Instanton

As a result of several complementary theoretical efforts, primarily the path integral centroid perspective [33, 34 and 35], the periodic orbit [36] or instanton [37] approach and the above crossover quantum activated rate theory [38], one possible candidate for a unifying perspective on QTST has emerged [39] from the ideas from [39, 40, 4T and 42]. In this theory, the QTST expression for the forward rate constant is expressed as [39]... 

It should be noted that in the cases where y"j[,q ) > 0, the centroid variable becomes irrelevant to the quantum activated dynamics as defined by (A3.8.Id) and the instanton approach [37] to evaluate based on the steepest descent approximation to the path integral becomes the approach one may take. Alternatively, one may seek a more generalized saddle point coordinate about which to evaluate A3.8.14. This approach has also been used to provide a unified solution for the thennal rate constant in systems influenced by non-adiabatic effects, i.e. to bridge the adiabatic and non-adiabatic (Golden Rule) limits of such reactions. 

In addition to the trivial solutions, there is a /S-periodic upside-down barrier trajectory called instanton, or bounce [Langer 1969 Callan and Coleman 1977 Polyakov 1977]. At jS oo the instanton dwells mostly in the vicinity of the point x = 0, attending the barrier region (near x ) only during some finite time (fig. 20). When jS is raised, the instanton amplitude... 

Unlike the trivial solution x = 0, the instanton, as well as the solution x(t) = x, is not the minimum of the action S[x(t)], but a saddle point, because there is at least one direction in the space of functions x(t), i.e. towards the absolute minimum x(t) = 0, in which the action decreases. Hence if we were to try to use the approximation of steepest descents in the path integral (3.13), we would get divergences from these two saddle points. This is not surprising, because the partition function corresponding to the unbounded Hamiltonian does diverge. 

A more vexing issue is that one of the e in (3.37) equals zero. To see this, note that the function Xins(t) [where Xjns is the instanton solution to (3.34)] can be readily shown to satisfy (3.37) with o = 0. Since the instanton trajectory is closed, it can be considered to start arbitrarily from one of its points. It is this zero mode which is responsible for the time-shift invariance of the instanton solution. Therefore, the non-Gaussian integration over Cq is expected to be the integration over... 

Callan and Coleman [1977] have obtained this formula in the limit -r cc by summing up all multi-instanton contributions to Im F, i.e., taking into account the trajectories that pass through the barrier more than twice. These trajectories enter into Im F with factors exp( — nSinsj/n (where n is the number of passes) and, therefore, they can be neglected when Sins > 1-... 

The identity of eqs. (2.6) (at T = 0) and (3.47) for the cubic parabola is also demonstrated in appendix A. Although at first glance the infinite determinants in (3.46) might look less attractive than the 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. 

Each extremal trajectory includes n kink-antikink pairs, where n is an arbitrary integer, and the kink centers are placed at the moments 0 < tj < < tin < P forming the instanton gas (fig. 22). Its contribution to the overall path integral may be calculated in exactly the same manner as was done in the previous subsection, with the assumption that the instanton gas is dilute, i.e., the kinks are independent of each other. 

In the nonadiabatic regime A is proportional to the adiabatic splitting 2 Fd. The instanton trajectory crosses the barrier twice, each time bringing the factor A/A a associated with the probability to cross the nonadiabaticity region remaining on the same adiabatic term (and thus... 

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... 

These equations form a fourth-order system of differential equations which cannot be solved analytically in almost all interesting nonseparable cases. Further, according to these equations, the particle slides from the hump of the upside-down potential — V(Q) (see fig. 24), and, unless the initial conditions are specially chosen, it exercises an infinite aperiodic motion. In other words, the instanton trajectory with the required periodic boundary conditions,... 

Once the instanton trajectory has been numerically found, one proceeds to the calculation of prefactor, which amounts to finding determinants of differential operators. The direct two-dimensional generalization of (3.46) is... 

This formula, however, tacitly supposes that the instanton period depends monotonically on its amplitude so that the zero-amplitude vibrations in the upside-down barrier possess the smallest possible period 2nla>. This is obvious for sufficiently nonpathological one-dimensional potentials, but in two dimensions this is not necessarily the case. Benderskii et al. [1993] have found that there are certain cases of strongly bent two-dimensional PES when the instanton period has a minimum at a finite amplitude. Therefore, the cross-over temperature, formally defined as the lowest temperature at which the instanton still exists, turns out to be higher than that predicted by (4.7). At 7 > Tc the trivial solution Q= Q Q is the saddle-point coordinate) emerges instead of instanton, the action equals S = pV (where F " is the barrier height at the saddle point) and the Arrhenius dependence k oc exp( — F ") holds. 

The most remarkable feature of expression (4.8) is that it does not contain any cross terms 8x 8s. This is a consequence of time-shift invariance of the instanton solution (d s/dt = dVjds, x = 0). This fact can be expressed as invariance of the action with respect to the infinitesimal transformation s s -I- cs, c 0 [cf. eq. (3.42)]. In the new coordinates the determinants break up into longitudinal and transverse parts and (4.4) becomes... 

It is noteworthy that eq. (4.15a) is nothing but the linearized classical upside-down barrier equation of motion (8S/8x = 0) for the new coordinate x. Therefore, while x = 0 corresponds to the instanton, the nonzero solution to (4.15a) describes how the trajectory escapes from the instanton solution, when it deviates from it. The parameter X, referred to as the stability angle [Gutzwil-ler 1967 Rajaraman 1975], generalizes the harmonic-oscillator phase co, which would appear in (4.15), if CO, were a constant. The fact that X is real indicates the aforementioned instability of the instanton in two dimensions. Guessing that the determinant det( — -I- co, ) is a function of X only,... 

As a simple illustration of this technique consider the case of high frequency co, viz. co, 5co,/5t 1 for the instanton trajectory (but co, is still small compared to the total barrier... 

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]... 

The practical way of calculating 2 is different from that used in the derivation of (4.18). Since 2 is invariant with respect to canonical transformations, it is preferable to seek it in the initial coordinate system. Writing the linearized equation for deviations from the instanton solution 6Q,... 

In Benderskii et al. [1993] the numerical instanton analysis of tunneling escape out of the metastable well with the Hamiltonian... 

In fig. 26 the Arrhenius plot ln[k(r)/coo] versus TojT = Pl2n is shown for V /(Oo = 3, co = 0.1, C = 0.0357. The disconnected points are the data from Hontscha et al. [1990]. The solid line was obtained with the two-dimensional instanton method. One sees that the agreement between the instanton result and the exact quantal calculations is perfect. The low-temperature limit found with the use of the periodic-orbit theory expression for kio (dashed line) also excellently agrees with the exact result. Figure 27 presents the dependence ln(/Cc/( o) on the coupling strength defined as C fQ. The dashed line corresponds to the exact result from Hontscha et al. [1990], and the disconnected points are obtained with the instanton method. For most practical purposes the instanton results may be considered exact. 

Fig. 26. Arrhenius plot [ln(fc/a>o) against a>o /2it] for the PES (4.28) with Q = 0.1, C = 0.0357, = 1, F /a>o = 3. Solid line shows instanton result separate points, numerical calculation data from Hontscha et al. [1990] and dashed line, low-temperature limit using (3.32) for fc,D.

Fig. 28. Contour plot and instanton trajectory for PES (4.28) with the parameters of fig. 25, ojqP = 35.

P Q-) =p Q-,Q-,p), which in the harmonic approximation is described by (3.16), PhiQ-iQ-,P) exp(— CO Q1 tanh co ). Having reached the point Q, the particle is assumed to suddenly tunnel along the fast coordinate Q+ with probability A id(Q-), which is described in terms of the usual one-dimensional instanton. The rate constant comes from averaging the onedimensional tunneling rate over positions of the slow vibration mode,... 

The dimensionless upside-down barrier frequency equals = 2(1 — and the transverse frequency Qf = Q. The instanton action at = oo in the one-dimensional potential (4.41) equals [cf. eq. (3.68)]... 

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