# The time-dependent Born-Oppenheimer approximation

Gianluca Panati; Herbert Spohn; Stefan Teufel

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

- Volume: 41, Issue: 2, page 297-314
- ISSN: 0764-583X

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topPanati, Gianluca, Spohn, Herbert, and Teufel, Stefan. "The time-dependent Born-Oppenheimer approximation." ESAIM: Mathematical Modelling and Numerical Analysis 41.2 (2007): 297-314. <http://eudml.org/doc/250020>.

@article{Panati2007,

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

author = {Panati, Gianluca, Spohn, Herbert, Teufel, Stefan},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Schrödinger equation; Born-Oppenheimer approximation; adiabatic methods; almost-invariant
subspace.},

language = {eng},

month = {6},

number = {2},

pages = {297-314},

publisher = {EDP Sciences},

title = {The time-dependent Born-Oppenheimer approximation},

url = {http://eudml.org/doc/250020},

volume = {41},

year = {2007},

}

TY - JOUR

AU - Panati, Gianluca

AU - Spohn, Herbert

AU - Teufel, Stefan

TI - The time-dependent Born-Oppenheimer approximation

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2007/6//

PB - EDP Sciences

VL - 41

IS - 2

SP - 297

EP - 314

AB -
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.

LA - eng

KW - Schrödinger equation; Born-Oppenheimer approximation; adiabatic methods; almost-invariant
subspace.

UR - http://eudml.org/doc/250020

ER -

## References

top- 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 ${\mathrm{D}+\mathrm{H}}_{2}(v,j)\to \mathrm{HD}(v,j)+\mathrm{H}$. J. Chem. Phys.97 (1992) 7323–7341.
- M.V. Berry and R. Lim, The Born-Oppenheimer electric gauge force is repulsive near degeneracies. J. Phys. A23 (1990) L655–L657.
- 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).
- M. Born and R. Oppenheimer, Zur Quantentheorie der Molekeln. Ann. Phys. (Leipzig)84 (1927) 457–484.
- R. Brummelhuis and J. Nourrigat, Scattering amplitude for Dirac operators. Comm. Partial Differential Equations24 (1999) 377–394.
- Y. Colin de 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.
- C. Emmerich and A. Weinstein, Geometry of the transport equation in multicomponent WKB approximations. Commun. Math. Phys.176 (1996) 701–711.
- C. Fermanian-Kammerer and P. Gérard, Mesures semi-classiques et croisement de modes. Bull. Soc. Math. France130 (2002) 123–168.
- C. Fermanian-Kammerer and C. Lasser, Wigner measures and codimension 2 crossings. J. Math. Phys.44 (2003) 507–527.
- G.A. Hagedorn, A time dependent Born-Oppenheimer approximation. Commun. Math. Phys.77 (1980) 1–19.
- G.A. Hagedorn, High order corrections to the time-dependent Born-Oppenheimer approximation. I. Smooth potentials. Ann. Math.124 (1986) 571–590.
- 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.
- G.A. Hagedorn, High order corrections to the time-dependent Born-Oppenheimer approximation. II. Coulomb systems. Comm. Math. Phys.117 (1988) 387–403.
- G.A. Hagedorn, Molecular propagation through electron energy level crossings, Memoirs of the American Mathematical Society 111 (1994).
- G.A. Hagedorn and A. Joye, A time-dependent Born-Oppenheimer approximation with exponentially small error estimates. Commun. Math. Phys.223 (2001) 583–626.
- T. Kato, On the adiabatic theorem of quantum mechanics. Phys. Soc. Jap.5 (1950) 435–439.
- 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.
- C. Lasser and S. Teufel, Propagation through conical crossings: an asymptotic transport equation and numerical experiments, Commun. Pure Appl. Math.58 (2005) 1188–1230.
- R.G. Littlejohn and W.G. Flynn, Geometric phases in the asymptotic theory of coupled wave equations. Phys. Rev. A44 (1991) 5239–5255.
- A. Martinez and V. Sordoni, A general reduction scheme for the time-dependent Born-Oppenheimer approximation. C. R. Acad. Sci. Paris, Sér. I334 (2002) 185–188.
- 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.
- 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.
- J. von Neumann and E.P. Wigner. Z. Phys.30 (1929) 467.
- G. Panati, H. Spohn and S. Teufel, Space-adiabatic perturbation theory in quantum dynamics. Phys. Rev. Lett.88 (2002) 250405.
- G. Panati, H. Spohn and S. Teufel, Space-adiabatic perturbation theory. Adv. Theor. Math. Phys.7 (2003) 145–204.
- J. Sjöstrand, Projecteurs adiabatiques du point de vue pseudodifferéntiel. C. R. Acad. Sci. Paris, Sér. I317 (1993) 217–220.
- V. Sordoni, Reduction scheme for semiclassical operator-valued Schrödinger type equation and application to scattering. Comm. Partial Differential Equations28 (2003) 1221–1236.
- H. Spohn and S. Teufel, Adiabatic decoupling and time-dependent Born-Oppenheimer theory. Commun. Math. Phys.224 (2001) 113–132.
- S. Teufel, Adiabatic perturbation theory in quantum dynamics, Lecture Notes in Mathematics 1821. Springer (2003).
- S. Weigert and R.G. Littlejohn, Diagonalization of multicomponent wave equations with a Born-Oppenheimer example. Phys. Rev. A47 (1993) 3506–3512.
- 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.
- 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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