The NEO-TDDFT methodYang:2018 is a multicomponent extension of the TDDFT method within the NEO framework. It allows the simultaneous calculation of the electronic and proton vibrational excitation energies. In the NEO-TDDFT method, the linear response of the NEO Kohn-Sham system to perturbative external fields is computed. The NEO-TDDFT working equation is
Here, the occupied electronic orbitals are denoted with indices and , whereas the unoccupied electronic orbitals are denoted with indices and . The upper case indices denote protonic orbitals. The solution of Eq. (13.48) provides the electronic and proton vibrational excitation energies , as well as the transition excitation and de-excitation amplitudes, and , respectively. Analogous to the TDDFT method, the Tamm-Dancoff approximation (TDA) can be imposed within the NEO framework, defining the NEO-TDDFT-TDA method that is represented by
The extension of the NEO-TDDFT and NEO-TDDFT-TDA approaches to open-shell electron systems is straightforward.Culpitt:2019 NEO-TDHF and NEO-CIS take the similar form as NEO-TDDFT and NEO-TDA without electron-proton correlation treatment and pure coulomb treatment for proton and electronic exchange correlation.