  We develop an extension of the original Reiss-Frisch-Lebowitz scaled particle
theory that can serve as a predictive method for the hard sphere pair
correlation function g(r). The reversible cavity creation work is analyzed both
for a single spherical cavity of arbitrary size, as well as for a pair of
identical such spherical cavities with variable center-to-center separation.
These quantities lead directly to prediction of g(r). Smooth connection
conditions have been identified between the small-cavity situation where the
work can be exactly and completely expressed in terms of g(r), and the
large-cavity regime where macroscopic properties become relevant. Closure
conditions emerge which produce a nonlinear integral equation that must be
satisfied by the pair correlation function. This integral equation has a
structure which straightforwardly generates a solution that is a power series
in density. The results of this series replicate the exact second and third
virial coefficients for the hard sphere system via the contact value of the
pair correlation function. The predicted fourth virial coefficient is
approximately 0.6 percent lower than the known exact value. Detailed numerical
analysis of the nonlinear integral equation has been deferred to the sequel
(following paper)
