Summation of free-energy diagrams of an anharmonic crystal and equation of state for a Lennard-Jones solid

We present a method of calculating the Helmholtz free energy of an anharmonic crystal. The exact expression for F, obtained by summing an infinite series of contributions, from all the loops and bubbles (quartic and cubic contributions to the self-energy of the Green's function), is evaluated numerically and the equation of state results for a Lennard-Jones solid are compared with the lambda(2) perturbation theory (PT) which contains only the lowest order cubic and quartic contributions. It is shown that the infinite sum results are considerably improved over the lambda(2) PT results for higher temperatures. Next we have presented a powerful ansatz approach of evaluating the same sum. The numerical results from this method are shown to be identical to the exact sum except at near T-m where they are very slightly different. The ansatz method is then extended to the higher order lambda(4) diagrams and here too the numerical results are found to be improved over the results from the lambda(4) PT. The ansatz procedure is then extended to thr propagator renormalization and the numerical results obtained seem to have the best agreement with the results of classical Monte Carlo simulations. [S0163-1829(98)04729-8].