How to Find Instanton Trajectory

The practical method to find the instanton trajectory is not to solve the classical equations of motion but to directly find the instanton path by minimizing the Euclidean action. Let us consider the Lagrangian with the inverted potential 

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 

Instead of using the time variable t, it is more convenient to introduce some other parametrization. Without loss of generality, we introduce a new parameter z that spans the interval [—1, 1] and call qo(z) a path to distinguish from trajectory qo(r). There is naturally one-to-one correspondence between path and classical trajectory and the correspondence can be established by the energy conservation, which defines the velocity of the parameter z as 

The main idea of improvement is to find a direction in the space of trajectories along which the classical action. Equation (6.109), decreases. In order to do so, we use z(t) to define the r-dependent functions [z(t)] that form the basis of expansion in the vicinity of qo[z(r)]. Namely, we look for a better instanton trajectory in the form 

Inserting Equation (6.113) into Equation (6.109), we obtain the classical action 5o( C ) as a function of the coefficients C . Minimization of this function gives a new trajectory and accomplishes one step of iteration. The shape of the obtained path is again given by Equation (6.111) with the modified coefficients C -I- SC .  

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

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