Instanton Trajectories

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... 

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... 

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... 

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... 

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

In the case of a symmetric (or Just slightly asymmetric) potential the instanton trajectory consists of kink and antikink, which are separated by infinite time and do not interact with each other. In other words, we may change the boundary conditions, namely, suppose that the time spans from — 00 to -t- 00 for a single kink, and then multiply the action in (5.72) by factor 2. 

Fig. 38, Contour plot, MEP and instanton trajectory for isomerization of malonaldehyde (6.4). The instanton is drawn for large but finite in the limit = oo it emanates from the potential minimum.

Fig. 39. Contour plot, MEP and instanton trajectory for isomerization of hydrogenoxalate anion (6.5).

The functional B[(2(r)] actually depends only on the velocity dQ/dr at the moment when the non-adiabaticity region is crossed. If we take the path integral by the method of steepest descents, considering that the prefactor B[(2(r)] is much more weakly dependent on the realization of the path than Sad[Q(A]> we shall obtain the instanton trajectory for the adiabatic potential V a, then B[(2(t)] will have to be calculated for that trajectory. Since the instanton trajectory crosses the dividing surface twice, we finally have... 

How to Find Instanton Trajectory and How to Incorporate Accurate ab initio Quantum Chemical Calculations. 

In order to solve Eq. (34), we use the method of characteristics and consider a family of classical trajectories on the inverted potential q(p, x), p(P, x), where P is an (A — 1)-dimensional parameter to characterize the trajectory and x is the time running for the infinite interval along the trajectory, where x = — oo corresponds to the minimum of the potential q(p, —oo) = q ,p(p, —oo) = 0. The solution we want is the trajectory that connects the two potential minima and along which the action becomes minimum. This is called the instanton trajectory and belongs to the above mentioned family qo(x) = q(Po At q close to the potential minimum q , the momentum p(q) is linear with respect to the deviation (q — q ) and Wo(q) is quadratic. 

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. 

The instanton trajectory qo( t) to be found satisfies the boundary conditions. 

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... 

Figure 13. Comparison of the ab initio instanton trajectory by the CCSD(T)/(aug-)cc-pVDZ method with those obtained by (b) the global MP2 method and (c) analytic potential energy function.

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].

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... 

The basis functions ( ) (z) used to define the instanton trajectory is... 

Figure 17. Iteration process of the calculation of instanton trajectory in the cubic potential for N = 2 and Ci = 0.6 in Eq. (98). The parameter A iter is the number of iteration. Taken from Ref. [31],...

To the extent one can consider an instanton trajectory as a physical process (in fact only the initial and final states of the instanton trajectory have a physical sense ) only the function g has a physical meaning and therefore the physical requirements for boundary conditions should be applied to the function g. 

The boundary conditions formulated in Eqs. (65, 67, 68) contain all the physical requirements for the instanton trajectory. Now we need to figure out the boundary conditions for the functions g 1, 2 that are consistent with Eqs. (65, 67, 68). We will show below that the function 2 — 0 in the superconducting region. Thus, at r Tj the Green function g has the same boundary conditions with those defined by Eq. (65) for function gk... 

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