  The bandgap of insulating materials is renormalized in various ways by the
electron-phonon interaction owing to the dynamical and quantum fluctuations of
nuclei. These fluctuation effects are considered in the perturbative
Allen-Heine-Cardona theory using the formulae for the Fan-Migdal and
Debye-Waller terms. However, the material dependence is not clear in the
formulae. Thus, in this study, we focus on the analytical form of the
Debye-Waller term and find that the term can be reformulated using the momentum
matrix. In addition, the optical selection rule is found to play a role. For
diamond-type materials, the Debye-Waller term can be approximately decomposed
into a product of the optical transition energy, the mean square displacement
of nuclei, and the dipole transition probability. The decomposition can also be
applied with an additional approximation to zinc-blende-type materials, as
revealed by our first-principles calculation. The magnitudes of the
Debye-Waller term of several materials can thus be estimated using basic
physical quantities prior to performing the calculation of the electron-phonon
interaction.
