Development of analytic energy gradient method in nuclear orbital plus molecular orbital theory

This study formulates the analytic energy gradients in the Hartree-Fock calculations of the NOMO theory, which simultaneously determines nuclear and electronic wave functions without the Born-Oppenheimer approximation. The formulations correspond to the translation- and rotation-contaminated (TRC), translation-free (TF), and translation- and rotation-free (TRF) treatments. The optimizations of the orbital centers for several diatomic molecules, which have been performed by using the analytic energy gradients, have given the averaged nuclear distances [R₀] reflecting the quantum effects of nuclei and the anharmonicity of the potential energy surfaces. The numerical assessments have clarified that the effects of eliminating the translational and rotational contaminations, i.e., the TRF effects are important to improve the evaluations of {R₀}, especially for the molecules including hydrogen atoms.