  The quantum mechanical formula for Mayer's second cluster integral for the
gas of relativistic particles with hard-core interaction is derived. The proper
pion volume calculated with quantum mechanical formula is found to be an order
of magnitude larger than its classical evaluation.
  The second cluster integral for the pion gas is calculated in quantum
mechanical approach with account for both attractive and hard-core repulsive
interactions. It is shown that, in the second cluster approximation, the
repulsive pion-pion-interactions as well as the finite width of resonances give
important but almost canceling contributions. In contrast, an appreciable
deviation from the ideal gas of pions and pion resonances is observed beyond
the second cluster approximation in the framework of the Van der Waals
excluded-volume model.
