Improved RISM-CASSCF Optimization via State-Average Treatment and Damping for Characterizing Excited Molecules in Solution with Multireference Perturbation Theory

Determination of excited states of near-infrared (NIR) bioimaging dyes is a challenging theoretical task because of their energy levels with a small gap and the presence of solvation. In the previous study, we showed that the development of the reference interaction site model coupled with the complete active space second-order perturbation theory, the RISM-CASPT2 method, and its extension with the density matrix renormalization group enabled high accuracy prediction of the photochemical properties of bioimaging-related fluorescent molecules in solution (Shimizu et al., J. Chem. Theory Comput. 2018, 14, 5673–5679). This method, however, has a technical issue in convergence of CASSCF optimization, which was encountered when applying the method to a wider variety of systems; thus, practical applications have been hindered. Here, we present an improved scheme of CASSCF optimization with and without the density matrix renormalization group treatment. Detailed derivations and analysis of the second-order orbital optimization scheme with the inclusion of solvation through RISM revealed the requirement of a correction term to the orbital Hessian matrix. As a practical approach, the state-average RISM-CASPT2 method with damping treatment for solvation is presented for improving the convergence of the calculation under reasonable computational cost. The improved scheme allows for performing accurate and numerically stable theoretical analysis of the bioimaging-related excited state with various types of solvation for a PO-bridged rhodol derivative, which is recently highlighted as a promising photostable NIR dye molecule.