The CMP-invariance for symmetrical Coulomb four-body systems with unit charges

The bound state spectra in the symmetrical Coulomb four-body systems with unit charges X(+)X(+)Y(-)Y(-) are considered. The total energy and other properties for such systems depend only on the mass ratio s = m(Y)/m(X). It is shown that there is an additional charge-mass-permutation invariance in these systems. By applying this invariance we can determine the total energies for the systems with s greater than or equal to 1 if the energy dependence is known for s less than or equal to 1, and vice versa. To produce quite accurate interpolation formulas for the total energies we have made calculations for a number of exotic systems, including Ps(2), p(+)p(+)mu(-)mu(-), d(+)d(+)pi(-)pi(-) and eight others. For all these systems the variational results are the best values to date; This is the first study based on numerical results of bound state spectra for Coulomb four-body systems with unit charges.