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The polarizabilities describing the response of the multipole moments of a charged harmonic oscillator to a field and field gradient are calculated exactly, without recourse to perturbation theory. Generalizations of the Kirkwood and Unsöld treatments for multipole polarizabilities are given, and found to be exact for the harmonic oscillator. Changes in the multipole polarizabilities on confinement of the oscillator to a spherical box are also calculated.
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