Stockholder projector analysis: a Hilbert-space partitioning of the molecular one-electron density matrix with orthogonal projectors

A previously introduced partitioning of the molecular one-electron density matrix over atoms and bonds [D. Vanfleteren et al., J. Chem. Phys. 133, 231103 (2010)] is investigated in detail. Orthogonal projection operators are used to define atomic subspaces, as in Natural Population Analysis. The orthogonal projection operators are constructed with a recursive scheme. These operators are chemically relevant and obey a stockholder principle, familiar from the Hirshfeld-I partitioning of the electron density. The stockholder principle is extended to density matrices, where the orthogonal projectors are considered to be atomic fractions of the summed contributions. All calculations are performed as matrix manipulations in one-electron Hilbert space. Mathematical proofs and numerical evidence concerning this recursive scheme are provided in the present paper. The advantages associated with the use of these stockholder projection operators are examined with respect to covalent bond orders, bond polarization, and transferability.     

Influence of electron correlation and degeneracy on the Fukui matrix and extension of frontier molecular orbital theory to correlated quantum chemical methods

The Fukui function is considered as the diagonal element of the Fukui matrix in position space, where the Fukui matrix is the derivative of the one particle density matrix (1DM) with respect to the number of electrons. Diagonalization of the Fukui matrix, expressed in an orthogonal orbital basis, explains why regions in space with negative Fukui functions exist. Using a test set of molecules, electron correlation is found to have a remarkable effect on the eigenvalues of the Fukui matrix. The Fukui matrices at the independent electron model level are mathematically proven to always have an eigenvalue equal to exactly unity while the rest of the eigenvalues possibly differ from zero but sum to zero. The loss of idempotency of the 1DM at correlated levels of theory causes the loss of these properties. The influence of electron correlation is examined in detail and the frontier molecular orbital concept is extended to correlated levels of theory by defining it as the eigenvector of the Fukui matrix with the largest eigenvalue. The effect of degeneracy on the Fukui matrix is examined in detail, revealing that this is another way by which the unity eigenvalue and perfect pairing of eigenvalues can disappear.     

Can one oxidize an atom by reducing the molecule that contains it?

Negative values for the condensed Fukui function are identified as the key to designing molecules in which reduction of the molecule is associated with oxidation of one of the atomic centers, or vice versa. Sufficient conditions for negative condensed Fukui functions are derived, and metal complexes are identified as likely candidates for this exotic redox chemistry. Based on our theoretical understanding of where negative values of the Fukui function occur [P. W. Ayers, R. C. Morrison and R. K. Roy. J. Chem. Phys., 2002, 116, 8731], molecular-orbital diagrams for molecules where molecular oxidation is coupled to atomic reduction (or vice versa) are sketched. Whether one could design a metal complex with these properties is an open question but, if one could, then that compound would have fascinating redox chemistry and interesting magnetic properties. Candidate molecules for this property include metal complexes with small metal-to-ligand and/or ligand-to-metal charge transfer excitation energies.     

Observation of the aggregation behavior of silica sols using laser speckle contrast measurements

We report on a new optical method to observe the onset of aggregation in alcoholic tetraethoxysilane (TEOS) sols using laser speckle contrast measurements. The contrast in a speckle image produced by coherent light provides information about the internal contrast of the medium being studied. For silica sols, changing the amount of acid or base catalyst was the most important factor in determining the aggregation behavior of the sol. We investigated this effect by varying the TEOS/base ratio by a factor of 6. This shifted the onset of aggregation as determined by speckle contrast from 46% of the gel time for the lowest amount of base to 74% for the highest. Conversely, varying TEOS/acid ratio by a factor of 3, shifted the onset of aggregation from 74% of the gel time for the lowest amount of acid to 64% for the highest. Measurements of this type provide information that can be used to test models of sol aggregation and gel formation.     

Structurally Diverse Bench-Stable Nickel(0) Pre-Catalysts: A Practical Toolkit for In Situ Ligation Protocols**

A flurry of recent research has centered on harnessing the power of nickel catalysis in organic synthesis. These efforts have been bolstered by contemporaneous development of well-defined nickel (pre)catalysts with diverse structure and reactivity. In this report, we present ten different bench-stable, 18-electron, formally zero-valent nickel–olefin complexes that are competent pre-catalysts in various reactions. Our investigation includes preparations of novel, bench-stable Ni(COD)(L) complexes (COD=1,5-cyclooctadiene), in which L=quinone, cyclopentadienone, thiophene-S-oxide, and fulvene. Characterization by NMR, IR, single-crystal X-ray diffraction, cyclic voltammetry, thermogravimetric analysis, and natural bond orbital analysis sheds light on the structure, bonding, and properties of these complexes. Applications in an assortment of nickel-catalyzed reactions underscore the complementary nature of the different pre-catalysts within this toolkit.