Avoiding negative populations in explicit Poisson tau-leaping

The explicit tau-leaping procedure attempts to speed up the stochastic simulation of a chemically reacting system by approximating the number of firings of each reaction channel during a chosen time increment tau as a Poisson random variable. Since the Poisson random variable can have arbitrarily large sample values, there is always the possibility that this procedure will cause one or more reaction channels to fire so many times during tau that the population of some reactant species will be driven negative. Two recent papers have shown how that unacceptable occurrence can be avoided by replacing the Poisson random variables with binomial random variables, whose values are naturally bounded. This paper describes a modified Poisson tau-leaping procedure that also avoids negative populations, but is easier to implement than the binomial procedure. The new Poisson procedure also introduces a second control parameter, whose value essentially dials the procedure from the original Poisson tau-leaping at one extreme to the exact stochastic simulation algorithm at the other; therefore, the modified Poisson procedure will generally be more accurate than the original Poisson procedure.     

The Physical Basis of the VSEPR Model

The VSEPR model is a consequence of the correlation of same-spin electrons resulting from the operation of the Pauli exclusion principle. Although the VSEPR rules can be interpreted in terms of an orbital model they do not provide the physical basis for the model.     

Enantioselective aziridination using copper complexes of biaryl Schiff bases

Racemic 2,2'-diamino-6,6'-dimethylbiphenyl is resolved using simulated moving bed chromatography, and the absolute configuration of the enantiomers is confirmed via the X-ray crystal structure of a derivative. The diamine is condensed with a range of aldehydes to give bidentate aldimine proligands L. Molecular structures of the complexes formed between L and Cu(I) fall into two classes; bimetallic double helices ([Cu(2)L(2)](2+)) and monometallic ([CuL](+)). The latter are strikingly more efficient in the aziridination of alkenes than are the former in terms of rate, turnover, and enantioselection. In particular, the imine ligand formed from the diamine and 2,6-dichlorobenzaldehyde gives, in combination with Cu(I) or Cu(II), up to 99% ee in the aziridination of 6-acyl-2,2-dimethylchromene and 88-98% ee for a range of cinnamate esters. Styrenic and other alkenes are converted with lower selectivities (5-54%). The catalytic system shows a linear response in product ee to catalyst ee, and the product ee does not vary significantly during the reaction. UV spectrophotometric investigations indicate that conversion of Cu(I) to Cu(II) is not essential for catalysis but that Cu(II) is probably also a competent system.     

Ligand close packing and the geometry of the fluorides of the nonmetals of periods 3, 4, and 5

This paper discusses the geometry of the fluorides of the nonmetals of periods 3, 4, and 5 in terms of the ligand close packing (LCP) model according to which molecular geometry is determined primarily by ligand-ligand repulsions (Pauli closed shell repulsions) rather than by the bonding and lone pair Pauli repulsions of the VSEPR model. The LCP model becomes the dominant factor in determing geometry when the ligands are sufficiently crowded that they may be regarded as essentially incompressible. Ligand close packing is a modification of the VSEPR model in which ligand-ligand repulsion (Pauli closed shell repulsion) is given more emphasis than bonding and nonbonding electron pair Pauli repulsion. The nonmetals of period 3 are large enough to form octahedral six coordinated molecules in which the ligands are close packed. The larger nonmetals of period 4 also have a maximum coordination number of six and an octahedral geometry although the ligands are not close packed. Ligand radii derived from the interligand distances in the molecules of period 3 depend only on the charge of the fluorine ligands and are consistent with the previously derived radii obtained from the fluorides of the close packed tetrahedral molecules of the period 2 elements. Although the ligands in the molecules of the period 4 nonmetals are not close packed, these elements are not large enough to form molecules with a higher coordination number. However, the larger period 5 nonmetals may have coordination numbers of seven and eight. The seven coordinated molecules have a pentagonal bipyramidal geometry in which the equatorial ligands are close packed. The eight coordinated molecules have a square antiprism geometry, which is not a close packed geometry although the fluorine interligand distances are only a little larger than expected for close packing. The difference between the axial and equatorial bond lengths in the trigonal bipyramidal pentafluorides and the pentagonal bipyramidal pentafluorides can be understood on the basis of ligand close packing. Ligand packing prevents the lone pair in AF(6)E molecules from fully entering the valence shell and thereby exerting its full stereochemical effect so that these molecules have a C(3)(v)() distorted octahedral geometry rather than a geometry based on pentagonal bipyramidal seven coordination.     

Tuning the optical properties of fluorinated indolylfulgimides

Photochromic fluorinated indolylfulgides have been identified as potential candidates for a wide range of applications including optical switches, photoregulators of biological processes, and optical memory media. In humid environments or biological systems, hydrolytic stability is essential. In an effort to improve hydrolytic stability, a series of indolylfulgimides has been synthesized from a parent trifluoromethyl-substituted indolylfulgide. The nitrogen of the succinimide moiety is linked to either a dimethyl amino or one of seven substituted phenyl groups. The phenyl groups feature substituents with increasing electron-withdrawing ability. The spectral characteristics of each compound have been examined, revealing that the wavelength absorption maxima of each form increases with increasing electron-withdrawing ability of the substituted N-phenyl ring. The quantum yields of the photoreactions have been determined with the N-(phenyl)fulgimide showing a ring closure value of nearly 0.30 in toluene. In addition, the hydrolytic, thermal, and photochemical stabilities of each compound have been measured. The fulgimides exhibit at least a 200-fold enhancement of hydrolytic stability for the Z-form and over a 1000-fold enhancement for the C-form in comparison to the same form of the parent fulgide. The N-(2,3,5,6-tetrafluoro-4-trifluoromethylphenyl)fulgimide can undergo up to 3000 photochemical cycles (coloration followed by bleaching) before losing 20% of its initial absorbance at photostationary state.     

Maximal estimates on measure spaces for weak-type multipliers

We describe sufficient conditions for transferring from locally compact abelian groups to measure spaces the weak-type bounds of maximal operators defined by multipliers of weak type. This leads to homomorphism theorems for maximal multiplier operators. Communicated by Guido Weiss     

Geometry of the Fluorides, Oxofluorides, Hydrides, and Methanides of Vanadium(V), Chromium(VI), and Molybdenum(VI): Understanding the Geometry of Non-VSEPR Molecules in Terms of Core Distortion

This paper describes a study of the topology of the electron density and its Laplacian for the molecules VF(5), VMe(5), VH(5), CrF(6), CrMe(6), CrOF(4), MoOF(4), CrO(2)F(2,) CrO(2)F(4)(2)(-) and CrOF(5)(-) all of which, except VF(5,) CrF(6), and CrOF(5)(-) have a non-VSEPR geometry. It is shown that in each case the interaction of the ligands with the metal atom core causes it to distort to a nonspherical shape. In particular, the Laplacian of the electron density reveals the formation of local concentrations of electron density in the outer shell of the core, which have a definite geometrical arrangement such as four in a tetrahedral arrangement or five in a square pyramidal or trigonal bipyramidal and six in an octahedral arrangement. Ligands that are predominately covalently bonded are found opposite regions of charge depletion between these core charge concentrations. In VH(5), VMe(5), CrOF(4), and MoOF(4), these core charge concentrations have a square pyramidal arrangement, and the regions of charge depletions have the corresponding inverse square pyramidal arrangement so that these molecules have a square pyramidal geometry rather than a trigonal prism geometry. In CrMe(6), there are five core charge concentrations with a trigonal bipyramidal arrangement so that the regions of charge depletion have a trigonal prismatic arrangement and the molecule has the corresponding trigonal prism geometry rather than an octahedral geometry. In contrast, molecules in which the only ligand is the more ionically bound fluorine are less affected by core distortion and have VSEPR-predicted structures. The unexpected bond angles in CrO(2)F(2) and the preference of CrO(2)F(4)(2)(-) for a cis structure are also discussed in terms of the pattern of core charge concentrations.