The Library

Superadiabatic transitions in quantum molecular dynamics

Tools

Betz, Volker, Goddard, B. (Benjamin) and Teufel, Stefan, 1970-. (2009) Superadiabatic transitions in quantum molecular dynamics. Proceedings of the Royal Society of London. A, Mathematical, Physical and Engineering Sciences, Vol.465 (No.2111). pp. 3553-3580. ISSN 1364-5021

Preview

PDF
WRAP_Betz_SUPERADIABATIC.pdf - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Download (392Kb)

Official URL: http://dx.doi.org/10.1098/rspa.2009.0337

Abstract

We study the dynamics of a molecule’s nuclear wave function near an avoided crossing of two electronic energy levels for one nuclear degree of freedom. We derive the general form of the Schrödinger equation in the nth superadiabatic representation for all n є N. Using these results, we obtain closed formulas for the time development of the component of the wave function in an initially unoccupied energy subspace when a wave packet travels through the transition region. In the optimal superadiabatic representation, which we define, this component builds up monotonically. Finally, we give an explicit formula for the transition wave function away from the avoided crossing, which is in excellent agreement with high-precision numerical calculations.

Item Type:	Journal Article
Subjects:	Q Science > QC Physics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Quantum theory -- Mathematical models, Wave functions, Schrödinger equation, Eigenvalues
Journal or Publication Title:	Proceedings of the Royal Society of London. A, Mathematical, Physical and Engineering Sciences
Publisher:	The Royal Society Publishing
ISSN:	1364-5021
Date:	8 November 2009
Volume:	Vol.465
Number:	No.2111
Number of Pages:	28
Page Range:	pp. 3553-3580
Identification Number:	10.1098/rspa.2009.0337
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	Engineering and Physical Sciences Research Council (EPSRC)
Grant number:	EP/D07181X/1 (EPSRC)
References:	[1] M. Berry. Histories of adiabatic quantum transitions. P Roy Soc Lond A Mat, 429(1876):61–72, Jan 1990. [2] M. Berry and R. Lim. Universal transition prefactors derived by superadiabatic renormalization. J Phys A-Math Gen, 26(18):4737–4747, Jan 1993. [3] V. Betz and B. Goddard. Superadiabatic representations for Born-Openheimer transitions through an avoided Landau-Zener crossing. in preparation. [4] V. Betz and S. Teufel. Precise coupling terms in adiabatic quantum evolution. Ann. Henri Poincar´e, 6(2):217–246, 2005. [5] V. Betz and S. Teufel. Precise coupling terms in adiabatic quantum evolution: the generic case. Comm. Math. Phys., 260(2):481–509, 2005. [6] V. Betz and S. Teufel. Landau-Zener formulae from adiabatic transition histories. Lecture Notes in Physics 690, p. 19-32, Springer, 2006. [7] M. Born and R. Oppenheimer. Zur Quantentheorie der Molekeln. Ann. Phys. (Leipzig), 84:457–484, 1927. [8] L. S. Cederbaum, E. Gindensperger, and I. Burghardt. Short-time dynamics through conical intersections in macrosystems. Physical Review Letters, 94(11):113003, 2005. [9] Y. N. Demkov, V. N. Ostrovskii, and E. A. Solov’ev. Two-state approximation in the adiabatic and sudden-perturbation limits. Phys. Rev. A, 18(5):2089–2096, Nov 1978. [10] M. Dimassi and J. Sj¨ostrand, Spectral Asymptotics in the Semi-Classical Limit, Cambridge University Press, 1999. [11] G. A. Hagedorn. A time dependent Born-Oppenheimer approximation. Comm. Math. Phys., 77(1):1– 19, 1980. [12] G. A. Hagedorn. Molecular propagation through electron energy level crossings. Mem. Amer. Math. Soc., 111(536):vi+130, 1994. [13] G. A. Hagedorn and A. Joye. Landau-Zener Transitions Through Small Electronic Eigenvalue Gaps in the Born-Oppenheimer Approximation. Ann. Inst. H. Poincare Sect. A 68:85–134 (1998). [14] G. A. Hagedorn and A. Joye. A time-dependent Born-Oppenheimer approximation with exponentially small error estimates. Comm. Math. Phys., 223(3):583–626, 2001. [15] G. A. Hagedorn and A. Joye. Time development of exponentially small non-adiabatic transitions. Comm. Math. Phys., 250(2):393–413, 2004. [16] G. A. Hagedorn and A. Joye. Determination of non-adiabatic scattering wave functions in a bornoppenheimer model. Ann. Henri Poincar´e, 6(5):937–990, 2005. [17] G. A. Hagedorn and A. Joye. Recent results on non-adiabatic transitions in quantum mechanics. In N. Chernov, Y. Karpeshina, I. Knowles, R. Lewis, and R. Weikard, editors, Recent Advances in Differential Equations and Mathematical Physics., volume 412 of AMS Contemporary Mathematics Series, pages 183–198. to appear, 2006. [18] G. A. Hagedorn and A. Joye. Mathematical analysis of Born-Oppenheimer approximations. In Proceedings of the ’Spectral Theory and Mathematical Physics’ Conference in Honor of Barry Simon, AMS Proc. of Symposia in Pure Math. to appear, 2007. [19] A. Joye. Proof of the Landau-Zener Formula. Asympt. Anal. 9:209–258 (1994). [20] L. D. Landau. Collected Papers of L.D. Landau. Pergamon Press, Oxford, 1965. [21] C. Lasser and S. Teufel. Propagation through conical crossings: an asymptotic semigroup. Comm. Pure Appl. Math., 58(9):1188–1230, 2005. [22] R. Lim andM. Berry. Superadiabatic tracking of quantumevolution. J Phys A-Math Gen, 24(14):3255– 3264, Jan 1991. [23] F. London. ¨Uber den Mechanismus der Hom¨oopolaren Bindung. In P. Debye, editor, Probleme der Modernen Physik. Herzel, Leipzig, 1928. [24] A. Martinez. An Introduction to Semiclassical and Microlocal Analysis. Springer, 2002. [25] A. Martinez and V. Sordoni. A general reduction scheme for the time-dependent Born-Oppenheimer approximation. C. R. Math. Acad. Sci. Paris, 334(3):185–188, 2002. [26] A. Martinez and V. Sordoni. Twisted Pseudodi?erential Calculus and Application to the Quantum Evolution of Molecules. to appear in Memoirs of the AMS. [27] V. Rousse. Landau-Zener transitions for eigenvalue avoided crossings, Asympt. Anal. 37:293-328 (2004) [28] S. Teufel. Adiabatic perturbation theory in quantum dynamics, volume 1821 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2003. [29] D. Zener. Non-adiabatic crossings of energy levels. Proc. Roy. Soc. London, 137:696–702, 1932. [30] A. H. Zewail. Femtochemistry: Ultrafast Dynamics of the chemical bond. World Scientific, New York, 1994.
URI:	http://wrap.warwick.ac.uk/id/eprint/2462