A first-order boundary value problem with boundary condition on a countable set of points

LetE={E _n } be the family of subspaces spanning the eigenfunctions and adjoint functions of the boundary-value problem $$ - i\frac{{dy}}{{dx}} = \lambda y, - \alpha \leqslant x \leqslant \alpha , U(y) \equiv \int_{ - a}^a {y(t)} d\sigma (t) = 0$$ that correspond to “close” eigenvalues (in the sense of the distance defined as the maximal of the Euclidean and the hyperbolic metrics). For a purely discrete measuredσ, it is shown that the systemE does not form an unconditional basis of subspaces inL ²(?a, a) if at least one of the end points ±a is mass-free.     

Transmission of electronic effects in certain ployenes

The effect of substituent variation in compounds of the general type R(CHCH)_nCHO (1) or p-RC₆H₄ (CHCH)_nCHO (2) on the stretching vibrations of the CHO group (ν_co), the dipole moments (μ), the long wave maxima (λ_max), and the halfwave potentials (E_{$\text{1}\text{2}$}) for the polarographic reduction of the CHO group have been studied as well as the influence of variation of substituent R on the rate constants (k) of the alkaline hydrolysis of polyenic esters R(CHCH)_nCOOEt (3). A linear relationship between the σ⁺, σ_c⁰ or σ_c⁺ values and the experimental values of ν_co, μ, gl_max, E_{$\text{1}\text{2}$} and k has been established, which indicates the importance of the mesomeric mechanism of the transmission of electronic effects through the chain of conjugated double bonds in compounds 1, 2 and 3. Transmission coefficients (π′) for one (π′_1F), two (π′_2F), three (π′_2F), and four (π′_4F) double bonds were calculated, and it was found that these values are bound together by the ratio: (π′_1F):(π′_2F):(π′_3F):(π′_4F) = (π′_1F):(π′_1F)²:(π′_1F)³:(π′_1F⁴, i.e. the influence of substituents R is decreased m a geometrical progression when the number of double bonds increases. The distribution of the electronic density in 1 was calculated by the LCAO SCF method in the Pariser-Parr-Pople approximation, and it was found that the π-electronic density values at the O atoms of the CHO group correlates well with σ_c⁰ constants and the calculated values of π′_1F, π′_2F, and π′_3F are bound together by the above ratio.     

Hypercoordinate atoms of second-row elements in dodecahedrane endohedral complexes

The energy characteristics and geometric parameters of the dodecahedrane endohedral complexes X@C₂₀H₂₀ (X = C⁴⁻, N³⁻, O²⁻, F⁻, Ne) were studied by the density functional theory B3LYP method with the 6-311G(d,p), 6-311+G(d,p), and 6-311G(df,p)) basis sets. In all structures the central atoms X are characterized by a coordination number of 20. The energy of formation of the complexes decreases in the order X = C⁴⁻, N³⁻, O²⁻, F⁻, Ne. The coordination number of the central atom remains unchanged upon adding Li⁺ counterions to anionic systems. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 5, pp. 824–830, May, 2007.     

Double proton shifts in associates of formic acid with hydrides of third period elements

The mechanisms of double proton shift in associates HC(O)OH ... X of formic acid with hydrides (X = SiH₄, PH₃, PH₅, H₂S, SH₄, CIH, and CIH₃) were studied by theab initio method (SCF/3G*). The activation barriers to this reaction in associates with PH₃, H₂S, SH₄, CIH, and CIH₃ are equal to 68.3, 10.0, 26.0, 1.0, and 0.4 kcal mol^–1, respectively. For X = SiH₄ and PHS₅ transition states for the double proton shift were not determined, and in all of the other cases studied they are synchronous (concerted or one-step).Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 4, pp. 817–822, April, 1996.     

Polydentate coordination of the thallium atom in complexes with chiral amidinylcyclopentadienyl ligands

The chiral thallium amidinium cyclopentadiene-N-ylide complexes [C₅(CO₂Me)₄{ArNC(Ar")NAr}]Tl were synthesized and structurally characterized by X-ray diffraction analysis and NMR spectroscopy. In these complexes, an unusual mode of coordination of the thallium atom was found, viz., the thallium atom is coordinated by both the side-chain nitrogen atom (N—Tl, 2.833(6) ) and the system of the cyclopentadienyl ring (Tl—Cp, 2.887(4) ⁵-bonding).     

Basicities and structures of 2-(N-arylaminomethylene)-3(2H)-benzo[b]thiophenones,-benzo[b]furanones,and -benzo[b]selenophenones

The ionization constants of the conjugate acids of the reaction series 2-(N-arylaminomethylene)-3(2H)-benzo[b]thiophenones,-benzo[b]furanones, and -benzo[b]selenophenones were determined by potentiometric titration in anhydrous acetonitrile. The electronic and vibrational spectroscopic data showed that the protonation center in the molecules of these compounds is the carbonyl oxygen atom. It was established by correlation analysis that N-aryl substituents affect the protonation center primarily via an induction mechanism.Communication XVIII from the series Basicities and structures of azomethines and their structural analogs. See [1] for communication XVII.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. pp. 619–623, May, 1977.     

Synthesis and Structure of New 2-(2-Quinolyl)-1,3-tropolone Derivatives

4,6-Di-tert-butyl-3-nitro-1,2-benzoquinone reacts with substituted 2-methylquinolines to give the corresponding 2-(2-quinolyl)-4-nitro-1,3-tropolones and 2-(2-quinolyl)-1,3-tropolones.     

Quantum-chemical Investigation of the Hyprevalent Intramolecular Coordination X←N (X = C,Si, Ge) in Quasimonocyclic Models of IVa Group Atranes

The structure and the pentacoordination effect in quasimonocyclic models of IVa group atranes were investigated by ab initio[MP2 (full) /6-311+G **] and the density functional [B3LYP/6-311+G **] quantum chemical calculations. The calculations revealed considerable stabilization of the quasimonocyclic conformations relative to their free-of-strain trans-s-transconformations, which is caused by the formation of secondary (R)XN (X=C, Si, Ge) bonds of the hypervalent type. The strength of the intramolecular (R)XN coordination increases in the order X=C, Si, Ge. The nature of attractive (R)XN coordination is determined by donor-acceptor interaction of the nitrogen lone electron pair and antibonding orbital which primary localize at the X-R bond. Energy of X ··· N (X=Si, Ge) contact is about 3-7 kcal mol^-1.     

Reaction of 1-h-1-oxo-2,4,6,8-tetrakis(tert-butyl)phenoxazine or the potassium salt of 1-hydroxy-2,4,6,8-tetrakis(tert-butyl)-phenoxazin-10-yl radical with certain platinum metal derivatives

Conclusions The reaction of 1-H-1-oxo-2,4,6,8-tetrakis(tert-butyl)phenoxazine or the K salt of 1-hydroxy-2,4,6,8-tetrakis(tert-butyl)phenoxazin-10-yl radical with platinum metal compounds forms paramagnetic complexes in which the unpaired electron is delocalized in the ligand. In the case of Pd and Pt allyl compounds the complexes are quite stable and can be isolated.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 8, pp. 1841–1847, August, 1987.