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The success of the R-matrix with pseudo-state (RMPS) method to model singlephotoionization processes, benchmarked against dedicated synchrotron lightsource measurements, is exploited and extended to investigate, single photondouble ionization cross-sections for the He-like, Li$^+$ ion. We investigatethese processes from both the ground state and the excited $\rm 1s2s^{1,3}S$metastable levels of this He-like system. Comparisons of the results from theR-matrix plus pseudo-state (RMPS) method are made with other state-of-the-arttheoretical approaches such as time-dependent close-coupling (TDCC), B-splinesand the convergent close-coupling (CCC). Excellent agreement with various othertheoretical approaches are achieved but differences occur which arehigh-lighted and discussed. For the ground state the peak of the cross sectionis $\sim$ 2 kilo-barns (Kb), that for the $\rm 1s2s^{1}S$ state is $\sim$ 6 Kband $\sim$ 1 Kb for the corresponding $\rm 1s2s^{3}S$ state. All the crosssections for single photon double ionization are extremely small, being in theregion of 2Kb -- 10 Kb, or less, rendering their experimental determinationextremely challenging.

An R-matrix with pseudo-state (RMPS) approach to the single photon double ionization and excitation of the He-like Li$^{+}$ ion