  Like adiabatic time-dependent density-functional theory (TD-DFT), the
Bethe-Salpeter equation (BSE) formalism of many-body perturbation theory, in
its static approximation, is "blind" to double (and higher) excitations, which
are ubiquitous, for example, in conjugated molecules like polyenes. Here, we
apply the spin-flip \textit{ansatz} (which considers the lowest triplet state
as the reference configuration instead of the singlet ground state) to the BSE
formalism in order to access, in particular, double excitations. The present
scheme is based on a spin-unrestricted version of the $GW$ approximation
employed to compute the charged excitations and screened Coulomb potential
required for the BSE calculations. Dynamical corrections to the static BSE
optical excitations are taken into account via an unrestricted generalization
of our recently developed (renormalized) perturbative treatment. The
performance of the present spin-flip BSE formalism is illustrated by computing
excited-state energies of the beryllium atom, the hydrogen molecule at various
bond lengths, and cyclobutadiene in its rectangular and square-planar
geometries.
