Symmetry calculation for molecules and transition states

The symmetry of molecules and transition states of elementary reactions is an essential property with important implications for computational chemistry. The automated identification of symmetry by computers is a very useful tool for many applications, but often relies on the availability of three‐dimensional coordinates of the atoms in the molecule and hence becomes less useful when these coordinates are a priori unavailable. This article presents a new algorithm that identifies symmetry of molecules and transition states based on an augmented graph representation of the corresponding structures, in which both topology and the presence of stereocenters are accounted for. The automorphism group order of the graph associated with the molecule or transition state is used as a starting point. A novel concept of label‐stereoisomers, that is, stereoisomers that arise after labeling homomorph substituents in the original molecule so that they become distinguishable, is introduced and used to obtain the symmetry number. The algorithm is characterized by its generic nature and avoids the use of heuristic rules that would limit the applicability. The calculated symmetry numbers are in agreement with expected values for a large and diverse set of structures, ranging from asymmetric, small molecules such as fluorochlorobromomethane to highly symmetric structures found in drug discovery assays. The new algorithm opens up new possibilities for the fast screening of the degree of symmetry of large sets of molecules. © 2014 Wiley Periodicals, Inc.     

A unique computer representation for molecular structures

Converting a non-unique connection table to a unique (canonical) name can be accomplished by assigning unique sequence numbers on the basis of topological (constitutional) properties. An algorithm is reported which performs this task by perceiving the topological symmetry of the molecule. A convention for assigning a single name to topologically unique structures which vary only in the position of π-electrons (resonance forms) is also presented.     

Ein neues programmierbares Schema zur Bestimmung der Molekülsymmetrie

The symmetry of molecules can be determined, when their structure is known, using schemes of questions (algorithms), by means of which the existence of certain elements of symmetry is checked. In this paper, a new algorithm is presented, which renders feasible the prompt determination of symmetry point groups and visualizes relations between different types of symmetry. The application of the algorithm is demonstrated by means of a graphic scheme and a Pascal computer version.

     

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     

Molecular symmetry perception

An algorithm for molecular symmetry perception is presented. The method identifies the full set of molecular symmetry elements (proper and improper) and determines their coordinates. The algorithm eliminates the necessity to explore the entire graph automorphism group; as a result its computer application is extremely effective. Application to several dendrimers and fullerenes with high topological symmetry is presented.     

Representations and descriptors unifying the study of molecular and bulk systems

Establishing a unified framework for describing the structures of molecular and periodic systems is a long-standing challenge in physics, chemistry, and material science. With the rise of machine learning methods in these fields, there is a growing need for such a method. This perspective aims to discuss the development and use of three promising approaches—topological, atom-density, and symmetry-based—for the prediction and rationalization of physical, chemical, and mechanical properties of atomistic systems across different scales and compositions.     

Symmetrizer: Algorithmic determination of point groups in nearly symmetric molecules

Symmetry is an extremely useful and powerful tool in computational chemistry, both for predicting the properties of molecules and for simplifying calculations. Although methods for determining the point groups of perfectly symmetric molecules are well‐known, finding the closest point group for a “nearly” symmetric molecule is far less studied, although it presents many useful applications. For this reason, we introduce Symmetrizer, an algorithm designed to determine a molecule's symmetry elements and closest matching point groups based on a user‐adjustable tolerance, and then to symmetrize that molecule to a given point group geometry. In contrast to conventional methods, Symmetrizer takes a bottom‐up approach to symmetry detection by locating all possible symmetry elements and uses this set to deduce the most probable point groups. We explain this approach in detail, and assess the flexibility, robustness, and efficiency of the algorithm with respect to various input parameters on several test molecules. We also demonstrate an application of Symmetrizer by interfacing it with the WebMO web‐based interface to computational chemistry packages as a showcase of its ease of integration. © 2012 Wiley Periodicals, Inc.     

Shape and symmetry of heptacoordinate transition-metal complexes: structural trends

The stereochemistries of heptacoordinate transition-metal complexes are analyzed by using continuous symmetry and shape measures of their coordination spheres. The distribution of heptacoordination through the transition-metal series is presented based on structural database searches including organometallic and Werner-type molecular complexes, metalloproteins, and extended solids. The most common polyhedron seems to be the pentagonal bipyramid, while different preferences are found for specific families of compounds, as in the complexes with three or four carbonyl or phosphine ligands, which prefer the capped octahedron or the capped trigonal prism rather than the pentagonal bipyramid. The symmetry maps for heptacoordination are presented and shown to be helpful for detecting stereochemical trends. The maximal symmetry interconversion pathways between the three most common polyhedra are defined in terms of symmetry constants and a large number of experimental structures are seen to fall along those paths.     

The use of symmetry in direct Møller-Plesset second-order calculations

Summary An algorithm for utilising abelian point group symmetry in direct MP2 energy calculations is presented. This is based upon the direct MP2 method of Head-Gordon, Pople and Frisch. The method uses the petite atomic orbital integral list as in conventional transformations coupled with a symmetry adaption of the three quarter transformed integrals. Representative calculations for ethylene and benzene are presented which demonstrate the potential of the method.     

Group of symmetry of the periodic system of chemical elements

The concurrence of the group of symmetry of the periodic system of elements with the group of dynamical symmetry of a hydrogenlike atom is employed in the theoretical investigation of atoms. The character of the degeneracy of the eigenvalues of a hydrogenlike atom Hamiltonian, without changing its eigenfunctions, was changed by introducing into this Hamiltonian a term which violates the symmetry in relation to transformations from the subgroup O(4) of the group SO(4, 2). In consequence, it was realized that such “reorganization” of the states of a hydrogenlike atom, which form a representation of the group SO(4, 2), effects the splitting of this representation into finite‐dimensional multiplets, first of which are in full agreement with the experimentally observable composition of electron shells of atoms, and retains the physical meaning of quantum numbers that define electron states. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 499–508, 1999     

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.     

计算机辅助结构解析中自同构群的生成算法

在结构解析过程中,自同构群的生成是必须的。通过全通道拓扑等价性算法,得到了化合物的自同构群,并比较了几种拓扑等价性算法,结果发现,全通道算法能够正确划分化合物的结构图,从而为ＥＳＥＳＯＣ系统的立体异构体穷举生成奠定了基础。     

SASS: a symmetry adapted stochastic search algorithm exploiting site symmetry

A simple symmetry adapted search algorithm (SASS) exploiting point group symmetry increases the efficiency of systematic explorations of complex quantum mechanical potential energy surfaces. In contrast to previously described stochastic approaches, which do not employ symmetry, candidate structures are generated within simple point groups, such as C2, Cs, and C2v. This facilitates efficient sampling of the 3N-6 Pople's dimensional configuration space and increases the speed and effectiveness of quantum chemical geometry optimizations. Pople's concept of framework groups [J. Am. Chem. Soc. 102, 4615 (1980)] is used to partition the configuration space into structures spanning all possible distributions of sets of symmetry equivalent atoms. This provides an efficient means of computing all structures of a given symmetry with minimum redundancy. This approach also is advantageous for generating initial structures for global optimizations via genetic algorithm and other stochastic global search techniques. Application of the SASS method is illustrated by locating 14 low-lying stationary points on the cc-pwCVDZ ROCCSD(T) potential energy surface of Li5H2. The global minimum structure is identified, along with many unique, nonintuitive, energetically favorable isomers.     

CAESAR: a new conformer generation algorithm based on recursive buildup and local rotational symmetry consideration

A highly efficient conformer search algorithm based on a divide-and-conquer and recursive conformer build-up approach is presented in this paper. This approach is combined with consideration of local rotational symmetry so that conformer duplicates due to topological symmetry in the systematic search can be efficiently eliminated. This new algorithm, termed CAESAR (Conformer Algorithm based on Energy Screening and Recursive Buildup), has been implemented in Discovery Studio 1.7 as part of the Catalyst Component Collection. CAESAR has been validated by comparing the conformer models generated by the new method and Catalyst/FAST. CAESAR is consistently 5-20 times faster than Catalyst/FAST for all data sets investigated. The speedup is even more dramatic for molecules with high topological symmetry or for molecules that require a large number of conformers to be sampled. The quality of the conformer models generated by CAESAR has been validated by assessing the ability to reproduce the receptor-bound X-ray conformation of ligands extracted for the Protein Data Bank (PDB) and assessing the ability to adequately cover the pharmacophore space. It is shown that CAESAR is able to reproduce the receptor-bound conformation slightly better than the Catalyst/FAST method for a data set of 918 ligands retrieved from the PDB. In addition, it is shown that CEASAR covers the pharmacophore space as well or better than Catalyst/FAST.     

三梯级和四梯级圆筒和Mbius分子的拓扑特性研究

Walba等以其卓越的工作,合成了三-THYME(C_(42)H_(72)O_(18))和四-THYME(C_(56)H_(96)O_(24))圆筒及其Mbius扭曲环带分子,被誉为拓扑学进入有机化学领域的奇迹,成为迄今为止拓扑立体化学研究的重要内容,但从拓扑学的观点探索分子图拓扑结构特性尚缺乏深入研究,本文作者考虑到一般性,曾将扭曲数T为偶数(0,2,4)的定义为Hckel型,扭曲数     

Symmetry operation measures

We introduce a new mathematical tool for quantifying the symmetry contents of molecular structures: the Symmetry Operation Measures. In this approach, we measure the minimal distance between a given structure and the structure which is obtained after applying a selected symmetry operation on it. If the given operation is a true symmetry operation for the structure, this distance is zero; otherwise it gives an indication of how different the transformed structure is from the original one. Specifically, we provide analytical solutions for measures of all the improper rotations, S n p, including mirror symmetry and inversion, as well as for all pure rotations, C n p. These measures provide information complementary to the Continuous Symmetry Measures (CSM) that evaluate the distance between a given structure and the nearest structure which belongs to a selected symmetry point-group.     

Symmetry specified enumeration of substituted derivatives: an easy solution to the complex problem

Several sophisticated methods to solution of symmetry specified enumeration problems are available in the modern literature. In this paper we propose a simple technique that allows one to manually compute the exact numbers of fixed-symmetry derivatives for a given structure either with inclusion or ignoring the substitution patterns. The basic idea of the method suggested consists in the derivation of Pólya-like cycle indices for the automorphism groups of specially constructed orbit partition graphs; the expansion of these indices and subsequent simple calculations result in the desired numbers of substituted derivatives with achiral substituents. Limitations of the new technique (and a method suggested earlier) depend on the relevance of the orbit partitions for particular subgroups of the point symmetry group. For illustration purposes, the results obtained for the prismane (D _3h ) and adamantane (T _d ) structures are discussed. In the former case the numbers of substituted derivatives can be found for all subgroups of the D _3h group, whereas in the latter case these numbers can be determined for eight out of eleven subgroups of the T _d point symmetry group. This work is based on the text of the lecture presented by the authors at the 5th All-Russia Conference on Molecular Modeling (Moscow, April 2007). The paper deals with the methodology and detailed treatment of applied aspects related to solution of enumeration problems for substituted derivatives with prescribed symmetry groups. Unlike the known methods of symmetry specified enumeration, the technique suggested is simple enough and may be regarded as generalization of the Pólya methodology, which is widely used by chemists. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 2, pp. 227–245, February, 2008.