  We determine the frequency dependence of the four independent Cartesian
tensor elements of the dielectric function for monoclinic symmetry Y$_2$SiO$_5$
using generalized spectroscopic ellipsometry from 40-1200 cm$^{-1}$. Three
different crystal cuts, each perpendicular to a principle axis, are
investigated. We apply our recently described augmentation of lattice
anharmonicity onto the eigendielectric displacement vector summation approach
[A. Mock et al., Phys. Rev. B 95, 165202 (2017)], and we present and
demonstrate the application of an eigendielectric displacement loss vector
summation approach with anharmonic broadening. We obtain excellent match
between all measured and model calculated dielectric function tensor elements
and all dielectric loss function tensor elements. We obtain 23 A$_{\mathrm{u}}$
and 22 B$_{\mathrm{u}}$ symmetry long wavelength active transverse and
longitudinal optical mode parameters including their eigenvector orientation
within the monoclinic lattice. We perform density functional theory
calculations and obtain 23 A$_{\mathrm{u}}$ symmetry and 22 B$_{\mathrm{u}}$
transverse and longitudinal optical mode parameters and their orientation
within the monoclincic lattice. We compare our results from ellipsometry and
density functional theory and find excellent agreement. We also determine the
static and above reststrahlen spectral range dielectric tensor values and find
a recently derived generalization of the Lyddane-Sachs-Teller relation for
polar phonons in monoclinic symmetry materials satisfied [M. Schubert, Phys.
Rev. Lett. 117, 215502 (2016)].
