Mode Expansion in Equilibrium Statistical Mechanics. III. Optimized Convergence and Application to Ionic Solution Theory

The mode expansion is an infinite series for the Helmholtz free energy of a system of classical particles with spherically symmetric interparticle potentials. A systematic procedure is presented for making the mode expansion converge as quickly as possible for particles whose potential includes a strong repulsion. This optimization procedure is physically related to the excluded volume effects produced by the repulsive forces. When the mode expansion is truncated after the one‐mode term (the random phase approximation), the optimized result is related to the spherical model integral equation. The optimized mode expansion is applied to the restricted primitive model of aqueous 1–1 and 2–2 electrolyte solutions and compared with the results obtained from Monte Carlo calculations for salt concentrations up to $\text{2M}$ . The comparison indicates that in this concentration range, the optimized mode expansion converges fairly rapidly. Truncation of the expansion for the 1–1 electrolyte after the two‐mode term yields values for the Helmholtz free energy, osmotic coefficient, and activity coefficients which are in almost perfect agreement with the Monte Carlo calculations up to $\text{2M}$ . Accurate values of the excess internal energy are also obtained. For 2–2 electrolytes, the convergence of the optimized mode expansion is not as rapid as that for 1–1 electrolytes. Even so, when truncated after the two‐mode term, the optimized mode expansion for the 2–2 electrolyte predicts configurational energies which differ by at most 6% from the Monte Carlo results for concentrations between $\text{0.5M}$ and $\text{2M}$ .