  Bounded excitons in transition metal dichalcogenides monolayers lead to
numerous opto-electronic applications, which require a detailed understanding
of the exciton dynamics. The dynamical properties of excitons with finite
momentum transfer $\textbf{Q}$ can be investigated experimentally using
electron energy-loss (EEL) spectroscopy. The EEL spectrum depends on the
response function of the material which in turn is determined by the exciton
energies and eigenvectors in the exciton band structure. In this work, we
utilize ab initio density-functional theory plus Bethe-Salpeter equation
(DFT+BSE) approach to explore the exciton band structure and also
$\textbf{Q}$-resolved EEL spectrum in monolayer ${\mathrm{WSe}}_{2}$. In
particular, we carefully examine the discrepancies and connections among the
existing EEL spectrum formulas for quasi-two-dimensional (2D) systems, and
establish a proper definition of the EEL spectrum, which is then used to
calculate the EEL spectra of monolayer ${\mathrm{WSe}}_{2}$. We find that
remarkably, the dispersion of the calculated lowest-energy EELS peaks for the
in-plane momentum transfer follows almost precisely the non-parabolic upper
band of the lowest bright A exciton, and also agrees well with the previous
experiment. Furthermore, we show that only the bright exciton with its electric
dipole being parallel to the direction of the transfered momentum is excited,
i.e., EEL spectroscopy selectively probes bright exciton bands. This explains
why only the upper band of the A exciton, which is a longitudinal exciton with
an in-plane dipole moment, was observed in the previous experiment. Our
findings will stimulate further EEL experiments to measure other branches of
the exciton band structure, such as the parabolic lower band of the A exciton,
and hence will lead to a better understanding of the exciton dynamics in
quasi-2D materials.
