A-Dimensional Separable Potential Model

Before dealing with real polyatomic molecules, let us first check the theory presented in Chapter 6 and explain the numerical procedure by taking a simple exactly solvable Al-dimensional model. We consider a particle of mass m in the A -dimensional separable potential 

In this case the instanton trajectory is just a straight line connecting the points Xi = Xo with Xi =0 i = 2. N), and the tunneling splitting Aq is given by the onedimensional answer for the quartic potential Vb(xf—x ) (see Section 2.5). To test the present theory we perform the following nonlinear coordinate transformation x q  

FIGURE 7.1 Iteration process of calculating theinstanton trajectory in the separable potential model. Niter is the number of iterations. The inset shows the resnlt after 25 iterations. (Taken from Reference [43] with permission.) 

For solving the matrix differential equation. Equation (6.97), we make use of the analytical form of the instanton. Equation (6.111), and rewrite Equation (6.97) in terms of the parameter z as 

This significantly improves the accuracy of W. We have used C2 = 0.025 and 0.05, and confirmed three stable digits in the tunneling splitting Aq = 0.624 cm . The analytical one-dimensional answer gives Aq = 0.6238 cm  

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