\textit{Ab Initio} Adiabatic Potential Energy Surfaces and Non-adiabatic Couplings for O$_3$: Construction of Four State Diabatic Hamiltonian

Abstract

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 ($θ$,$φ$) at a fixed hyperradius $ρ= 4$ Bohr for a four state sub-Hilbert space.

Figures (20)

• Figure 1: Schematic diagram of the Jacobi coordinates for the O$_3$ system.
• Figure 2: Four low lying adiabatic PESs of O$_3$ [Figures (a) to (d)] at $\gamma = 90^\circ$ computed using 7S-SA-MCSCF followed by ic-MRCI(Q) with a CAS(18e,12o) and AVQZ basis set. In reference to the reactant channel, O + O$_2$ setting at 0, C$_{2v}$ and D$_{3h}$ minima are at -1.101 eV and -0.264 eV respectively, respectively and the dissociation energy of O$_2$ is indicated as 5.10 eV.
• Figure 3: Figure (a) represents the ground adiabatic $\tilde{X}^1A'$ (u$_1$) PES computed using 7-SA-MCSCF method with CAS(18e,12o) and the AVQZ basis set. Three marked PECs on the PES correspond to $r = 4.1$ Bohr (blue), $r = 2.8$ Bohr (green), and $R = 8.0$ Bohr (red), representing the C$_{2v}$, D$_{3h}$ and O$_2$ dissociation regions via the B2 process, respectively. Figures (b), (c), and (d) presents the comparison between the 7-SA-MCSCF and ic-MRCI(Q) data over the marked PECs using same CAS and basis. All energies are referenced to the asymptotic $\text{O}_2(^3\Sigma_g^-)$ + $\text{O}(^3P)$ channel.
• Figure 4: Comparison of the Dawes et. al. (brown), Schinke et. al. (blue), Mankodi et. al. (purple) PECs with present PEC along the MEP as a function of one of the O-O bond length, R$_1$ (Bohr) while keeping other O-O bond (R$_2$) fixed at 2.3 Bohr. All energies are referenced to the asymptotic $\text{O}_2(^3\Sigma_g^-)$ + $\text{O}(^3P)$ channel.
• Figure 5: Schematic diagram of a contour (red circle) around a CI in Jacobi coordinates for the O$_3$ system.