We have taken K-band spectra covering seven cooling flow clusters. The spec
tra show many of the 1-OS transitions of molecular hydrogen, as well as som
e of the higher vibrational transitions, and some lines of ionized hydrogen
. The line ratios allow us to conclude that the rotational states of the fi
rst excited vibrational state are in approximate local thermodynamic equili
brium (LTE), so that densities above 10(5) cm(-3) are likely, but there is
evidence that the higher vibrational states are not in LTE. The lack of pre
ssure balance between the molecular gas and the ionized components emphasiz
es the need for dynamic models of the gas. The ratios of the ionized to mol
ecular lines are relatively constant but lower than from starburst regions,
indicating that alternative heating mechanisms are necessary.