Double excitations and state-to-state transition dipoles in pi-pi* excited singlet states of linear polyenes: Time-dependent density-functional theory versus multiconfigurational methods

The effect of static and dynamic electron correlation on the nature of excited states and state-to-state transition dipole moments is studied with a multideterminant wave function approach on the example of all-trans linear polyenes (C4H6, C6H8, and C8H10). Symmetry-forbidden singlet nA(g) states were found to separate into three groups: purely single, mostly single, and mostly double excitations. The excited-state absorption spectrum is dominated by two bright transitions: 1B(u)-2A(g) and 1B(u)-mA(g), where mA(g) is the state, corresponding to two-electron excitation from the highest occupied to lowest unoccupied molecular orbital. The richness of the excited-state absorption spectra and strong mixing of the doubly excited determinants into lower-nA(g) states, reported previously at the complete active space self-consistent field level of theory, were found to be an artifact of the smaller active space, limited to pi orbitals. When dynamic sigma-pi correlation is taken into account, single- and double-excited states become relatively well separated at least at the equilibrium geometry of the ground state. This electronic structure is closely reproduced within time-dependent density-functional theory (TD DFT), where double excitations appear in a second-order coupled electronic oscillator formalism and do not mix with the single excitations obtained within the linear response. An extension of TD DFT is proposed, where the Tamm-Dancoff approximation (TDA) is invoked after the linear response equations are solved (a posteriori TDA). The numerical performance of this extension is validated against multideterminant-wave-function and quadratic-response TD DFT results. It is recommended for use with a sum-over-states approach to predict the nonlinear optical properties of conjugated molecules.