That second quantity is known as the configuration integral and is the constant of proportionality between the 1- and 2-particle partition functions, so this quantity itself is given by minus kinetic temperature times log partition function.
For a weakly coupled ensemble, this quantity contains an interaction term proportional to the logarithm of a quantity that can be written as a series via the technique of Mayer cluster expansion. That second quantity is known as the configuration integral and is the constant of proportionality between the 1- and 2-particle partition functions, so this quantity itself is given by minus kinetic temperature times log partition function. This variable is the natural potential in the temperature, volume, particle number basis. It is the Legendre transform of the internal energy with respect to entropy. It represents the amount of useful work that can be obtained from a closed system by reducing it to the standard state. For 10 points, name this quantity given macroscopically by internal energy minus entropy times temperature.
For a weakly coupled ensemble, this quantity contains an interaction term proportional to the logarithm of a quantity that can be written as a series via the technique of Mayer cluster expansion.
Helmholtz free energy [or Helmholtz function; prompt on free energy or F or A]