  In this work, we provide a detailed derivation of Foldy-Wouthuysen (FW)
transformation for two-dimensional (2D) gapped Dirac fermions under external
fields and apply the formalism to study valley excitons in 2D semiconducting
materials. Similar to relativistic quantum few-body problem, the gapped Dirac
equation can be transformed into a Schr\"{o}dinger equation with "relativistic"
correction terms. In this 2D materials system, the correction terms can be
interpreted as the Berry-curvature effect. The Hamiltonian for a valley exciton
in external fields can be written based on the FW transformed Dirac
Hamiltonian. Various valley-dependent effects on excitons, such as
fine-structure splittings of exciton energy levels, valley-selected exciton
transitions, and exciton valley Zeeman effect are discussed within this
framework.
