  All-electron configuration interaction theory in the framework of the
Dirac-Coulomb Hamiltonian has been applied to the TaN molecule, a promising
candidate in the search for Beyond-Standard-Model physics in both the hadron
and the lepton sector of matter. We obtain in the first excited {$^3\Delta_1$}
state a ${\cal{P,T}}$-odd effective electric field of $36.0 \left[\frac{\rm
GV}{\rm cm}\right]$, a scalar-pseudoscalar ${\cal{P,T}}$-odd interaction
constant of $32.8$ [kHz], and a nuclear magnetic-quadrupole moment interaction
constant of $0.74$ [$\frac{10^{33} {\text{Hz}}}{e\, {\text{cm}}^2}$]. The
latter interaction constant has been obtained with a new approach which we
describe in detail. Using the same highly correlated all-electron wavefunctions
with up to $2.5$ billion expansion terms, we obtain a parallel magnetic
hyperfine interaction constant of $-2954$ [MHz] for the $\rm {^{181}{T}a}$
nucleus, a very large molecule-frame electric dipole moment of $-4.91$ [Debye],
and spectroscopic constants for the four lowest-lying electronic states of the
molecule.
