High-level coupled-cluster computations of the two lightest trihalogen cations (F-3(+) and Cl-3(+)) predict the ground electronic state to be (X) over tilde (1)A(1). As expected, the trifluorine cation is even less stable than the trichlorine cation, which has been detected and studied experimentally. The Brueckner-reference coupled-cluster doubles and perturbatively connected triples method with a basis set of beyond triple-zeta quality predicts the classical (X) over tilde (1)A(1) F-3(+) --> P-2 F + (II)-I-2 F-2(+) dissociation energy to be 15 kcal/mol. We expect that more complete basis sets and higher levels of theoretical treatment will not qualitatively change this dissociation barrier, and thus the trifluorine cation should be a viable species. The lowest linear triplet states of both F-3(+) and Cl-3(+) at the correlated levels of theory are bound by only 2-3 kcal/mol. The electronic wave function for the (X) over tilde (1)A(1) state of F-3(+) exhibits substantial multireference character and, similar to (X) over tilde (1)A(1) O-3, proves to be a difficult case for single-reference ab initio methods based on a spin-restricted Hartree-Fock (RHF) determinant. More specifically, RHF-based coupled-cluster singles and doubles method and its extension with connected triple excitations predict different orderings of the (X) over tilde (1)A(1) F-3(+) stretching frequencies (omega(1) and omega(3)). Reliable predictions for the harmonic vibrational frequencies of this system are obtained through the use of two Brueckner-reference coupled-cluster methods and a large basis set of beyond triple-zeta quality [our best predictions are omega(1)(A(1)) = 825 cm(-1), omega(2)(A(1)) = 376 cm(-1), omega(3)(B-2) = 752 cm(-1)]. Comparison with the previous ab initio analyses of F: stresses the need for a very high level of treatment of dynamic electron correlation to obtain chemically accurate results. The issue of inversion symmetry breaking in a possible dissociation product of the trifluorine molecular cation, F-2(+), is also addressed and it is shown that a "symmetry dilemma" in the region near the equilibrium F-F distance (similar to 1.3 Angstrom) can be resolved through the use of coupled-cluster methods based on a Brueckner-reference determinant, which has a reference instability shifted away from its position in spin-restricted open-shell and spin-unrestricted Hartree-Fock determinants.