Infrared spectra and structure of carbanions : XIII. Dimeric dianions derived from α,β-diaryl-acrylonitriles

The IR spectra of the products of electrochemical and chemical (alkali metals, naphthalene anion-radical) reduction of substituted α,βdiaryl-acrylonitriles (32 compounds) were studied. It was found that the anion-radicals initially formed dimerize into the corresponding dimeric dianions. Correlation of v(CN) (dimeric dianions) with v(CN) (sodium methoxide adducts, ref 26), as well as with the σ^? constants of the substituents in the α-aromatic rings were found. The effects of the β-substituents, counter ions and solvents on v(CN) of the dimeric dianions are discussed.     

Infrared spectra and structure of carbanions: XVIII. A semiempirical study of carbanions of cyanoacetic acid derivatives

The IR spectra of cyanoacetic acid, ethyl cyanoacetate and cyanoacetamide as well as those of related mono- and, whenever possible, dianions have been studied in dimethyl-sulphoxide (DMSO) and DMSO-d₆.The observed nitrile and carbonyl absorption frequencies correlate linearly with the corresponding Wiberg bond indices given by CNDO/2 calculations with full geometry optimization. These calculations predict carbanionic structures throughout except in the case of the dinegative ion of cyanoacetamide, which could be considered as originating from the aminoacetylenic tautomer of NCCH₂CONH₂. Parallel MINDO/3 calculations, however, predict that the latter dianion is again a carbanion. This result is in reasonable agreement with normal coordinate calculations and the experimental isotopic shifts of vibrational frequencies of the dianion ¹⁵NCCH^?CONH^?.     

Infrared spectra and structure of carbanions—XIII: A study of the carbanions generated from acetonitrile,acetonitrile-d3 and acetonitrile-15n

The study of the IR spectral data for metallated acetonitrile (counter ions Li⁺, Na⁺, K⁺ in solvents tetrahydrofuran, THF, and hexamethylphosphotriamide, HMPT) and its D_3? and ¹⁵N- derivatives together with $\text{CNDO}\text{2}$ and normal coordinate calculations showed that the mesomeric ion H₂CCN has a favoured planar structure and the carbon-metal bond has a pronounced ionic character.     

Ab initio HF, density functional and experimental studies on the IR spectra and structure of 1,2-benzisothiazol-3-(2H)-thione-1,1-dioxide (thiosacchanin) and its nitranion

The spectral and structural changes taking place in the course of the conversion of 1,2-benzisothiazol-3-(2H)-thione-1,1-dioxide (thiosaccharin) into a nitranion have been studied on the basis of both IR spectra and ab initio HF 6-31G(d) and BLYP 6-31G(d,p) force field calculations. The conversion causes nu(as)SO2 and nu(s)SO2 frequency decreases of 47 and 13 cm(-1), respectively, and other spectral changes. The nuC=S coordinate is strongly delocalized. The ab initio geometries of the isolated molecule and nitranion agree well with the single-crystal X-ray ones, determined for thiosaccharin and its sodium (potassium) monohydrate salts, respectively. The nitranionic charge is delocalized almost uniformly within the thiocarbonyl (0.29 e-), sulfonyl (0.24 e-), and phenylene (0.24 e-) groups, and the nitranionic center (0.23 e-).     

The conversion of phenylpropanedinitrile (phenylmalononitrile) into the carbanion, followed by IR spectra, ab initio and DFT force field calculations

The spectral and structural changes, caused by the conversion of phenylpropanedinitrile (phenylmalononitrile) into the carbanion, have been followed by IR spectra, ab initio HF, MP2 and DFT BLYP force field calculations. In agreement between theory and experiment, the conversion is accompanied with strong frequency decreases (with 114 cm(-1), mean value) of the cyano stretching bands nu(C triple bond N), dramatic increases in the corresponding integrated intensities (136-fold, total value), strong enhancement of the nu(C triple bond N) vibrational coupling and other essential spectral changes. According to the calculations, the strongest structural changes take place at the carbanionic center: (i) shortenings of the Cz-Ph and Cz-CN bonds with 0.064-0.092 A, and increases in the corresponding bond orders with 0.14-0.21 U; (ii) simultaneous enlargements of the bond angles at the same carbon atom with 7.6 degrees -9.7 degrees, as from tetrahedral its configuration becomes trigonal. The carbanionic charge is distributed between the two cyano groups (0.44-0.52 e(-)), phenyl ring (0.31-0.34 e(-)) and carbanionic center (0.14-0.25 e(-)). The formation of moderately strong (CH(3))(2)S=O...H-C(CN)(2)C(6)H(5) hydrogen bonds has been found experimentally.     

Universal Algorithms for Learning Theory. Part II: Piecewise Polynomial Functions

This paper is concerned with estimating the regression function f_ρ in supervised learning by utilizing piecewise polynomial approximations on adaptively generated partitions. The main point of interest is algorithms that with high probability are optimal in terms of the least square error achieved for a given number m of observed data. In a previous paper [1], we have developed for each β > 0 an algorithm for piecewise constant approximation which is proven to provide such optimal order estimates with probability larger than 1- m^-β. In this paper we consider the case of higher-degree polynomials. We show that for general probability measures ρ empirical least squares minimization will not provide optimal error estimates with high probability. We go further in identifying certain conditions on the probability measure ρ which will allow optimal estimates with high probability.     

Adaptive Finite Element Methods with convergence rates

Summary. Adaptive Finite Element Methods for numerically solving elliptic equations are used often in practice. Only recently [12], [17] have these methods been shown to converge. However, this convergence analysis says nothing about the rates of convergence of these methods and therefore does, in principle, not guarantee yet any numerical advantages of adaptive strategies versus non-adaptive strategies. The present paper modifies the adaptive method of Morin, Nochetto, and Siebert [17] for solving the Laplace equation with piecewise linear elements on domains in ² by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm (which is equivalent to the H¹ norm). Namely, it is shown that whenever s>0 and the solution u is such that for each n1, it can be approximated to accuracy O(n^–s) in the energy norm by a continuous, piecewise linear function on a triangulation with n cells (using complete knowledge of u), then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process. Moreover, the number of arithmetic computations in the proposed method is also of order O(n) for each n1. The construction and analysis of this adaptive method relies on the theory of nonlinear approximation.Mathematics Subject Clasification (2000): 65N30, 65Y20, 65N12, 65N50, 68W40, 68W25.This work has been supported by the Office of Naval Research Contract Nr. N00014-03-10051, the Army Research Office Contract Nr. DAAD 19-02-1-0028, the National Science Foundation Grants DMS 0221642, DMS 9872890 the Deutsche Forschungsgemeinschaft grant SFB 401, the European Communitys Human Potential Programme under Contract HPRN-CT-2002-00286, Breaking Complexity.