LCAO–MO similarity measures and taxonomy

Similarity measures between pairs of molecular wave functions are described. They are based on the geometrical structure of the LCAO–MO framework and upon multivariate analysis ideas. The theoretical framework is presented, and formulae for some integrals needed are given. Two main measures, distance and correlation coefficients, are used. Distance and correlation matrices induce relationships in the whole MO set, which can be depicted through minimal spanning tree techniques. Furthermore, principal component analysis allows a two-dimensional visualization of the Mo manifold geometrical relationships. Various examples are given in order to obtain information on how basis set, environment, excitation, bending, stretching, and electronegativity affect the induced order. For this purpose “ab initio” SCF–LCAO–MO calculations with double- and single-zeta quality basis sets have been used for various simple molecular structures: H₂O, NH₃, CH₄, N₂, O₂, C₂, NO, CN, and CO. The results obtained can open the way to LCAO–MO taxonomy. Using this information, other areas of interest are connected with similarity measures (SCF and CI , localization procedures, etc.), proving in this manner their potential utility.     

Rigidity of harmonic maps of maximum rank

Thehomotopical rank of a mapf:M →N is, by definition, min{dimg(M) ¦g homotopic tof}. We give upper bounds for this invariant whenM is compact Kähler andN is a compact discrete quotient of a classical symmetric space, e.g., the space of positive definite matrices. In many cases the upper bound is sharp and is attained by geodesic immersions of locally hermitian symmetric spaces. An example is constructed (Section 9) to show that there do, in addition, exist harmonic maps of quite a different character. A byproduct is construction of an algebraic surface with large and interesting fundamental group. Finally, a criterion for lifting harmonic maps to holomorphic ones is given, as is a factorization theorem for representations of the fundamental group of a compact Kähler manifold. The technique for the main result is a combination of harmonic map theory, algebra, and combinatorics; it follows the path pioneered by Siu in his ridigity theorem and later extended by Sampson.     

On Transition Structures for Hydride Transfer Step in Enzyme Catalysis. A Comparative Study on Models of Glutathione Reductase Derived from Semiempirical, HF, and DFT Methods

As a model of the chemical reactions that take place in the active site of gluthatione reductase, the nature of the molecular mechanism for the hydride transfer step has been characterized by means of accurate quantum chemical characterizations of transition structures. The calculations have been carried out with analytical gradients at AM1 and PM3 semiempirical procedures, ab initio at HF level with 3-21G, 4-31G, 6-31G, and 6-31G basis sets and BP86 and BLYP as density functional methods. The results of this study suggest that the endo relative orientation on the substrate imposed by the active site is optimal in polarizing the C4-Ht bond and situating the system in the neighborhood of the quadratic region of the transition structure associated to the hydride transfer step on potential energy surface. The endo arrangement of the transition structure results in optimal frontier HOMO orbital interaction between NADH and FAD partners. The geometries of the transition structures and the corresponding transition vectors, that contain the fundamental information relating reactive fluctuation patterns, are model independent and weakly dependent on the level of theory used to determine them. A comparison between simple and complex molecular models shows that there is a minimal set of coordinates describing the essentials of hydride transfer step. The analysis of transition vector components suggests that the primary and secondary kinetic isotope effects can be strongly coupled, and this prompted the calculation of deuterium and tritium primary, secondary, and primary and secondary kinetic isotope effects. The results obtained agree well with experimental data and demonstrate this coupling.     

Duality and intersection theory in complex manifolds. I

We introduce the concept of a twisting cochain and a twisted complex associated to a coherent sheaf. For sheaves of submanifolds these twisted complexes are used to construct on cochain level the Grothendieck theory of dual class and Gysin map. These explicit constructions give, for instance, a local formula for dual class of higher codimensional submanifolds. We prove a refined version of the Hirzebruch Riemann Roch using such local formulas. We also prove a theorem on when global analytic intersection classes can be computed from first order geometric data. This theory will be used to prove the Holomorphic Lefschetz formula (in Part II) and the Hirzebruch Riemann Roch for analytic coherent sheaves.The first author is supported in part by NSF grants GP-36418X1 and MCS 76-08478. The second by MCS 75-07986 and Sonderforschungsbereich Theoretische Mathematik at Bonn University     

Eigenvectors and cyclic vectors of bilateral weighted shifts. II: Simply invariant subspaces

Let T be an injective bilateral weighted shift onl ² thought as "multiplication by λ" on a space of formal Laurent series L²(β). (a) If L²(β) is contained in a space of quasi-analytic class of functions, then the point spectrum σ_p(T^?) of T^? contains a circle and the cyclic invariant subspaceM _f of T generated by f is simply invariant (i.e., ∩{(T^k M _f)^?: k ≥ 0}= {0}) for each f in L²(β); (b) If L²(β) contains a non-quasi-analytic class of functions (defined on a circle г) of a certain type related with the weight sequence of T, then there exists f in L²(ß) such thatM _f is a non-trivial doubly invariant subspace (i.e., (TM _f)^? =M _f); furthermore, if г ? σ_p(T^*), then σ_p (T^*) = г and f can be chosen so that σ_p([T∣M _f]^*) = г?{α}, for some α ε г. Several examples show that the gap between operators satisfying (a) and operators satisfying (b) is rather small.     

A Molecular Electron Density Theory Study of the [3+2] Cycloaddition Reaction of an Azomethine Ylide with an Electrophilic Ethylene Linked to Triazole and Ferrocene Units

The [3+2] cycloaddition (32CA) reaction of an azomethine ylide (AY) with an electrophilic ethylene linked to triazole and ferrocene units has been studied within the Molecular Electron Density Theory (MEDT) at the ωB97X-D/6-311G(d,p) level. The topology of the electron localization function (ELF) of this AY allows classifying it as a pseudo(mono)radical species characterized by the presence of two monosynaptic basins, integrating a total of 0.76 e, at the C1 carbon. While the ferrocene ethylene has a strong electrophilic character, the AY is a supernucleophile, suggesting that the corresponding 32CA reaction has a high polar character and a low activation energy. The most favorable ortho/endo reaction path presents an activation enthalpy of 8.7 kcal·mol⁻¹, with the 32CA reaction being exergonic by −42.1 kcal·mol⁻¹. This reaction presents a total endo stereoselectivity and a total ortho regioselectivity. Analysis of the global electron density transfer (GEDT) at the most favorable TS-on (0.23 e) accounts for the high polar character of this 32CA reaction, classified as forward electron density flux (FEDF). The formation of two intermolecular hydrogen bonds between the two interacting frameworks at the most favorable TS-on accounts for the unexpected ortho regioselectivity experimentally observed.     

Structural Diversity from the Transannular Cyclizations of Natural Germacrone and Epoxy Derivatives: A Theoretical–Experimental Study

Treatment of germacrone ( 1 ) with different electrophiles, and of its epoxy derivatives germacrone‐4,5‐epoxide ( 2 ), germacrone‐1,10‐epoxide ( 3 ) and isogermacrone‐4,5‐epoxide ( 4 ) with Brönsted/Lewis acids and Ti^III, gives rise to a great structural diversity. Thus, by using a maximum of two steps, the production of more than 40 compounds corresponding to 14 skeletons is described. Computational calculations rationalizing the structural divergence produced are also described. Finally, since some of the compounds generated are bioactive natural sesquiterpenes, the mechanisms of formation of these substances will provide new insights in their biosynthesis.     

Access to Optically Active 3‐Substituted Piperidines by Ring Expansion of Prolinols and Derivatives

The ring expansion of prolinols via an aziridinium intermediate gives C3‐substituted piperidines in good yields and enantiomeric excess, the substituent at the C3 position being derived from the most reactive nucleophile in the reaction mixture. Depending on the nucleophile, the reaction proceeds under thermodynamic or kinetic control. The regioselectivity of attack of nucleophiles on the aziridinium intermediate depends on the nature of the substituents on the nitrogen atom and the C4 position of the starting prolinols.