  We investigate the applicability of the two existing versions of a
time-dependent Gutzwiller approximation (TDGA) beyond the frequently used limit
of infinite spatial dimensions. To this end, we study the two-particle response
functions of a two-site Hubbard model where we can compare the exact results
and those derived from the TDGA. It turns out that only the more recently
introduced version of the TDGA can be combined with a diagrammatic approach
which allows for the evaluation of Gutzwiller wave functions in finite
dimensions. For this TDGA method we derive the time-dependent Lagrangian for
general single-band Hubbard models.
