Molecular geometry and electronic structures of stable organic derivatives of divalent germanium and tin [(\operatorname{Me} _3 \operatorname{Si} )_2 \operatorname{N} {\kern 1pt} ---{\kern 1pt} Me_2 ]_n (M = Ge, n = 1; M = Sn, n = 2): a theoretical study

The molecular geometry and electronic structure of stable organic derivatives of divalent germanium and tin, [(Me₃Si)₂N-M-OCH₂CH₂NMe₂]_n (M = Ge (4), n = 1; M = Sn (5), n =2) and their isomers with broken (4a, 5a) and closed (4b, 5b) intramolecular coordination bonds M←NMe₂, were studied by the density functional (PBE/TZ2P/SBK-JC) and NBO methods. Factors responsible for stability of their dimers 4c and 5c were established. Dimerization of 5b in the gas phase is a thermodynamically favorable process (ΔG ⁰ = ?2.1 kcal mol^?1) while that of 4b is thermally forbidden (ΔG ⁰ = 10.1 kcal mol^?1), which is consistent with experimental data. The M←NMe₂ coordination bond energies, ΔE ⁰, were found to be ?5.3 and ?8.6 kcal mol^?1 for M = Ge and Sn, respectively. NBO analysis showed that the metal atoms M in molecules 4 and 5 are weakly hybridized. The lone electron pairs of the M atoms have strong s-character while vacant orbitals of these atoms, LP* M, are represented exclusively by the metal np_z-AOs. The strongest orbital interactions between subunits in dimers 4c and 5c involve electron density donation from the lone electron pairs of oxygen atoms (LP O) to the LP* M orbitals.     

Microscopic description of the reactions d + d → p + 3H and d + d → n + 3He at low energies

A collective adiabatic approach is used to explore the total and differential cross sections for the reactions d + d → p + ³H and d + d → n + ³He at incident-deuteron energies of up to 6 MeV. All substantially contributing partial waves of order not higher than that of G waves are taken into account. The experimental value of the difference of the cross sections for the above mirror reactions is reproduced theoretically under the assumption that nuclear forces obey the condition of isotopic invariance. The positions and amplitudes of the maxima in the calculated total cross sections virtually coincide with those of the corresponding experimental values. It is shown that, around the maxima of the cross sections under study, dominant contributions to them come from the P wave. The sensitivity of observables to the parameters of nucleon-nucleon interaction is analyzed.     

Heteroorganic betaines

The structures of 6,6-dimethyl-6-silafulvene C₅H₄SiMe₂ (3), its donor-acceptor complex with ammonia. C₅H₄SiMe₂·NH₃, dimethylfulvene, a number of cyclopentadienylides, methylenetrimethylphosphorane (6), and silicon-containing organophosphorus betaine⁻C₅H₄SiMe₂CH₂PMe₃ ⁺ (13), the product of nucleophilic addition of6 to3, were calculated using the density functional approach. For compound13, the potential energy minimum corresponds to the conformation withgauche-arrangement of the cyclopentadienyl anionie and trimethylphosphonium cationic centers and a C−Si−C−P dihedral angle of 30.5°, which is due to the Coulomb attraction between these centers. According to calculations, betaine13 is rather stable toward decomposition into3 and6 (ΔH ^o=42 kcal mol⁻¹, ΔG ^Δ=30 kcal mol⁻¹). The main channel of thermal decomposition of compound13 involves an intramolecular nucleophilic substitution, which proceeds with elimination of trimethylphosphine and results in 1,1-dimethyl-1-silaspiro[2,4]hepta-4,6-diene, which then undergoes a ready and irreversible isomerization into 6,6-dimethyl-6-silabicyclo[3.2.0]hepta-1,3-diene owing to the [1.5]-sigmatropic shift of the C−Si bond. For Part 4, see Ref. 1. Published inIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 11, pp. 1850–1857, November, 2000.     

Complexity of a determinate algorithm for the discrete logarithm

Translated from Matematicheskie Zametki, Vol. 55, No. 2, pp. 91–101, February, 1994.     

Electronic structure and stability relations of CaF2 and NaCl phases in a Ti-H system

Calculations are made of the electronic structures of TiH_0.73 and TiH hydrides having a NaCl crystal structure and TiH_1.5, TiH_0.73, and TiH having a CaF₂ structure. Estimates of their superconductivity temperature indicate that this should be significantly higher for the stoichiometric NaCl phase compared to nonstoichiometric phases. The results are compared with the characteristics of the χ phase of titanium hydride TiH_0.73 which undergoes a superconducting transition at 4.3 K. Fiz. Tverd. Tela (St. Petersburg) 40, 195–197 (February 1998)     

Spin-Dependent Vibrational Spectra of the Gd@C60 Endohedral Metallofullerene

Optics and Spectroscopy - We present an ab initio calculation of the structures and the IR spectra of the endofullerene Gd@C60 in different spin states. The lines in the IR spectra of Gd@C60...     

Expansion and Destruction of a Relativistic Electron Beam in a Uniform Magnetostatic Field under the Influence of a Symmetric TM Wave

We solve the problem of the motion of a flow of relativistic electrons guided by a uniform magnetostatic field in the channel of interaction with a symmetric TM wave. The problem is related to the analysis of operation of powerful oscillators of Cherenkov microwave radiation and can also have other applications. This problem is of special interest for a possible reduction of the magnetostatic field within the permissible limits of the beam expansion. For synchronous interaction of electrons with a TM wave, we obtain a simple estimate of the possible expansion of the beam through the energy argument and consider the characteristic features of motion in some cases on the basis of more accurate calculations including those for the case of violation of the radial finiteness of motion (microwave defocusing of the beam). Attention is drawn to the inconsistency of the known models of expansion of the beam with a fixed maximum velocity of the radial drift of electrons.     

Anharmonic resonances in the vibrational spectra of pyrazine

The quartic force field of pyrazine has been calculated in the B3LYP/6-31G(d) hybrid density-functional approximation. Based on the results of this calculation, the total IR (250–3800 cm^–1) and Raman (400–3200 cm^–1) spectra of pyrazine have been interpreted with consideration for the Fermi and Darling-Dennison resonances and their spectral manifestations. A precision method is proposed for anharmonic analysis of the vibrational states of polyatomic molecules on the basis of consideration of their theoretical anharmonicity constants in combination with the corresponding experimental frequencies. The method of linear scaling of frequencies has been theoretically substantiated.__________Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 72, No. 1, pp. 13–22, January–February, 2005.     

On the structure of finite commutative rings with an identity

We describe the structure of the finite primary rings of principal ideals; we prove that every such ring is the factor-ring of the ring of integers of a finite extension of the field of rational p-adic numbers; we touch on the problem of the number of nonisomorphic rings of this type with a fixed number of elements.Translated from Matematicheski Zametki, Vol. 10, No. 6, pp. 679–688, December, 1971.The author wishes to express his deep thanks to A. I. Uzkov and V. P. Elizarov for their regular help and interest in this study.