In theories which describe the response of freely rotating molecules to externally imposed stimuli it is frequently necessary to average rotationally a product of direction cosines relating space-fixed and molecular coordinate frames. In this paper a systematic method for deriving the required tensor averages is presented, and results up to the seventh rank are explicitly shown. Where appropriate both reducible and irreducible expressions are given and their equivalence is demonstrated. Finally, some useful identities relating rotational averages of different ranks are noted.

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