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 N-dimensional model. We consider a particle of mass m in the N-dimensional separable potential

V (x) = V0(x21 − x20 )2 + N∑


2 ω2i x

2 i . (7.1)

In this case the instanton trajectory is just a straight line connecting the points x1 = ±x0 with xi = 0(i = 2, . . . , N ), and the tunneling splitting 0 is given by the onedimensional answer for the quartic potential V0(x21 −x20 )2 (see Section 2.5). To test the present theory we perform the following nonlinear coordinate transformation x → q:

qk = Tk1x1 + N∑

i=2 Tki (xi + 1 + x21 )2, (k = 1, 2, . . . , N ), (7.2)

where Ti j is the (i, j) element of the following N × N matrix T . T = T 1nT 1(n−1) · · · T 12, (7.3)

where T 1 j is also an N × N orthogonal matrix and represents the rotational matrix in the (x1, x j ) plane. In other words, the diagonal elements of T 1 j are all unity except for the 1st and the j th elements, which are cos(α), and the off-diagonal elements are all zero except for the (T 1 j )1 j = sin(α) and (T 1 j ) j1 = − sin(α), where α is the rotation angle. In the new coordinates q the Hamiltonian operator has the form of Equation (6.82) and the corresponding tunneling splitting Equation (6.100) is invariant under any coordinate transformation. The numerical computations in terms of the new coordinates allow checking the numerical procedure by comparison with the analytical answer. It should be noted that all the factors in Equation (6.100),

g detA

, ( pTA−1p

) , W0, W1, (7.4)

are invariant separately so that the numerical comparison can be made separately for each term.