Liquid water is not currently stable on the surface of Mars; however, transient liquid water (ice melt) may occur if the surface temperature is between the melting and boiling points. Such conditions are met on Mars with current surface pressures and obliquity due to the large diurnal range of surface temperatures. This yields the potential for transient, nonequilibrium liquid water. A general circulation model is used to undertake an initial exploration of the variation of this transient liquid water potential (TLWP) for different obliquities and over a range of increased pressures representing progressively earlier phases of Martian geological history. At higher obliquities and slightly higher surface pressures (<50 mbar), TLWP conditions are met over a very large fraction of the planet. As the surface pressure is increased above about 50--100 mbar, however, increased atmospheric thermal blanketing reduces the diurnal surface temperature range, essentially eliminating the possibility of even transient liquid water. At high enough pressures, the mean temperature is sufficiently elevated to allow stable liquid water. Thus the potential for liquid water on Mars has not decreased monotonically over planetary history as the atmosphere was lost. Instead, a distinct minimum in TLWP (the dead zone) will have occurred during the extended period for which pressures were in the middle range between about 0.1 and 1 bar. This has direct and restrictive implications for chemical weathering and life. The fundamental conclusion of this study is largely insensitive to invocation of brines and to more detailed treatment of atmospheric radiative processes.