Fig. 2: Loss function \(-{{{{{{{\rm{Im}}}}}}}}[{\varepsilon }^{-1}]\) of liquid water at selected transferred momenta.

The transferred momenta correspond to (a) q = 0.32 Å−1 and (b) q = 0.95 Å−1. The results are obtained by solving the time-dependent density functional theory (TDDFT) equation with different exchange-correlation kernels fxc: no fxc (blue), the bootstrap fxc (orange), and the modified long-range corrected kernel \({f}_{{{{{{{{\rm{xc}}}}}}}}}^{*}\) (purple) introduced in this work. Experimental curves from Hayashi and Hiraoka9 are shown for comparison.