Continuous symmetry numbers and entropy

Traditionally, entropy changes are corrected for rotational permutability only if the molecule is perfectly rotationally symmetric. By this approach, only a small fraction of all known molecules must be evaluated in terms of symmetry numbers, while all other molecules are totally exempt of these considerations. A general approach which encompasses all molecules, symmetric or not, is proposed here. It is based on introducing the notion of continuity to symmetry numbers and on allowing noninteger values. In the first part of the account, we provide arguments as to why continuity is needed and what difficulties one may encounter by adopting the "black-or-white" approach to symmetry. In the second part, we provide a working methodology of how to evaluate the symmetry number content of any molecule, symmetric or not. Finally, in the third part, we demonstrate the implications of this approach on entropy issues involving melting points, Jahn-Teller distortions (of fullerene) upon ionization, molecular distortion due to overcrowdedness, permutability of isotopes, and the structure of proton sponges. It is shown that continuous symmetry numbers provide entropy values, which better agree with experimental observations, and that they are capable of identifying correlations between symmetry and physical/chemical measurables.     

Effect of molecular symmetry on melting temperature and solubility

Molecular symmetry has a pronounced effect on the melting properties and solubility of organic compounds. As a general rule, symmetrical molecules in crystalline form have higher melting temperatures and exhibit lower solubilities compared with molecules of similar structure but with lower symmetry. Symmetry in a molecule imparts a positive amount of residual entropy in the solid phase (i.e., more possible arrangements leading to the same structure). This means that the entropy of a crystal of symmetric molecules is greater than the entropy of crystal of a similar, but non-symmetric molecule. An analysis is presented relating the enthalpy, entropy and temperature of melting for an idealised system of structural isomers of different molecular symmetries. The analysis presented helps explain why often, yet not always, the crystal of a more symmetric molecule, which has greater entropy to start (closer to that of the liquid), also exhibits a greater gain in entropy upon melting, compared with the crystal of a less symmetrical molecule. The residual entropy due to molecular symmetry has the direct effect of reducing the entropy gain upon melting (a negative effect). However, molecular symmetry also exerts indirect effects on both the entropy and enthalpy of melting. These indirect effects, imposed by the condition of equilibrium melting, are positive, such that it is the balance between the direct and indirect effects what determines the value observed for the entropy of melting of the symmetric molecules. When the indirect effect of molecular symmetry is greater than its direct effect, the observed entropy gain upon melting of the more symmetrical molecule is greater than that of a less symmetrical one.     

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.     

Continuous symmetry analysis of NMR chemical shielding anisotropy

Molecular symmetry is a key parameter which dictates the NMR chemical shielding anisotropy (CSA). Whereas correlations between specific geometrical features of molecules and the CSA are known, the quantitative correlation with symmetry--a global structural feature--has been unknown. Here we demonstrate a CSA/symmetry quantitative relation for the first time: We study how continuous deviation from exact symmetry around a nucleus affects its shielding. To achieve this we employed the continuous symmetry measures methodology, which allows one to quantify the degree of content of a given symmetry. The model case we use for this purpose is a population of distorted SiH(4) structures, for which we follow the (29)Si CSA as a function of the degree of tetrahedral symmetry and of square-planar symmetry. Quantitative correlations between the degree of these symmetries and the NMR shielding parameters emerge.     

The use of site symmetry in constructing symmetry adapted functions

It is shown that the application of a projection operator from a given group to a function is equivalent to the successive application of projection operators from factor groups of the starting group to that function. When used with the factor groups representing the site symmetry of a position and the simplest group of interchanges of positions, this concept provides a very simple method for obtaining symmetry adapted linear combinations of basis functions.     

Molecules and crystals with both icosahedral and cubic symmetry

Notwithstanding the apparent incompatibility between octahedral and icosahedral symmetries, fragments with the two types of symmetry coexist in many molecules and crystals, as evidenced by continuous shape and symmetry measures. A geometric analysis of Platonic and Archimedean polyhedra and of a variety of molecular and crystal structures strongly suggests that octahedral symmetry is latent in icosahedral polyhedra and vice versa. In this Feature Article, new concepts and structural data from the literature combine to offer a perspective view of complex molecular and extended structures. Its influence on the common cubic packing of icosahedral molecules is discussed for a variety of examples, including water clathrates, dodecahedrane, Buckminsterfullerene, the Pd145 and Mo132 clusters and several intermetallic phases.     

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.     

Chiral polymerization: symmetry breaking and entropy production in closed systems

We solve numerically a kinetic model of chiral polymerization in systems closed to matter and energy flow, paying special attention to its ability to amplify the small initial enantiomeric excesses due to the internal and unavoidable statistical fluctuations. The reaction steps are assumed to be reversible, implying a thermodynamic constraint among some of the rate constants. Absolute asymmetric synthesis is achieved in this scheme. The system can persist for long times in quasi-stationary chiral asymmetric states before racemizing. Strong inhibition leads to long-period chiral oscillations in the enantiomeric excesses of the longest homopolymer chains. We also calculate the entropy production σ per unit volume and show that σ increases to a peak value either before or in the vicinity of the chiral symmetry breaking transition.     

Stability,continuity, and symmetry of variational wave-functions

Interrelations between the local and the global aspects of the stability, continuity, and symmetry properties of variational wave-functions are discussed. The spherical limit of one-electron diatomic molecules and the Hartree–Fock approximation of the ground state of the two-electron atom are shown to exhibit the various concepts involved in an ab initio, yet sufficiently simple, manner.     

Determining the metric and the symmetry group of finite point sets in space with an application to cyclohexane

To examine molecular geometry, we ask two questions: (1) What are the metric properties of a finite set of points in space, i.e., what are the relations of the distances between the points? (2) What is the symmetry or point group of the point set? These questions are answered by applying algebraic and algorithmic tools. Results from distance geometry are used to describe the metric. In combination with a so-called minimizing algorithm, distance geometry is also used to develop a procedure that allows generation of the symmetry group without referring to geometric intuition. The general methods presented here are applied to cyclohexane, facilitating a complete geometrical analysis of all its conformers.     

Continuous symmetry measures: A note in proof of the folding/unfolding method

The measurement of the degree of symmetry proved to be a useful tool in the prediction of quantitative structural–physical correlations. These measurements have been based, in the most general form, on the folding/unfolding algorithm, for which we provide here a new and simpler proof. We generalize this proof to the case of objects composed of more than one (full) orbit. An important practical issue we consider is the division of the graph into symmetry orbits and the mapping of the symmetry group elements onto the points of the graph. The logical constraints imposed by the edges of the graph are reviewed and used for the successful resolution of the coupling between different orbits.     

General method for symmetry orbitals and tensors in electronic structure calculations

The symmetry orbital tensor (SOT) method, which makes full use of symmetries in all point groups and can be applied to the self-consistent field (SCF) and post-SCF calculations, is introduced. The principal feature of this method is the definition of the symmetry orbitals (SOs). Any element in a molecular point group will transform one SO to another equivalent SO or simply to itself, and no mixture among SOs exists. Thus, although the SOs for non-Abelian point groups may adapt to reducible representations, their transformation properties are much simpler than in conventional treatments. This article also presents a general scheme to generate SOs for all point groups. The direct products of N SOs form an Nth-rank SOT group, and each matrix element between SOTs is the product of a physical factor and a geometric factor. Compared with the canonical molecular orbitals, the use of SOs can noticeably reduce the computation efforts by decreasing the number of integrals needed in the SCF calculations or the number of configurations needed in the configuration interaction (CI) calculations. The SOT-SCF and SOT-CI approaches are formulated and a preliminary SOT-SCF program is written. Pilot calculations demonstrate the value of the SOT approach, at least at the closed-shell Hartree–Fock level. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 305–321, 1999     

The fundamental syntopy of quasi-symmetric systems: Geometric criteria and the underlying syntopy of a nuclear configuration space

Point symmetry is a discrete concept; A nuclear configuration for a given stoichiometry either has or has not a particular point symmetry. By contrast, both static and dynamic properties of actual molecules exhibit continuous features. Using the formalism of fuzzy-set theory, we had previously proposed the concept of syntopy as a continuous extension of the symmetry concept for quasi-symmetric systems: This was based on an energetic criterion taking into account the energy costs of nuclear rearrangements. This extension of symmetry was necessarily dependent on the considered electronic state: For a given geometric arrangement of the nuclei, the energy cost of some rearrangement is dependent on the actual potential surface, that is, on the electronic state, in the Born–Oppenheimer approximation. In the extension of the syntopy model reported in the present work, we consider a syntopy criterion that is common to all electronic states. The syntopy thus defined—called the fundamental syntopy of the reduced nuclear configuration space—is independent of the potential surface and of the electronic state: It is defined only in terms of a geometric condition, which makes it more appropriate to rationalize mesoscopic structures. This new syntopy model provides a connection between all possible syntopies generated by the various potential-energy surfaces supported by the considered family of atomic nuclei. © 1993 John Wiley & Sons, Inc.     

Internal symmetry groups of non-rigid molecules

Hougen has established, for quasi-rigid molecules, the relationship between permutationinversions acting on the molecular Hamiltonians written in Cartesian co-ordinates and permutation-rotations (perrotations) of symmetry acting on nuclear equilibrium configurations. We extend these relations to the case of non-rigid molecules. For this, we introduce kinetic perrotations which act on nuclear equilibrium configurations in the same way as do Altmann's isodynamic operators. We show that isodynamic operators do not always form a group. Moreover, their action cannot be extended to the electrons. They cannot be used for the classification of molecular wave functions. This classification is achieved by using the group of Longuet-Higgins and the group of the corresponding feasible perrotations.     

Quasispin symmetry for the derivation of coupled cluster equations for the Hubbard model of benzene

The concept of quasispin is applied to a special case of the Pariser–Parr–Pople (PPP ) model of the benzene molecule, namely, the Hubbard Hamiltonian. Added to the spin, space, and alternancy symmetries already taken into account in the PPP Hamiltonian, this new symmetry, called quasispin symmetry, has the effect of reducing the size of the CI matrix. Coupled cluster (CC ) equations are then obtained after applying the CC approach with doubles as well as its extension that accounts for triexcited clusters (CCSDT -1). The derivation of these equations following the use of quasispin to the Hubbard model of benzene constitutes the most simple nontrivial example of CC results. In addition, the CC equations can be written in explicit algebraic form using the symbolic computation language MAPLE.     

On discerning symmetry properties of graphs

A procedure is outlined which allows the symmetry properties of graphs to be systematically and rigorously investigated. It is based on a search for all the automorphisms of a graph and this is accompanied by suitably applying the procedure for recognizing identical graphs. It consists in finding all the distinctive labeling of the vertices of graph associated with the smallest binary code derived by a particular interpretation of the associated adjacency matrix. No prior cognizance of symmetry operations is required which is in contrast to the usual discussions of the symmetry properties of molecules which are based on the knowledge of pertinent symmetry operations. This is important since in graphs it is neither apparent nor generally possible to simply enlist those permutations of labels which leave the connectivity invariant (i.e., do not alter the form of the adjacency matrix). The procedure is applied to the Petersen graphs and the Desargues-Levi graph, both associated with isomerizations of trigonal bipyramidal complex and other chemical transformations. It is shown that these graphs of high symmetry belong to symmetry groups of order 120 and 240 respectively. The approach can also provide a basis for the development of the symmetry properties of non-rigid molecules in which connectivity is preserved.     

Full non-rigid group theory and symmetry of melamine

The non-rigid molecule group (NRG) theory in which the dynamic symmetry operations are defined as physical operations is a new field in chemistry. Smeyers, in a series of papers, applied this notion to determine the character table of restricted NRG of some molecules. For example, Smeyers and Villa computed the r-NRG of the triple equivalent methyl rotation in pyramidal trimethylamine with inversion and proved that the r-NRG of this molecule is a group of order 648, containing two subgroups of order 324 without inversion [5]. In this work, a simple method is described, through which it is possible to calculate character tables for the symmetry group of molecules. We study the full NRG of melamine, and prove that it is a groups of order 48, with 27 and 10 conjugacy classes. Also, we compute the symmetry of melamine and prove that it is a non-abelian groups of order 6. The method can be generalized to apply to other non-rigid molecules.