The challenge

Accurate descriptions of muonic molecule properties are essential to an understanding of $\mu$CF processes. The energy levels of the loosely bound state (J,v)=(1,1) affect the temperature dependence of molecular formation rates; since 1 meV $\sim$ 10 K, it is desirable to achieve an accuracy of better than 1 meV for the energy level, which should be compared to the three body break up energy of muonic molecular ions of $\sim 3$ keV. In addition to the energy levels, the molecular formation matrix elements are sensitive to details of the $d\mu t$ wave function, particularly in its asymptotic region, the formation rates being proportional to the square of a wave function parameter C (see Section 2.2). Furthermore, the accuracy of the three-body wave function is important for $\mu \alpha$sticking as well. The convergence in variational calculations of the sticking probability $\omega _{s}$ is much slower than that of the energy biding energy $\epsilon$; the general trend is that when $\epsilon$ is converged to n_cdigits, $\omega _{s}$ is accurate only to n_s/3 significant figures [3,84]. Thus, understanding two of the most important processes in $\mu$CF, i.e., molecular formation and sticking, demands the solution of the three-body problem to an accuracy of better than 10^-7, a challenging task to the theorists. Indeed, in 1993, Szalewicz, one of the experts in the field, asserted these calculations to be ``some of the most demanding few-body calculations, taxing the most powerful supercomputers'' [85].

Because the three body problem is central to processes involved in this thesis, we shall review its theoretical framework in some detail in the sections that follow, focusing on understanding the underlying physical concepts.