  Nuclei with an octupole deformation have a non-zero electric octupole moment,
electric dipole moment (EDM), Schiff moment and magnetic quadrupole moment
(MQM) in the intrinsic frame which rotates with the nucleus. In a state with
definite angular momentum in the laboratory frame, these moments are forbidden
by parity (P) and time reversal invariance (T) conservation, meaning their
expectation values vanish due to nuclear rotation. However, nuclei with an
octupole deformation have doublets of close opposite parity rotational states
with the same spin, which are mixed by T,P-odd nuclear forces. This mixing
produces the orientation of the nuclear axis along nuclear spin and all moments
existing in the intrinsic frame appear in the laboratory frame (provided the
nuclear spin I is sufficiently large to allow such moment). Such a mechanism
produces enhanced T,P-violating nuclear moments. This enhancement also takes
place in nuclei with a soft octupole vibration mode. Schiff moments in such
nuclei have been calculated in previous works. In the present paper we consider
the magnetic quadrupole moment which appears in isotopes with nuclear spin $I
\geq 1$. Magnetic interaction between the nuclear MQM and electrons produces an
atomic EDM and T,P-violating nuclear spin - molecular axis interaction
constants for molecules in electronic states with non-zero electron angular
momentum. Measurements of these constants may be used to test CP-violation
theories and search for axion dark matter in atomic, molecular and solid state
experiments. Potential candidate nuclei include 153Eu, 161Dy, 221Fr, 223Fr,
223Ra, 223Rn, 225Ac, 227Ac, 229Th, 229Pa, 233U and 235U. We subsequently
consider molecules containing these nuclei (EuO, EuN+, RaF, AcO, AcN+, AcF+,
ThO and ThF+).
