  We present a range-separated linear-response time-dependent
density-functional theory (TDDFT) which combines a density-functional
approximation for the short-range response kernel and a frequency-dependent
second-order Bethe-Salpeter approximation for the long-range response kernel.
This approach goes beyond the adiabatic approximation usually used in
linear-response TDDFT and aims at improving the accuracy of calculations of
electronic excitation energies of molecular systems. A detailed derivation of
the frequency-dependent second-order Bethe-Salpeter correlation kernel is given
using many-body Green-function theory. Preliminary tests of this
range-separated TDDFT method are presented for the calculation of excitation
energies of the He and Be atoms and small molecules (H2, N2, CO2, H2CO, and
C2H4). The results suggest that the addition of the long-range second-order
Bethe-Salpeter correlation kernel overall slightly improves the excitation
energies.
