  In previous work devoted to {\it ab initio} calculations of hyperfine
structure constants in nitrogen and fluorine atoms, we observed sizeable
relativistic effects, a priori unexpected for such light systems, that can even
largely dominate over electron correlation. We observed that the atomic wave
functions calculated in the Breit-Pauli approximation describe adequately the
relevant atomic levels and hyperfine structures, even in cases for which a
small relativistic $LS$-term mixing becomes crucial. In the present work we
identify new levels belonging to the spectroscopic terms $2p^4(^3\!P) 3d ~ \;
^{2,4}\!(P,D,F)$ of the fluorine atom, for which correlation effects on the
hyperfine structures are small, but relativistic $LS$-term admixtures are
decisive to correctly reproduce the experimental values. The Breit-Pauli
analysis of the hyperfine matrix elements nails cases with large cancellation,
either between $LS$ pairs for individual hyperfine operators, or between the
orbital and the spin-dipole contributions. Multiconfiguration
Dirac-Hartree-Fock calculations are performed to support the Breit-Pauli
analysis.
