Hong Zhang, Sean Smith
1
Quantum Mechanical Unimolecular Dissociation Rates: Computing Resonances for the Full Thermal Range of Angular Momenta Hong Zhang, Sean Smith Hong Zhang, Sean Smith Centre for Computational Molecular Science, University of Queensland, Brisbane QLD 4072 Australia 1 Introduction •Exact calculations of quantum resonances are essential for a complete understanding of unimolecular reaction dynamics, as well as for benchmarking and further developing unimolecular reaction rate theory. 1 For example, in unimolecular dissociation, to understand the temperature variation of rate constants, it is important to implement many large angular momentum (J > 0) calculations as accurately as possible. However, these J > 0 calculations are still very challenging even for triatomic reactions, especially when dealing with long- lived complex forming systems. The major reason for this situation is the so called ‘angular momentum catastrophe’: 2 many J > 0 calculations have to be performed, and the size of the Hamiltonian matrix increases linearly with J. For these non-zero J calculations, it is apparently impractical to employ conventional direct diagonalisation methods due to the requirement of a significant computer core memory. 2 Methodology 2.1 Representation: In general, we treat the three internal Jacobi coordinates (R, r, ) in discrete variable representation (DVR), while the three Eulerian angles (, , ) are described in a basis set. 3 Results We have d eveloped several Lanczos eveloped several Lanczos representation methods and designed a parallel representation methods and designed a parallel computing model in both Lanczos and Chebyshev computing model in both Lanczos and Chebyshev iterative approaches. iterative approaches. Combination of both efficient methodologies Combination of both efficient methodologies and parallel computing strategy has made and parallel computing strategy has made rigorous quantum mechanical calculations rigorous quantum mechanical calculations possible for challenging full thermal range of possible for challenging full thermal range of total angular momenta in HO total angular momenta in HO 2 . . For most of the energies, the reported For most of the energies, the reported statistical adiabatic channel method / classical statistical adiabatic channel method / classical trajectory rate constants agree well with the trajectory rate constants agree well with the average of the fluctuating quantum mechanical average of the fluctuating quantum mechanical rates. rates. Fig. 2. (a) Plot of the quantum logarithmic rates versus resonance energies for J = 10 from RCFD method. Thin dotted line - QM results; red line - Troe et al. SACM/CT calculations; green line – quantum average. (b) comparison between RCFD method and LHFD method (see green dotted line). HO HO 2 2 : : J J = 0-50 resonance energies and widths using = 0-50 resonance energies and widths using both Lanczos and Chebyshev parallel computing both Lanczos and Chebyshev parallel computing methods. methods. Comparisons between quantum mechanical rate Comparisons between quantum mechanical rate constants and statistical theories /classical constants and statistical theories /classical trajectory results. trajectory results. 9 •In recent years, we have developed several quantum mechanical methods based upon Lanczos and Chebyshev iterations to solve this large-scale eigenvalue problem. 3-8 •For larger angular momenta the computational tasks are still too heavy using a conventional single- processor algorithm. Hence, we adopt a parallel computing strategy in our methods, which have been applied to compute resonance states in the HO 2 system for angular momentum as high as J = 50. 1.Smith & Gilbert, Theory of Unimolecular and Recombination Reactions (Blackwell Scientific Publisher, Oxford, 1990). 2.D. G. Truhlar, Comput Phys Commun, 84: 78 (1994). 3.Zhang & Smith, Phys Chem Chem Phys, 6: 884, 2004 (invited review). 4. Zhang & Smith, J Chem Phys, 123: 014308, 2005. 5. Zhang & Smith, J Chem Phys 120: 9583, 2004. 6. Zhang & Smith, J Chem Phys 118: 10042, 2003. 7. Zhang & Smith, J Chem Phys 115: 5751, 2001. 8. Zhang & Smith, Phys Chem Chem Phys 3: 2282, 2001. 9. Harding, Maergoiz, Troe & Ushakov, J Chem.Phys, 113 : 11019, 2000. Fig. 1. Plot of the quantum logarithmic rates versus resonance energies for J = 0 from (a) LHFD method and (b) RCFD method. Thin dotted line - QM results; red line - Troe et al. SACM/CT calculations; green line – quantum average. Fig. 3. Plot of the quantum logarithmic rates versus resonance energies from RCFD method. (a) for J = 20; (b) for J = 30; (c) for J = 40; and (d) for J = 50. 4 Conclusions 5 References 2.2 Propagation: In both iterative Lanczos and Chebyshev methods, the basic propagation is a three term recursion. 2.3 Final analysis: 1 1 1 , 1 1 1 , 2 2 2 2 2 2 2 2 2 2 2 2 ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 2 1 2 1 2 1 2 1 , , 1 2 1 2 j Jj j j j Jj j j j j j T t T T t T J J R T j j T r R r R V r r r R R R H Parallel computing: in propagation, the most time consuming part is the matrix-vector multiplication. We use Message-passing interface (MPI) to perform parallel computation. 2 1 0 2 1 0 22 21 12 11 10 01 00 0 0 0 0 0 0 H H H H H H H Lanczos homogenous filter diagonalisation Lanczos homogenous filter diagonalisation (LHFD). (LHFD). Real Chebyshev filter diagonalisation Real Chebyshev filter diagonalisation (RCFD). (RCFD). 2.10 2.15 2.20 2 .25 2.30 2.35 2.40 2.4 5 2.50 6 7 8 9 10 11 12 13 14 (b ) RCFD log 10 k(s -1 ) E (eV ) 2.1 0 2.15 2.2 0 2 .25 2.3 0 2.35 2.40 2.45 2.50 6 7 8 9 10 11 12 13 14 (a ) L H FD log 10 k(s -1 ) E (eV ) 2.1 2.2 2.3 6 7 8 9 10 11 12 13 14 (b) log 10 k(s -1 ) E (eV ) 2.1 2.2 2.3 2.4 2.5 6 7 8 9 10 11 12 13 14 (a ) log 10 k(s -1 ) E (eV ) 2.1 2.2 2.3 2.4 2.5 2.6 6 7 8 9 10 11 12 13 14 (a ) J = 20 log 10 k(s -1 ) E (eV ) 2.1 2 .2 2.3 2.4 2 .5 2 .6 7 8 9 10 11 12 13 14 (b ) J = 30 log 10 k(s -1 ) E (eV ) 2.1 2.2 2.3 2.4 2.5 2.6 6 7 8 9 10 11 12 13 14 (c) J = 40 log 10 k(s -1 ) E (eV ) 2.1 2.2 2.3 2.4 2.5 2.6 6 7 8 9 10 11 12 13 14 (d ) J = 50 log 10 k(s -1 ) E (eV ) Scaling: d Scaling: d ue to the communications and loading ue to the communications and loading balance issues, the model doesn’t scale ideally balance issues, the model doesn’t scale ideally with ( with ( J J +1) for even spectroscopic symmetry or +1) for even spectroscopic symmetry or J J for odd spectroscopic symmetry. for odd spectroscopic symmetry. However, one can However, one can achieve wall clock times (e.g., for even symmetry achieve wall clock times (e.g., for even symmetry J J = 10 HO = 10 HO 2 2 case) that are within about a factor of case) that are within about a factor of 3 of 3 of J J = 0 calculations. For non parallel = 0 calculations. For non parallel computing, the wall clock times will computing, the wall clock times will approximately be a factor of 11 of approximately be a factor of 11 of J J = 0 = 0 calculations calculations processes processes
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Quantum Mechanical Unimolecular Dissociation Rates: Computing Resonances for the Full Thermal Range of Angular Momenta. Hong Zhang, Sean Smith. Centre for Computational Molecular Science, University of Queensland, Brisbane QLD 4072 Australia. 1 Introduction. - PowerPoint PPT Presentation
Transcript of Hong Zhang, Sean Smith
Quantum Mechanical Unimolecular Dissociation Rates: Computing Resonances for the Full Thermal Range of Angular Momenta
Hong Zhang, Sean SmithHong Zhang, Sean Smith
Centre for Computational Molecular Science, University of Queensland, Brisbane QLD 4072 Australia
1 Introduction
•Exact calculations of quantum resonances are essential for a complete understanding of unimolecular reaction dynamics, as well as for benchmarking and further developing unimolecular reaction rate theory.1 For example, in unimolecular dissociation, to understand the temperature variation of rate constants, it is important to implement many large angular momentum (J > 0) calculations as accurately as possible. However, these J > 0 calculations are still very challenging even for triatomic reactions, especially when dealing with long-lived complex forming systems. The major reason for this situation is the so called ‘angular momentum catastrophe’:2 many J > 0 calculations have to be performed, and the size of the Hamiltonian matrix increases linearly with J. For these non-zero J calculations, it is apparently impractical to employ conventional direct diagonalisation methods due to the requirement of a significant computer core memory.
2 Methodology
2.1 Representation: In general, we treat the three internal Jacobi coordinates (R, r, ) in discrete variable representation (DVR), while the three Eulerian angles (, , ) are described in a basis set.
3 Results
We have developed several Lanczos representation methods and eveloped several Lanczos representation methods and designed a parallel computing model in both Lanczos and Chebyshev designed a parallel computing model in both Lanczos and Chebyshev iterative approaches.iterative approaches. Combination of both efficient methodologies and parallel computing Combination of both efficient methodologies and parallel computing strategy has made strategy has made rigorous quantum mechanical calculations possible rigorous quantum mechanical calculations possible for challenging full thermal range of total angular momenta in HOfor challenging full thermal range of total angular momenta in HO22. . For most of the energies, the reported statistical adiabatic channel For most of the energies, the reported statistical adiabatic channel method / classical trajectory rate constants agree well with the average method / classical trajectory rate constants agree well with the average of the fluctuating quantum mechanical rates. of the fluctuating quantum mechanical rates.
Fig. 2. (a) Plot of the quantum logarithmic rates versus resonance energies for J = 10 from RCFD method. Thin dotted line - QM results; red line - Troe et al. SACM/CT calculations; green line – quantum average. (b) comparison between RCFD method and LHFD method (see green dotted line).
HOHO22: : JJ = 0-50 resonance energies and widths using both Lanczos and = 0-50 resonance energies and widths using both Lanczos and
Chebyshev parallel computingChebyshev parallel computing methods.methods. Comparisons between quantum mechanical rate constants and statistical Comparisons between quantum mechanical rate constants and statistical theories /classical trajectory results.theories /classical trajectory results.99
•In recent years, we have developed several quantum mechanical methods based upon Lanczos and Chebyshev iterations to solve this large-scale eigenvalue problem.3-8
•For larger angular momenta the computational tasks are still too heavy using a conventional single-processor algorithm. Hence, we adopt a parallel computing strategy in our methods, which have been applied to compute resonance states in the HO2 system for angular momentum as high as J = 50.
1. Smith & Gilbert, Theory of Unimolecular and Recombination Reactions (Blackwell Scientific Publisher, Oxford, 1990).
2. D. G. Truhlar, Comput Phys Commun, 84: 78 (1994). 3. Zhang & Smith, Phys Chem Chem Phys, 6: 884, 2004 (invited review). 4. Zhang & Smith, J Chem Phys, 123: 014308, 2005. 5. Zhang & Smith, J Chem Phys 120: 9583, 2004. 6. Zhang & Smith, J Chem Phys 118: 10042, 2003. 7. Zhang & Smith, J Chem Phys 115: 5751, 2001. 8. Zhang & Smith, Phys Chem Chem Phys 3: 2282, 2001. 9. Harding, Maergoiz, Troe & Ushakov, J Chem.Phys, 113 : 11019, 2000.
Fig. 1. Plot of the quantum logarithmic rates versus resonance energies for J = 0 from (a) LHFD method and (b) RCFD method. Thin dotted line - QM results; red line - Troe et al. SACM/CT calculations; green line – quantum average.
Fig. 3. Plot of the quantum logarithmic rates versus resonance energies from RCFD method. (a) for J = 20; (b) for J = 30; (c) for J = 40; and (d) for J = 50.
4 Conclusions
5 References
2.2 Propagation: In both iterative Lanczos and Chebyshev methods, the basic propagation is a three term recursion.
2.3 Final analysis:
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Scaling: dScaling: due to the communications and loading balance issues, the ue to the communications and loading balance issues, the model doesn’t scale ideally with (model doesn’t scale ideally with (JJ+1) for even spectroscopic symmetry +1) for even spectroscopic symmetry or or JJ for odd spectroscopic symmetry. for odd spectroscopic symmetry. However, one can achieve wall However, one can achieve wall clock times (e.g., for even symmetry clock times (e.g., for even symmetry JJ = 10 HO = 10 HO22 case) that are within case) that are within
about a factor of 3 of about a factor of 3 of JJ = 0 calculations. For non parallel computing, the = 0 calculations. For non parallel computing, the wall clock times will approximately be a factor of 11 of wall clock times will approximately be a factor of 11 of JJ = 0 = 0 calculationscalculations processesprocesses
