Implementation of Ensemble Density Functional Theory for excited states in the real-space Octopus code

Ensemble Density Functional Theory is a method to calculate excitation energies from an ensemble of ground and excited states, which can resolve some shortcomings – like multiplet splittings and double excitations – of Time-Dependent Density Functional Theory (TDDFT) approximations. Whereas many EDFT formulations require self-consistent ensemble calculations, the computationally efficient Direct Ensemble Correction (DEC) takes the form of a correction to the Kohn-Sham energy differences, as in linear-response TDDFT [Yang et al., Phys. Rev. Lett. 119, 033003 (2017)]. This theory, in the symmetry-eigenstate Hartree-exchange approximation, has already been applied to model systems and atoms, but now we implement DEC in the real-space Octopus code to be able to apply it to larger real systems. Octopus is well-suited for this study because it can handle model systems as well as real systems, and both finite and periodic boundary conditions. We compare the results from DEC to other standard excited-state approaches like Configuration Interaction Singles, and both linear-response and time-propagation Time-Dependent Hartree-Fock and TDDFT. We also analyze the computational cost and accuracy of DEC compared to these different approaches.