\textit{Ab Initio} Adiabatic Potential Energy Surfaces and Non-adiabatic Couplings for O$_3$: Construction of Four State Diabatic Hamiltonian
We compute highly accurate first principle based \textit{ab initio} adiabatic potential energy surfaces (PESs) using State-Averaged Multi-Configurational Self-Consistent Field (SA-MCSCF) followed by internally contracted Multi-Reference Configuration Interaction method incorporating fixed-reference Davidson corrections [ic-MRCI(Q)], where a full valence active space of 18 electrons in 12 orbitals and aug-cc-pVQZ basis set are employed for the low-lying four singlet electronic states of ozone ($\tilde{X}^1A'$, $1~^1A''$, $1~^1A'$ and $2^1A''$). It accurately reproduces the dissociation energies of ozone (1.101 eV) as well as the molecular oxygen (5.106 eV) along with vibrational frequencies of O$_3$ in comparison with experimental data. To ensure appropriate accuracy and proper convergence in the interaction as well as asymptotic regions, we (a) extend the number of electronic states in SA-MCSCF calculation (singlet as well as triplet and quintet); (b) systematically expand the active space [(12e,9o) $\rightarrow$ (18e,12o) $\rightarrow$ (24e,15o)] and basis set size (AVDZ $\rightarrow$ AV6Z $\rightarrow$ Complete Basis Set limit); (c) incorporate multi-reference character along with Davidson correction. Conical intersections between the adjacent electronic states (1-2, 2-3 and 3-4) are located at \textit{C}$_{2v}$, \textit{D}$_{3h}$ as well as \textit{C}$_{s}$ geometries through the four-state adiabatic-to-diabatic transformation of non-adiabatic coupling terms (NACTs) computed at Coupled-Perturbed Multi-Configurational Self-Consistent Field (CP-MCSCF) method along the circular contours. Finally, we present: (a) ic-MRCI(Q) calculated minimum energy path of incoming oxygen to the diatom (O$_2$) is devoid of any ``reef'' feature; (b) NACTs and diabatic PES matrix elements as function of hyperangles ($\theta$,$\phi$) at a fixed hyperradius $\rho = 4$ Bohr for a four state sub-Hilbert space.