Symmetry based properties of the transition metal dichalcogenide nanotubes

The full geometrical symmetry groups (the line groups) of the monolayered, 2Hb and 3R polytypes of the inorganic MoS₂ and WS₂ micro- and nanotubes of arbitrary chirality are found. This is used to find the coordinates of the representative atoms sufficient to determine completely the geometrical structure of the tubes. Then some physical properties which can be deduced from the symmetry are discussed: electron band degeneracies, selection rules, general forms of the second rank tensors and potentials, phonon spectra. Received 13 April 2000     

Space-time theories and symmetry groups

This paper addresses the significance of the general class of diffeomorphisms in the theory of general relativity as opposed to the Poincaré group in a special relativistic theory. Using Anderson's concept of an absolute object for a theory, with suitable revisions, it is shown that the general group of local diffeomorphisms is associated with the theory of general relativity as its local dynamical symmetry group, while the Poincaré group is associated with a special relativistic theory as both its global dynamical symmetry group and its geometrical symmetry group. It is argued that the two groups are of equal significance as symmetry groups of their associated theories.     

Classification of the states of nonrigid molecular systems with continuous axial symmetry groups

Construction of the stationary state classification is considered for nonrigid molecular systems whose geometrical symmetry of internal dynamics is determined by continuous axial groups. The construction is based on the methods of symmetry group chains.     

Geometrical approach to the atomic spectra theory. The helium atom

New equations for helium spectra calculations are obtained within geometrical interpretation of quantum mechanics suggested by the author earlier. The main idea of the above interpretation is that atoms can be considered not as the systems with many electrons but as a microscopic topological defect of the physical space-time without any point-like particles inside. The groups of symmetry transformations of such defects are suggested to be isomorphic to the symmetry groups of atoms with many identical electrons (permutation group, for example). New equations were derived within approximation that is similar to the one in the self consistent field theory of Hartree-Fok, but these equations differ strongly from Hartree-Fock equations. Numerical calculations of ionization potentials for para—and orto—helium lead to results that are in a good agreement with experiment.     

Operators of physical quantities of nonrigid molecular systems with continuous axial symmetry groups

The construction of the effective operators of physical quantities is considered for nonrigid molecular systems whose geometrical symmetry of internal dynamics is determined by continuous axial groups. The construction is based on the methods of a symmetry group chain.     

Coupling of matter to supergravity from the rheonomic symmetry point of view and the origin of the auxiliary fields

We write a first-order action for the Wess-Zumino supermultiplet regarding it as a 0-form on the graded Poincaré supergroup manifold. The standard supersymmetry transformations are reproduced by the rheonomic symmetry mechanism. When this action is coupled to the geometrical action of supergravity on the group manifold, the same rheonomic symmetry mechanism automatically generates the so-called minimal set of auxiliary fields.     

Geometrical spin symmetry and spin

Unification of General Theory of Relativity and Quantum Mechanics leads to General Quantum Mechanics which includes into itself spindynamics as a theory of spin phenomena. The key concepts of spindynamics are geometrical spin symmetry and the spin field (space of defining representation of spin symmetry). The essence of spin is the bipolar structure of geometrical spin symmetry induced by the gravitational potential. The bipolar structure provides a natural derivation of the equations of spindynamics. Spindynamics involves all phenomena connected with spin and provides new understanding of the strong interaction.     

Cylindrical Space–Time Model and Mirror Symmetry Violations

The symmetrical character of empty space–time is proposed as a putative reason for mirror symmetry violations in the bioorganic world and in the world of elementary particles, and a new geometrical model for the empty space–time is put forward. In contrast to flat four-dimensional Minkowski space, this model is described by a cylindrical four-dimensional space metrics. The geodesic lines of this space are helical. The helical structure of the model proposed shows that distortion of mirror symmetry may be a consequence of geometrical structure of the empty space rather than a characteristic of any specific interaction.     

On the definition of intramolecular motions

The restrictions imposed when defining separate types of internal motions in rigid and nonrigid molecules are considered. These restrictions are based on the symmetry properties of intramolecular dynamics. It is shown that the violation of these restrictions gives rise to physically incorrect consequences.     

The symmetry group paradox for non-rigid molecules

In many situations, the energy levels for a quantum system, whose Hamiltonian is invariant under a specific symmetry group, are split when the Hamiltonian is replaced by a new one with lower symmetry. In non-rigid molecules (NRM), fast quantum tunnelling processes allow the molecule to change between different geometrical configurations related by permutations of identical nuclei (or with inversion as well), resulting in the splitting of the energy levels for the rigid molecule (RM) case where tunnelling is absent. However, for NRM, there is apparently a paradoxical situation where although the original RM energy levels are associated with a symmetry group isomorphic to the point group for the geometrical configuration, the split NRM energy levels are associated with a symmetry group consisting of all permutations and inversions related to the fast quantum tunnelling processes between configurations, and for which the point group is a subgroup. The resolution of this paradox, where energy level splitting is evidently accompanied by an enlargement of the symmetry group, is the subject of this article.     

On the Dual Symrnetry in Two-dimensional Field Theory

Discrete and continuous dual symmetry is analysed and explained in its geometrical origin. Linear eigenvalue equations of the dual transforming operators U are deduced in different forms in a suitable gauge, under the gauge transformation. Different dual parameters are compared too.     

A geometrical setting for Lax equations associated to dynamical systems

We develop a geometrical framework for dealing with Lax equations associated to dynamical systems over a manifold M. We also show that this theory reproduces the global versions of Lax equations given before as well as the usual theory of reduced systems obtained from systems defined on Lie groups and with such group as a symmetry group.     

Problem of mixing of stereoisomers in the hydrazine molecule N2H4

It is shown that the important consequence of nonrigid motions observed in the hydrazine molecule N₂H₄ is mixing of stereoisomers, although the geometrical symmetry group, taking into account these motions, similarly to the point group of one equilibrium configuration, has no improper transformations of the molecule as a whole. Here, an important point is that the geometrical group of a nonrigid molecule is the so-called noninvariance dynamical group. The effect of mixing of stereoisomers on the total picture of nonrigid motions is considered using the symmetry methods. In particular, it is shown that a nonrigid molecule exhibits properties of a symmetric top, whereas a rigid molecule represents an asymmetric top.     

二十面体Sc13，Sc13¬+1，Sc13¬-1团簇的稳定性与电子结构研究

采用密度泛函理论框架下的广义梯度近似(DFT/GGA),对Sc13团簇进行了几何结构优化,得到13原子钪团簇的基态结构为正二十面体(Ih),在此基础上对二十面体(Ih)Sc13, Sc13¬+1和Sc13-1团簇的稳定性、电子结构磁矩进行研究.结果发现Sc13,Sc13¬+1和Sc13¬-1团簇都在Ih结构时最稳定,该尺寸团簇的稳定性主要由二十面体密堆积构型决定;带电能使团簇的结构稍稍收缩从而使团簇的稳定性有所增强;团簇的键长和对称性对团簇的磁矩有很明显的影响.     

四碘甲状腺素团簇结构与光谱性质的密度泛函理论研究

在B3LYP/Lanl2mb基组水平上,利用密度泛函理论(DFT)优化了四碘甲状腺素团簇的几何结构.基于该团簇的几何结构下,其吸收和发射光谱的研究使用相同的基组水平并采用极化连续介质模型(PCM)下用含时密度泛函理论(TDDFT).研究结果表明,优化所得甲状腺素团簇的几何结构对称性为C_1;在基态稳定结构基础上,得出其输运性质,即甲状腺素团簇为p型输运材料;通过含时密度泛函理论,在优化好的基态结构基础上,又计算了它的溶剂效应,进一步得出该分子在水溶剂中的吸收光谱和发射光谱特性.     

利用对称性原理推导几何光学三定律

从对称性的角度,分析了光波在无限大均匀介质中、两种媒质的分界面上传播必须遵从的时间和空间平移不变性,推导出光的直线传播、反射以及折射定律,揭示了支配几何光学三定律的更为深刻而底层的对称性原因,这种视角对于光学部分的教学将产生深刻的影响.     

Effect of the thickness of a dicarbocyanine dye layer on the conformational composition and spatial orientation of layer components

The conformational and aggregate structure and the spatial orientation of polyatomic organic molecules on solid substrates are investigated by the example of molecular layers of symmetric and asymmetric dicarbocyanine dyes. It is shown that the structure of a dicarbocyanine dye layer is determined by two mechanisms: (i) the change in the symmetry of the intramolecular charge distribution in the monomeric components of the layer as a result of the intermolecular interaction with the substrate, which leads to closer values of the equilibrium concentrations of two monomers with different spatial structure (rotational stereoisomers), and (ii) the association of monomer molecules with formation of dimers and J aggregates. The relative position of the spectra of the layer components, the spatial orientation of the components, and the concentration ratio of different monomeric forms depend on the structure and the electron-donating ability of terminal groups. In going from a monomolecular layer to multilayered structures, the asymmetry of the intramolecular charge distribution induced by the substrate and the ratio of different stereoisomeric forms first change and then become stabilized.     

Symmetry of torsional motion of identical tops in the CH2D-CH2D isotopic form of the ethane molecule

The symmetry of the torsional motion of identical tops in the CH₂D-CH₂D isotopical form of the ethane molecule is described by using the dynamic noninvariant group, i.e., the group that is larger than the symmetry group of the Hamiltonian of intramolecular motion in the Born-Oppenheimer approximation. As a result, the nuclear statistical weights of stationary states are determined with allowance for the motion of identical tops between energetically nonequivalent configurations and a procedure is developed for the qualitative construction of the operators of physical quantities describing the torsional-rotational spectrum of this system in an arbitrary vibronic state.