The self-trapping by the nondiagonal particle-phonon interaction between two quasi-degenerate energy levels of excitonic system, is studied. We propose this is realized in charge transfer exciton, where the directions of the polarization give the quasi-degeneracy. It is shown that this mechanism, unlike the conventional diagonal one, allows a coexistence and resonance of the free and self-trapped states even in one-dimensional systems and a quantitative theory for the optical properties (light absorption and time-resolved luminescence) of the resonating states is presented. This theory gives a consistent resolution for the long-standing puzzles in quasi-one-dimensional compound A-PMDA.PACS numbers: 71.38.+i, 78.66, 05.10.Ln The charge transfer (CT) exciton in one-dimension (1D) attracts great interests due to its peculiar features, e.g., strong interaction with the phonons, magnons, and large nonlinearity in the optical responses. However the theoretical understanding of it is rather limited compared with that of Wannier and Frenkel excitons, because of its intermediate radius and strong coupling nature. E.g., although the prototypical compound Anthracene-PMDA (A-PMDA), which shows unusual vibronic structure in the optical spectra, is thoroughly experimentally investigated from 70-th till nowdays [1][2][3][4][5][6][7][8], even qualitative theoretical understanding it's optical response is missing.The most dramatic phenomenon which occurs at intermediate or strong coupling with the phonons is the self-trapping where the free (F) state of polaron with small lattice deformation coexists and resonates with the state with strong lattice relaxation [9,10]. Although the latter state is Bloch invariant it is traditionally classified as being self-trapped (ST) to stress larger degree of ionic distortion around the quasiparticle. However, this self-trapping scenario was not considered yet since the F-ST resonance is prohibited by Rashba-Toyozawa theorem (RTT) in 1D systems [9,10]. Although the RTT is probably valid for one quasiparticle state in the relevant energy range, the neglect of it's domain of definition caused severe delusion in the physics of 1D systems because RTT was treated as the strict ban for F-ST coexistence in 1D. However, when there are two quasi-degenerate quasiparticle states, i.e. the conditions of the theorem are violated, one can consider the quasi-degenerate self-trapping (QDST) mechanism, when the F-ST resonance is driven by nondiagonal interaction with phonons with respect to quasiparticle levels. An example of QDST was observed in the mixed valence systems [11].In this paper we first demonstrate by exact diagrammatic Monte Carlo method [12] that QDST is general phenomenon pertinent for 1D lattice, study the optic properties of resonating QDST states, and resolve longstanding puzzle of the quasi-1D system A-PMDA [1][2][3][4][5][6][7][8]. The minimal model to demonstrate F-ST coexistence in 1D systems involves one optic phonon branch with frequency ω = 0.1, two quasiparticle branches with ener...