Robust estimation for non-homogeneous data and the selection of the optimal tuning parameter: the density power divergence approach

<div><p>The density power divergence (DPD) measure, defined in terms of a single parameter <i>α</i>, has proved to be a popular tool in the area of robust estimation [<a href="#CIT0001" target="_blank">1</a>]. Recently, Ghosh and Basu [<a href="#CIT0005" target="_blank">5</a>] rigorously established the asymptotic properties of the MDPDEs in case of independent non-homogeneous observations. In this paper, we present an extensive numerical study to describe the performance of the method in the case of linear regression, the most common setup under the case of non-homogeneous data. In addition, we extend the existing methods for the selection of the optimal robustness tuning parameter from the case of independent and identically distributed (i.i.d.) data to the case of non-homogeneous observations. Proper selection of the tuning parameter is critical to the appropriateness of the resulting analysis. The selection of the optimal robustness tuning parameter is explored in the context of the linear regression problem with an extensive numerical study involving real and simulated data.</p></div

Theoretical Photoelectron Spectroscopy of Quadruple-Bonded Dimolybdenum(II,II) and Ditungsten(II,II) Paddlewheel Complexes: Performance of Common Density Functional Theory Methods

We have revisited the gas-phase photoelectron spectra of quadruple-bonded dimolybdenum(II,II) and ditungsten(II,II) paddlewheel complexes with modern density functional theory methods and obtained valuable calibration of four well-known exchange–correlation functionals, namely, BP86, OLYP, B3LYP*, and B3LYP. All four functionals were found to perform comparably, with discrepancies between calculated and experimental ionization potentials ranging from <0.1 to ∼0.5 eV, with the lowest errors observed for the classic pure functional BP86. All four functionals were found to reproduce differences in ionization potentials (IPs) between analogous Mo2 and W2 complexes, as well as large, experimentally observed ligand field effects on the IPs, with near-quantitative accuracy. The calculations help us interpret a number of differences between analogous Mo2 and W2 complexes through the lens of relativistic effects. Thus, relativity results in not only significantly lower IPs for the W2 complexes but also smaller HOMO–LUMO gaps and different triplet states relative to their Mo2 counterparts

Stereochemical Diversity of {MNO}¹⁰ Complexes: Molecular Orbital Analyses of Nickel and Copper Nitrosyls

The great majority of {NiNO}¹⁰ complexes are characterized by short Ni–N­(O) distances of 1.60–1.65 Å and linear NO units. Against this backdrop, the {CuNO}¹⁰ unit in the recently reported [Cu­(CH₃NO₂)₅(NO)]²⁺ cation (<b>1</b>) has a CuNO angle of about 120° and a very long 1.96 Å Cu–N­(O) bond. According to DFT calculations, metal–NO bonding in <b>1</b> consists of a single Cu­(d_<i>z</i>²)–NO­(π*) σ-interaction and essentially no metal­(d_π)–NO­(π*) π-bonding, which explains both the bent CuNO geometry and the long, weak Cu–N­(O) bond. This σ-interaction is strongly favored by a ligand <i>trans</i> to the NO; indeed such a <i>trans</i> ligand may be critical for the existence and stability of a {CuNO}¹⁰ unit. By contrast, {NiNO}¹⁰ complexes exhibit a strong avoidance of such <i>trans</i> ligands. Thus, a five-coordinate {NiNO}¹⁰ complex appears to favor a trigonal-bipyramidal structure with the NO in an equatorial position, as in the case of [Ni­(bipy)₂(NO)]⁺ (<b>6</b>). An unusual set of Ni­(d)–NO­(π*) orbital interactions accounts for the strongly bent NiNO geometry for this complex

Theoretical Photoelectron Spectroscopy of Low-Valent Carbon Species: A ∼6 eV Range of Ionization Potentials among Carbenes, Ylides, and Carbodiphosphoranes

High-quality density functional theory calculations underscore a nearly 6 eV range for the ionization potentials (IPs) of neutral, low-valent carbon compounds, including carbenes, ylides, and zero-valent carbon compounds (carbones) such as carbodiphosphoranes (CDPs) and carbodicarbenes. Thus, adiabatic IPs as low as 5.5 ± 0.1 eV are predicted for CDPs, which are about 0.7–1.2 eV lower than those of simple phosphorus and sulfur ylides. In contrast, the corresponding values for N-heterocyclic carbenes are about 8.0 eV while those for simple singlet carbenes such as dichlorocarbene and difluorocarbene range from about 9.0 eV to well over 11.0 eV

Theoretical Photoelectron Spectroscopy of Metal–Metal Quintuple Bonds: Relativity-Driven Reordering of Frontier Orbitals

A recent reinvestigation of the gas-phase photoelectron spectra of Group 6 metal–metal quadruple-bonded complexes with scalar-relativistic DFT calculations showed that common exchange-correlation functionals reproduce the lowest ionization potentials in a semiquantitative manner. The finding encouraged us to undertake a DFT study of metal–metal quintuple bonds in a set of bisamidinato complexes with the formula MI2[HC(NR)2]2 (M = Cr, Mo, W; R = H, Ph, 2,6-iPr2C6H3) and idealized D2h symmetry. Scalar-relativistic OLYP/STO-TZ2P calculations indicated significant shifts in valence orbital energies among the three metals, which translate to lower first ionization potentials, higher electron affinities, and lower HOMO–LUMO gaps for the W complexes relative to their Cr and Mo counterparts. These differences are largely attributable to substantially larger relativistic effects in the case of tungsten relative to those of its lighter congeners

Energetics of Saddling versus Ruffling in Metalloporphyrins: Unusual Ruffled Dodecasubstituted Porphyrins

Presented herein is a first major density functional theory (BP86/D3/STO-TZ2P) survey of the energetics of saddling versus ruffling for a wide range of dodecasubstituted metalloporphyrins with M = Ni, Cu, Zn, Pd, and Pt. For the majority of X₈TPP (i.e., β-octasubstituted-<i>meso</i>-tetraphenylporphyrin), the calculations indicated a clear preference for the saddled conformation, consistent with a large body of experimental data. The preference for the saddled conformation relative to the ruffled conformation was found to vary from about ∼0.3–0.4 eV for Me₈TPP derivatives up to 1 eV for I₈TPP and (CF₃)₈TPP derivatives. For X = Ph, that is, dodecaphenylporphyrins, the saddled and the ruffled conformation are almost equienergetic, with even a slight preference for the ruffled conformation in some cases. This finding provides a satisfactory explanation for the X-ray crystallographic observation of both saddled and ruffled conformations for dodecaphenylporphyrin complexes as well as for spectroscopic evidence for conformational mobility of these complexes in solution. The calculations also indicate near-equienergetic saddled and ruffled conformations for <i>meso</i>-tetraacetylenyltetrabenzoporphyrins, again consonant with key crystallographic findings. By and large, both the energetics and nonplanar distortions of the metalloporphyrin derivatives correlated well with the Charton and Sterimol B₁ steric parameters of the peripheral substituents

Metallocorrole Interactions with Carbon Monoxide, Nitric Oxide, and NitroxylA DFT Study of Low-Energy Bound States

Presented herein is a first DFT survey of metallocorrole (M = Mn, Fe, Co) adducts of CO, NO, NO^–, and HNO. Bound states with relatively short M–N­(O) distances <1.9 Å are predicted for both [M­(Cor)­(NO)]⁻ and [M­(Cor)­(HNO)] derivatives for all three metals. The calculations also confirm the existence of moderately stable charge-neutral CoCO corrole adducts

Synthesis and Molecular Structure of a Copper Octaiodocorrole

Although rather delicate on account of their propensity to undergo deiodination, β-octaiodoporphyrinoids are of considerable interest as potential precursors to novel β-octasubstituted macrocycles. Presented herein are early results of our efforts to synthesize β-octaiodocorrole derivatives. Oxidative condensation of 3,4-diiodopyrrole and aromatic aldehydes failed to yield free-base octaiodocorroles. Treatment of copper <i>meso</i>-tris­(<i>p</i>-cyanophenyl)­corrole with <i>N</i>-iodosuccinimide and trifluoroacetic acid over several hours, however, yielded the desired β-octaiodinated product in ∼22% yield. Single-crystal X-ray structure determination of the product revealed a strongly saddled corrole macrocycle with metrical parameters very close to those of analogous Cu octabromocorrole complexes. The compound was also found to exhibit an exceptionally red-shifted Soret maximum (464 nm in dichloromethane), underscoring the remarkable electronic effect of β-octaiodo substitution

Molecular Structure of a Free-Base β‑Octaiodo<i>-meso</i>-tetraarylporphyrin. A Rational Route to <i>cis</i> Porphyrin Tautomers?

Although a <i>cis</i> tautomer has long been invoked as an intermediate in porphyrin tautomerism, the first such species was only recently isolated and structurally characterized in the form of a β-heptakis­(trifluoromethyl)-<i>meso</i>-tetraarylporphyrin. Reported herein is the molecular structure of a β-octaiodo-<i>meso</i>-tetraarylporphyrin solvate, which also exhibits a <i>cis</i> tautomeric structure. Both structures implicate two factors as critical to the stabilization of the <i>cis</i> tautomeric forma free-base porphyrin that is naturally strongly saddled on steric grounds and a hydroxylic or amphiprotic solvent that can provide hydrogen-bonded N–H···X-H···N (X = O in both the above examples) straps connecting the central NH groups with the antipodal unprotonated nitrogens. The results raise the prospect that a rational strategy affording <i>cis</i> porphyrin tautomers in a predictable manner may be within reach

Molecular Structures of Free-Base Corroles: Nonplanarity, Chirality, and Enantiomerization

The molecular structures of free-base corroles are illustrative of a variety of bonded and nonbonded interactions including aromaticity, intra- as well as intermolecular hydrogen bonding, steric interactions among multiple NH hydrogens within a congested central cavity, and the effects of peripheral substituents. Against this backdrop, an X-ray structure of 2,3,7,8,12,13,17,18-octabromo-5,10,15-tris­(pentafluorophenyl)­corrole, H₃[Br₈TPFPCor], corresponding to a specific tautomer, has been found to exhibit the strongest nonplanar distortions observed to date for any free-base corrole structure. Two adjacent <i>N</i>-protonated pyrrole rings are tilted with respect to each other by approximately 97.7°, while the remainder of the molecule is comparatively planar. Dispersion-corrected DFT calculations were undertaken to investigate to what extent the strong nonplanar distortions can be attributed to steric effects of the peripheral substituents. For <i>meso</i>-triphenylcorrole, DFT calculations revealed nonplanar distortions that are only marginally less pronounced than those found for H₃(Br₈TPFPCor). A survey of X-ray structures of sterically unhindered corroles also uncovered additional examples of rather strong nonplanar distortions. Detailed potential energy calculations as a function of different saddling dihedrals also emphasized the softness of the distortions. Because of nonplanar distortions, free-base corrole structures are chiral. For H₃[Br₈TPFPCor], DFT calculations led to an estimate of 15 kcal/mol (0.67 eV) as the activation barrier for enantiomerization of the free-base structures, which is significantly higher than the barrier for NH tautomerism calculated for this molecule, about 5 kcal/mol (0.2 eV). In summary, steric crowding of the internal NH hydrogens appears to provide the main driving force for nonplanar distortions of <i>meso</i>-triarylcorroles; the presence of additional β-substituents adds marginally to this impetus