  Excitons, trions, biexcitons, and exciton-trion complexes in two-dimensional
transition metal dichalcogenide sheets of MoS$_2$, MoSe$_2$, MoTe$_2$, WS$_2$
and WSe$_2$ are studied by means of density functional theory and path integral
Monte Carlo method in order to accurately account for the particle-particle
correlations. In addition, the effect of dielectric environment on the
properties of these exciton complexes is studied by modifying the effective
interaction potential between particles. Calculated exciton and trion binding
energies are consistent with previous experimental and computational studies,
and larger systems such as biexciton and exciton-trion complex are found highly
stable. Binding energies of biexcitons are similar or higher than those of
trions, but the binding energy of the trion depends significantly stronger on
the dielectric environment than that of biexciton. Therefore, as a function of
an increasing dielectric constant of the environment the exciton-trion complex
"dissociates" to a biexciton rather than to an exciton and a trion.
