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The $1^1D_2 (c\bar c)$ state is the ground state of spin-singlet D-wavecharmonia. Although it has not been found yet, the experimental data accumulaterapidly. This charmonium attracts more and more attention, especially when theBaBar Collaboration finds that the X(3872) particle has negative parity. Inthis paper we calculate the double-gamma and double-gluon annihilationprocesses of $^1D_2$ charmonia and bottomonia by using the instantaneousBethe-Salpeter method. We find the relativistic corrections make the decaywidths of $1^1D_2 (c\bar c)$ 2$\sim$5 times smaller than the non-relativisticresults. If this state is below the $D^0{D^{0}}^\ast$ threshold, we can use thesum of annihilation widths and EM transition widths to estimate the total decaywidth. Our result for $1{^1D_2} (c\bar c)$ with $m=3820$ GeV is $\Gamma= 432$keV. The dominant decay channel $1^1D_2 (c\bar c)\to h_c \gamma$, whosebranching ratio is about 90%, can be used to discover this state.

Annihilation rate of $2^{-+}$ charmonium and bottomonium