  We investigate, for the first time, the thermal signature of a
single-stranded helical molecule that is described beyond usual
nearest-neighbor electron hopping, by analyzing electronic specific heat.
Depending on the hopping of electrons, two different kinds of helical systems
are considered. In one case the hopping is confined within a few neighboring
lattice sites which is referred to as short-range hopping helix, while in the
other case, electrons can hop in all possible sites making the system a
long-range hopping one. These two helices accurately emulate the structures of
single-stranded DNA and protein molecules, respectively. Each helix geometry is
exposed to a transverse electric field applied perpendicular to the helix axis.
The interplay among the helicity, higher-order hopping, and the electric field
has significant impact on thermal response. Our comprehensive theoretical
analysis reveals that, under low-temperature conditions, the short-range
hopping helix exhibits greater sensitivity to temperature compared to the
long-range hopping helix system. Conversely, the scenario reverses in the
high-temperature limit. The thermal response of the helices can be modified
selectively by means of the electric field, and the difference between the
specific heats of the two helices gradually decreases with increasing the field
strength. The molecular handedness, on the other hand, does not have any
appreciable effect on the thermal signature. In addition, we also explore a
significant application of electronic specific heat (ESH). Finally, we discuss
the results of ESH by considering the spin degree of freedom and in the context
of real biological helical systems.
