Moment Theory Bounds for the Second-Order Optical Properties of Atoms and Molecules

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Abstract
A variety of oscillator‐strength‐distribution frequency moments are employed in conjunction with theorems from the theory of moments to construct bounded estimates for the dynamic polarizabilities and related optical properties of atoms and diatomic molecules. This technique circumvents the explicit construction of an accurate approximation to the dipole oscillator‐strength distribution appearing in the Kramers‐Heisenberg formula, and relates the dispersive optical properties of a system to the simpler frequency moments of its excitation spectrum. The bounds obtained follow from the theory of Tchebycheff systems of functions, although the Stieltjes nature of the polarizability insures that various other related formalisms, including the theory of Padé approximants, Gaussian quadratures, variational principles, and linear programming methods, can also be employed in their construction. Illustrative applications of the moment technique are given for the ground states of atoms and diatomic molecules employing oscillator‐strength sum‐rule moments, which can be determined from the ground state wavefunction of the system, and negative‐even‐integer (Cauchy) moments, determined from both theoretical and semi‐empirical procedures. Highly accurate bounds for the refractivities, Verdet coefficients, and Rayleigh scattering cross sections of atomic and molecular hydrogen and helium are obtained using a small number of sum‐rule moments and the static value of the polarizability. The sum‐rule moment bounds obtained for the optical properties of the heavier inert gases, which exhibit more structured and complex dipole excitation spectra, are somewhat weaker than those for the smaller systems, although the mean values of the bounds are generally in good agreement with experiment. The long wavelength Faraday rotation data for neon appears to be somewhat anomalous, however. Theoretical and semiempirical Cauchy moments for the polarizability components of diatomic hydrogen, nitrogen, and oxygen are employed in the construction of Cauchy moment bounds. The theoretical results for molecular hydrogen, in the time‐dependent Hartree‐Fock approximation, are in good agreement with previous theoretical determinations of the dispersion in hydrogen, and the semiempirical results aid in the interpretation of uv Rayleigh scattering measurements for molecular hydrogen and nitrogen. Comparisons of the Cauchy moment bounds for the molecular Verdet coefficients with measured Faraday rotation data shows that all three diatomic gases satisfy the ``modified'' Becquerel formula. The ease with which bounds for the second‐order optical properties of atoms and molecules can be constructed when the required frequency moments are available suggests that the moment technique should be one of continuing interest, with extensions of the approach to additional linear and nonlinear optical susceptibilities expected to be similarly rewarding.