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The time-dependent Born-Oppenheimer approximation

Published online by Cambridge University Press:  16 June 2007

Gianluca Panati ,
Herbert Spohn  and
Stefan Teufel
Gianluca Panati
Affiliation:
Zentrum Mathematik, TU München, Germany.
Herbert Spohn
Affiliation:
Zentrum Mathematik, TU München, Germany.
Stefan Teufel
Affiliation:
Mathematisches Institut, Universität Tübingen, Germany.
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Abstract

We explain why the conventional argument for deriving the time-dependent Born-Oppenheimer approximation is incomplete and review recent mathematical results, which clarify the situation and at the same time provide a systematic scheme for higher order corrections. We also present a new elementary derivation of the correct second-order time-dependent Born-Oppenheimer approximation and discuss as applicationsthe dynamics near a conical intersection of potential surfaces and reactive scattering.

Keywords

Schrödinger equationBorn-Oppenheimer approximationadiabatic methodsalmost-invariantsubspace.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 2: Special issue on Molecular Modelling , March 2007 , pp. 297 - 314
Copyright
© EDP Sciences, SMAI, 2007

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References

Adelman, D.E., Shafer, N.E., Kliner, D.A.V. and Zare, R.N., Measurement of relative state-to-state rate constants for the reaction ${\rm D+H}_2 (v,j)\rightarrow {\rm HD}(v,j)+{\rm H} $ . J. Chem. Phys. 97 (1992) 7323–7341. CrossRef
Berry, M.V. and Lim, R., The Born-Oppenheimer electric gauge force is repulsive near degeneracies. J. Phys. A 23 (1990) L655–L657. CrossRef
A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu and J. Zwanziger, The geometric phase in quantum systems. Texts and Monographs in Physics, Springer, Heidelberg (2003).
Born, M. and Oppenheimer, R., Zur Quantentheorie der Molekeln. Ann. Phys. (Leipzig) 84 (1927) 457–484. CrossRef
Brummelhuis, R. and Nourrigat, J., Scattering amplitude for Dirac operators. Comm. Partial Differential Equations 24 (1999) 377–394. CrossRef
Colin de, Y. Verdière, M. Lombardi and C. Pollet, The microlocal Landau-Zener formula. Ann. Inst. H. Poincaré Phys. Theor. 71 (1999) 95-127.
J.-M. Combes, P. Duclos and R. Seiler, The Born-Oppenheimer approximation, in Rigorous Atomic and Molecular Physics, G. Velo, A. Wightman Eds., New York, Plenum (1981) 185–212.
Emmerich, C. and Weinstein, A., Geometry of the transport equation in multicomponent WKB approximations. Commun. Math. Phys. 176 (1996) 701–711. CrossRef
Fermanian-Kammerer, C. and Gérard, P., Mesures semi-classiques et croisement de modes. Bull. Soc. Math. France 130 (2002) 123–168. CrossRef
Fermanian-Kammerer, C. and Lasser, C., Wigner measures and codimension 2 crossings. J. Math. Phys. 44 (2003) 507–527. CrossRef
Hagedorn, G.A., A time dependent Born-Oppenheimer approximation. Commun. Math. Phys. 77 (1980) 1–19. CrossRef
Hagedorn, G.A., High order corrections to the time-dependent Born-Oppenheimer approximation. I. Smooth potentials. Ann. Math. 124 (1986) 571–590. CrossRef
G.A. Hagedorn, High order corrections to the time-independent Born-Oppenheimer approximation. I. Smooth potentials. Ann. Inst. H. Poincaré Sect. A 47 (1987) 1–19.
Hagedorn, G.A., High order corrections to the time-dependent Born-Oppenheimer approximation. II. Coulomb systems. Comm. Math. Phys. 117 (1988) 387–403. CrossRef
G.A. Hagedorn, Molecular propagation through electron energy level crossings, Memoirs of the American Mathematical Society 111 (1994).
Hagedorn, G.A. and Joye, A., A time-dependent Born-Oppenheimer approximation with exponentially small error estimates. Commun. Math. Phys. 223 (2001) 583–626. CrossRef
Kato, T., On the adiabatic theorem of quantum mechanics. Phys. Soc. Jap. 5 (1950) 435–439. CrossRef
Klein, M., Martinez, A., Seiler, R. and Wang, X.P., On the Born-Oppenheimer expansion for polyatomic molecules. Commun. Math. Phys. 143 (1992) 607–639. CrossRef
C. Lasser and S. Teufel, Propagation through conical crossings: an asymptotic transport equation and numerical experiments, Commun. Pure Appl. Math. 58 (2005) 1188–1230.
Littlejohn, R.G. and Flynn, W.G., Geometric phases in the asymptotic theory of coupled wave equations. Phys. Rev. A 44 (1991) 5239–5255. CrossRef
A. Martinez and V. Sordoni, A general reduction scheme for the time-dependent Born-Oppenheimer approximation. C. R. Acad. Sci. Paris, Sér. I 334 (2002) 185–188.
Mead, C.A. and Truhlar, D.G., On the determination of Born-Oppenheimer nuclear motion wave functions including complications due to conical intersections and identical nuclei. J. Chem. Phys. 70 (1979) 2284–2296. CrossRef
Nenciu, G. and Sordoni, V., Semiclassical limit for multistate Klein-Gordon systems: almost invariant subspaces and scattering theory. J. Math. Phys. 45 (2004) 3676–3696. CrossRef
J. von Neumann and E.P. Wigner. Z. Phys. 30 (1929) 467.
Panati, G., Spohn, H. and Teufel, S., Space-adiabatic perturbation theory in quantum dynamics. Phys. Rev. Lett. 88 (2002) 250405. CrossRef
Panati, G., Spohn, H. and Teufel, S., Space-adiabatic perturbation theory. Adv. Theor. Math. Phys. 7 (2003) 145–204. CrossRef
J. Sjöstrand, Projecteurs adiabatiques du point de vue pseudodifferéntiel. C. R. Acad. Sci. Paris, Sér. I 317 (1993) 217–220.
Sordoni, V., Reduction scheme for semiclassical operator-valued Schrödinger type equation and application to scattering. Comm. Partial Differential Equations 28 (2003) 1221–1236. CrossRef
Spohn, H. and Teufel, S., Adiabatic decoupling and time-dependent Born-Oppenheimer theory. Commun. Math. Phys. 224 (2001) 113–132. CrossRef
S. Teufel, Adiabatic perturbation theory in quantum dynamics, Lecture Notes in Mathematics 1821. Springer (2003).
Weigert, S. and Littlejohn, R.G., Diagonalization of multicomponent wave equations with a Born-Oppenheimer example. Phys. Rev. A 47 (1993) 3506–3512. CrossRef
Wu, Y.-S.M. and Kupperman, A., Prediction of the effect of the geometric phase on product rotational state distributions and integral cross sections. Chem. Phys. Lett. 201 (1993) 178–186. CrossRef
Yin, L. and Mead, C.A., Magnetic screening of nuclei by electrons as an effect of geometric vector potential. J. Chem. Phys. 100 (1994) 8125–8131. CrossRef