Instanton Theory

In the first part of this section, the instanton theory [2] is explained by taking the motion of a particle of mass m in one-dimensional potential V x). Tunneling splitting in a symmetric double well potential and decay of metastable state by tunneling through a potential barrier are employed as examples. In the second subsection, it is shown that the results can be reproduced by the WKB method with slight modification. 

The instanton theory was invented in the field theory by introducing imaginary time to Minkowski space-time. The instanton is the classical object in the Euclidean space-time that gives a finite action. The instanton is also called pseudo-particle. Here the theory is explained by following Coleman [2]. See also References [17,39,43,46]. 

Let us consider the transition amplitude from position x,- to X/ under the Hamiltonian H, 

The reason we consider the amplitude given by Equation (2.80) is as follows. If we expand this amplitude by using a complete set of energy eigenstates. 

At r - 00, the leading term gives the lowest energy and the corresponding wave function. 

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Two-dimensional semiclassical studies described in section 4 and applied to some concrete problems in section 6 show that, when no additional assumptions (such as moving along a certain predetermined path) are made, and when the fluctuations around the extremal path are taken into account, the two-dimensional instanton theory is as accurate as the one-dimensional one, and for the tunneling problem in most cases its answer is very close to the exact numerical solution. Once the main difficulty of going from one dimension to two is circumvented, there seems to be no serious difficulty in extending the algorithm to more dimensions that becomes necessary when the usual basis-set methods fail because of the exponentially increasing number of basis functions with the dimension. 

The WKB theory we developed and briefly described below is formally theoretically equivalent to the instanton theory [56-58, 71-76], but is more straightforward and practical, and probably easier to understand. Let us start with the ID case in order to comprehend the basic ideas. The semiclassical wave function is given as usual by... 

This is also closely related to the semiclassical periodic orbit ( instanton ) theory W. H. Miller, J. Chem. Phys. 62 1899 (1975). 

The overall phenomenon effectively looks like the so-called deep tim-neling. [46, 140, 149, 151, 269] However, of course, they are independent phenomena. In the quantmn deep tmmeling, the paths go over the barrier through imaginary space and/or time, as in the instanton theory. [46] On the other hand, all the non-Born-Oppenheimer paths in this study are supposed to run in the real space, although the present theory of path branching can be generalized so that the paths can penetrate into complex spaces. 

The instanton theory of tunneling splittings in hydrogen-bonded systems and decay of metastable states in polyatomic molecules was studied by Nakamura et al. [182, 192, 195, 201-204, 216] They formulated a rigorous solution of the multidimensional Hamiltonian-Jacobi and transport equations, developed numerical methods to construct a multidimensional tunneling instanton path, and applied this method to HO [201], malonaldehyde [192, 195], vinyl radical [203], and formic acid dimer [202]. Coupled electron and proton transfer reactions were recently reviewed by Hammes-Schiffer and Stuchebrukhov [209]. 

In this subsection, we show that the same results as those obtained by the instanton theory in the previous subsection can be derived by the WKB theory with a slight modification [43,46]. We consider the tunneling splitting in a symmetric double well potential and as usual we use the asymptotic WKB wave function localized in one of the wells—say, left-side well ... 

The formulation here indicates that the WKB theory works well and gives the same results as the instanton theory, if we observe that the energy is the quantity proportional to fi. This modified WKB method is actually simpler than the instanton treatment and can be directly extended to multidimensional systems. This is discussed in Chapters 6 and 7. 

The WKB treatment and comparison with the instanton theory made by many authors are discussed in the book by Benderskii et al. [17] (see also [52]). We don t go into the details here. 

Generalization of the one-dimensional instanton theory to two and higher dimensions has been naturally tried by many authors [17,40-42]. Here the two-dimensional extension is briefly explained according to the work by Benderskii et al. [17]. First, let us consider the decay of a metastable state. The direct two-dimensional (Qi, Q2) generalization of Equations (2.142) and (2.143) is given by... 

In this section, generalizations of the instanton theory and the modified WKB theory described in Chapter 2 to multidimensional space are presented [43], Those who are not interested in the generalization of the instanton approach and the proof of its equivalence to the modified WKB theory can skip Section 6.1.1 and Section 6.1.2. In Section 6.1.3 a general WKB formulation for a general Hamiltonian in curved space is provided and its final expression of tunneling splitting can be directly applied to any real systems. 

The results obtained in the previous subsection of instanton theory can be reproduced in a much easier way with use of the WKB approximation. This is quite helpful to understand the physical meaning of the various quantities and the procedures used. In the same way as we did in the one-dimensional case in Section 2.5.2, we do not need to construct the complex valued Lagrange manifold [7,15,30,37], which constitutes the main obstacle of the conventional WKB theory. Without the energy term, the Hamilton-Jacobi equation can be easily solved and generalization for an... 

Let us introduce here the semiclassical instanton approach to reaction rate constant discussed by Kryvohuz [225]. Using the semiclassical instanton theory based on the... 

Further detailed discussions and applications to collinear chemical reactions are given in Reference [225]. Miller and co-workers have formulated the quantum instanton theory, starting from the expression of rate constant in terms of the flux operators, and applied it to various practical processes. Those who are interested in that should refer to References [226,227], The semiclassical instanton theory explained above can be conveniently generalized to the case of nonadiabatic chemical reaction. This is discussed in the next subsection. 

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