  An experiment utilizing a TlF molecular beam is being developed by the
CeNTREX collaboration to search for hadronic interactions that violate both
time-reversal (T) and parity (P) invariance. Here we propose to use the same
beam to look for a T-invariance conserving but P-nonconserving (PNC) effect
induced by the anapole moment of the Tl nucleus, via a vector coupling of the
two nuclear spins in TlF. To measure the nuclear anapole moment, the dc
electric and magnetic fields in CeNTREX are replaced by rf fields resonant with
a nuclear spin flip transition. We adapt the relativistic coupled cluster
method in a combination with relativistic density functional theory for the
calculation of the molecular PNC spin-spin vector coupling constant that links
the experimental signal with the anapole moment. The value of the P-conserving
spin-spin coupling constant calculated within the same approach is found to be
in good agreement with available experimental data.
