  Starting from the numerical solution of the 6-band \textbf{k.p} description
of a lattice-mismatched ellipsoidal quantum dot situated inside a nanowire,
including a spin Zeeman effect with values appropriate to a dilute magnetic
semiconductor, we propose and test phenomenological models of the effect of the
built-in strain on the heavy hole, light hole and exciton states. We test the
validity and the limits of a description restricted to a ($\Gamma_8$)
quadruplet of ground states and we demonstrate the role of the interactions of
the light-hole state with light-hole excited states. We show that the built-in
axial strain not only defines the character, heavy-hole or light-hole, of the
ground state, but also mixes significantly the light-hole state with the
split-off band's states: Even for a spin-orbit energy as large as 1 eV, that
mixing induces first-order modifications of properties such as the spin value
and anisotropy, the oscillator strength, and the electron-hole exchange, for
which we extend the description to the light-hole exciton. CdTe/ZnTe quantum
dots are mainly used as a test case but the concepts we discuss apply to many
heterostructures, from mismatched II-VI and III-V quantum dots and nanowires,
to III-V nanostructures submitted to an applied stress and to silicon
nanodevices with even smaller residual strains.
