Relations between lattice dynamics of simple metals and liquid state properties

The actual interaction between ions in a metal is divided into the direct interaction of the ion cores and the ion-electron-ion interaction. The ion-electron-ion interaction is treated according to the Hartree-Fock approximation modified by a screened exchange potential. Since until now the bare ion potential is not known very accurately, the observed phonon frequencies were used to determine a model pseudopotential with the least squares method. Using this model pseudopotential the electronic band structure, the density of states as well as the electrical resistivity of molten Na, K, Al and Pb were calculated.     

Analytic gradients for the combined sr-DFT/lr-MP2 method: application to weakly bound systems

The formalism for analytical gradients in short-range density functional schemes with long-range ab initio corrections (sr-DFT/lr-ab initio) is presented. On the density-functional side, both sr-LDA and sr-PBE are available, while on the ab initio side lr-HF and lr-MP2 are possible. Details of the implementation in the Molpro package are given. Results from test calculations using different basis sets on weakly bound systems of the HB6/04, DI6/04, CT7/04 and WI9/04 databases are presented.     

On Simultaneous Digital Expansions of Polynomial Values

Let s _q denote the q-ary sum-of-digits function and let \({P_1(X), P_2(X) \in \mathbb{Z}[X]}\) with \({P_1(\mathbb{N}), P_2(\mathbb{N}) \subset \mathbb{N}}\) be polynomials of degree \({h, l \geqq 1, h \neq l}\) , respectively. In this note we show that ( \({s_q(P_1(n))/s_q(P_2(n)))_{n \geqq 1}}\) is dense in \({\mathbb{R}^+}\) . This extends work by Stolarsky [9] and Hare, Laishram and Stoll [6].     

On a second conjecture of Stolarsky: the sum of digits of polynomial values

Let q, r ≥ 2 be integers, and denote by s _q the sum-of-digits function in base q. In 1978, K.B. Stolarsky conjectured that $$\lim_{N \to \infty} \frac{1}{N} \sum_{n \leq N} \frac{s_2(n^r)}{s_2(n)} \leq r.$$ In this paper we prove this conjecture. We show that for polynomials ${P_1(X), P_2(X) \in \mathbb{Z}[X]}$ of degrees r ₁, r ₂ ≥ 1 and integers q ₁, q ₂ ≥ 2, we have $$\lim_{N \to \infty} \frac{1}{N} \sum_{n \leq N}\frac{s_{q_1}(P_1(n))}{s_{q_2}(P_2(n))} = \frac{r_1 (q_1 - 1) {\rm log}q_2}{r_2(q_2 - 1) {\rm log} q_1}.$$ We also present a variant of the problem to polynomial values of prime numbers.     

Synthesis and Reactions of P-Rich Silylphosphanes

Abstract

The formation of P-rich compounds like (Me₃Si)₃P₇ 1 and P₄(SiMe₂)₃ 2 generated from Na/K-alloy, white phosphorus, Me₃SiCl and Me₂SiCl₂, resp., or the formation of 1 from P₄, lithiumalkyles and Me₃SiCl on the other hand occurs in several reaction steps₁.     

A redox series of aluminum complexes: characterization of four oxidation states including a ligand biradical state stabilized via exchange coupling

Electrophilic activation and subsequent reduction of substrates is in general not possible because highly Lewis acidic metals lack access to multiple redox states. Herein, we demonstrate that transition metal-like redox processes and electronic structure and magnetic properties can be imparted to aluminum(III). Bis(iminopyridine) complexes containing neutral, monoanionic, and dianionic iminopyridine ligands (IP) have been characterized structurally and electronically; yellow (IP)AlCl(3) (1), deep green (IP(-))(2)AlCl (2) and (IP(-))(2)Al(CF(3)SO(3)) (3), and deep purple [(IP(2-))Al](-) (5) are presented. The mixed-valent, monoradical complex (IP(-))(IP(2-))Al is unstable toward C-C coupling, and [(IP(2-))Al](2-)(μ-IP-IP)(2-) (4) has been isolated. Variable-temperature magnetic susceptibility and EPR spectroscopy measurements indicate that the biradical character of the ligand-based triplet in 2 is stabilized by strong antiferromagnetic exchange coupling mediated by aluminum(III): J = -230 cm(-1) for ? = -2J(?(L(1))·?(L(2))). Coordination geometry-dependent (IP(-))-(IP(-)) communication through aluminum(III) is observed electrochemically. The cyclic voltammogram of trigonal bipyramidal 2 displays successive ligand-based oxidation events for the two IP(1-/0) processes, at -0.86 and -1.20 V vs SCE. The 0.34 V spacing between redox couples corresponds to a conproportionation constant of K(c) = 10(5.8) for the process (IP(-))(2)AlCl + (IP)(2)AlCl → 2(IP(-))(IP)AlCl consistent with Robin and Day Class II mixed-valent behavior. Tetrahedral 5 displays localized, Class I behavior as indicated by closely spaced redox couples. Furthermore, CV's of 2 and 5 indicate that changes in the coordination environment of the aluminum center shift the potentials for the IP(1-/0) and IP(2-/1-) redox couples by up to 0.9 V.     

Oxygen‐Promoted CH Bond Activation at Palladium

[Pd(P(Ar)(tBu)₂)₂] ( 1 , Ar=naphthyl) reacts with molecular oxygen to form Pd^II hydroxide dimers in which the naphthyl ring is cyclometalated and one equivalent of phosphine per palladium atom is released. This reaction involves the cleavage of both C? H and O? O bonds, two transformations central to catalytic aerobic oxidizations of hydrocarbons. Observations at low temperature suggest the initial formation of a superoxo complex, which then generates a peroxo complex prior to the C? H activation step. A transition state for energetically viable C? H activation across a Pd? peroxo bond was located computationally.     

Towards an incremental expansion of strong correlation effects in solids

A scheme based on local correlation-energy increments, which has been applied successfully to the investigation of dynamical correlation effects in semiconductors and ionic crystals, is extended here to include non-dynamical (static) electron correlation as well. It is shown how to use abinitio multi-configuration self-consistent-field and multi-reference configuration-interaction calculations for this purpose, and test applications are reported for potential curves of molecules at the dissociation limit.