Molecular vibrations of doublet states obtained using Koopmans’ theorem

We report the Lagrangian-based Koopmans ionization potential (IP)/electron affinity (EA) method, denoted Koopmans-IP/EA, for the gradients and Hessians of singly ionized and electron-attached states starting from a closed-shell restricted Hartree-Fock (RHF) reference. By exploiting Koopmans' theorem, the method avoids spin contamination inherent to unrestricted approaches and provides a description of pure doublet states without the need for an explicit open-shell treatment. Analytic first and second derivatives of the Koopmans-IP/EA Lagrangian with respect to nuclear displacements and external electric fields are derived and implemented in the koala program package, enabling geometry optimizations, harmonic vibrational frequencies, and infrared intensities of doublet states at a computational cost comparable to a ground-state Hartree-Fock Hessian. The method is benchmarked against unrestricted Hartree-Fock, restricted open-shell Hartree-Fock, and equation-of-motion IP coupled cluster singles-doubles (EOM-IP-CCSD) reference data for a set of open-shell systems, including C2H4+, CH2O+, NH3+, NO2, and NO3. Qualitative disagreement with EOM-IP-CCSD is only found for certain cases with pronounced orbital relaxation, e.g., for CH2O+. The computational overhead of the Koopmans-IP/EA Hessian relative to the ground-state RHF Hessian is assessed for a chloro bay-substituted CF3-tetraazaperopyrene-Cl4 anion, yielding an overhead of ∼121%, demonstrating that the method is applicable to systems of realistic size, avoiding spin-contamination.