Instanton path

From the very simple WKB considerations it is clear that the tunneling rate is proportional to the Gamov factor exp —2j[2(F(s(0) — )] ds, where s Q) is a path in two dimensions Q= 61)62 ) connecting the initial and flnal states. The most probable tunneling path , or instanton, which renders the Gamov factor maximum, represents a compromise of two competing factors, the barrier height and its width. That is, one has to optimise the instanton path not only in time, as has been done in the previous section, but also in space. This complicates the problem so that numerical calculations are usually needed. 

Again we use the ImF method in which the tunneling rate is determined by the nontrivial instanton paths which extremize the Eucledian action in the barrier. Let for deflniteness the potential V Q) have a single minimum at = 0, F(0) = 0, separated from the continuous spectrum... 

The functional (4.8) permits one to study the set of paths which actually contribute to the partition-function path integral thereby leading to the determinant (4.17). Namely, the symmetric Green function for the deviation from the instanton path x(t) is given by [Benderskii et al. 1992a]... 

Minimization of this quantity gives a set of new coefficients and the improved instanton trajecotry. The second and third terms in the above equation require the gradient and Hessian of the potential function V(q)- For a given approximate instanton path, we choose Nr values of the parameter zn =i 2 and determine the corresponding set of Nr reference configurations qo(2n) -The values of the potential, first and second derivatives of the potential at any intermediate z, can be obtained easily by piecewise smooth cubic interpolation procedure. 

Figure 15, Iterative calculation of the instanton path. The labels 1-9 show gradual improvement of the instanton trajectory shape using the MP2/cc-pVDZ ab initio data. After switching to the CCSD(T)/(aug-)cc-pVDZ ab initio method, only two more steps needed to achieve convergence and obtain the final results. Taken from Ref. [104].

For the purpose of illustration, let us next consider the simple 2D case. Introducing the local variables (i, ) where ie[—runs along the instanton path Qo(i) and is the coordinate perpendicular to the instanton, we obtain... 

Note that there are removable singularities at both ends of the instanton path, namely, at z = 0 and z = 1. Thus, in the practical numerical computations some modihcations to remove the singularities are necessary. The details are not given here and the reader should refer to Ref. [30, 31]. Finally, the numerically stable... 

Fig. 7. The tunneling paths in a double minimum potential like that of Fig. 5 may be classified as having one or more (odd) numbers of instantons, or tunneling segments. Three such traverses of the classically forbidden region are shown above. The classification of all paths according to the number N of instantons is the basis for evaluating the path integral as the sum in Eq. 29 note however that the therein is constructed to include non-harmonic (beyond semiclassical) fluctuations around the minimum action instanton paths, which are evaluable by Metropolis Monte Carlo...

The conclusion we have arrived at is what may be called a generalized Fukui theorem, which states that the reaction path near the minimum goes along the lowest eigenfrequency direction (see Tachibana and Fukui [1979]). Further consideration of the Fukui theorem is addressed in Section 8.3. Here we note only that this theorem is not valid when the instanton path is constrained by the symmetry of potential to lie totally along the direction of one of initial state modes. 

This is quite different from the first two. Due to the reflection symmetry of this potential, the instanton path always remains directed along the Q axis. The transverse q vibration changes only the width of the reactive channel according to Eqs. (4.23) and (4.24). When C > 0, the vibrational-ly adiabatic squeezed barrier is greater than the bare one. This case of dynamically induced formation of the barrier was studied by Auerbach and Kivelson [1985] in context of nuclear physics. The opposite case C < 0, corresponding to the vibration-assisted tunneling, will be considered in Section 8.3. 

Note that in the present case, neither the Fukui theorem [Fukui, 1970] mentioned in Section 8.4 nor the similar statement concerning the initial direction of the instanton path (see Section 4.1) are valid. Both state that the reaction coordinate (IRC in the case of Fukui theorem or instanton) near the minimum of the surface is directed along the coordinate of the vibration with the lowest frequency. That would mean for our case that... 

The double line represents the minimum energy path (MEP), which is the reaction path assumed by TST. The single line represents the instanton trajectory for proton tunneling and the dashed line the instanton trajectory for deuteron tunneling. The heavier deuteron tunnels closer to the MEP, where the barrier is lower. These distinct instanton paths are the reason for the lowering of the KIE by the promoting vibration that we mentioned earlier. 

The preexponential factor. 4(r) in the rate expression (29.19) consists of a longitudinal component, 4 and a perpendicular component A , representing the contributions of fluctuations that are, respectively, parallel and perpendicular to the instanton path. If in the one-dimensional potential U( (x) the long turmeling action evaluated at = 0 for T = 0 is replaced by the short action evaluated at the zero-point energy hQ(,/2, we can use the approximation, 4 — Q.q/1k for the longitudinal component. The perpendicular component is treated in the adiabatic approximation, which yields = 1 if [/ (x) is replaced by its vibrationally adiabatic counterpart. Using these approximations, we obtain... 

Figure 4. Instanton path for tunneling between pairs of minima in effective potential at Z oo limit for with R = 5. Panel at left show contour maps of effective potential in spheroidal coordinates, at right maps in cylindrical coordinates. Heavy dots show exact instanton paths long-dashes indicate parabolic approximation short-dashes straight-line approximation.

Figure 4 shows contour plots of the effective potential and compares the exact instanton path obtained from our algorithm with the straight-line approximation [16] and with a paraboUc approximation. 

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]. 

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