Let $X\hookrightarrow S$ be an embedding of smooth schemes which splits to first order. Arinkin and C\u{a}ld\u{a}raru constructed an HKR isomorphism between the derived self-intersection $X\times^R_SX$ and the shifted normal bundle $\mathbb{N}_{X/S}[-1]$ from a fixed first order splitting. One can obtain two HKR isomorphisms from two first order splittings. We define the generalized Atiyah class associated to a closed embedding and two first order splittings. We give a sufficient and necessary condition for when the two HKR isomorphisms are equal. In the case of the diagonal embedding, we recover a result of Grivaux.