ESAIM: Mathematical Modelling and Numerical Analysis

Research Article

The time-dependent Born-Oppenheimer approximation

Panati, Gianlucaa1, Spohn, Herberta1 and Teufel, Stefana2

a1 Zentrum Mathematik, TU München, Germany.

a2 Mathematisches Institut, Universität Tübingen, Germany.

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.

(Received September 6 2005)

(Online publication June 16 2007)

Key Words:

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

Mathematics Subject Classification:

  • 81Q05;
  • 81Q15;
  • 81Q70
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