In this work, we present an excitonic method, denoted as excitonic configuration interaction (ECI), which employs an excitonic basis comprising not only GS and LE products but also multilocal excitation (MLE) products, i.e., those with more than one excited site state. Furthermore, it treats the ground and the excited full-system states on an equal footing, which is not characteristic for the exciton models. For these reasons, ECI is a generalization of the exciton approach, which is why we call it an excitonic (or exciton-like) method rather than an exciton method. The full-system Hamiltonian matrix in the excitonic basis is evaluated within GFT, without additional approximations. This makes ECI somewhat more expensive than the FEM, but-as the evaluation of the GFT formulas is rather cheap-still significantly cheaper than "exact matrix element" methods like the LASSI or ASD methods. Hence, ECI can be viewed as a bridge between exciton models and exact fragment-based methods. We also propose a modification of the GFT formulas so that the site states can be calculated with embedding via arbitrary point charges placed on the positions of the nuclei of the other chromophores. We show that these charges can be varied to optimize the energy of the full-system ground state and used in the calculation of the excited site states. Further, we propose a simple algorithm to exclude the site states that might lead to the violation of SOA, and show that the GFTbased approach can be used even for systems with strong interactions between chromophores. The entire ECI methodology will be presented in detail in Section 3.