  The mean spherical approximation (MSA) can be solved semi-analytically for
the Gaussian core model (GCM) and yields - rather surprisingly - exactly the
same expressions for the energy and the virial equations. Taking advantage of
this semi-analytical framework, we apply the concept of the self-consistent
Ornstein-Zernike approximation (SCOZA) to the GCM: a state-dependent function K
is introduced in the MSA closure relation which is determined to enforce
thermodynamic consistency between the compressibility route and either the
virial or energy route. Utilizing standard thermodynamic relations this leads
to two different differential equations for the function K that have to be
solved numerically. Generalizing our concept we propose an
integro-differential-equation based formulation of the SCOZA which, although
requiring a fully numerical solution, has the advantage that it is no longer
restricted to the availability of an analytic solution for a particular system.
Rather it can be used for an arbitrary potential and even in combination with
other closure relations, such as a modification of the hypernetted chain
approximation.
