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Transcript of Thomas Heine Email: [email protected] Fakultät Mathematik und Naturwissenschaften,...
Thomas Heine
Email: [email protected]
Fakultät Mathematik und Naturwissenschaften, Institut für Physikalische Chemie und Elektrochemie
Simulation of processes on nano scales using the DFTB method
Email: [email protected]
Off-topic: DFTxTB: A quantum mechanical hybrid method
Joint LCAO ansatz:
D T
D
D N N N
ki kk
i
N
i k1 N
kk 1
C C
MO AO or cGTO
AO
ND: Number if DFT basis functionsNT: Number of TB basis functions
TD
D
and use the same type of primitives
Theor. Chem. Acc. 2005, 114, 68
Email: [email protected]
Kohn-Sham matrix:DT
TD T
DD
T
F
FF
F
F
TT T T TTkl k K(k) L(l) l kl
TL(l
DD D DDkl k eff l k k l
T
DT TDkl k l lk
l
D DTk ef )f kl
D
T
k l
l
T
q
q
1F T V V S
2
V
1F T V S
2
1V SF
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
Email: [email protected]
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
Email: [email protected]
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.
Email: [email protected]
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
Email: [email protected]
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
Email: [email protected]
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
Email: [email protected]
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
Email: [email protected]
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
Email: [email protected]
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)
2
2
2
,2
DFTBii
DFTBii i j j
j
d RM E
dt
dF
dt
Email: [email protected]
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
21
*
21
( ) 2 ( ) ( 2 )
( ) 2 ( ) ( 2 ) ( )
|ij ij ij i j
t ER t R t t R t t
R
t EC t C t t C t t S XC t t
C
twith X S and S
Email: [email protected]
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
Email: [email protected]
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
Email: [email protected]
Which basis cage?
dark: 36:14 based
J. Chem. Soc., Perkin Trans. 2, 2001, 487–490
light: 36:15 based
Email: [email protected]
Which are the stable isomers?
side view top view top viewside view
point group
relative energy [kJ/mol]
(1,4) positions atequatorial hexagons!
J. Chem. Soc., Perkin Trans. 2, 1999, 707–711
Email: [email protected]
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
Email: [email protected]
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
Email: [email protected]
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
Email: [email protected]
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
6e-
Email: [email protected]
Confirmation by 13C NMR fingerprint
Nature 408 (2000) 427-428
J. Phys. Chem. A 2005, 109, 7068-7072
Email: [email protected]
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
Email: [email protected]
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
Email: [email protected]
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
Email: [email protected]
Electromechanical properties of CNTs
armchairzigzag
Electronic transmission probability T(E) depends strongly on L/L!
Small 2005, 1, 399
Email: [email protected]
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
Email: [email protected]
Mechanical properties of MoS2 nanotubes - experiment
Breaking a WS2 nanotube with an AFM, in-situ SEM
Proc. Natl. Acad. Sci. USA 2006, 103, 523.
Email: [email protected]
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!
Email: [email protected]
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
Email: [email protected]
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
Email: [email protected]
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
Email: [email protected]
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