Instanton trajectory, multidimensional

Now, the general formulation of the problem is finished and ready to be applied to real systems without relying on any local coordinates. The next problems to be solved for practical applications are (1) how to find the instanton trajectory qo( t) efficiently in multidimensional space and (2) how to incorporate high level of accurate ab initio quantum chemical calculations that are very time consuming. These problems are discussed in the following Section III. A. 2. 

One might think that it would be easy to find the instanton trajectory by running classical trajectories even in a multidimensional space. This is actually not true at all. Instead of doing that, we introduce a new parameter z, which spans the interval [—1,1] instead of using the time x and employ the variational principle using some basis functions to express the tarjectory. The 1 1 correspondence between x and z can be found from the energy conservation and the time variation of z is expressed as... 

The theory developed for tunneling splitting can be easily extended to the decay of the metastable state through multidimensional tunneling, namely, tunneling predissociation of polyatomic molecules. In the case of predissociation, however, the instanton trajectory cannot be fixed at both ends, but one end should be free (see Fig. 17). The boundary conditions are... 

In a multidimensional case, however, it would be hopeless, as mentioned above, to find the right instanton trajectory that satisfies the boundary conditions. Equation (6.108), at both ends by simply shooting classical trajectories. The practical method... 

When the instanton trajectory X, jf(T)(T = 0 t = t ) is determined from this condition, the one-dimensional effective potential Veff Xi st) is obtained for the given temperature p. Thus the original multidimensional problem is reduced to the onedimensional transmission through this effective potential along the instanton trajectory and the rate constant k T) is given by... 

In particular, the high-temperature regime T>TC, where the trivial trajectory jc(t) = x is the only solution contributing to Im F, is correctly described by the instanton formalism. Furthermore, the equivalence of (2.6) (at T = 0) and (3.68) for the cubic parabola is demonstrated in Appendix B. Although at first the infinite determinants in (3.67) might look less attractive than simple formulas (2.6) and (2.7) or the direct WKB solution by Schmid, it is the instanton approach that permits direct generalization to dissipative tunneling and to the multidimensional problem. 

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