Thomas Heine Email: thomas.heine@chemie.tu-dresden.de Fakultät Mathematik und Naturwissenschaften,...

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Thomas Heine

Email: thomas.heine@chemie.tu-dresden.de

Fakultät Mathematik und Naturwissenschaften, Institut für Physikalische Chemie und Elektrochemie

Simulation of processes on nano scales using the DFTB method

Off-topic: DFTxTB: A quantum mechanical hybrid method

Joint LCAO ansatz:

D N N N

i k1 N

MO AO or cGTO

ND: Number if DFT basis functionsNT: Number of TB basis functions

and use the same type of primitives

Theor. Chem. Acc. 2005, 114, 68

Kohn-Sham matrix:DT

TT T T TTkl k K(k) L(l) l kl

DD D DDkl k eff l k k l

DT TDkl k l lk

D DTk ef )f kl

1F T V V S

1F T V S

2T q F

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

For ca. 5000 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 http://www.demon-software.com

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.

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

Calculate gradients

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

Calculate gradients

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

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)

DFTBii

DFTBii i j j

d RM E

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”

( ) 2 ( ) ( 2 )

( ) 2 ( ) ( 2 ) ( )

|ij ij ij i j

t ER t R t t R t t

t EC t C t t C t t S XC t t

twith X S and S

Illustrative applications of the DFTB method as implemented in deMon

1. Optimisation of many (~500,000) isomers2. Long-time MD trajectories (ns region)3. Doing nasty things with nano-scale systems4. Explore complicated potential energy surfaces

Local minima of many isomers

36:14 36:15

x Total Distinct Non-radical

Total Distinct Non-radical

2 630 90 90 630 41 41

4 58 905 7 461 7 317 58 905 2 608 2 553

6 1 947 793 243 985 221 665 1 947 793 82 123 74 549

• 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

Which basis cage?

dark: 36:14 based

light: 36:15 based

Which are the stable isomers?

side view top view top viewside view

point group

relative energy [kJ/mol]

(1,4) positions atequatorial hexagons!

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

13C and 45Sc NMR gives information on symmetry

Graph theory: 11 isomers (point groups D3 and S6) outof 6332 are compatible withone 45Sc and 11+1 13C signalsNature 408 (2000) 427-428

Which Sc3N@C68 isomer has been found?

Nature 408 (2000) 427-428

•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

Simple explanation using Hückel and MO theory

aromatic (4N+2 rule)

not aromatic (hole in system)

antiaromatic (8 membered ring)

•Sc3N@C68: 3 adjacent pentagons connected to Sc•~2 electrons per adjacent pentagon •isoelectronic with 10 membered ring (aromatic)

-0.2276

-0.2067

-0.1888

-0.1703

-0.1213

-0.2279

-0.1051

-0.1084

-0.1376

Confirmation by 13C NMR fingerprint

Nature 408 (2000) 427-428

J. Phys. Chem. A 2005, 109, 7068-7072

13C NMR in Sc3N@C80

TMS [ppm]

Magn. Res. Chem. 2004, 42,199

IR spectrum of Sc3N@C80

unpublished

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

Elastic properties of SWCNT’s

zigzag armchair

•Independent on temperature•Rupture at L/L≈0.15•Hooke-like behaviour up to DL/L≈0.1

300K: full circles600K: squares1000K: empty circles

Small 2005, 1, 399

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

Electromechanical properties of CNTs

armchairzigzag

Electronic transmission probability T(E) depends strongly on L/L!

Small 2005, 1, 399

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

Mechanical properties of MoS2 nanotubes - experiment

Breaking a WS2 nanotube with an AFM, in-situ SEM

Proc. Natl. Acad. Sci. USA 2006, 103, 523.

Mechanical properties of MoS2 nanotubes - simulation

Breaking a MoS2 nanotube with an AFM

Proc. Natl. Acad. Sci. USA 2006, 103, 523.

Almost harmonic behaviour until rupture!

Speeding up the exploration of reaction mechanisms

• Standard technique: 1. Get an idea of the transition state(s) (TS)2. optimise each TS3. 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/DFTB2. Optimise TS with GGA-DFT3. Compute IRC with GGA-DFT4. Compute entropy corrections using GGA-DFT and

harmonic approximation5. Refine computations with higher level theory (MP2,

CCSD(T), MR methods

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

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

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

Thomas Heine Email: thomas.heine@chemie.tu-dresden.de Fakultät Mathematik und Naturwissenschaften,...

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