The first principle based theory [1-2] is presented for the construction of highly “accurate” potentially coupled Hamiltonian to explore photoelectron (PE) spectra of NO3 - and 1,3,5-C6H6F3 [3-4], reactive scattering processes of D+ + H2, He + H2 + and F + H2 [5-10], and phase transition of perovskites, LaMnO3 and BaTiO3 [11-12]. Jahn-Teller (JT), Pseudo Jahn-Teller (PJT) and Renner-Teller (RT) types of interactions are investigated. [3-5,7,9,11-12] Theoretically calculated photoelectron spectra show peak by peak correspondence with the experimental and other theoretical findings. [3-4, 11-12] Cross-sections and/or rate constants of D+ + H2 [6], He + H2 + [8] and F + H2 [10] processes obtained from scattering dynamics over global diabatic surfaces of H3 + [5], He + H2 + [7] and F + H2 [9] systems, respectively, exhibit good accord with the experimentally measured ones. The surface phonon modes and the bosonic distribution of their states for H2/D2-Cu(111) dissociative chemisorption process exhibits significant quantum effect [13] as observed in experiments [14]. Optical spectra of LaMnO3 and ferroelectric properties of BaTiO3 show anomalous as well as sharp temperature dependence due to JT/PJT effect both in ground and excited states. Such observations have been investigated theoretically using model [15] as well as ab initio adiabatic as well as diabatic potential energy surfaces (PESs) of MnO6 9- [11] and TiO6 8- [12] to interpret the experimental spectra due to the excitations of the quantum rotors.