Charge‐transfer transitions and optic spectra of chromites: A model computation

In the cluster approach, we consider the peculiarities of charge‐transfer (CT) states and CT O 2p → Cr 3d transitions in the octahedral (CrO₆)^9? complex. We have computed the reduced matrix elements of electric‐dipole transition operator on many‐electron wave functions — the initial and final states of CT transitions. We have parameterized the obtained results and computed the relative intensities of various allowed CT transitions in the absence of the mixing of CT configurations having the same symmetry. Using the Tanabe‐Sugano technique, we have taken into account this mixing and obtained the energies of many‐electron CT transitions and their actual intensities as well. We have also allowed for the Coulomb interaction between the 2p‐electrons of the O^2? ligands and the 3d‐electrons of the central Cr³⁺ ion in the (CrO₆)^9? cluster. This interaction proved insignificant for the optic spectra. Modeling the optic spectrum of chromium‐based oxides has yielded a complicated CT band consisting of 33 lines with the main maximum at about 7 eV and satellites in the range of 4–5 and 8–9 eV. The total extent of the CT band is about 8 eV. The model spectrum is in satisfactory agreement with experimental data, which shows the limited validity of the generally accepted notion of a simple structure of CT spectra. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Wreath products of the groups of monotone permutations

We introduce the concept of wreath product of the m-groups of permutations and prove that an m-transitive group of permutations with an m-congruence is embeddable into the wreath product of the suitable m-transitive m-groups of permutations. This implies that an arbitrary m-transitive group in the product of two varieties of m-groups embeds into the wreath product of the suitable m-transitive groups of these varieties.     

Charge-transfer transitions and optical spectra of chromites

Specific features of charge-transfer states and charge-transfer transitions of the O_2p → Cr_3d type in octahedral complexes (CrO₆)^9? have been considered in the cluster approximation. Reduced matrix elements of the electric-dipole moment operator on many-electron wave functions, which are the initial and final states for charge-transfer transitions, are calculated. The results are parameterized, and the relative intensities of different allowed charge-transfer transitions in the absence of mixing of different charge-transfer configurations with identical symmetry are calculated. This mixing is taken into account within the Tanabe-Sugano theory, and the true energies and intensities of many-electron charge-transfer transitions are obtained. The Coulomb interaction between 2p electrons of O^2? ligands and 3d electrons of the central Cr³⁺ ion in (CrO₆)^9? cluster is considered. The influence of this interaction on the optical spectra is found to be insignificant. Simulation of the optical spectra of chromium oxides has shown the presence of a band of complex charge-transfer transitions composed of 33 lines with a total width of about 8 eV. The model spectrum is in adequate agreement with the experimental data, which indicates limited applicability of the widespread view that charge-transfer transition spectra have a simple structure.     

A Fiber Bundle Approach to the Transpositional Relations in Nonholonomic Mechanics

The equations of nonholonomic mechanics may be derived using a number of variational principles. This paper studies some of these principles from the contemporary geometric point of view, taking into account various bundle structures that are intrinsically present in the nonholonomic setting.     

The geometry of the Routh problem

Summary In this paper the motion without sliding of a homogeneous ball on a surface of revolution is studied. It is shown that the necessary condition for stability of stationary periodic motions of the ball obtained by Routh is also a sufficient one. We prove that the nondegenerate invariant manifolds are diffeomorphic to unions of invariant tori filled with quasiperiodic motions.     

On congruences of m-groups

We indicate a way for constructing m-congruences of an arbitrary m-transitive representation, introduce the notions of m-2-transitive and m-primitive representations, and describe the m-transitive primitive representations in terms of stabilizers. Also we give necessary and sufficient conditions for m-2-transitivity and study some properties of these representations.     

On intersections of Sylow 2-subgroups in automorphism groups of finite simple groups of exceptional Lie type

We consider the problem of interpolation of finite sets of numerical data bounded in L _p -norms (1 ≤ p < ∞) by smooth functions that are defined in an n-dimensional Euclidean ball of radius R and vanish on the boundary of the ball. Under some constraints on the location of interpolation nodes, we obtain two-sided estimates with a correct dependence on R for the L _p -norms of the Laplace operators of the best interpolants.