Padé Approximants to the Normal Dispersion Expansion of Dynamic Polarizabilities

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Abstract
A procedure for investigating the dispersion of dynamic polarizabilities and associated molecular excitation spectra is presented. The time‐dependent Schrödinger equation is replaced through the expedient of an ansatz by a frequency‐dependent equivalent and the pertinent first‐order perturbation function expanded in a power series in the applied frequency. There results a set of coupled inhomogeneous differential equations which can be solved sequentially by utilizing the methods for static polarizability calculations, providing the coefficients for a power‐series expansion in frequency of the polarizability (Cauchy series). The latter is identified as a series of Stieltjes, which can be summed within and continued outside its radius of convergence by Padé approximants, furnishing an approximation to the polarizability throughout the entire (cut‐) complex plane. The power‐series coefficients from the expansion of the polarizability further provide a test of the convegence of the computation, which is independent of applied frequency. Applications are made in the dipole case to atomic hydrogen, for which exact solution for the expanded perturbation function is calculated, and to the helium atom and hydrogen molecule in the coupled Hartree–Fock approximation (time‐dependent Hartree–Fock theory), for which the equations are solved using a convenient variational procedure. Lower‐bounded rapid convergence to the correct polarizabilities and associated refractivities in the normal dispersion region is obtained using a small number of Cauchy coefficients and the lowest‐order Padé approximants. Further, accurate approximations to the lowest‐lying transition frequencies and oscillator strengths are obtained from the poles and residues, respectively, of the Padé approximants. These results indicate that variational computations of optical susceptibilities in the normal dispersion region, combined with subsequent summation and analytical continuation in frequency, can provide a practical procedure for investigating the optical dispersion and excitation spectra in atomic and molecular systems.