Padé Approximants for Two- and Three-Body Dipole Dispersion Interactions

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Abstract
A new method is presented for using theoretical or experimental optical dispersion data to construct bounded estimates to dispersion force coefficients. The bounds are obtained from the Casimir–Polder integral formula and an analytic continuation of the power series expansion of the dynamic multipole polarizabilities for the interacting species. Recognition of the dynamic polarizabilities at imaginary frequency as series of Stieltjes allows construction of the necessary bounded continuations with Padé approximants expressed directly in terms of Cauchy dispersion coefficients. The sequence of bounds obtained incorporates the London and Slater–Kirkwood values in lowest order, and provides improvements to these two well‐known bounds in higher order. The method is applied to the two‐ and three‐body dipole interactions of atomic and molecular hydrogen and of helium using theoretical Cauchy dispersion coefficients, and to the inert gases, alkali atoms, and molecular hydrogen, nitrogen, and oxygen using coefficients obtained from optical dispersion and absorption data. Comparison is made with dispersion force estimates obtained from ab initio calculation, semiempirical methods, and molecular beam and low‐temperature gas kinetic measurements. In addition to supplying accurate numerical bounds for dispersion force coefficients, the Padé procedure provides the basis for a discussion of previously devised techniques for estimating dispersion force coefficients. The importance of utilizing a proper analytic continuation of the dynamic polarizability in the successful application of these procedures is emphasized. Finally, the accuracy of well‐known dispersion force combination rules is discussed within the framework of the Padé procedure, and the relation of the latter to alternative bounding methods which have recently appeared is discussed.