  Exciton spectra of monolayer transition metal dichalcogenides (TMDs) in
various dielectric environments are studied.
  The screened hydrogen model (SHM) [Phys. Rev. Lett. 116, 056401 (2016)] is
examined by comparing its exciton spectra with the radial equation (RE)
solutions. While the SHM is found to describe the nonhydrogenic exciton Rydberg
series reasonably well, it fails to account for the linear decrease of the
exciton energy with the orbital quantum number $m$. The exciton Bohr orbit
shrinks as $\lvert m\rvert$ becomes larger resulting in increased strength of
the electron-hole interaction and a decrease of the exciton energy. The exciton
effective radius expression of the SHM can characterize the exciton radius's
dependence on $n$, but it cannot properly describe the exciton radius's
dependence on $m$, which is the cause of the SHM's poor description of the
exciton energy's $m$-dependence. For monolayer WS$_2$ on the SiO$_2$ substrate,
our calculated $s$ exciton Rydberg series agrees closely with that measured by
optical reflection spectroscopy [Phys. Rev. Lett. 113, 076802 (2014)], while
the calculated $p$ excitons offer an explanation for the two broad features of
a two-photon absorption spectrum [Nature 513, 214 (2014)]. Our calculated
exciton energies for monolayer TMDs in various dielectric environments compare
favourably with experimental data. Variational wave functions are obtained for
a number of strongly bound exciton states and further used to study the Stark
effects in monolayer TMDs, an analytical expression being deduced which yields
a redshift of the ground state energy to a good accuracy. The numerical
solution of the RE combined with the variational method provides a simple and
effective approach for the study of excitons in monolayer TMDs.
