  In pump-probe spectroscopy, two laser pulses are employed to garner dynamical
information from the sample of interest. The pump initiates the optical process
by exciting a portion of the sample from the electronic ground state to an
accessible electronic excited state, an exciton. Thereafter, the probe
interacts with the already excited sample. The change in the absorbance after
pump provides information on transitions between the excited states and their
dynamics. In this work we study these exciton-exciton transitions by means of
an ab initio real time propagation scheme based on dynamical Berry phase
formulation. The results are then analyzed taking advantage of a Fermi-golden
rule approach formulated in the excitonic basis-set and in terms of the
symmetries of the excitonic states. Using bulk LiF and 2D hBN as two prototype
materials, we discuss the selection rules for transitions involving strongly
bound excitons, for which the hydrogen model cannot be used.
