  The NV$^{-}$ color center in diamond has been demonstrated as a powerful
nanosensor for quantum metrology, due to the sensitivity of its optical and
spin properties to external electric, magnetic, and strain fields. In view of
these applications, we use quantum embedding to derive a many-body description
of strain and charge induced Stark effects on the NV$^{-}$ center. We quantify
how strain longitudinal to the axis of NV$^{-}$ shifts the excited states in
energy, while strain with a component transverse to the NV axis splits the
degeneracies of the $^{3}E$ and $^{1}E$ states. The largest effects are for the
optically relevant $^{3}E$ manifold, which splits into $E_{x}$ and $E_{y}$ with
transverse strain. From these responses we extract strain susceptibilities for
the $E_{x/y}$ states within the quasi-linear regime. Additionally, we study the
many-body dipole matrix elements of the NV$^{-}$ and find a permanent dipole
1.64 D at zero strain, which is $\sim 30 \%$ smaller than that obtained from
recent density functional theory calculations. We also determine the transition
dipole between the $E_{x}$ and $E_{y}$ and how it evolves with strain.
