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**Molecular propagation through electron energy level crossings.**
*(English)*
Zbl 0833.92025

Mem. Am. Math. Soc. 536, 130 p. (1994).

In the standard time-dependent Born-Oppenheimer approximation, the electrons and nuclei are treated separately, but in a way that respects the coupling of their motions. The electron evolution depends adiabatically on the configuration of the nuclei, and the electrons produce an effective potential in which the nuclei move semiclassically. Traditional Born-Oppenheimer approximations are valid only under the basic assumption that the electrons are in a discrete energy level that is isolated from the rest of the spectrum of the electronic Hamiltonian.

The purpose of the present paper is to study the phenomena that arise when this basic assumption is violated by allowing electron level crossings. The paper obtains the classification and structure theory for generic, minimal multiplicity level crossings, pointing out 11 distinct types of crossings and proving that the relevant part of the electron Hamiltonian can be put into a normal form near a generic crossing point. Another basic result is that the crossing manifold of the nuclear configuration space has generically codimension 1, 2, 3, or 5 for minimal multiplicity crossings, this codimension depending on the crossing type. Two numerical simulations of evolution through codimension 2 crossings are exposed. The paper represents the synthesis of the results, particularly the author’s ones, of more than a decade of study on the molecular systems.

The purpose of the present paper is to study the phenomena that arise when this basic assumption is violated by allowing electron level crossings. The paper obtains the classification and structure theory for generic, minimal multiplicity level crossings, pointing out 11 distinct types of crossings and proving that the relevant part of the electron Hamiltonian can be put into a normal form near a generic crossing point. Another basic result is that the crossing manifold of the nuclear configuration space has generically codimension 1, 2, 3, or 5 for minimal multiplicity crossings, this codimension depending on the crossing type. Two numerical simulations of evolution through codimension 2 crossings are exposed. The paper represents the synthesis of the results, particularly the author’s ones, of more than a decade of study on the molecular systems.

Reviewer: S.Curteanu (Iaşi)

### MSC:

92E20 | Classical flows, reactions, etc. in chemistry |

81V55 | Molecular physics |

35B25 | Singular perturbations in context of PDEs |

92E99 | Chemistry |

81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |

92-02 | Research exposition (monographs, survey articles) pertaining to biology |