The deexcitation process of an atom in an optically allowed excited state by a collision with an axially symmetric molecule is considered. In order to make a precise comparison with experimental data that have recently been obtained, we extend previous work for a Penning ionization process [T. Watanabe and K. Katsuura, J. Chem. Phys. 47, 800 (1967)]. Using the straight-line trajectory impact parameter method, the probability of deexcitation in the incident atom is described by a discrete-continuum excitation transfer mechanism. The effects of the ionization yield eta in molecular target and of the molecular anisotropic property of the optical transition dipole are considered. The cross-section formula sigma is presented by a similar formula for Penning ionization of the atomic target by introducing a stereo factor C(lambda) as sigma=C(lambda)[e(4)mu(2)mu(E,perpendicular to)(2)/(4piepsilon(0))(2)(h) over barv](2/5). Here, v is the relative velocity of the colliding system and lambda is given by the ratio lambda=(mu(Eparallel to)/mu(Eperpendicular to)), where mu, mu(Eperpendicular to), mu(Eparallel to) are the transition dipole moments of an excited atom, A-->A*, those of a molecule at energy E for the perpendicular component and the parallel component with respect to molecular axis. Applications to He*(2(1)P)+H-2 (or D-2), Ne*[2p(5)(P-2(1/2))3s P-1(1)]+H-2 (or D-2) systems and systems of the same projectiles on C6H6, (or C6D6) molecules are made. The results for hydrogen molecules are compared with the experimental data. (C) 2004 American Institute of Physics.