Configuration interaction singles, time-dependent Hartree-Fock, and time-dependent density functional theory for the electronic excited states of extended systems

A general formalism for time-dependent linear response theory is presented within the framework of linear-combination-of-atomic-orbital crystalline orbital theory for the electronic excited states of infinite one-dimensional lattices (polymers). The formalism encompasses those of time-dependent Hartree-Fock theory (TDHF), time-dependent density functional theory (TDDFT), and configuration interaction singles theory (CIS) (as the Tamm-Dancoff approximation to TDHF) as particular cases. These single-excitation theories are implemented by using a trial-vector algorithm, such that the atomic-orbital-based two-electron integrals are recomputed as needed and the transformation of these integrals from the atomic-orbital basis to the crystalline-orbital basis is avoided. Convergence of the calculated excitation energies with respect to the number of unit cells taken into account in the lattice summations (N) and the number of wave vector sampling points (K) is studied taking the lowest singlet and triplet exciton states of all-trans polyethylene as an example. The CIS and TDHF excitation energies of polyethylene show rapid convergence with respect to K and they are substantially smaller than the corresponding Hartree-Fock fundamental band gaps. In contrast, the excitation energies obtained from TDDFT and its modification, the Tamm-Dancoff approximation to TDDFT, show slower convergence with respect to K and the excitation energies to the lowest singlet exciton states tend to collapse to the corresponding Kohn-Sham fundamental band gaps in the limit of K → ∞. We consider this to be a consequence of the incomplete cancellation of the self-interaction energy in the matrix elements of the TDDFT matrix eigenvalue equation, and to be a problem inherent to the current approximate exchange-correlation potentials that decay too rapidly in the asymptotic region.