Description: This article discusses an abstract index theorem on non-compact Riemannian manifolds. Abstract: We prove an abstract index theorem for essentially self-adjoint Fredholm supersymmetric first-order elliptic differential operators on Hermitian vector bundles over complete oriented Riemannian manifolds. According to our main result the supersymmetric L2-index of such an operator can be expressed as the sum of a "local contribution" (the familiar Atiyah-Singer index form, suitably restricted to and integrated over a finite region) and a "boundary contribution" (which depends only on the restriction of the operator at large distances). This is done by splicing together local parametrices and Green's operators defined "at infinity". The result yields (in fact is equivalent to) a generalisation of the relative index theorem of Gromov and Lawson.

Description: This article discusses small molecule elimination from group IVB (Ti, Zr, Hf) amido complexes. An ab initio quantum chemical analysis of HX (X = H, CH₃, Cl, NH₂, SiH₃) elimination by group IVB (Ti, Zr, Hf) amidos (H₂(X)M - NH₂ → H₂M = NH + HX), of interest in the context of CVD precursor design, is reported. Several deductions may be drawn from the calculations. First, in the transition state (TS) for HX elimination, electropositive and electroneutral X give rise to metal-transannular hydrogen (Ht) distances only slightly longer than normal metal-terminal hydride bonds lengths, while electronegative X groups yield substantially longer MHt distances. Second, the HX elimination barrier (∆Hǂelim) is lower when HX is polarized Hδ- • Hδ+ (X = SiH₃) or nonpolar (X = H). Third, a plot of calculated ∆Hǂelim versus MHt distances in the TS. Fourth, analysis of the electronic structure along the intrinsic reaction coordinate (IRC) supports the importance of N-H•••M agostic interactions preceding N-H scission. Fifth, the IRC shows the MHt distance decreasing as Ht is transferred from N to X, reaching a minimum when the transfer is roughly half complete, and then increasing once more is HX is eliminated. These results point to the ...