Identification of symmetries in molecules and complexes

An algorithm is presented that quickly detects local and global symmetries of single molecules and complexes. Based upon the Morgan Naming Algorithm, the algorithm involves traversing the molecule from a starting atom and building up a molecule name based upon the names of the atoms encountered along the traversal. Additional molecule names are generated from other starting atoms, and name-name matches are identified as corresponding to symmetry operations. A number of enhancements relative to prior methods yield increased efficiency and extended functionality. In particular, the present method detects not only global symmetries but also local symmetries associated with bond rotations as well as symmetries that are only apparent when alternate resonance forms are considered. Importantly, the present method works not only for single molecules but also for multimolecular complexes. As a consequence, it is well, and perhaps uniquely, suited to applications in supramolecular and host-guest chemistry. Applications include filtering out redundant conformations during conformational searching and free energy calculations; accelerating ligand-receptor docking calculations by reducing the sampling ranges of rotatable bonds linked to locally symmetric groups, such as phenyls; and automating the calculation of symmetry numbers for thermochemical applications.     

Fuzzy space periodic symmetries for polyynes and their cyano-compounds

In our previous papers on the molecular fuzzy symmetry, we analyzed the basic characterization in connection with the fuzzy point group symmetry. In this paper, polyynes and their cyano-derivatives are chosen as a prototype of linear molecules to probe the one-dimensional fuzzy space group of parallel translation. It is notable that the space group is an infinite group whereas the point group is a finite group. For the fuzzy point group, we focus on considering the fuzzy characterization introduced due to the difference of atomic types in the monomer through point symmetry transformation in the beginning; and then we consider the difference between the infinity of space group and the finite size of real molecules. The difference between the point group and the space group lies in the translation symmetry transformation. This is the theme of this work. Starting with a simple case, we will only analyze the one-dimensional translation transformation and space fuzzy inversion symmetry transformation in this paper. The theory of the space group is often used in solid state physics; and some of its conclusions will be referred to. More complicated fuzzy space groups will be discussed in our future papers.     

Practical modeling of molecular systems with symmetries

A new method for efficient modeling of macromolecular systems with symmetries is presented. The method is based on a hierarchical representation of the molecular system and a novel fast binary tree‐based neighbor list construction algorithm. The method supports all types of molecular symmetry, including crystallographic symmetry. Testing the proposed neighbor list construction algorithm on a number of different macromolecular systems containing up to about 200,000 of atoms shows that (1) the current binary tree‐based neighbor list construction algorithm scales linearly in the number of atoms for the central subunit, and sublinearly for its replicas, (2) the overall computational overhead of the method for a system with symmetry with respect to the same system without symmetry scales linearly with the cutoff value and does not exceed 50% for all but one tested macromolecules at the cutoff distance of 12 Å. (3) the method may help produce optimized molecular structures that are much closer to experimentally determined structures when compared with the optimization without symmetry, (4) the method can be applied to models of macromolecules with still unknown detailed structure. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Discrete spatial symmetries in energy derivatives

Based upon the invariance of N-particle systems under discrete operations of reflection, inversion, and rotation by 2π/n, a method for obtaining complete sets of relations among energy derivatives of all orders has been presented. The method is based on the criterion that, for a discrete symmetry operation such as reflection across a plane; the change in energy due to simultaneous arbitrary displacements of each particle is equal to another set of “conjugate” displacements of all particles. Applications of the above relations for particular molecular systems possessing a combination of symmetry operations is also presented. Here, via the row echelon analysis, the number of independent derivatives are found, and simple symmetry relations that allow determination of the remaining derivatives are presented. For example, for a homonuclear diatomic molecule with D_2h symmetry only 1 of the 6 first derivatives and 1 of the 21 second derivatives need to be independently calculated.     

Symmetries and fuzzy symmetries of Carbon nanotubes

Carbon nanotubes (CNTs) possess the fuzzy cylinder group characteristic. Comparing with the linear and planer molecules, there are included the fuzzy symmetry of the cylinder screw rotation (CSR) in relation to some higher ( $>$ 2) fold rotation axis. The CSR may be noted as the product of translation (T) and rotation (C). The CSR symmetry will be imperfect owing to the introduction of T. As the extent of whole translation is more than 10-fold than every time, the membership function of CNT in relation to CSR will be more than about 0.9, and such CNT may be seems as provided with the perfect CSR symmetry. For analyse the CNT we may using the cylindrical orthogonal curvilinear coordinate system. The MO ought to be provided with a pure irreducible representation, but the component of symmetry adapted atomic orbital (SA-AO) set may be not sole, and it is difficult to get and analyse the ‘pure’ $\uppi $ -MO. There are some various AO (1S-, 2S-, 2Pz-, 2Pr-, 2Pt of carbon and 1S- for hydrogen)-set components in a certain MO. For the CNT with the same diameter and different length, the MO energy and the SA-AO component versus the relative serial number will be with the similar distribution. The MOs of CNT with higher fold C symmetry may be provided with two-dimensional irreducible representation. For the molecular skeleton and the MO which belong to one-dimensional irreduable representation, their membership functions in relation to the CSR with the product of the same T and different C would be equality. However, for the single MO which belong to two-dimensional irreducible representation that may be somewhat difference. The torus carbon nanotube (TCNT) may be provided the symmetry with the torus group and torus screw rotation (TSR), such symmetry would be or near be not rare in nature. Similar as the planer rectangle (called as the MH rectangle) may composed the Hückel- or Möbius-strip band, the more MH rectangles in the cylinder CNT may be composed the more Hückel- or Möbius-strip bands, such strip bands set may be called strip tube, meanwhile the fuzzy CSR symmetry will be transform to the perfect TSR symmetry. The intersecting line (Z-axis) of the MH rectangles will be transform to the common basic circle of these strip bands. When the CNT to form a TCNT, as one of the MH rectangle form a Hückel-strip band or an $n(t)$ -twisted Möbius-strip band itself, the other MH rectangle will be form the strip band with the same topological structure synchronously, and the set of these strip bands may be called the strip tube. The boundary closed curves of the strip band may reflect the torus group symmetrical characteristic of the relative strip bands. The closed curve may correspond to a cyclical group or subgroup. The number of carbon atomic pairs on the closed curve denoted the order of such group or subgroup. As the CNT to form the TCNT, it is different as the single MH rectangle, they may be to form the fractal-twisted Möbius-strip tube synchronously, in which the single Möbius-strip band may be formed from more one MH rectangle, however, single MH rectangle may enter into only one Möbius-strip band. As the hetero-CNT with the helical-structure distribution, such hetero-CNT may form the relative torus hetero-CNT, but according to the continuity of CNT tube side, a certain twisted to form Möbius-strip tube may often be required. There is some interaction between the distributional helical-structure and twisting way, such interaction may touch to the degree of tightness of the helical-structure distribution in torus hetero-CNT.     

Group-theoretical analysis of the possible symmetries of the Jahn-Teller systems

The total classification of the possible symmetries of the extremal points of adiabatic potential surfaces of the Jahn—Teller systems for all molecular symmetry point groups was elaborated on the basis of group-theoretical analysis. The method proceeds from the consecutive split of degenerate irreducible representations of the corresponding electron terms as a consequence of the symmetry descent.     

Clifford algebra,symmetries, and vectors

The realization of a Clifford algebra in laboratory space is considered and it is demonstrated that the elements of the algebra cannot, as often assumed, be directly identified with vectors in this space, but, rather, that they form the parametric space of the symmetry operations of the Euclidean group as performed in the laboratory space. Details of this parametrization are established and expressions are given that determine the action of the Euclidean-group operations (screws included) on laboratory-space vectors in terms of the elements of the Clifford algebra. A discussion of Clifford vectors, bivectors, and pseudoscalars and their relation to the Gibbs vectors is provided. The correct definition of axial and polar vectors within the Clifford algebra is carefully discussed. It is shown how simple it is to generate finite point groups in 4-dimensional space by means of the Clifford algebra. © 1996 John Wiley & Sons, Inc.     

Special entangled fermionic systems and exceptional symmetries

Special fermionic systems entered the realm of quantum chemistry in the seventies in the work of Borland and Dennis in the form of a toy model. This work was leading to a detailed study of the N-representability problem by Klyachko. The topic then has been reconsidered in the light of entanglement theory boiling down to the notion of entanglement polytopes. Recently building on certain properties of such special fermionic systems, a connection between the coupled cluster method and entanglement has been established. In this paper we show that precisely such a special class of systems also provides an interesting physical realization for structures related to the Lie algebras of exceptional groups. This result draws such exotic symmetry structures under the umbrella of entangled systems of physical relevance.

     

Biaxial nematics: symmetries,order domains and field-induced phase transitions †

We studied the symmetry and spatial uniformity of the orientational order of the biaxial nematic phase in the light of recent experimental observations of phase biaxiality in thermotropic bent-core and calamitic-tetramer nematics. Evidence is presented supporting monoclinic symmetry, instead of the usually assumed orthorhombic symmetry. The use of deuterium nuclear magnetic resonance to differentiate between the possible symmetries is described. The spatial aspects of biaxial order are presented in the context of the cluster model, wherein macroscopic biaxiality can result from the field-induced alignment of biaxial and possibly polar domains. The implications of different symmetries on the alignment of biaxial nematics and on the measurements of biaxial order are discussed in conjunction with the microdomain structure of the biaxial phase.     

High symmetries in quantum chemistry

A method of simplifying the solution of secular equations occurring in electronic structure, normal mode and nuclear spin state calculations for molecules possessing symmetry is illustrated by applying it to the π-electrons of the recently discovered C₆₀ cluster. In contrast to the usual procedure, the method of reduction to characters does not require actual matrix realizations of the irreducible representations of the symmetry group but only the information contained in the character and multiplication tables. In the case of C₆₀ it leads to explicit algebraic solutions for all the π-orbital energies.     

直链共轭多烯的模糊ta/2对称性

近年来关于分子模糊对称性的工作多属于模糊点对称性的研究．关于模糊空间对称性探讨较少．只曾对线状一维模糊周期分子进行过一些分析．本文在此基础上进一步对于较复杂的平面一维模糊周期分子——直链共轭多烯（简称为共轭多烯）分子进行了较仔细的探讨．除模糊平移变换外,这里还将涉及模糊的螺旋旋转和滑移反映等空间变换．此外,还讨论了存在其中的其他模糊点对称变换．对于点对称元素的变动导致的模糊对称性特征,往往和某种空间对称变换的模糊对称性特征相关．对于分子轨道,除模糊对称变换的隶属函数外,分析了所属不可约表示成分．对这些分子的某些性质和其模糊对称性特征之间的相关性进行探讨．     

Fuzzy symmetries of two classes of linear polyacene molecules

The fuzzy symmetries of two kinds of linear polyacene molecules are probed into in the paper. In these molecules, any one of the benzene rings abreast connects at most other two rings in two ways: either its two opposite C–C bonds combine with two other rings, respectively, or its two meta-position C–C bonds connect two rings in cis- and trans-form, respectively. The former is called p-polyacenes (or straight polyacenes), and the latter is m-polyacenes (or kinked ones). It can be thought as the planar molecule with approximate one-dimensional space periodic transformation (parallel translation) symmetry, namely, group G₁²{{\rm G}_{1}^{2}} symmetry, when the number of its benzene ring is very large; on the other hand, it can be considered as the fuzzy group G₁²{{\rm G}_{1}^{2}} symmetry, if the benzene ring number is not large enough. The p-polyacene and m-polyacene with 20 benzene rings are analyzed as typical examples, and the energies of the π-molecular orbital (MO) and the fuzzy symmetry characters related to the space symmetry transformations are carefully examined. Moreover, the π-MOs of the p-polyacenes and m-polyacenes with different numbers of benzene ring are investigated to obtain the related rules.     

One- and two-particle fractional parentage for arbitrary point groups,configurations, and coupling schemes

The fractional parentage coefficients (CFPS ) for arbitrary symmetry (including non-simply reducible point groups), different coupling schemes, and several open shells are discussed with emphasis on the common features. The differences between the coupling schemes arise merely from a different interpretation of the relevant symmetry group. The formulation uses the particle-number representation (so-called second quantization), in which the CFPS appear as the reduced matrix elements of the creation or annihilation operators. This shows, that there is no principal difference in the fractional parentage scheme of one or several open shells. For the latter case the theory of adjective CFPS is worked out and applied to an example of octahedral symmetry.     

Variational localized-site cluster expansions. VIII. Projection of spin symmetries

We investigate the total spin structure of an approximate localized-site wave function for a collection of paramagnetic sites interacting so as to favor a singlet ground state. As the number of sites becomes infinite we obtain the distribution of weights of the different symmetry components of the localized site wave function; further, although only a very small fraction of such nonsymmetric wave functions is actually singlet, we find that it generally yields the same bulk property expectation values as its singlet-projected component.     

Fuzzy ta/2 symmetries of straight chain conjugate polyene molecules

On the basis of our recent studies on the molecular fuzzy point group symmetry, we further probe into the more complicated planar one-dimensional fuzzy periodic molecules—straight chain conjugate polyene. Except for the fuzzy translation transformation, the space transformation of the fuzzy screw rotation and the glide plane will be referred to. In addition, other fuzzy point symmetry transformation lain in the space transformation is discussed. Usually there is a correlation between the fuzzy symmetry characterization caused by the transition of the point symmetry elements and by certain space symmetry transformation. For the molecular orbital, the irreducible representation component is analyzed besides the membership function of the fuzzy symmetry transformation. Also, we inquire into the relativity between some molecular property and the fuzzy symmetry characterization.     

Polarization dependent resonant x-ray emission spectroscopy of D2O and H2O water: assignment of the local molecular orbital symmetry

The polarization dependence of the split two peaks in the lone-pair region in the x-ray emission spectra has been determined at several different excitation energies for both D(2)O and H(2)O water. In contrast to predictions based on a narrow range of local water structures where the two peaks would be of different molecular orbital symmetry and arise from, respectively, intact and dissociated molecules, we show that the two peaks in the lone-pair region are both of lone-pair 1b(1) orbital symmetry. The results support the interpretation that the two peaks appear due to fluctuations between two distinct different main structural environments.     

Continuous symmetry measures for complex symmetry group

Symmetry is a fundamental property of nature, used extensively in physics, chemistry, and biology. The Continuous symmetry measures (CSM) is a method for estimating the deviation of a given system from having a certain perfect symmetry, which enables us to formulate quantitative relation between symmetry and other physical properties. Analytical procedures for calculating the CSM of all simple cyclic point groups are available for several years. Here, we present a methodology for calculating the CSM of any complex point group, including the dihedral, tetrahedral, octahedral, and icosahedral symmetry groups. We present the method and analyze its performances and errors. We also introduce an analytical method for calculating the CSM of the linear symmetry groups. As an example, we apply these methods for examining the symmetry of water, the symmetry maps of AB₄ complexes, and the symmetry of several Lennard‐Jones clusters. © 2014 Wiley Periodicals, Inc.     

Electrostatic interaction for arbitrary point groups,configurations, and coupling schemes

The electrostatic interaction for arbitrary symmetry (including non-simply reducible point groups), different coupling schemes, and several open shells is discussed from a common viewpoint using the particle-number representation (second quantization). The different coupling schemes simply arise from a different interpretation of the relevant symmetry group. Using the adjective CFPS of the preceding paper, the interaction within several open shells can be calculated. In the case of one-center expansions like ligand field theory the many-particle matrix elements are directly expressed by the radial Slater integrals using isoscalar factors. An example of the formalism is worked out. By the way the unitary transformations between strong and weak field coupling schemes are expressed recursively in the number of particles.     

Optimized structures of Cr2(CO)6+ with three different symmetries by density functional calculation

Density functional calculations have been made on a binuclear metal carbonyl ion Cr₂(CO)₆⁺ found in our laser ablation–molecular beam (LAMB) experiment. Optimized structures are calculated for three different conformations: T33 of D_3d symmetry with three terminal carbonyl groups on each chromium atom, B2T22 of D_2h symmetry with two bridging carbonyl groups and two terminal carbonyl groups on each chromium atom, and B4T11 of D_4h symmetry with four bridging carbonyl groups and one terminal carbonyl group on each chromium atom. The most stable conformation is T33 which is 36.76 and 286.44 kJ mol⁻¹ lower in energy than B2T22 and B4T11, respectively. The difference of conformation exerts a significant influence on the internuclear distance between chromium and the carbon of terminal CO, but hardly on the Cr–Cr bond length. For B2T22 and B4T11, longer C–O distances for bridging carbonyls compared with those for terminal ones indicate effective π*-back donation from the chromium atom to the bridging carbonyl groups. Furthermore, the relative abundance of Cr₂(CO)_n⁺ (n = 0–6) observed in our previous experimental study can be explained qualitatively by comparison of the excess energy produced in the formation of a Cr⁺–Cr bond with the CO dissociation energy of Cr₂(CO)₆⁺. © 1998 John Wiley & Sons, Ltd.     

EPR STUDIES OF CUBANE-LIKE TETRANUCLEAR MOLYBDENUM CLUSTER COMPOUNDS

<正> The polycrystalline EPR spectra of three cubane-type tetranuclear molybdenum cluster compounds Mo4S4(μ-OAc)2(dtp)4(1), Mo4S4(μ-OAc)2(dtp)4-·H (2), Mo4OS3(μ-dtp)3(dtp)3(3) are reported. By comparison of the EPR parameters at room temperature and at 77K, it is found that the EPR spectra of (1) and (2) should be attributed to the whole molecule of the cluster compounds while that of (3) to the molybdenum atoms under two different local environments. According to the results of quantum chemistry calculations by means of SCC-EHMO method and the data of X-ray crystal structure analysis, it is shown that the frontier orbital (HOMO), which is composed mainly of the multiple centre d-pπ bonds botween the metal atoms and the bridge (or terminal) groups and occupied by the unpaired electrons of the clusters, is similar to that of the π-conjugated system of free radicals. In many circumstances, the local environment around the Mo atom in cluster distors from ideal Td symmetry to a lower symmetry as to make th