  The success of the R-matrix with pseudo-state (RMPS) method to model single
photoionization processes, benchmarked against dedicated synchrotron light
source measurements, is exploited and extended to investigate, single photon
double ionization cross-sections for the He-like, Li$^+$ ion. We investigate
these 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 the
R-matrix plus pseudo-state (RMPS) method are made with other state-of-the-art
theoretical approaches such as time-dependent close-coupling (TDCC), B-splines
and the convergent close-coupling (CCC). Excellent agreement with various other
theoretical approaches are achieved but differences occur which are
high-lighted and discussed. For the ground state the peak of the cross section
is $\sim$ 2 kilo-barns (Kb), that for the $\rm 1s2s^{1}S$ state is $\sim$ 6 Kb
and $\sim$ 1 Kb for the corresponding $\rm 1s2s^{3}S$ state. All the cross
sections for single photon double ionization are extremely small, being in the
region of 2Kb -- 10 Kb, or less, rendering their experimental determination
extremely challenging.
