Symmetry adaption

The matrix of force constants of a symmetric molecule is reduced to its invariants (nearly diagonal form). For the case of distance dependent potentials these invariants are expressed by the derivations of the potentials and geometric factors. For the purpose of parametrization the inversions of these formulae are derived. The general equilibrium condition and the elimination of the translational and rotational coordinates are discussed. The example of the tetrahedral AB₄ structure is worked out.     

浅谈分子对称性中独立的对称元素

通过图示和矩阵两方面来阐述独立的对称元素和非独立的对称元素之间的区别与联系,利用比较法帮助读者正确理解独立对称元素的内涵。     

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.     

RMSD and Symmetry

A common approach for comparing the structures of biomolecules or solid bodies is to translate and rotate one structure with respect to the other to minimize the pointwise root-mean-square deviation (RMSD). We present a new, robust numerical algorithm that computes the RMSD between two molecules or all the mutual RMSDs of a list of molecules and, if desired, the corresponding rotation matrix in a minimal number of operations as compared to previous algorithms. The RMSD gradient can also be computed. We address the problem of symmetry, both in alignment (possible alternative alignments due to indistinguishable atoms) as well as geometry. In the latter case, it is possible to have degenerate superposition. A necessary condition is optimal superimposability to one's mirror image. Double (respectively, triple) degeneracy results in a one- (respectively, two)-parameter family of rotations leaving the superposition invariant. The software, frmsd , is freely available at http://www.ams.stonybrook.edu/~coutsias/codes/frmsd.tgz . © 2019 Wiley Periodicals, Inc.     

分子的模糊对称性

将模糊数学方法引入对分子对称点群的研究,建立描述具有不完整分子对称性的模糊点对称群(集合).建立具有模糊对称性分子轨道的模糊表示及其模糊特征标(模糊广义宇称).通过对典型的线状分子、平面分子以及非平面的立体分子等进行分析,展示了一个新的理论化学园地.初步探讨了具有模糊对称性的动态反应体系.从模糊对称性出发,探讨了分子轨道对称守恒原理的半定量特征.     

Microscopic Symmetry Imposed by Rotational Symmetry Boundary Conditions in Molecular Dynamics Simulation

A large number of viral capsids, as well as other macromolecular assemblies, have icosahedral structure or structures with other rotational symmetries. This symmetry can be exploited during molecular dynamics (MD) to model in effect the full viral capsid using only a subset of primary atoms plus copies of image atoms generated from rotational symmetry boundary conditions (RSBC). A pure rotational symmetry operation results in both primary and image atoms at short range, and within nonbonded interaction distance of each other, so that nonbonded interactions can not be specified by the minimum image convention and explicit treatment of image atoms is required. As such an unavoidable consequence of RSBC is that the enumeration of nonbonded interactions in regions surrounding certain rotational axes must include both a primary atom and its copied image atom, thereby imposing microscopic symmetry for some forces. We examined the possibility of artifacts arising from this imposed microscopic symmetry of RSBC using two simulation systems: a water shell and human rhinovirus 14 (HRV14) capsid with explicit water. The primary unit was a pentamer of the icosahedron, which has the advantage of direct comparison of icosahedrally equivalent spatial regions, for example regions near a 2-fold symmetry axis with imposed symmetry and a 2-fold axis without imposed symmetry. Analysis of structural and dynamic properties of water molecules and protein atoms found similar behavior near symmetry axes with imposed symmetry and where the minimum image convention fails compared with that in other regions in the simulation system, even though an excluded volume effect was detected for water molecules near the axes with imposed symmetry. These results validate the use of RSBC for icosahedral viral capsids or other rotationally symmetric systems.     

Symmetry and ferromagnetic clusters

Isomers of pure Fe₁₃ and icosahedral Fe₁₂X clusters are studied using the all-electron linear-combination-of-Gaussian-type-orbital (LCGTO) local-density-functional (LDF) methods that allow the spin and geometry of the cluster to be determined self-consistently. The Fe₁₃ ground state is icosahedral. The icosahedral cluster also has the greatest magnetic moment because of increased symmetry-required orbital degeneracy for electrons of different spins. The central atom of the icosahedral iron cluster has been varied to optimize the spin of the cluster keeping the oribital contribution to the magnetic moment quenched. Varying the central atom under this constraint can alter the magnetic moment by more than 20%. Similar studies have begun on 55-atom icosahedral iron clusters.     

Symmetry of metal chelates

Is a metal chelate symmetric, with the motion of the metal described by a single-well potential, or is it asymmetric, in a double-well potential? For hydrogen, this is the familiar question of the symmetry of a hydrogen bond. The molecular symmetry of MLn complexes (M = Li, Na, K, Al, Pd, Rh, Si, Sn, Ge, Sb, etc.; L is the anion of 3-hydroxy-2-phenylpropenal) in solution is now probed with the method of isotopic perturbation of equilibrium. A statistical mixture of 3-hydroxy-2-phenylpropenal-d0, -1-d, and -1,3-d2 was synthesized and converted to various metal complexes. Some complexes show two aldehydic signals, which means that their ligands are monodentate. For LiL, NaL, and KL, the 13C NMR isotope shifts, delta CH(D) - delta CH(H), for the aldehydic CH groups are small and negative, consistent with L- being a resonance hybrid. They are small and positive for AlL3, PdL2, Rh(CO)2L, SiX3L, SiL3+X-, (CF3)3GeL, SbCl4L, (EtO)4TaL, and (EtO)4NbL. The positive isotope shifts are unusual, but since they are small and temperature independent, they are intrinsic and indicate that these metal chelates are symmetric, as expected. Large positive isotope shifts, up to 400 ppb, are observed for Ph3GeL, Me3GeL, Ph2GeL2, Bu3SnL, and Ph4SbL. However, it is likely that these are monodentate complexes undergoing rapid metal migration, as judged from the X-ray crystal structures of Ph3SnL and Ph4SbL. NMR experiments indicate an intermolecular mechanism for exchange, which may be a biomolecular double metal transfer. It is remarkable that the isotope shifts in these five complexes demonstrate that they are asymmetric structures, even though they appear from other NMR evidence to be symmetric chelates.     

Symmetry Numbers of Activated Complexes

The conventional symmetry numbers σ_≠~' of activated complexes may lead to error in the rate constant expression of transition state theory, whereas the statistical factor ι~≠ or ι may violate the principle of detailed balance. A mathematically precise definition of the symmetry number σ_≠ of activated complex is given, i.e. σ_≠=_iN_4(?)/m, m is the number of physically distinct configurations of labelled transition state and N_i is the identical atoms in the activated complex. The identical atoms must belong to the same molecule of reactants and products. The present symmetry numbers σ_≠ of activated complexes assure not only obtaining correct rate constant expressions but also obeying the principle of detailed balance. It can be used with the statistical factor ι to construct the structures of transition states for unimolecular and bimolecular exchange reactions.     

Molecular Symmetry and Related Properties

Abstract

The molecular rotational symmetry number, [sgrave], is defined and evaluated for a wide range of rigid organic compounds. Molecular symmetry is an important determinant of a variety of properties of crystalline substances. It is shown that the entropy of melting, ΔS_m, is related to symmetry by ΔS_m = 13.5–R In[sgrave]. Since molecular symmetry plays an important role in the determination of the entropy of melting it is also important in determining vapor pressure and ideal solubility, both of which depend upon entropy of melting.     

对称性轨道的构造

提出1种以配体轨道的旋转性质为基础,并利用群生成元表示矩阵来构造对称性轨道的新方法。讨论了对称性系数之间的相互关系,同时给出群重迭积分的计算过程。     

Symmetry of Kekulé structures

The symmetry of Kekulé structures for aromatic hydrocarbons is studied by group theory. The general problem of deducing the distribution over irreducible representations (Λ_Kek) or characters of the representation based on the Kekulé structures (χ_Kek) has not been solved. A partial solution is given for two classes of molecules, namely (a) the “straight chain” aromatics (polyacenes): naphthalene, anthracene, naphthacene, etc., and (b) the “zig-zag chain” aromatics: phenanthrene, chrysene, picene, etc. As a part of this solution the number of Kekulé structures (K) in the two cases was found to be (a) K = Q + 1 and (b) K = F_Q+1, respectively. Here Q is the number of benzene rings in the molecule in question, and F_i denotes the i-th member of the Fibonacci series. Symmetrical structures (Λ_Kek) or characters (χ_Kek) are given for a number of additional molecules as examples: benzene (D_6h), tetraphene (C_s), benzo[c] phenanthrene (C_2v), pyrene (D_2h), triphenylene (D_3h), perylene (D_2h), pentaphene (C_2v), dibenzophenanthrene (C_2v), heptaphene (C_2v) and coronene (D_6h). Here the appropriate symmetry groups are given in parentheses.     

Symmetry analysis of sigmatropic rearrangements

Two small groups (E, σ_ν) and (E, C₂) are proposed to describe the orbital symmmetry of the whole reaction paths with respect to suprafacial and antarafacial sigmatropic rearrangements. Furthermore, the symmetries of the two small groups are extended to the systems as a many body treatment performed by Matsen.     

Symmetry groups of chemical graphs

The symmetry groups of all trees are shown to be expressible as generalized wreath products by a tree pruning algorithm. The symmetry groups of certain cyclic graphs which can be expresssed as generalized compositions are also shown to be generalized wreath products. The symmetry groups of complete multipartite graphs can be obtained in a similar manner. Character tables of symmetry groups of certain chemical graphs are also obtained.     

Orbital Correspondence Analysis in Maximum Symmetry

The use of Orbital Correspondence Analysis in Maximum Symmetry (OCAMS) for determining the allowed course of a reaction is described in terms of the familiar “A, S” notation. Several well known thermal and photochemical reactions are used as examples: Polyene cyclization; the fragmentation and dimerization of cyclobutadiene; [_π4 + _π2] cycloaddition; the isomerization of Dewar-benzene. The relation of OCAMS to the classical correlation procedures of Woodward and Hoffmann and of Longuet-Higgins and Abrahamson is brought out, as are the sources of the occasionally contradictory predictions of OCAMS and of the classical procedures.     

丙二烯分子的模糊对称性

利用作者提出的探讨分子及其轨道模糊对称性的方法分析了丙二烯在内旋转过程中模糊对称性特征. 考虑到此过程中经历不同状态所属的对称点群, 即D2h、D2d与D2. 利用包含这些点群中所有元素的最小点群D4h进行分析. 将D4h中的元素分为四组: (i) G0——包含在D2点群中的元素, 也是所有上述点群中都存在的元素; (ii) G1——包含在D2h点群中, 但不包含在D2d点群中的元素; (iii) G2——包含在D2d点群中, 但不包含在D2h点群中的元素; (iv) G3——包含在D4h点群中, 但不包含在D2h与D2d点群中的元素. 分别分析在内旋转过程中各个分子轨道(MO)相应每一组元素的隶属函数的共性变化规律性.     

Symmetry measures of the electron density

In this communication we define electronic symmetry operation and symmetry group measures, eSOM and eSGM, respectively, develop the basic algorithms to obtain them, and give some examples of the possible applications of these new computational tools. These new symmetry measures based on the electron density have been tested in an analysis of (a) the inversion symmetry for heteronuclear diatomic molecules, for the eclipsed and staggered conformations of ethane and tetrafluoroethane, and for a series of octahedral sulfur halides; (b) the reflection symmetry of three different conformers of tetrafluoroethene; and (c) the loss of C₆ symmetry along the B_2u distortion mode of benzene and an analysis of rotational symmetry for different six‐member ring heterocycles. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Symmetry adaption reduced to tabulated quantities

The aim of this paper is to give an algebraic formula for symmetry-adapted linear combinations avoiding intuitive or laborious projection operator techniques. By utilization of the tabulated Clebsch-Gordan coefficients and surface harmonics of the point-groups the symmetry-adapted linear combinations are given in formula (4). A five-step algorithm is proposed and the example of a tetrahedron worked out. The relation to methods using site symmetry is discussed.     

定域分子轨道对称性

提出了利用定域分子轨道重心确定分子轨道的对称性。确定了线型的、非线型的、环状的分子定域分子轨道对称性。结果表明非线型分子和其中双中心键具有相同对称性,但含孤对或三中心键的分子则存在一些差别。还讨论了定域分子轨道的特征。