A very appealing method in calculating three body problems, known as the
non-adiabatic coupled rearrangement channel method [111], was
developed by Kamimura. The method has been applied to many muonic and other
three-body
problems [112,113,114,48,115,44,47,45,46].
In his approach, the total three body-wave function is expanded in basis
functions spanned over three rearrangement channels in the Jacobian
co-ordinate system, which is illustrated in Fig. 2.2 for the
case of the
system.
Gaussian gemials are used as basis set functions
to describe radial dependences of the wave function.
For each channel c, the basis has the form
In addition to the philosophical appeal of treating the muon on the same footing as the nuclei, as mentioned above, this method possesses some practical advantages, which include the following. First, because the basis spans three channels, linear dependencies among the basis functions is smaller, i.e. non-orthogonality is not too severe, compared to the single-channel basis function. This permitted calculations in double precision (64 bits, 1416 decimal digits), which resulted in a very short computation time. Second, due to the explicit use of Jacobian co-ordinates with the rearrangement channels, the method is suitable for scattering calculations, where correct boundary conditions are satisfied (c.f. recall the problems with the Adiabatic Representation). This feature is also useful in describing the wave function in the calculation of molecular formation rates (see Section 2.2).
The relative contributions of each channel can be tested with this method
by removing one channel from the calculations, which proves the
relative importance of the channels, in the order (c=1) > (c=2) >(c=3), as expected above, but also shows that to achieve the 10^{-3}eV level of accuracy in
,
it is necessary to include
c=3. With the basis set of
,
the final accuracy of about
eV was achieved, which includes the estimated uncertainty in
extrapolation to
[111].
Among other approaches for the three-body Coulomb problem, two of them, which have received recent attention in CF theory, are briefly discussed below.