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### Nonadiabatic transitions through tilted avoided crossings

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Betz, Volker and Goddard, Benjamin D..
(2011)
*Nonadiabatic transitions through tilted avoided crossings.*
SIAM Journal on Scientific Computing, Vol.33
(No.5).
pp. 2247-2276.
ISSN 1064-8275

**Full text not available from this repository.**

Official URL: http://dx.doi.org/10.1137/100802347

## Abstract

We investigate the transition of a quantum wave-packet through a one-dimensional avoided crossing of molecular energy levels when the energy levels at the crossing point are tilted. Using superadiabatic representations, and an approximation of the dynamics near the crossing region, we obtain an explicit formula for the transition wave function. Our results agree extremely well with high precision ab-initio calculations.

Item Type: | Journal Article |
---|---|

Subjects: | Q Science > QA Mathematics Q Science > QC Physics |

Divisions: | Faculty of Science > Mathematics |

Library of Congress Subject Headings (LCSH): | Born-Oppenheimer approximation, Molecular dynamics, Phase transformations (Statistical physics), Quantum theory |

Journal or Publication Title: | SIAM Journal on Scientific Computing |

Publisher: | Society for Industrial and Applied Mathematics |

ISSN: | 1064-8275 |

Date: | 13 September 2011 |

Volume: | Vol.33 |

Number: | No.5 |

Page Range: | pp. 2247-2276 |

Identification Number: | 10.1137/100802347 |

Status: | Peer Reviewed |

Publication Status: | Published |

Access rights to Published version: | Restricted or Subscription Access |

References: | [1] M. V. Berry and R. Lim, Universal transition prefactors derived by superadiabatic renormalization, J. Phys. A, 26 (1993), pp. 4737–4747. [2] V. Betz and B. D. Goddard, Accurate prediction of nonadiabatic transitions through avoided crossings, Phys. Rev. Lett., 103 (2009), 213001. [3] V. Betz, B. D. Goddard, and S. Teufel, Superadiabatic transitions in quantum molecular dynamics, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), pp. 3553–3580. [4] V. Betz and S. Teufel, Precise coupling terms in adiabatic quantum evolution, Ann. Henri Poincar´e, 6 (2005), pp. 217–246. [5] V. Betz and S. Teufel, Precise coupling terms in adiabatic quantum evolution: The generic case, Commun. Math. Phys., 260 (2005), pp. 481–509. [6] C. Fermanian Kammerer and C. Lasser, Propagation through generic level crossings: A surface hopping semigroup, SIAM J. Math. Anal., 40 (2008), pp. 103–133. [7] G. A. Hagedorn, Semiclassical quantum mechanics. III. The large order asymptotics and more general states, Ann. Physics, 135 (1981), pp. 58–70. [8] G. A. Hagedorn, Molecular propagation through electron energy level crossings, Mem. Amer. Math. Soc., 111 (1994), pp. vi+130. [9] G. A. Hagedorn and A. Joye, Landau-Zener transitions through small electronic eigenvalue gaps in the Born-Oppenheimer approximation, Ann. Inst. H. Poincar´e-Phys. Theor., 68 (1998), pp. 85–134. [10] G. A. Hagedorn and A. Joye, Molecular propagation through small avoided crossings of electron energy levels, Rev. Math. Phys., 11 (1999), pp. 41–101. [11] G. A. Hagedorn and A. Joye, Time development of exponentially small non-adiabatic transitions, Comm. Math. Phys., 250 (2004), pp. 393–413. [12] G. A. Hagedorn and A. Joye, A time-dependent Born-Oppenheimer approximation with exponentially small error estimates, Comm. Math. Phys., 223 (2001), pp. 583–626. [13] G. A. Hagedorn and A. Joye, Determination of non-adiabatic scattering wave functions in a Born-Oppenheimer model, Ann. Henri Poincar´e, 6 (2005), pp. 937–990. [14] C. Lasser and T. Swart, Single switch surface hopping for a model of pyrazine, J. Chem. Phys., 129 (2008), 034302. [15] C. Lasser, T. Swart, and S. Teufel, Construction and validation of a rigorous surface hopping algorithm for conical crossings, Commun. Math. Sci., 5 (2007), pp. 789–814. [16] C. Lubich, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, Zur. Lect. Adv. Math., European Mathematical Society, Z¨urich, 2008. [17] A. Martinez and V. Sordoni, A general reduction scheme for the time-dependent Born- Oppenheimer approximation, C. R. Math. Acad. Sci. Paris, 334 (2002), pp. 185–188. [18] H. Nakamura, Nonadiabatic Transition, World Scientific Publishing, Singapore, 2002. [19] G. Nenciu and V. Sordoni, Semiclassical limit for multistate Klein-Gordon systems: Almost invariant subspaces, and scattering theory, J. Math. Phys., 45 (2004), pp. 3676–3696. [20] T. S. Rose, M. J. Rosker, and A. H. Zewail, Femtosecond real-time probing of reactions. IV. The reactions of alkali halides, J. Chem. Phys., 91 (1989), pp. 7415–7436. [21] V. Rousse, Landau-Zener transitions for eigenvalue avoided crossings in the adiabatic and Born-Oppenheimer approximations, Asymptot. Anal., 37 (2004), pp. 293–328. [22] S.-I. Sawada, R. Heather, B. Jackson, and H. Metiu, A strategy for time dependent quantum mechanical calculations using a Gaussian wave packet representation of the wave function, J. Chem. Phys., 83 (1985), pp. 3009–3027. [23] S. Teufel, Adiabatic Perturbation Theory in Quantum Dynamics, Lecture Notes in Math. 1821, Springer-Verlag, Berlin, 2003. [24] J. C. Tully, Molecular dynamics with electronic transitions, J. Chem. Phys., 93 (1990), pp. 1061–1071. [25] A. I. Voronin, J. M. C. Marques, and A. J. C. Varandas, Trajectory surface hopping study of the li + li2(x1σ+ g ) dissociation reaction, J. Phys. Chem. A, 102 (1998), pp. 6057–6062. [26] J. Von Neumann and E. Wigner, ¨Uber das Verhalten von Eigenwerten bei Adiabatischen Prozessen, Phys. Z, 30 (1929), p. 467. [27] Q. Wang, R. W. Schoenlein, L. A. Peteanu, R. A. Mathies, and C. V. Shank, Vibrationally coherent photochemistry in the femtosecond primary event of vision, Science, 266 (1994), pp. 422–424. [28] D. Zener, Non-adiabatic crossings of energy levels, Proc. Roy. Soc. London, 137 (1932), pp. 696–702. |

URI: | http://wrap.warwick.ac.uk/id/eprint/39866 |

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