no. 2
The time-dependent Born-Oppenheimer approximation
Panati, Gianluca ; Spohn, Herbert ; Teufel, Stefan1
1 Mathematisches Institut, Universität Tübingen, Germany. stefan.teufel@uni-tuebingen.de
ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 2, pp. 297-314.

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 applications the dynamics near a conical intersection of potential surfaces and reactive scattering.

DOI : 10.1051/m2an:2007023
Classification : 81Q05, 81Q15, 81Q70
Mots clés : Schrödinger equation, Born-Oppenheimer approximation, adiabatic methods, almost-invariant subspace
Panati, Gianluca  ; Spohn, Herbert  ; Teufel, Stefan 1

1 Mathematisches Institut, Universität Tübingen, Germany. stefan.teufel@uni-tuebingen.de 
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Panati, Gianluca; Spohn, Herbert; Teufel, Stefan. The time-dependent Born-Oppenheimer approximation. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 2, pp. 297-314. doi : 10.1051/m2an:2007023. http://www.numdam.org/articles/10.1051/m2an:2007023/

[1] D.E. Adelman, N.E. Shafer, D.A.V. Kliner and R.N. Zare, Measurement of relative state-to-state rate constants for the reaction D+H 2 (v,j)→ HD (v,j)+H. J. Chem. Phys. 97 (1992) 7323-7341.

[2] M.V. Berry and R. Lim, The Born-Oppenheimer electric gauge force is repulsive near degeneracies. J. Phys. A 23 (1990) L655-L657.

[3] 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). | MR | Zbl

[4] M. Born and R. Oppenheimer, Zur Quantentheorie der Molekeln. Ann. Phys. (Leipzig) 84 (1927) 457-484. | JFM

[5] R. Brummelhuis and J. Nourrigat, Scattering amplitude for Dirac operators. Comm. Partial Differential Equations 24 (1999) 377-394. | Zbl

[6] Y. Colin De Verdière, M. Lombardi and C. Pollet, The microlocal Landau-Zener formula. Ann. Inst. H. Poincaré Phys. Theor. 71 (1999) 95-127. | Numdam | MR | Zbl

[7] 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.

[8] C. Emmerich and A. Weinstein, Geometry of the transport equation in multicomponent WKB approximations. Commun. Math. Phys. 176 (1996) 701-711. | Zbl

[9] C. Fermanian-Kammerer and P. Gérard, Mesures semi-classiques et croisement de modes. Bull. Soc. Math. France 130 (2002) 123-168. | Numdam | Zbl

[10] C. Fermanian-Kammerer and C. Lasser, Wigner measures and codimension 2 crossings. J. Math. Phys. 44 (2003) 507-527. | Zbl

[11] G.A. Hagedorn, A time dependent Born-Oppenheimer approximation. Commun. Math. Phys. 77 (1980) 1-19. | Zbl

[12] G.A. Hagedorn, High order corrections to the time-dependent Born-Oppenheimer approximation. I. Smooth potentials. Ann. Math. 124 (1986) 571-590. | Zbl

[13] 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. | Zbl

[14] G.A. Hagedorn, High order corrections to the time-dependent Born-Oppenheimer approximation. II. Coulomb systems. Comm. Math. Phys. 117 (1988) 387-403.

[15] G.A. Hagedorn, Molecular propagation through electron energy level crossings, Memoirs of the American Mathematical Society 111 (1994). | MR | Zbl

[16] G.A. Hagedorn and A. Joye, A time-dependent Born-Oppenheimer approximation with exponentially small error estimates. Commun. Math. Phys. 223 (2001) 583-626.

[17] T. Kato, On the adiabatic theorem of quantum mechanics. Phys. Soc. Jap. 5 (1950) 435-439.

[18] M. Klein, A. Martinez, R. Seiler and X.P. Wang, On the Born-Oppenheimer expansion for polyatomic molecules. Commun. Math. Phys. 143 (1992) 607-639. | Zbl

[19] C. Lasser and S. Teufel, Propagation through conical crossings: an asymptotic transport equation and numerical experiments, Commun. Pure Appl. Math. 58 (2005) 1188-1230.

[20] R.G. Littlejohn and W.G. Flynn, Geometric phases in the asymptotic theory of coupled wave equations. Phys. Rev. A 44 (1991) 5239-5255.

[21] 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. | Zbl

[22] C.A. Mead and D.G. Truhlar, 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.

[23] G. Nenciu and V. Sordoni, Semiclassical limit for multistate Klein-Gordon systems: almost invariant subspaces and scattering theory. J. Math. Phys. 45 (2004) 3676-3696. | Zbl

[24] J. Von Neumann and E.P. Wigner. Z. Phys. 30 (1929) 467.

[25] G. Panati, H. Spohn and S. Teufel, Space-adiabatic perturbation theory in quantum dynamics. Phys. Rev. Lett. 88 (2002) 250405. | MR

[26] G. Panati, H. Spohn and S. Teufel, Space-adiabatic perturbation theory. Adv. Theor. Math. Phys. 7 (2003) 145-204.

[27] J. Sjöstrand, Projecteurs adiabatiques du point de vue pseudodifferéntiel. C. R. Acad. Sci. Paris, Sér. I 317 (1993) 217-220. | Zbl

[28] V. Sordoni, Reduction scheme for semiclassical operator-valued Schrödinger type equation and application to scattering. Comm. Partial Differential Equations 28 (2003) 1221-1236.

[29] H. Spohn and S. Teufel, Adiabatic decoupling and time-dependent Born-Oppenheimer theory. Commun. Math. Phys. 224 (2001) 113-132. | Zbl

[30] S. Teufel, Adiabatic perturbation theory in quantum dynamics, Lecture Notes in Mathematics 1821. Springer (2003). | MR | Zbl

[31] S. Weigert and R.G. Littlejohn, Diagonalization of multicomponent wave equations with a Born-Oppenheimer example. Phys. Rev. A 47 (1993) 3506-3512.

[32] Y.-S.M. Wu and A. Kupperman, Prediction of the effect of the geometric phase on product rotational state distributions and integral cross sections. Chem. Phys. Lett. 201 (1993) 178-186.

[33] L. Yin and C.A. Mead, Magnetic screening of nuclei by electrons as an effect of geometric vector potential. J. Chem. Phys. 100 (1994) 8125-8131.

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