The partition function for the isotropic isothermal-isobaric ensemble (NPT) is given by [39, 5]
where 
is the canonical partition function at volume V,
is its discrete path integral counterpart Eq. (2.1),
and 
is the external applied pressure.
If the transformation
is applied to Eq. (2.1)
to make the volume dependence explicit, then the expression for the
partition function becomes:
In Appendix B, the general problem of writing a path integral expression for a system in a finite size parallelpiped is reviewed. Inserting Eq. (3.2) into Eq. (3.1) yields the discrete Feynman path integral expression for the isotropic isothermal-isobaric partition
Again, the path integral formalism effectively replaces each classical point particle with a cyclic chain ``molecule'' [30]. Therefore, the quantum mechanical isothermal isobaric ensemble is analogous to the classical isothermal isobaric treatment of cyclic chain molecules.
The standard thermodynamic connection formula,
, can be used
to prove two important identities in the isothermal
isobaric ensemble. The first is the pressure virial theorem
[39, 15]
and the second is work virial theorem[39, 15]
Note, the internal work plus kT balances the external work.
The excess factor of kT can be thought of as the
work done by the ``piston'' to maintain the pressure balance.
(Others favor a formulation where the work balances and the
pressures are different by an additive factor of 
.)
The pressure connection formula can also be used to derive a discrete path integral expression for the average pressure in a quantum mechanical system. Using the relation
and performing the volume derivative of 
Eq.(3.2), it
is straightforward to show
where 
is the derivative of the potential with
respect to the explicit volume dependence
present in the potential energy, only and
Equation (3.8) will be referred to as the primitive
pressure estimator.
Note, this formula must be made consistent with periodic
boundary conditions before it can be applied,
in analogy with standard classical procedures[32].
Also, the specific definition of
is complicated by
the approximations made in truncating the potential energy function
(see Appendix C).
The primitive pressure estimator can be manipulated to produce another pressure estimator in the same manner as the path integral virial kinetic energy estimator is obtained from the primitive kinetic energy estimator [13, 6]. The identity which can be proved via integration by parts [13, 6]
is inserted into Eq.(3.8) to yield
where where 
is the path centroid and 
is the centroid force.
This pressure estimator, the virial pressure estimator,
also must be made consistent with periodic
boundary conditions before it can be applied.
The path integral manifests quantum mechanical effects
by replacing classical point particles with cyclic polymeric
pseudomolecules [30].
The virial pressure estimator,
, is the center of mass pressure virial
of the pseudomolecular system[32, 35]. Finally, the derivation
assumes the ``size'' of the pseudomolecules is small compared
to the volume (i.e. surface terms are neglected).
This approximation is already embodied in the form of path integral expression
utilized in the derivation. If this approximation is not valid,
the image approximation to the free particle propagator
must be introduced (see Appendix B)[40, 41].
Finally, the virial expression can also be derived by assuming
that only the particle centroid is scaled by the volume
or the limits of integration of the other normal modes
are not bound by the volume (see below).
