  Frequency dependent specific heat, introduced by Grest and Nagel, offers
valuable insight into the vitrification of supercooled liquid. We calculate
this quantity and other thermodynamic properties of supercooled liquid water by
varying temperature and density across the "no man's land" all the way to the
formation of amorphous ice. The calculations are aided by very long computer
simulations, often more than 50 $\mu s$ long. Density fluctuations that arise
from the proximity to a putative liquid-liquid (LL) transition at 228 K, cast a
long shadow on the properties of water, both above and below the LL transition.
We carry out the calculation of the quantum mechanical static and
frequency-dependent specific heats by combining seminal works by Lebowitz,
Percus, and Verlet and Grest and Nagel with the harmonic approximation for the
density of states. The obtained values are in quantitative agreement with all
available experimental and numerical results of specific heats for both
supercooled water and ice. We calculate the entropy at all the state points by
integrating the specific heat. We find that the quantum corrected-contributions
of intermolecular vibrational entropy dominate the excess entropy of amorphous
phases over the crystal over a wide range of temperature. Interestingly, the
vibrational entropy lowers the Kauzmann temperature, $T_{\rm K}$, to 130 K,
just below the experimental glass-to-liquid water transition temperature,
$T_{\rm g}$, of 136 K and the calculated $T_{\rm g}$ of 135 K in our previous
study. A straightforward extrapolation of high temperature entropy from 250 K
to below however would give a much higher value of $T_{\rm K}$ $\sim$ 190 K.
The calculation of Lindemann ratios places the melting of amorphous ice $\sim$
135 K. The amorphous state exhibits an extremely short correlation length for
the distance dependence of orientational correlation.
