  (Ultra) Luminous Infrared Galaxies ((U)LIRGs) are objects characterized by
their extreme infrared (8-1000 $\mu$m) luminosities ($L_{LIRG}>10^{11}
$L$_\odot$ and $L_{ULIRG}>10^{12}$ L$_\odot$). The Herschel Comprehensive ULIRG
Emission Survey (HerCULES; PI van der Werf) presents a representative
flux-limited sample of 29 (U)LIRGs that spans the full luminosity range of
these objects (10$^{11}\leq L_\odot \geq10^{13}$). With the \emph{Herschel
Space Observatory}, we observe [CII] 157 $\mu$m, [OI] 63 $\mu$m, and [OI] 145
$\mu$m line emission with PACS, CO J=4-3 through J=13-12, [CI] 370 $\mu$m, and
[CI] 609 $\mu$m with SPIRE, and low-J CO transitions with ground-based
telescopes. The CO ladders of the sample are separated into three classes based
on their excitation level. In 13 of the galaxies, the [OI] 63 $\mu$m emission
line is self absorbed. Comparing the CO excitation to the IRAS 60/100 $\mu$m
ratio and to far infrared luminosity, we find that the CO excitation is more
correlated to the far infrared colors. We present cooling budgets for the
galaxies and find fine-structure line flux deficits in the [CII], [SiII], [OI],
and [CI] lines in the objects with the highest far IR fluxes, but do not
observe this for CO $4\leq J_{upp}\leq13$. In order to study the heating of the
molecular gas, we present a combination of three diagnostic quantities to help
determine the dominant heating source. Using the CO excitation, the CO J=1-0
linewidth, and the AGN contribution, we conclude that galaxies with large CO
linewidths always have high-excitation CO ladders, and often low AGN
contributions, suggesting that mechanical heating is important.
