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.
Mots clés : Schrödinger equation, Born-Oppenheimer approximation, adiabatic methods, almost-invariant subspace
@article{M2AN_2007__41_2_297_0,
author = {Panati, Gianluca and Spohn, Herbert and Teufel, Stefan},
title = {The time-dependent {Born-Oppenheimer} approximation},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {297--314},
publisher = {EDP-Sciences},
volume = {41},
number = {2},
year = {2007},
doi = {10.1051/m2an:2007023},
mrnumber = {2339630},
zbl = {1135.81338},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2007023/}
}
TY - JOUR AU - Panati, Gianluca AU - Spohn, Herbert AU - Teufel, Stefan TI - The time-dependent Born-Oppenheimer approximation JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2007 SP - 297 EP - 314 VL - 41 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2007023/ DO - 10.1051/m2an:2007023 LA - en ID - M2AN_2007__41_2_297_0 ER -
%0 Journal Article %A Panati, Gianluca %A Spohn, Herbert %A Teufel, Stefan %T The time-dependent Born-Oppenheimer approximation %J ESAIM: Modélisation mathématique et analyse numérique %D 2007 %P 297-314 %V 41 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2007023/ %R 10.1051/m2an:2007023 %G en %F M2AN_2007__41_2_297_0
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] ,, and, Measurement of relative state-to-state rate constants for the reaction . J. Chem. Phys. 97 (1992) 7323-7341.
[2] and, The Born-Oppenheimer electric gauge force is repulsive near degeneracies. J. Phys. A 23 (1990) L655-L657.
[3] ,,, and, The geometric phase in quantum systems. Texts and Monographs in Physics, Springer, Heidelberg (2003). | MR | Zbl
[4] and, Zur Quantentheorie der Molekeln. Ann. Phys. (Leipzig) 84 (1927) 457-484. | JFM
[5] and, Scattering amplitude for Dirac operators. Comm. Partial Differential Equations 24 (1999) 377-394. | Zbl
[6] , and, The microlocal Landau-Zener formula. Ann. Inst. H. Poincaré Phys. Theor. 71 (1999) 95-127. | Numdam | MR | Zbl
[7] , and, The Born-Oppenheimer approximation, in Rigorous Atomic and Molecular Physics, G. Velo, A. Wightman Eds., New York, Plenum (1981) 185-212.
[8] and, Geometry of the transport equation in multicomponent WKB approximations. Commun. Math. Phys. 176 (1996) 701-711. | Zbl
[9] and, Mesures semi-classiques et croisement de modes. Bull. Soc. Math. France 130 (2002) 123-168. | Numdam | Zbl
[10] and, Wigner measures and codimension 2 crossings. J. Math. Phys. 44 (2003) 507-527. | Zbl
[11] , A time dependent Born-Oppenheimer approximation. Commun. Math. Phys. 77 (1980) 1-19. | Zbl
[12] , High order corrections to the time-dependent Born-Oppenheimer approximation. I. Smooth potentials. Ann. Math. 124 (1986) 571-590. | Zbl
[13] , 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] , High order corrections to the time-dependent Born-Oppenheimer approximation. II. Coulomb systems. Comm. Math. Phys. 117 (1988) 387-403.
[15] , Molecular propagation through electron energy level crossings, Memoirs of the American Mathematical Society 111 (1994). | MR | Zbl
[16] and, A time-dependent Born-Oppenheimer approximation with exponentially small error estimates. Commun. Math. Phys. 223 (2001) 583-626.
[17] , On the adiabatic theorem of quantum mechanics. Phys. Soc. Jap. 5 (1950) 435-439.
[18] ,, and, On the Born-Oppenheimer expansion for polyatomic molecules. Commun. Math. Phys. 143 (1992) 607-639. | Zbl
[19] and, Propagation through conical crossings: an asymptotic transport equation and numerical experiments, Commun. Pure Appl. Math. 58 (2005) 1188-1230.
[20] and, Geometric phases in the asymptotic theory of coupled wave equations. Phys. Rev. A 44 (1991) 5239-5255.
[21] and, A general reduction scheme for the time-dependent Born-Oppenheimer approximation. C. R. Acad. Sci. Paris, Sér. I 334 (2002) 185-188. | Zbl
[22] and, 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] and, Semiclassical limit for multistate Klein-Gordon systems: almost invariant subspaces and scattering theory. J. Math. Phys. 45 (2004) 3676-3696. | Zbl
[24] and. Z. Phys. 30 (1929) 467.
[25] , and, Space-adiabatic perturbation theory in quantum dynamics. Phys. Rev. Lett. 88 (2002) 250405. | MR
[26] , and, Space-adiabatic perturbation theory. Adv. Theor. Math. Phys. 7 (2003) 145-204.
[27] , Projecteurs adiabatiques du point de vue pseudodifferéntiel. C. R. Acad. Sci. Paris, Sér. I 317 (1993) 217-220. | Zbl
[28] , Reduction scheme for semiclassical operator-valued Schrödinger type equation and application to scattering. Comm. Partial Differential Equations 28 (2003) 1221-1236.
[29] and, Adiabatic decoupling and time-dependent Born-Oppenheimer theory. Commun. Math. Phys. 224 (2001) 113-132. | Zbl
[30] , Adiabatic perturbation theory in quantum dynamics, Lecture Notes in Mathematics 1821. Springer (2003). | MR | Zbl
[31] and, Diagonalization of multicomponent wave equations with a Born-Oppenheimer example. Phys. Rev. A 47 (1993) 3506-3512.
[32] and, 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] and, Magnetic screening of nuclei by electrons as an effect of geometric vector potential. J. Chem. Phys. 100 (1994) 8125-8131.
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