Continuity in symmetry and in distinguishability of states: the symmetry numbers of nonrigid molecules

We establish a connection between the concept of distinguishability of quantum states and the concept of continuity in geometric symmetry. For this purpose, we employ the continuous symmetry measure and the nuclear wave functions of a molecule, and evaluate the physical effects of deviation from geometric symmetry. We apply this tool in presenting a unified approach to assigning symmetry numbers to rigid and nonrigid molecules, and readdress, as a specific example, the evaluation of the temperature dependence of the entropy of cyclobutane and cyclohexane. We believe we show that the concept of continuity in geometric symmetry provides a more natural and practical approach to the inherent link between symmetry and entropy, compared with the classical approach.     

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.     

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.     

Algorithms for computer detection of symmetry elements in molecular systems

Simple procedures for the location of proper and improper rotations and reflexion planes are presented. The search is performed with a molecule divided into subsets of symmetrically equivalent atoms (SEA) which are analyzed separately as if they were a single molecule. This approach is advantageous in many aspects. For instance, in those molecules that are symmetric rotors, the number of atoms and the inertia tensor of the SEA provide one straight way to find proper rotations of any order. The algorithms are invariant to the molecular orientation and their computational cost is low, because the main information required to find symmetry elements is interatomic distances and the principal moments of the SEA. For example, our Fortran implementation, running on a single processor, took only a few seconds to locate all 120 symmetry operations of the large and highly symmetrical fullerene , belonging to the I_h point group. Finally, we show how the interatomic distances matrix of a slightly unsymmetrical molecule is used to symmetrize its geometry. © 2013 Wiley Periodicals, Inc.     

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.     

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.     

Quadrupole versus dipole photomotors: Symmetry constraints and operating properties

The operating properties of different-type Brownian photomotors have been compared in relation to the symmetry interplay between their moving (particle/molecule) and immovable (substrate) units. As found, essentially different symmetry constraints and hence different motor behavior are characteristic of molecules if their electron density redistribution on excitation involves only odd-order or only even-order multipole moments. The two respective types of molecules give rise to dipole and quadrupole photomotors. Purely dipole photomotors can operate only on substrates with symmetric charge distribution and their ensemble with random molecular orientations undergoes equidirectional diffusion rather than directed motion. Contrary to this, purely quadrupole photomotors need substrates with antisymmetric charge distribution for directed motion to occur and they can move directionally in an ensemble. Also, quadrupole photomotors provide much smaller velocities of directed motion than their dipole counterparts (as a result of the much weaker interaction of quadrupoles than dipoles with an electric field). The generic distinctions between the two photomotor types are exemplified by donor-acceptor-substituted stilbenoids (dipole molecules) and symmetric squaraines (quadrupole molecules). The model used affords the rational design of photomotors by the selection of promising moving molecules and by the continuous wide-range tuning of charge-distribution symmetry in substrates.     

New N-representability results with symmetry in molecular quantum chemistry

We prove a new type of N-representability result: given a totally symmetric density function ρ, we construct a wavefunction Ψ such that the totally symmetric part of $\rho \Psi $ (its projection over the totally symmetric functions) be equal to ρ, and, furthermore, such that Ψ belongs to a given class of symmetry associated to the symmetry group of a molecule. Our proof uses deformations of density functions and which are solutions of a “Jacobian problem”. This allows us to formalize rigorously an idea of A. Görling (Phys. Rev. A 47 (1993) 2783), for Density-Functional Theory in molecular quantum chemistry, by defining a density functional that takes into account the symmetry of the molecule under study.     

Vibronic coupling and symmetry breaking in core electron ionization

It is shown that for highly symmetric molecules the ionization of a core electron leads quite generally to a lowering of the symmetry. The breaking of the symmetry is a consequence of the vibronic coupling between nearly degenerate core orbitals of different symmetry. The vibronic coupling leads to strong excitation of non-totally symmetric vibrational modes in addition to the usually observed excitation of totally symmetric modes. As an example, the vibrational structure of the Ols line of the CO₂ molecule is computed on the one-particle level.     

Initial conformations of macrocyclic compounds with rotational symmetry generated from a molecular fragment

An algorithm is described for generating atomic Cartesian coordinates of conformations of macrocyclic molecules possessing exact rotational or rotation-reflection symmetries. A fragment representing the asymmetric unit of the molecule is suitably oriented in space, and then a symmetry operator is applied to generate the initial coordinates of the molecule. An empirical force field of interatomic interactions is used to generate equilibrium conformations. Results of calculations performed on two cyclic polylactones and one crown ether using this approach are given. They reveal that symmetric conformations of these molecules are often preferred. Since the latter conformations are probably responsible for the specialized properties of these molecules, this method should facilitate doing theoretical studies on these kinds of compounds.     

First principles NMR calculations by fragmentation

The nuclear magnetic shielding tensor is a molecular property that can be computed from first principles. In this work we show that by utilizing the fragmentation approach, one is able to accurately compute this property for a large class of molecules. This is of great significance because the computational expense required in the evaluation of the shielding tensor for all nuclei in a large molecule is now subject to near linear scaling. On the basis of previous studies and this work, it is also very likely that all molecular properties that can be expressed as derivatives of the total energy of the system are also amenable to accurate evaluation via fragmentation. If only the chemical shifts for nuclei in a small part of a large molecule are of interest, then only those molecular fragments containing those nuclei need to have their shielding tensors evaluated. Further, the fragmentation approach allows one to construct a database of molecular fragments that could, in principle, be used in the NMR characterization of molecules and at the same time provide possible three-dimensional representations of these molecules.     

Energy-entropy prediction of octanol–water logP of SAMPL7 N-acyl sulfonamide bioisosters

Partition coefficients quantify a molecule’s distribution between two immiscible liquid phases. While there are many methods to compute them, there is not yet a method based on the free energy of each system in terms of energy and entropy, where entropy depends on the probability distribution of all quantum states of the system. Here we test a method in this class called Energy Entropy Multiscale Cell Correlation (EE-MCC) for the calculation of octanol–water logP values for 22 N-acyl sulfonamides in the SAMPL7 Physical Properties Challenge (Statistical Assessment of the Modelling of Proteins and Ligands). EE-MCC logP values have a mean error of 1.8 logP units versus experiment and a standard error of the mean of 1.0 logP units for three separate calculations. These errors are primarily due to getting sufficiently converged energies to give accurate differences of large numbers, particularly for the large-molecule solvent octanol. However, this is also an issue for entropy, and approximations in the force field and MCC theory also contribute to the error. Unique to MCC is that it explains the entropy contributions over all the degrees of freedom of all molecules in the system. A gain in orientational entropy of water is the main favourable entropic contribution, supported by small gains in solute vibrational and orientational entropy but offset by unfavourable changes in the orientational entropy of octanol, the vibrational entropy of both solvents, and the positional and conformational entropy of the solute.

     

Enantiodiscrimination in deuterium NMR spectra of flexible chiral molecules with average axial symmetry dissolved in chiral liquid crystals: the case of tridioxyethylenetriphenylene

Flexible chiral molecules undergoing fast interconversion (on the NMR time scale) between different conformational enantiomers may yield "average" axial species with enantiotopically related sites. Contrary to the situation observed for rigid axial molecules, signals from these enantiotopic sites in NMR spectra recorded in chiral liquid-crystalline solvents can be resolved. In the present work, we studied the deuterium NMR spectra of tridioxyethylenetriphenylene (compound 4) statistically deuterated to 10% in the flexible side chains and dissolved in chiral and achiral lyotropic liquid crystals based on poly(gamma-benzylglutamate). The fast chair-chair flipping of the side chains in 4 on average renders the molecule axially symmetric ( D 3 h ) with pairs of enantiotopic ethylene deuterons. These deuterons exhibit unusually large enantiodiscrimination. To explain this observation, we first describe how the average symmetry of flexible molecules can be derived from the symmetry of the "frozen" conformers and the nature of the averaging process. The procedure is then applied to 4 and used to analyze the NMR results. It is shown that the large enantiodiscrimination in the present case reflects a large difference in the orientational ordering of the conformational enantiomers participating in the interconversion processes as well as a large geometrical factor due to the special shape of the dioxyethylene side groups. (1)H and (13)C NMR spectra of 4 in the same lyotropic liquid crystalline solvent are analyzed to determine its ordering characteristics. Several related cases are also discussed.     

Modified feed-forward neural network structures and combined-function-derivative approximations incorporating exchange symmetry for potential energy surface fitting

The classical interchange (permutation) of atoms of similar identity does not have an effect on the overall potential energy. In this study, we present feed-forward neural network structures that provide permutation symmetry to the potential energy surfaces of molecules. The new feed-forward neural network structures are employed to fit the potential energy surfaces for two illustrative molecules, which are H(2)O and ClOOCl. Modifications are made to describe the symmetric interchange (permutation) of atoms of similar identity (or mathematically, the permutation of symmetric input parameters). The combined-function-derivative approximation algorithm (J. Chem. Phys. 2009, 130, 134101) is also implemented to fit the neural-network potential energy surfaces accurately. The combination of our symmetric neural networks and the function-derivative fitting effectively produces PES fits using fewer numbers of training data points. For H(2)O, only 282 configurations are employed as the training set; the testing root-mean-squared and mean-absolute energy errors are respectively reported as 0.0103 eV (0.236 kcal/mol) and 0.0078 eV (0.179 kcal/mol). In the ClOOCl case, 1693 configurations are required to construct the training set; the root-mean-squared and mean-absolute energy errors for the ClOOCl testing set are 0.0409 eV (0.943 kcal/mol) and 0.0269 eV (0.620 kcal/mol), respectively. Overall, we find good agreements between ab initio and NN prediction in term of energy and gradient errors, and conclude that the new feed-forward neural-network models advantageously describe the molecules with excellent accuracy.     

Photoinduced charge generation and recombination dynamics in model donor/acceptor pairs for organic solar cell applications: a full quantum-chemical treatment

This work focuses on two fundamental processes in organic solar cells-exciton dissociation and charge recombination-and describes how quantum-chemical calculations can be exploited to estimate the molecular parameters that determine the rates of these processes. The general concepts behind our approach are illustrated by considering a donor-acceptor complex made of a phthalocyanine (electron donor) molecule and a perylene (acceptor) molecule. The results highlight how the relative rates of the two processes depend on the dimensionality of the molecules, their relative positions, the symmetry of the relevant electronic levels, and the polarity of the medium. It is shown, for instance, that highly symmetric configurations of the complex can strongly limit charge recombination; this emphasizes the need for a fine control of the supramolecular organization at organic-organic interfaces in donor-acceptor blends.     

Abnormal inversion splitting in NH2D: rotational analysis of the nu5 bending vibrational band system

A high-resolution Fourier transform infrared spectrum of the nu(5) bending vibrational band system region of the partially deuterated ammonia molecule NH(2)D has been measured and rotationally analyzed. The spectrum consists of strong a-type transitions between the states of same vibrational symmetry and weaker c-type transitions between the states of different vibrational symmetry. The Hamiltonian model used includes interaction terms between the rotational states of both upper and lower inversion doublets. The vibrational term values for the symmetric and the antisymmetric component of the upper-inversion doublet are 1,605.637 965(620) cm(-1) and 1,590.993 82(100) cm(-1), respectively, where the numbers in parentheses are one-standard deviations in the least significant digit. These figures are close to the corresponding values 1,605.62 cm(-1) and 1,590.72 cm(-1) obtained recently from results based on high-level ab initio calculations. The order of the vibrational term values is abnormal in the ammonia family, as typically the symmetric state is lower in wavenumber than the antisymmetric one.     

X-ray structural investigation of gossypol and its derivatives XXIII autoclathrates based on a mixed gossypol — 1,4-Dioxane matrix

Gossypol forms various complexes with the isomeric dioxanes. The clathrate with 1,4-dioxane is the only complex of gossypol in which the intrinsic symmetry of the gossypol molecule — the symmetry of a twofold axis — is retained. In this complex, two out of the three 1,4-dioxane molecules belonging to each gossypol molecule participate in the construction of a mixed H-bound gossypol -dioxane matrix, while the third molecule plays the part of guest, the guest molecules having no H-bonds with the host matrix and undergoing desolvation at 108–110°C.     

Symmetry numbers and chemical reaction rates

This article shows how to evaluate rotational symmetry numbers for different molecular configurations and how to apply them to transition state theory. In general, the symmetry number is given by the ratio of the reactant and transition state rotational symmetry numbers. However, special care is advised in the evaluation of symmetry numbers in the following situations: (i) if the reaction is symmetric, (ii) if reactants and/or transition states are chiral, (iii) if the reaction has multiple conformers for reactants and/or transition states and, (iv) if there is an internal rotation of part of the molecular system. All these four situations are treated systematically and analyzed in detail in the present article. We also include a large number of examples to clarify some complicated situations, and in the last section we discuss an example involving an achiral diasteroisomer.     

Critical analysis and extension of the Hirshfeld atoms in molecules

The computational approach to the Hirshfeld [Theor. Chim. Acta 44, 129 (1977)] atom in a molecule is critically investigated, and several difficulties are highlighted. It is shown that these difficulties are mitigated by an alternative, iterative version, of the Hirshfeld partitioning procedure. The iterative scheme ensures that the Hirshfeld definition represents a mathematically proper information entropy, allows the Hirshfeld approach to be used for charged molecules, eliminates arbitrariness in the choice of the promolecule, and increases the magnitudes of the charges. The resulting "Hirshfeld-I charges" correlate well with electrostatic potential derived atomic charges.