Orthogonally constrained CASSCF framework: Newton-Raphson orbital optimization and nuclear gradients

In a recent work [S. Yalouz and V. Robert, J. Chem. Theory Comput. 19, 1381 (2023)], we introduced the foundations of an orthogonally constrained complete active space self-consistent field (OC-CASSCF) framework that produces state-specific molecular orbitals for mutually orthogonal multiconfigurational electronic states. In the present study, we extend this approach by incorporating a Newton-Raphson orbital-optimization scheme, for which we derive the analytical expressions of the orbital gradient and Hessian. Furthermore, we outline a practical route toward the evaluation of analytical nuclear gradients, enabling geometry optimizations within the OC-CASSCF formalism. Benchmark calculations on the three lowest singlet states of LiH and H2O molecules demonstrate a systematic improvement as compared to conventional state-averaged CASSCF, even when using modestly sized active spaces.