A double Yukawa (DY) potential, fitted on the carbon - carbon (CC) van der Waals (vdW) Lennard-Jones 12,6 (LJ) potential and integrated on two facing spheres, yields another DY function, whose parameters are simply related to those of the original CC LJ. We exploit this fact to obtain a new analytic expression of the DY type that describes the high-temperature physical interactions between molecules in good agreement with the well-known Girifalco potential, which was obtained from the direct spherical integration of the LJ potential. The main advantage of having a DY pair potential comes from the fact that analytic expressions of thermodynamic functions can be obtained within the Percus - Yevick approximation of the Ornstein - Zernike equation. As an example, an analytic expression of the Helmholz free energy is derived, and the double tangent construction on its vdW loop allows the determination of the liquid - vapour coexistence line and an estimation of the critical parameters ( K, ). This is in good agreement with the results predicted from various other theories with the Girifalco potential.

In order to assess the accuracy of the supersymmetric WKB (SWKB) approximation for the calculation of phase shifts, the SWKB and WKB (with the Langer modification) phase shifts up to O(h 2 ) are compared numerically to the exact calculations for the Coulomb potential B/r, the inverse-square potential A/r 2 and their combinations A/r 2 +B/r, which include the Kratzer (A > 0, B < 0) and inverted Kratzer (A < 0, B > 0) potentials. For the inverse-square potential both the SWKB and the WKB methods yield the exact phase shifts. For the other two potentials considered, the results show that the SWKB phase shifts are in many cases superior; however, it is found that the = 0 WKB Coulomb phase shifts are more accurate for 0.8 < κ < 1.3 with κ = mB/h 2 k. In addition, the excellent agreement of the SWKB phase shifts up to O(h 4 ) with the exact ones shows that the SWKB series expansion in powers of h is rapidly convergent for the potentials considered.

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Exact analytical expressions for the classical vibrational-rotational partition function of the Lennard-Jones (m, n) and Morse potentials are derived. The values obtained from such expressions are compared to accurate evaluations of the corresponding quantum-mechanical partition function for the LJ (10, 6) and Morse potentials. Both sets of values are found to be in close agreement over a wide range of temperatures.

The analytical formula, derived by Iglesias-Groth et al for the interaction energy between shells of spherical carbon onions using the surface continuum approximation, is shown to be in very good agreement with the numerical values obtained by Lu and Yang from the discrete summation over the van der Waals Lennard-Jones LJ(12, 6) CC potential as well as the density functional results of Pis Diez and Iñiguez. The empirical law, V j,j +1 = Kj (j + 1), with K = −3.0 eV, giving the interaction energy between the j th and (j + 1)th shell derived from the discrete summation results, is recovered with an analytical formula for K in terms of the LJ(12, 6) parameters.