In the case of tunneling in chemical reactions [see ﬁgure1.1(c)], the total energy of reagents cannot be generally considered to be small; treating it as the ﬁrst-order term of h¯ as was successfully done in Chapters 6 and 7 does not lead to any sensible semiclassical theory. The semiclassical mechanics requires (1) construction of Lagrange manifolds, (2) determination of amultidimensional caustic surface, and (3) connection of wave functions between classically allowed and forbidden regions . Unfortunately, these problems have not yet been solved mathematically. As was discussed in Section 3.1, in the framework of the ordinary WKB approach we can construct Lagrange manifolds and caustics in the case of two dimensions and the connection problem can be treated relatively well by assuming the local separability. Extension of this method to higher dimensions, however, cannot be practical. The complexvalued (complex coordinate and/or momentum) WKB theory has been devised, as brieﬂy explained there, but cannot be simply extended to multidimensions (see, for instance, ). This formidable task can hardly be accomplished mainly because of the complexity of the Lagrange manifold and the fact that analytical properties of any realistic potential energy surfaces extended into the complex-valued coordinate space cannot be guaranteed. The use of a complex trajectory was nicely materialized by the double-ended complex trajectory in the classical S-matrix theory , in which the quantized boundary conditions are satisﬁed at both (initial and ﬁnal) ends. Unfortunately, this method turned out not to be practically useful, because ﬁnding such trajectories is also a formidable task in a multidimensional complex coordinate (and/or momentum) space. We can, however, make a break through the difﬁculty of detecting caustics in multidimensional space. An efﬁcient way of doing that has been devised along classical trajectory . This is not a trivial task at all, since it is not possible to ﬁnd caustics in high-dimensional space by just shooting classical trajectories. Since the caustics can be found along each classical trajectory, chemical dynamics with tunneling effects incorporated can now be treated by the on-the-ﬂy method. In this subsection, this method is explained.