  We theoretically investigate the dynamic structure factor of a strongly
interacting Fermi gas at the crossover from Bardeen-Cooper-Schrieffer
superfluids to Bose-Einstein condensates, by developing an improved random
phase approximation within the framework of a density functional theory - the
so-called superfluid local density approximation. Compared with the previous
random-phase-approximation studies based on the standard Bogoliubov-de Gennes
equations, the use of the density functional theory greatly improves the
accuracy of the equation of state at the crossover, and leads to a better
description of both collective Bogoliubov-Anderson-Goldstone phonon mode and
single-particle fermionic excitations at small transferred momentum. Near
unitarity, where the s-wave scattering length diverges, we show that the
single-particle excitations start to significantly contribute to the spectrum
of dynamic structure factor once the frequency is above a threshold of the
energy gap at $2\Delta$. The sharp rise in the spectrum at this threshold can
be utilized to measure the pairing gap $\Delta$. Together with the sound
velocity determined from the phonon branch, the dynamic structure factor
provides us some key information of the crossover Fermi superfluid. Our
predictions could be examined in experiments with $^{6}$Li or $^{40}$K atoms
using Bragg spectroscopy.
