Approximate time-dependent density functional theory

In this contribution, we summarize recent advances in the development of the time-dependent density functional based tight-bind method (TD-DFTB). The scheme is characterized by (i) the use of a limited, usually minimal basis, (ii) a two-center approximation for the Kohn-Sham Hamiltonian, (iii) a second-order functional expansion of the total energy and (iv) the simplification of two-electron integrals in the Mulliken approximation. The range of validity of these approximations is assessed by comparison to first principles time-dependent density functional theory (TDDFT) calculations in converged basis sets. Within the TD-DFTB framework, several strategies to solve the time-dependent Kohn-Sham equations are discussed. These include a linear response treatment in the frequency domain, direct propagation in the time domain leading to a non-adiabatic molecular dynamics scheme, and an O(N) implementation. The numerical efficiency is investigated and several illustrative applications of the method are reviewed. Results are reported for the absorption and luminescence spectra of passivated and functionalized Silicon nanocrystals with hundreds of atoms, the potential energy surfaces of azulene and Retinal chromophores, and the vibrational response of C60 due to electronic excitation with femtosecond laser pulses. Within its well characterized range of validity we find that TD-DFTB provides an excellent ratio of accuracy over computational cost.