A Note on Characterizing the k-OPTNeighborhood Via Group Theory

Group theory can be used to model and synthesize the neighborhood of Traveling Salesman tours reachable through k-OPT exchanges. A primary concept is that a dihedral group action partitions the sets of cut arcs so that k-OPT exchanges of orbital elements are conjugate. Also presented is a method to produce all k-OPT exchanges for a given set of cut arcs.     

Synthesis and Structure of a Chromium(III) Complex Supported by a β-diketiminate and an Enediolate Ligand

Abstract  

The complex (bis(2,6-diisopropylphenyl)pentane-2,4-ketiminato)(9,10-dihydro-9,10-(ethenediolato)anthracene)(tetrahydrofuran)chromium(III) (1) has been synthesized and structurally characterized. Complex 1 crystallizes in the spacegroup Cc with the crystal cell parameters a = 22.285 (3) ?, b = 9.3317 (13) ?, c = 40.782 (6) ?, β = 99.338 (3)°, V = 8369 (2) ?³, and Z = 8. Two symmetry unique complex molecules were located in the asymmetric unit making Z′ = 2. The chromium atoms display square pyramidal coordination with THF located in the axial position. The previously neutral α-diketone moiety has been reduced by two electrons to an enediolate group with a corresponding increase of oxidation state of the chromium center.     

The Nicolas and Robin inequalities with sums of two squares

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality σ(n) < e ^γ n log log n holds for every integer n > 5040, where σ(n) is the sum of divisors function, and γ is the Euler–Mascheroni constant. We exhibit a broad class of subsets of the natural numbers such that the Robin inequality holds for all but finitely many . As a special case, we determine the finitely many numbers of the form n = a ² + b ² that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality n/φ(n) < e ^γ log log n; since σ(n)/n < n/φ(n) for n > 1 our results for the Robin inequality follow at once.      

The Nicolas and Robin inequalities with sums of two squares

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality σ(n) < e ^γ n log log n holds for every integer n > 5040, where σ(n) is the sum of divisors function, and γ is the Euler–Mascheroni constant. We exhibit a broad class of subsets S{\mathcal {S}} of the natural numbers such that the Robin inequality holds for all but finitely many n ? S{n \in \mathcal {S}} . As a special case, we determine the finitely many numbers of the form n = a ² + b ² that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality n/φ(n) < e ^γ log log n; since σ(n)/n < n/φ(n) for n > 1 our results for the Robin inequality follow at once.     

Adiabatic approximations to internal rotation

A number of subtle and confusing issues are addressed concerning large amplitude motion (LAM) coordinates (chi) for internal molecular motions, using the methyl rotation in acetaldehyde (CH(3)CHO) as a model problem. If the LAM coordinate is chosen to be one of the H-C-C-O dihedral angles rho(1), rho(2), or rho(3), it lacks the required 2pi3 periodicity, and its use is thus undesirable. An excellent local internal coordinate for this model problem is tau(3)=13(rho(1)+rho(2)+rho(3)-2pi). A similarly good but nonlocal coordinate for the adiabatic approximation of internal rotation is provided by the intrinsic reaction coordinate s. Comparison of the mass-independent V(0)(tau(3)) and the mass-dependent V(0)(s) internal rotation curves shows that the two are virtually identical for the parent isotopolog of acetaldehyde. A unified internal coordinate projection scheme for determining complementary vibrational frequencies and subsequently V(ZPVE)(chi) along a path for LAM has been formulated, where V(ZPVE)(chi) is the zero-point vibrational energy correction to the internal rotation curve. In addition to its simplicity, the projection scheme developed for a distinguished reaction path generated by constrained optimizations is appealing because the vibrational frequencies along the LAM path are invariant to chemically meaningful choices of the internal coordinates for the complementary modes.     

Effective Equations for Repulsive Quasi‐One Dimensional Bose–Einstein Condensates Trapped with Anharmonic Transverse Potentials

One‐dimensional nonlinear Schrödinger equations are derived to describe the axial effective dynamics of cigar‐shaped atomic repulsive Bose‐Einstein condensates trapped with anharmonic transverse potentials. The accuracy of these equations in the perturbative, Thomas‐Fermi, and crossover regimes were verified numerically by comparing the ground‐state profiles, transverse chemical potentials and oscillation patterns with those results obtained for the full three‐dimensional Gross‐Pitaevskii equation. This procedure allows us to derive different patterns of 1D nonlinear models by the control of the transverse confinement even in the presence of an axial vorticity.     

磁瓶式光电子能谱和速度成像光电子能谱对APS-(A=C14H10,蒽)的研究:电子结构,APS·的自旋轨道耦合和APS-的自旋偶极态

Gaseous dibenzo-7-phosphanorbornadiene P-sulfide anions APS^-(A=C₁₄H₁₀ or anthracene) were generated via electrospray ionization, and characterized by magnetic-bottle photoelectron spectroscopy, velocity-map imaging (VMI) photoelectron spectroscopy, and quantum chemical calculations. The electron affinity (EA) and spin-orbit (SO) splitting of the APS^· radical are determined from the photoelectron spectra and Franck-Condon factor simulations to be EA=(2.62±0.05) eV and SO splitting=(43±7) meV. VMI photoelectron images show strong and sharp peaks near the detachment threshold with an identical electron kinetic energy (eKE) of 17.9 meV at three different detachment wavelengths, which are therefore assigned to autodetachment from dipole-bound anion states. The B3LYP/6-31++G(d,p) calculations indicate APS^· has a dipole moment of 3.31 Debye, large enough to support a dipole-bound electron.     

Phenylamido and Diphenylamido Complexes of Chromium(II)

Abstract  Addition LiNHPh to a [(NacNac)Cr(μ-I)]₂ [NacNac = 2,4 pentane-N-N′-bis(2,6-diisopropylphenyl) ketiminato] in THF yields a dimeric anilido complex, 1, that crystallizes in P2₁/n with unit cell parameters a = 13.390(2) ?, b = 26.298(5) ?, c = 21.326(4) ?, β = 93.905(4)°, V = 7492(2) ?³ and Z = 4. The same reaction in the presence of aniline yields a monomeric anilido complex, 2, with a coordinated aniline molecule. Complex 2 crystallizes in P2 ₁ /c with unit cell parameters a = 28.530(5) ?, b = 10.8116(18) ?, c = 26.492(5) ?, β = 115.685(3)°, V = 7364(2) ?³ and Z = 8. Addition of aniline to isolated 1 yields 2. Addition of LiNPh₂ to [(NacNac)Cr(μ-I)]₂ in THF yields a monomeric complex that crystallizes in P2₁/n with the unit cell parameters a = 11.152(6) ?, b = 23.820(12) ?, c = 17.915(9) ?, β = 106.223(9)°, V = 4569(4) ?³ and Z = 4. All complexes have been spectroscopically and structurally characterized. Graphical Abstract  Addition of LiNHPh or LiNPh₂ to [(NacNac)Cr(μ-I)]₂ yields a dimeric or a monomeric complex whereas addition of LiNHPh in the presence of aniline yields the monomeric anilido complex.