  We compare the Frequency-Resolved Frozen Phonon Multislice (FRFPMS) method,
introduced in Phys. Rev. Lett. 124, 025501 (2020), with other theoretical
approaches used to account for the inelastic scattering of high energy
electrons, namely the first-order Born approximation and the quantum excitation
of phonons model. We show, that these theories lead to similar expressions for
the single inelastically scattered intensity as a function of momentum transfer
for an anisotropic quantum harmonic oscillator in a weak phase object
approximation of the scattered waves, except for a too small smearing of the
scattering potential by the effective Debye-Waller factor (DWF) in the FRFPMS
method. We propose that this issue can be fixed by including an explicit DWF
smearing into the potential and demonstrate numerically, that in any realistic
situation, a FRFPMS approach revised in this way, correctly accounts for the
single inelastically scattered intensity and the correct elastic scattering
intensity. Furthermore our simulations illustrate that the only requirement for
such a revised FRFPMS method is the smallness of mean squared displacements for
all atomic species in all frequency bins. The analytical considerations for the
FRFPMS method also explain the $1/\omega^2$-scaling of FRFPMS spectra observed
in Phys. Rev. B 104, 104301 (2021) by the use of classical statistics in the
molecular dynamics simulation. Moreover, we find that the FRFPMS method
inherently adds the contributions of phonon loss and gain within each frequency
bin. Both of these issues related to the frequency-scaling can be fixed by a
system-independent post-processing step.
