Continued Factorization Method for van der Waals Interactions

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Abstract
A continued factorization procedure is introduced to telescope the infinite series of the dynamic polarizability into a finite sum. The effective oscillator strengths can also be continually factorized. Both the upper and the lower bounds of the dynamical polarizability can be obtained from expressions identical in form. A single change of the last effective excitation energy of the continually factorized series turns it from an upper to a lower bound. The rest of the effective excitation energies are the natural resonance frequencies of the system. When the continued factorized series is substituted into the Casimir–Polder formula, we obtained the usual London formula for the van der Waals force with finite number of terms. Expressed in terms of oscillator strength sums and the natural resonance frequencies, all quantities in the London formula are given in closed forms. While it has the same formal structure of the Padé approximant method, the present method eliminates the entire process of constructing the approximant and gives tighter bounds.