The ansatz method of infinite summation of higher order diagrams
given in Shukla and Cowley, Phys. Rev. **58**, 2596, (1998), is
extended to the self-consistent phonon theory. We demonstrate the high
accuracy of this approach with respect to the first order
self-consistent (SC1) and improved self-consistent (ISC) phonon theories,
by comparing the results from the ansatz method with their
exact counterparts. The ISC theory is then extended to include the
remaining diagrams of
,
which could not be included in
its earlier formulation. This makes the ISC theory consistent, at
least to
.
The new ISC theory offers a substantial
improvement over the current ISC theory. The results of the equation
of state for a face centered cubic (fcc) nearest neighbor interaction
Lennard-Jones solid from the new ISC theory are shown to be in
excellent agreement with the results of the classical Monte Carlo
method also obtained for the same model.

*Phil. Mag. B*, ** ** August, 2000

We present a critical assessment of the equation of state results
for a fcc Lennard-Jones (LJ) solid calculated from two entirely
different summation procedures for an infinite set of free energy
diagrams. The first is the recent procedure given by Shukla and Cowley
(R.C. Shukla and E.R. Cowley, Phys. Rev. B **58**, 2596 (1998))
where the diagrams of the same order of magnitude generated from the
Van Hove ordering scheme, but arising in different orders of
perturbation theory (PT), are summed to infinity. The second procedure
is the self-consistent phonon theory (SC) which has been in use for
some time. In the first order version of this theory (SC1) only the
first order PT diagrams are summed and in the improved self-consistent
(ISC) theory the first important contribution (cubic term) arising from
the second order PT, omitted in SC1, is included as a correction to the
SC1 free energy. We have calculated the equation of state results from
the ISC theory by averaging the cubic tensor force constant and also
without averaging this constant (ISCU). This brings out the effect of
averaging which is a necessary requirement in the SC1 theory but
not in ISC. The results from SC1 and ISCU are poor. The results from
ISC and Shukla-Cowley summation procedures agree with each other
at low temperature. At high temperatures, the ISC results are in
poor agreement with the classical Monte Carlo (MC) results whereas
the Shukla-Cowley procedure yields results in excellent agreement
with MC.

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