Extended Born-Oppenheimer equations for non-Abelian situations: A study on NO₃ radical and 1,3,5-C₆H₃F₃⁺ radical cation

Abstract

Formulation of single-surface extended Born-Oppenheimer (EBO) equation is a long-standing problem for three or higher dimensional sub-Hilbert space mainly due to the difficulty to include the contribution of off-diagonal elements of anti-symmetric nonadiabatic coupling matrix (τ→) as the diagonal ones. Diagonalization of a vector matrix is possible only when its components commute with each other (i.e. curl τ→ = 0→) and thereby, that vector matrix (τ→) can be expressed as a product of a vector function and an antisymmetric scalar matrix. For two electronic state sub-Hilbert space (Abelian case), since curl τ→ = 0→ naturally, such factorization is readily obtained. In non-Abelian cases, for systems with three or more than three sub-Hilbert space, such factorization of nonadiabatic coupling matrix (NACM) may be achievable approximately (Sarkar and Adhikari, 2006; Sarkar and Adhikari, 2008), which will be explored further both theoretically and numerically. In this article, we take an attempt to generalize the single-surface EBO equations for higher dimensional sub-Hilbert space by selecting two realistic molecular systems containing Jahn-Teller (JT) conical intersections (CIs). The curl equations and gauge invariance (GI) of the eigenvalues of the NACM for NO₃ radical and 1,3,5-C₆H₃F₃⁺ (TFBZ+⁺) radical cation are computed numerically, which clearly indicates the possibility to formulate single-surface EBO equations.