Random matrix theory for transition strength densities in finite quantum systems: Results from embedded unitary ensembles

Embedded random matrix ensembles are generic models for describing statistical properties of finite isolated interacting quantum many-particle systems. For the simplest spinless fermion (or boson) systems, with say m$m$ fermions (or bosons) in N$N$ single particle states and interacting via k$k$-body interactions, we have EGUE(k$k$) [embedded GUE of k$k$-body interactions] with GUE embedding and the embedding algebra is U(N)$U\left(N\right)$. A finite quantum system, induced by a transition operator, makes transitions from its states to the states of the same system or to those of another system. Examples are electromagnetic transitions (then the initial and final systems are same), nuclear beta and double beta decay (then the initial and final systems are different), particle addition to or removal from a given system and so on. Towards developing a complete statistical theory for transition strength densities (transition strengths multiplied by the density of states at the initial and final energies), we have derived formulas for the lower order bivariate moments of the strength densities generated by a variety of transition operators. Firstly, for a spinless fermion system, using EGUE(k$k$) representation for a Hamiltonian that is k$k$-body and an independent EGUE(t$t$) representation for a transition operator that is t$t$-body and employing the embedding U(N)$U\left(N\right)$ algebra, finite-N$N$ formulas for moments up to order four are derived, for the first time, for the transition strength densities. Secondly, formulas for the moments up to order four are also derived for systems with two types of spinless fermions and a transition operator similar to beta decay and neutrinoless beta decay operators. In addition, moments formulas are also derived for a transition operator that removes k₀$k_0$ number of particles from a system of m$m$ spinless fermions. In the dilute limit, these formulas are shown to reduce to those for the EGOE version derived using the asymptotic limit theory of Mon and French (1975). Numerical results obtained using the exact formulas for two-body (k=2$k=2$) Hamiltonians (in some examples for k=3$k=3$ and 4$4$) and the asymptotic formulas clearly establish that in general the smoothed (with respect to energy) form of the bivariate transition strength densities take bivariate Gaussian form for isolated finite quantum systems. Extensions of these results to bosonic systems and EGUE ensembles with further symmetries are discussed.     

Iron‐ and Bismuth‐Catalyzed Asymmetric Mukaiyama Aldol Reactions in Aqueous Media

We have developed asymmetric Mukaiyama aldol reactions of silicon enolates with aldehydes catalyzed by chiral Fe^II and Bi^III complexes. Although previous reactions often required relatively harsh conditions, such as strictly anhydrous conditions, very low temperatures (?78 °C), etc., the reactions reported herein proceeded in the presence of water at 0 °C. To find appropriate chiral water‐compatible Lewis acids for the Mukaiyama aldol reaction, many Lewis acids were screened in combination with chiral bipyridine L1 , which had previously been found to be a suitable chiral ligand in aqueous media. Three types of chiral catalysts that consisted of a Fe^II or Bi^III metal salt, a chiral ligand ( L1 ), and an additive have been discovered and a wide variety of substrates (silicon enolates and aldehydes) reacted to afford the desired aldol products in high yields with high diastereo‐ and enantioselectivities through an appropriate selection of one of the three catalytic systems. Mechanistic studies elucidated the coordination environments around the Fe^II and Bi^III centers and the effect of additives on the chiral catalysis. Notably, both Brønsted acids and bases worked as efficient additives in the Fe^II‐catalyzed reactions. The assumed catalytic cycle and transition states indicated important roles of water in these efficient asymmetric Mukaiyama aldol reactions in aqueous media with the broadly applicable and versatile catalytic systems.     

Osmium(III) and Osmium(V) Complexes Bearing a Macrocyclic Ligand: A Simple and Efficient Catalytic System for cis‐Dihydroxylation of Alkenes with Hydrogen Peroxide

A simple protocol that uses [Os^III(OH)(H₂O)(L ‐N₄Me₂)](PF₆)₂ ( 1 ; L ‐N₄Me₂=N,N′‐dimethyl‐2,11‐diaza[3.3](2,6)pyridinophane) as a catalyst and H₂O₂ as a terminal oxidant for efficient cis‐1,2‐dihydroxylation of alkenes is presented. Unfunctionalized (or aliphatic) alkenes and alkenes/styrenes containing electron‐withdrawing groups are selectively oxidized to the corresponding vicinal diols in good to excellent yields (46–99 %). In the catalytic reactions, the stoichiometry of alkene:H₂O₂ is 1:1, and thus the oxidant efficiency is very high. For the dihydroxylation of cyclohexene, the catalytic amount of 1 can be reduced to 0.01 mol % to achieve a very high turnover number of 5500. The active oxidant is identified as the Os^V(O)(OH) species ( 2 ), which is formed via the hydroperoxide adduct, an Os^III(OOH) species. The active oxidant 2 is successfully isolated and crystallographically characterized.