A note on theorems of Bott and Samelson

Abstract

Introduction.* ) This work was supported by the Sakko-kai Foundation. * * ) Th e helpful suggestions of the referee are grateful acknowledged.

Relativistic Heavy-Neighbor-Atom Effects on NMR Shifts: Concepts and Trends Across the Periodic Table

“…Here we summarize the qualitative understanding of HALA effects on nuclear shieldings as of 2004. The analogy between SO-HALA effects and the Fermi-contact (FC) mechanism of indirect spin−spin coupling, implied qualitatively by Nakagawa et al, 20 was computationally tested in detail in a 1998 paper. 26 DFT calculations on a series of organoiodine complexes showed closely parallel behavior of the SO-HALA contributions to 13 C and 1 H NMR shieldings for different positions in iodobenzene and the corresponding I−C and I−H reduced spin−spin coupling constants, an enhancement of both quantities with carbon s-character in the C−I bond in iodoethane, iodoethene, and iodoethyne, as well as the Karplus-type dependence of the three-bond SO-HALA effects on the 1 H NMR shieldings on the I−C−C−H dihedral angle in iodoethane.…”

“…Group 17: Halogens. The first mentioned 19,20 and most intensively studied SO-HALA effect is the normal halogen dependence (NHD), which refers to the increasing SO-HALA shielding at the LA with increasing Z HA of an adjacent halide substituent for both organic and inorganic compounds. While we saw analogous dependencies throughout groups 14−17 (see Figure 29 in section 4.3.1), the SO-HALA effects and thus the NHD-type behavior tends to be most pronounced for HAs from group 17.…”

“…The first attempts by Nakagawa and co-workers to use perturbation theory (PT) to extend the theory of nuclear shielding in diamagnetic systems beyond the nonrelativistic (NR) Ramsey equation (see also section 2) were published in an organic chemistry journal 19 (and in the proceedings of a domestic Japanese NMR conference). 20 Note also that these ideas were preceded by the first theories of pseudocontact shifts in paramagnetic systems, 21−25 which are also known to have an SO origin. Nakagawa et al laid out the third-order perturbationtheory (PT3) framework that resulted from including SO coupling as an additional perturbation in Ramsey's theory and pointed out the important analogy of the SO-induced HALA effects with the Fermi-contact mechanism of indirect spin−spin coupling constants, which was further exploited based on quantitative computations almost 30 years later.…”

How does relativity affect magnetically induced currents?

“…This is a key mechanism of the so-called heavy-atom–light-atom (HALA) effect. 17–19 In the presence of an external magnetic field B and under consideration of SOC a non-zero and non-constant spin-density is induced into the wave function. The spin-current that directly corresponds to this induced spin-density is, in very good approximation, the only spin–orbit contribution to the currents (see the ESI† for details of term partitioning) j FR B − j non-SOC B ≈ j spin B ,SOC = −( e / m ) rot ρ spin B ,SOC (where we have used the same definitions as for eqn (4) and j non-SOC B = j Pauli B + j B ,scalar ).…”

Focal points and sup-norms of eigenfunctions

“…In [1] a manifold with such a point is denoted a F x0 ℓ (or Y x0 ℓ -)-manifold; if ℓ is the least common return time for all loops it is denoted by L x0 ℓ . If (M, g) has a focal point x 0 from which all geodesics are simple (non-intersecting) loops, then the integral cohomology ring H * (M, Z) is generated by one element [15]. For an F x0 ℓ manifold, H * (M, Q) has a single generator (Theorem 4 of [1]).…”