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We study H. Dao's invariant $\eta_c^R$ of pairs of modules defined over acomplete intersection ring $R$ of codimension $c$ having an isolatedsingularity.  Our main result is that $\eta_c^R$ vanishes for all pairs of modules when $R$is a {\em graded} complete intersection ring of codimension $c > 1$ having anisolated singularity. A consequence of this result is that all pairs of modulesover such a ring are $c$-$\Tor$-rigid.

The vanishing of a higher codimension analog of Hochster's theta invariant