Highly ordered N-heterocyclic carbene monolayers on Cu(111)

The benzannulated N-heterocyclic carbene, 1,3-dibenzylbenzimidazolylidene (NHCDBZ) forms large, highly ordered domains when adsorbed on a Cu(111) surface under ultrahigh vacuum conditions. A combination of scanning tunneling microscopy (STM), high resolution electron energy loss spectroscopy (HREELS) and density functional theory (DFT) calculations reveals that the overlayer consists of vertical benzannulated NHC moieties coordinating to Cu adatoms. Long range order results from the placement of the two benzyl units from a single NHC on opposite sides of the benzimidazole moiety, with the planes of their aromatic rings approximately parallel to the surface. The organization of three surface-bound benzyl substituents from three different NHCs into a triangular array controls the formation of a highly ordered Kagome-like lattice on the surface. This study illustrates the importance of dispersive interactions to control the binding geometry and self-assembly of the NHC.

Highly ordered N-heterocyclic carbene monolayers on Cu(111)

The benzannulated N-heterocyclic carbene, 1,3-dibenzylbenzimidazolylidene (NHCDBZ) forms large, highly ordered domains when adsorbed on a Cu(111) surface under ultrahigh vacuum conditions. A combination of scanning tunneling microscopy (STM), high resolution electron energy loss spectroscopy (HREELS) and density functional theory (DFT) calculations reveals that the overlayer consists of vertical benzannulated NHC moieties coordinating to Cu adatoms. Long range order results from the placement of the two benzyl units from a single NHC on opposite sides of the benzimidazole moiety, with the planes of their aromatic rings approximately parallel to the surface. The organization of three surface-bound benzyl substituents from three different NHCs into a triangular array controls the formation of a highly ordered Kagome-like lattice on the surface. This study illustrates the importance of dispersive interactions to control the binding geometry and self-assembly of the NHC.

Discrepancy of stratified samples from partitions of the unit cube

We extend the notion of jittered sampling to arbitrary partitions and study the discrepancy of the related point sets. Let $${\varvec{\Omega }}=(\Omega _1,\ldots ,\Omega _N)$$ Ω = ( Ω 1 , … , Ω N ) be a partition of $$[0,1]^d$$ [ 0 , 1 ] d and let the ith point in $${{\mathcal {P}}}$$ P be chosen uniformly in the ith set of the partition (and stochastically independent of the other points), $$i=1,\ldots ,N$$ i = 1 , … , N . For the study of such sets we introduce the concept of a uniformly distributed triangular array and compare this notion to related notions in the literature. We prove that the expected $${{{\mathcal {L}}}_p}$$ L p -discrepancy, $${{\mathbb {E}}}{{{\mathcal {L}}}_p}({{\mathcal {P}}}_{\varvec{\Omega }})^p$$ E L p ( P Ω ) p , of a point set $${{\mathcal {P}}}_{\varvec{\Omega }}$$ P Ω generated from any equivolume partition $${\varvec{\Omega }}$$ Ω is always strictly smaller than the expected $${{{\mathcal {L}}}_p}$$ L p -discrepancy of a set of N uniform random samples for $$p>1$$ p > 1 . For fixed N we consider classes of stratified samples based on equivolume partitions of the unit cube into convex sets or into sets with a uniform positive lower bound on their reach. It is shown that these classes contain at least one minimizer of the expected $${{{\mathcal {L}}}_p}$$ L p -discrepancy. We illustrate our results with explicit constructions for small N. In addition, we present a family of partitions that seems to improve the expected discrepancy of Monte Carlo sampling by a factor of 2 for every N.