Authors’ introduction: This is the second of a pair of papers on the propagation of molecular wave packets through avoided crossings of electronic energy levels in a limit where the gap between the levels shrinks as the nuclear masses are increased. Generic, minimal multiplicity avoided crossings can be classified into six types [G. A. Hagedorn, J. Phys. A, Math. Gen. 31, 369-383 (1998; Zbl 0956.81014)]. Our first paper [G. A. Hagedorn and A. Joye, Ann. Inst. Henri Poincaré Phys, Théor. A 68, 85-134 (1998; Zbl 0915.35090)] deals with Type 1 and 2; the present paper deals with Type 3, 4, 5, and 6.
In Type 1 and 2 avoided crossings, the electron energy levels essentially depend on only one nuclear configuration parameter. Because of rotational symmetry, this is the case for all diatomic molecules. The results of Part I show that in Type 1 and 2 avoided crossings, transitions between the levels are correctly described by the Landau-Zener formula.
In Type 3, 4, 5, and 6 avoided crossings, the electron energy levels essentially depend (respectively) on 2, 2, 3, md 4 nuclear configuration parameters. In practice, these arise in polyatomic molecules, or in diatomic or polyatomic systems in external fields. Molecular propagation through these more complicated avoided crossings is not governed directly by the Landau-Zener formula, and electronic transition probabilities depend on the shape of the nuclear wave packet. Intuitively, this is because different pieces of the wave packet feel different size minimum gaps between the electronic levels. As we explain in more detail below, the correct transition probability can be determined by decomposing the nuclear wave packet into infinitesimal pieces, using a different Landau-Zener formula for each infinitesimal piece, and then doing an integration.