Effects of the finite size of the ion [Formula Presented] on the energy levels of the molecules [Formula Presented] and [Formula Presented]

The energy shift due to the finite size of the pseudonucleus [Formula Presented] in the molecules [Formula Presented] and [Formula Presented], the subscripts indicating the first excited state with total angular momentum of one unit, is of importance in the theoretical estimation of the rate of [Formula Presented]-[Formula Presented] fusion catalyzed by negative muons. The energy shift in the molecule [Formula Presented] is calculated using perturbation theory up to second order. The finite-size shift is found to be 1.46 meV. This is significantly larger than the value of 0.7 meV for this energy shift calculated by Bakalov [Muon Catalyzed Fusion 3, 321 (1988)] by a method similar to the present method; recently found excellent agreement of theory with experimental results for the formation rate of the molecule [Formula Presented] was based on Bakalov’s value with some modifications. The results of a direct calculation of the finite-size energy shifts in [Formula Presented] using first-order perturbation theory are presented. The contribution from the quadrupole component of the [Formula Presented] charge distribution, which is not taken into account in the conventional scaling procedure based on the finite-size energy shifts of [Formula Presented], is found to be of the order of 1 meV and to depend on the angular-momentum states of [Formula Presented]. Sources of uncertainty in the current theoretical estimates are also discussed.