Such a basic idea of the instanton theory that the energy E is treated as the ﬁrstorder term of h¯ is useful for the problems of tunneling splitting in symmetric double well potential and decay of metastable state through tunneling. With use of this basic idea, the modiﬁed WKB theory was formulated and demonstrated to be applicable virtually to any real molecular systems with high accuracy. There are two key points that made this possible. The ﬁrst is that we can ﬁnd the instanton trajectory relatively easily and accurately in multidimensional space. The second is that the high level of ab initio quantum chemical data along the instanton trajectory can be efﬁciently used to calculate desirable physical quantities with no need to construct a global potential energy surface in multidimensional space. Although the high-level quantum chemical calculations-including ﬁrst-and second-order derivatives of potential energies-require a bit too much CPU time, this can be alleviated by using lower-level non-time-consuming quantum chemical calculations in an appropriate way. This recipe was explained in detail. Numerical applications to real polyatomic molecules have been carried out and demonstrated to reproduce the highly accurate spectroscopic experimental data. Furthermore, insufﬁciency of low (one and two) dimensional treatments and the effects of multidimensionality were clearly pointed out and numerically demonstrated. Vibrational excitations do not necessarily enhance tunneling, and tunneling probabilities may decrease or oscillate with the excitation, depending on the topology of potential energy surface. Although full-dimensional treatment is possible, from the viewpoint of comprehending the tunneling dynamics and avoiding unnecessary heavy computations, it is desirable to ﬁnd outwhich degrees of freedom are crucial to be included in order to comprehend the tunneling dynamics properly. An example was provided in Section 7.4 in which the out-of-plane wagging motion is found to play important roles in the case of malonaldehyde. Analyses along this line should be further carried out. It is also necessary to develop theories further so that we can deal with rovibrationally excited states in general. Recently, the ring-polymer instanton method has been developed by Althorpe and co-workers for calculating tunneling splitting in multidimensional systems . Some detailed studies would be necessary to compare with other methods such as the one presented here.