Implementation of Time-dependent Hartree-Fock in the real-space Octopus code

Time-dependent Hartree-Fock (TDHF), although an old approach, still finds its place in electronic structure theory because of its relatively cheap cost compared to more accurate post Hartree-Fock methods like higher order Configuration Interaction (CI) or Coupled Cluster. Its version under the Tamm-Dancoff approximation is Configuration Interaction Singles, which is widely used and particularly applicable to big molecules where more accurate methods may be unfeasibly expensive. TDHF also has similar building blocks as compared to other approaches which allows us to implement other approaches using the framework from TDHF. As its application is mostly aimed towards atoms and molecules, TDHF is mostly implemented using Gaussian basis sets. Real-space implementation of TDHF is rarely seen because of the additional cost of solving exchange integrals in grid-based approaches, which is also the case for ground-state Hartree-Fock and hybrid DFT functionals.

We have implemented TDHF in Octopus, a real-space code, and performed a benchmark study of excitation energies from full TDHF and TDHF-TDA on a set of molecules, comparing to Gaussian implementation [Gould et al., Phys. Chem. Lett. 13, 2452 (2022)] and also to the “Theoretical Best Estimate” of Loos et al., J. Chem. Theory Comput. 14 4360 (2018). We also compare the absorption spectrum from full TDHF to that from real-time TDHF. We show solutions to the difficulties in solving the ground-state HF in real-space, which is the first step to TDHF, in terms of the starting guess for SCF, mixing schemes for faster convergence, and the Adaptively Compressed Exchange (ACE) formalism to calculate the exact exchange.