  We introduce an individually fitted screened-exchange interaction for the
time-dependent Hartree-Fock (TDHF) method and show that it resolves the missing
binding energies in polymethine organic dye molecules compared to
time-dependent density functional theory (TDDFT). The interaction kernel, which
can be thought as a dielectric function, is generated by stochastic fitting to
the screened-Coulomb interaction of many-body perturbation theory (MBPT),
specific to each system. We test our method on the flavylium (Flav) and
indocyanine green (ICG) dye families with a modifiable length of the
polymethine bridge, leading to excitations ranging from the visible to
short-wave infrared (SWIR). Our approach validates earlier observations on the
importance of inclusion of medium range exchange for the exciton binding
energy. Our resulting method, TDHF@$v_W$, also achieves a mean absolute error
on par with MBPT at a computational cost on par with local-functional TDDFT.
