The identity used to derive the virial form of the path integral pressure estimator [13, 6]
assumes that the surface terms generated by the
implied integration by parts vanish.
This assumption is only valid if the size of the path integral
polymer chain is small
compared to the system parallelpiped. If this assumption is
not valid, additional normal modes beyond the cyclic polymer centers
of masses, the 
, must be introduced
where 
is the 
mode
and 
is the force on that mode
( 
).
This more general form of the identity can also be expressed using
multiple level staging variables [1] by simply taking
to be the ``end point bead''
[1] as opposed to the normal mode.
The one end point
bead staging form of the identity is exactly Eq. (5.1)
with the centroid replaced by the single end point bead, 
,
of the one stage transformation. Finally, if the polymer
chain molecule is large compared to the system parallelpiped,
the image approximation to the free particle propagator also
must be introduced (see Appendix B)[40, 41].
The disadvantage of working with the primitive form of the
pressure estimator as opposed to the virial form
is that the variance of the primitive estimator diverges as the number of
discretizations, M, goes to infinity[13].
In practice, M is finite and
indeed is taken just large enough that the path integral can be said to
have converged. Therefore,
the variance of the virial estimator is generally found to be within a factor
of two of the primitive estimator even in problems involving large
numbers of discretizations, 
. In fact, it is
recommended that both estimators be tabulated. Deviations
of the virial from the primitive estimator can indicate violations
of the conditions of the proof (or more likely poor convergence of the
simulation).
In general, the size of the path integral polymer chains is small compared to the system parallelpiped. The preceding discussion should make clear that the complications of avoiding this limit are such that studying a larger system is the best recourse. It is, therefore, formally and practically possible to generate configurations in the isothermal isobaric ensemble by assuming the first mode, only (end point or centroid), is coupled to the volume. This approximation is equivalent to the finite polymer size approximation because limits of integration of the first mode, only, are, in this approximation, bound by the volume. That is, the dependence of the canonical partition function on the volume arises only through the spatial variation of the first mode of each psuedomolecule.
The equations of motion necessary to generate the isotropic isothermal-isobaric ensemble assuming that the polymer chains are small compared to the volume are
The Eqs. (5.3) have the conserved quantity
The equations of motion necessary to generate the full flexible isothermal-isobaric ensemble assuming that the polymer chains are small compared to the volume are
Eqs. (5.5) have the conserved quantity
At present, there is no overiding theoretical reason to prefer the reduced methods (first mode volumetric scaling) over the full methods (all mode volumetric scaling) unless M is taken excessively large. The computer time necessary to evolve a given system is the same for both types of schemes. However, the convergence of the reduced methods will, in general, be more rapid.
