Heavy quarkonia mass-splittings in QCD: test of the $1/m$-expansion and estimates of $<α_s G^2>$ and $α_s$

New double ratios of exponential moments of different two-point functions, which are less sensitive to the heavy quark mass and to the continuum effects than the commonly used ratio of moments, are presented for a more refined analysis of the mass-splittings between the different heavy quarkonia states. We show that at the $c$ and $b$ quark mass scales the $1/m$-expansion does not converge for these quarkonia channels, while a connection of our mass and width formulae, with the potential model ones is done. Using the present value of the QCD coupling $\alpha_s$, we deduce the value: $<\alpha_s G^2> = (7.5\pm 2.5)\times 10^{-2}$ GeV$^4$ of the gluon condensate from $M_\psi - M_{\eta_c}$ and $M_{\chi_b} - M_\Upsilon$, which we compare with the ones from different fits of the heavy and light quark channels. We also find that $M_{\chi_c(P^1_1)} - M_{\chi_c(P^3_1)}$ is gouverned by the radiative corrections and gives $\alpha_s$(1.3 GeV) = 0.64$^{+0.36}_{-0.18}$ for 4 flavours, implying $\alpha_s(M_Z) = 0.127\pm 0.011$. Our predictions for the splittings of different heavy quarkonia states are summarized in Table 2, where, in particular, we find $M_{\Upsilon}-M_{\eta_b}\approx 63^{-29}_{+51}$ MeV implying the possible observation of the $\eta_b$ in the $\Upsilon$-radiative decay.