 36:15) C 36 H x, 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

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

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

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Sc 3 68 : The first fullerene with adjacent pentagons mass spectrum: Sc 3 68 graph theory: C 68 must have adjacent pentagons earlier calculations: adjacent pentagons energetically unfavoured assumption: stabilisation by endohedral Sc 3 N molecule Nature 408 (2000)

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13 C and 45 Sc NMR gives information on symmetry Graph theory: 11 isomers (point groups D 3 and S 6 ) out of 6332 are compatible with one 45 Sc and C signals Nature 408 (2000)

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Which Sc 3 68 isomer has been found? Nature 408 (2000) minimum number of pentagon adjacencies: 6140 and is 120 kJ/mol more stable than all other isomers. Added excess electrons (2, 4, 6) to simulate charge transfer increase the energy gap

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Generation of starting topologies for fullerenes Polyhedron with v vertices, f faces and e edges Math: Euler equation: v + f = e + 2 Chemistry: Fullerenes C n are trivalent: e = 3 n/2, v = n (e : number of bonds, v : number of atoms) Faces: f = n/2 + 2 p pentagons, h hexagons: (5p + 6h)/3 = n and p + h = n/2 + 2 p = 12, h = n/2-10 for all classical fullerenes P.W. Fowler and D.E. Manolopoulos, An Atlas of Fullerenes, Oxford University Press, Oxford, f e v

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This is not the strcuture of a fullerene! Nor is it a soccer ball. © P.W. Fowler Exeter, U.K. 100 m from P.W.Fowler’s home, there are no pentagons graphene sheet!

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Spiral algorithm for C hexagons, 12 pentagons One out of 1812 ways to form a C 60 fullerene isomer P.W. Fowler and D.E. Manolopoulos, An Atlas of Fullerenes, Oxford University Press, Oxford, Not all spirals form a closed cage!

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Which Sc 3 68 isomer has been found? Nature 408 (2000) minimum number of pentagon adjacencies: 6140 and is 120 kJ/mol more stable than all other isomers. Added excess electrons (2, 4, 6) to simulate charge transfer increase the energy gap

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Simple explanation using Hückel and MO theory aromatic (4N+2 rule) not aromatic (hole in system) antiaromatic (8 membered ring) Sc 3 68 : 3 adjacent pentagons connected to Sc ~2 electrons per adjacent pentagon isoelectronic with 10 membered ring (aromatic) 6e -

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Confirmation by 13 C NMR fingerprint Nature 408 (2000) J. Phys. Chem. A 2005, 109,

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13 C NMR in Sc 3 80 TMS [ppm] Magn. Res. Chem. 2004, 42,199

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IR spectrum of Sc 3 80 unpublished

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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

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

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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

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

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Applications of inorganic nanotubes

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Axial tension of WS 2 and MoS 2 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 WS 2 /MoS 2 particularly interesting regarding the axial tension

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Mechanical properties of MoS 2 nanotubes - experiment Breaking a WS 2 nanotube with an AFM, in-situ SEM Proc. Natl. Acad. Sci. USA 2006, 103, 523.

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Computer Simulations 1.One long-time MD simulation stretching the nanotube 2.After each equilibration after stretching a simulated annealing simulation 50 successive equilibrations (5 ps) 50 simulated annealing simulations (~20 ps) Hardware: P4, 2.8 GHz, 1 GB memory

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Mechanical properties of MoS 2 nanotubes - simulation Breaking a MoS 2 nanotube with an AFM Proc. Natl. Acad. Sci. USA 2006, 103, 523. Almost harmonic behaviour until rupture!

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Speeding up the exploration of reaction mechanisms Standard technique: 1.Get an idea of the transition state(s) (TS) 2.optimise each TS 3.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: 1.Get an idea of the PES with NEB/DFTB 2.Optimise TS with GGA-DFT 3.Compute IRC with GGA-DFT 4.Compute entropy corrections using GGA-DFT and harmonic approximation 5.Refine computations with higher level theory (MP2, CCSD(T), MR methods

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

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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 (~10 n, 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

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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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