Free energy of nearly close packed strips of hard squares in a ``brickwall'' configuration

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Abstract
The exact Helmholtz free energy FN of strips of m rows and an infinite number of columns of hard squares in a nearly close packed ``brickwall'' configuration is calculated through m = 4. These results are used to estimate a more accurate value of the free energy parameter C in the expansion, FN/NkT = 2 ln(λ/σ) −2 ln(τ −1) + C + D(τ −1) + E(τ −1)2 +..., for an infinite two dimensional system of hard squares in a ``brickwall'' configuration. The quantity σ is the particle diameter, λ is the mean thermal de Broglie wavelength, and τ is the ratio of the system area to the close packed system area. The estimate for C is −0.1787, which is greater than −0.2604 [J. Chem. Phys. 53, 1532 (1970)] or −0.268 [J. Chem. Phys. 56, 5434, (1972)], the estimated values of C for the square lattice configuration by the modified cell cluster and Monte Carlo methods, respectively. This supports earlier conclusions [J. Chem. Phys. 56, 5434 (1972); J. Comput. Phys. 7,403 (1971)] that the square lattice configuration is more stable. The previous modified cell cluster estimate of C for the ``brickwall'' configuration, −0.0239, deviates considerably from the new estimate because the cell cluster expansion converges slowly for this configuration.