Large-Scale Spectroscopic Mapping of the $\rho$ Ophiuchi Molecular Cloud
  Complex I. The C$_{2}$H to N$_2$H$^+$ Ratio as a Signpost of Cloud
  Characteristics
  We present 2.5-square-degree C$_{2}$H N=1-0 and N$_2$H$^+$ J=1-0 maps of the
$\rho$ Ophiuchi molecular cloud complex. These are the first large-scale maps
of the $\rho$ Ophiuchi molecular cloud complex with these two tracers. The
C$_{2}$H emission is spatially more extended than the N$_2$H$^+$ emission. One
faint N$_2$H$^+$ clump Oph-M and one C$_{2}$H ring Oph-RingSW are identified
for the first time. The observed C$_{2}$H to N$_{2}$H$^{+}$ abundance ratio
([C$_{2}$H]/[N$_{2}$H$^{+}$]) varies between 5 and 110. We modeled the C$_{2}$H
and N$_2$H$^+$ abundances with 1-D chemical models which show a clear decline
of [C$_2$H]/[N$_2$H$^+$] with chemical age. Such an evolutionary trend is
little affected by temperatures when they are below 40 K. At high density
(n$_H$ $>$ 10$^5$ cm$^{-3}$), however, the time it takes for the abundance
ratio to drop at least one order of magnitude becomes less than the dynamical
time (e.g., turbulence crossing time $\rm \sim$10$^5$ years). The observed
[C$_2$H]/[N$_2$H$^+$] difference between L1688 and L1689 can be explained by
L1688 having chemically younger gas in relatively less dense regions. The
observed [C$_{2}$H]/[N$_{2}$H$^{+}$] values are the results of time evolution,
accelerated at higher densities. For the relative low density regions in L1688
where only C$_2$H emission was detected, the gas should be chemically younger.
