  We develop a time dependent second order Green's function theory (GF2) for
calculating neutral excited states in molecules. The equation of motion for the
lesser Green's function (GF) is derived within the adiabatic approximation to
the Kadanoff-Baym (KB) equation using the second order Born approximation for
the self-energy. In the linear response regime, we recast the time dependent KB
equation into a Bethe-Salpeter-like equation (GF2-BSE), with a kernel
approximated by the second order Coulomb self-energy. We then apply our GF2-BSE
to a set of molecules and atoms and find that GF2-BSE is superior to
configuration interaction with singles (CIS) and/or time dependent Hartree-Fock
(TDHF), particularly for charge transfer excitations, and is comparable to CIS
with perturbative doubles (CIS(D)) in most cases.
