Analytical methods for calculating Continuous Symmetry Measures and the Chirality Measure

We provide analytical solutions of the Continuous Symmetry Measure (CSM) equation for several symmetry point-groups, and for the associated Continuous Chirality Measure (CCM), which are quantitative estimates of the degree of a symmetry-point group or chirality in a structure, respectively. We do it by solving analytically the problem of finding the minimal distance between the original structure and the result obtained by operating on it all of the operations of a specific G symmetry point group. Specifically, we provide solutions for the symmetry measures of all of the improper rotations point group symmetries, S(n), including the mirror (S(1), C(S)), inversion (S(2), C(i)) as well as the higher S(n)s (n > 2 is even) point group symmetries, for the rotational C(2) point group symmetry, for the higher rotational C(n) symmetries (n > 2), and finally for the C(nh) symmetry point group. The chirality measure is the minimal of all S(n) measures.     

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.     

Quantitative symmetry and chirality--a fast computational algorithm for large structures: proteins, macromolecules, nanotubes, and unit cells

Symmetry is one of the most fundamental properties of nature and is used to understand and investigate physical properties. Classically, symmetry is treated as a binary qualitative property, although other physical properties are quantitative. Using the continuous symmetry measure (CSM) methodology one can quantify symmetry and correlate it quantitatively to physical, chemical, and biological properties. The exact analytical procedures for calculating the CSM are computationally expensive and the calculation time grows rapidly as the structure contains more atoms. In this article, we present a new method for calculating the CSM and the related continuous chirality measure (CCM) for large systems. The new method is much faster than the full analytical procedures and it reduces the calculation time dependency from N! to N(2), where N is the number of atoms in the structure. We evaluate the cost of the applied approximations, estimate the error of the method, and show that deviations from the analytical solutions are within an error of 2%, and in many cases even less. The method is applicable at the moment for the cyclic symmetry point groups- C(i), C(s), C(n), and S(n), and therefore it can be used also for chirality measures, which are the minimal of the S(n) measures. We demonstrate the application of the method for large structures across chemistry: proteins, macromolecules, nanotubes, and large unit cells of crystals.     

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.     

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     

Quantifying the symmetry content of the electronic structure of molecules: molecular orbitals and the wave function

A scheme to quantify the symmetry content of the electronic wave function and molecular orbitals for arbitrary molecules is developed within the formalism of Continuous Symmetry Measures (CSMs). After defining the symmetry operation expectation values (SOEVs) as the key quantity to gauge the symmetry content of molecular wavefunctions, we present the working equations to be implemented in order to carry out real calculations using standard quantum chemistry software. The potentialities of a symmetry analysis using this new method are shown by means of some illustrative examples such as the changes induced in the molecular orbitals of a diatomic molecule by an electronegativity perturbation, the breaking of orbital symmetry along the dissociation path of the H(2) molecule, the changes in the molecular orbitals upon a geometrical distortion of the benzene molecule, and the inversion symmetry content in the different spin states of the [Fe(CH(3))(4)](2-) complex.     

A path from I(h) to C1 symmetry for C20 cage molecule

The symmetry of the C20 cage is studied based on the intrinsical relationship among point groups (Bradley, C. J.; Cracknell, A. P. The Mathematical Theory of Symmetry in Solids; Claredon Press: Oxford, 1972). The structure of the C20 cage with I(h) symmetry is constructed, as are eight other structures with subgroup symmetry. A path from I(h) symmetry to C1 symmetry is obtained for the closed-shell electronic state, and the structure with D2h symmetry is the most stable on this path. Using the D2h structure the correlation energy correction is studied on the condition of restricted excitation space at the CCSD(T) level. We obtain curves on the relation between the orbital numbers and the total energy at the CCSD(T), CCSD, and MP2 level, respectively. The results of these curves obtained from MP2 and CCSD(T) methods have the same tendency, while the results of CCSD gradually diverge with an increase in orbital numbers. When the orbitals used in the calculation reach 460, the total energy is -759.644 hartree at MP2 level and is -759.721 hartree by the CCSD(T) method. From the calculation results, we find that a large basis set can improve the reliability of the MP2 method, and to restrict excitation space is necessary when using the CCSD(T) method.     

Symmetry‐Enthalpy Correlations in Diels–Alder Reactions

Woodward–Hoffmann (WH) rules provide strict symmetry selection rules: when they are obeyed, a reaction proceeds; when they are not obeyed, there is no reaction. However, the voluminous experimental literature provides ample evidence that strict compliance to symmetry requirements is not an obstacle for a concerted reaction to proceed, and therefore the idea has developed that it is enough to have a certain degree of the required symmetry to have reactivity. Here we provide quantitative evidence of that link, and show that as one deviates from the desired symmetry, the enthalpy of activation increases, that is, we show that concerted reactions slow down the further they are from the ideal symmetry. Specifically, we study the deviation from mirror symmetry (evaluated with the continuous symmetry measure (CSM)) of the [4+2] carbon skeleton of the transition state of a series of twelve Diels–Alder reactions in seven different solvents (and in the gas phase), in which the dienes are butadiene, cyclopentadiene, cyclohexadiene, and cycloheptadiene; the dienophiles are the 1‐, 1,1‐, and 1,1,2‐cyanoethylene derivatives; the solvents were chosen to sample a range of dielectric constants from heptane to ethanol. These components provide twenty‐four symmetry–enthalpy DFT‐calculated correlation lines (out of which only one case is a relatively mild exception) that show the general trend of increase in enthalpy as symmetry decreases. The various combinations between the dienophiles, cyanoethylenes, and solvents provide all kinds of sources for symmetry deviations; it is therefore remarkable that although the enthalpy of activation is dictated by various parameters, symmetry emerges as a primary parameter. In our analysis we also bisected this overall picture into solvent effects and geometry variation effects to evaluate under which conditions the electronic effects are more dominant than symmetry effects.     

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.     

Symmetry and periodicity of potential surfaces: a test for multicenter interactions

Symmetry and periodicity of potential energy surfaces of chemical reactions and conformational changes are determined by the symmetry properties of the nuclear frameworks of all possible nuclear configurations of the given overall stoichiometry. For example, a mirror plane of a nuclear configuration implies a mirror plane of the potential surface (or that of the potential energy hypersurface in higher dimensions), and a local rotational symmetry of substituents implies a translational symmetry, that is, periodicity of the potential surface, if the latter is defined in terms of the usual bond length/bond angle internal coordinates. Such symmetry relations on potential surfaces are rather trivial consequences of molecular symmetry properties; however, when taken collectively for entire domains of nuclear configurations, they lead to nontrivial conclusions. Whereas symmetry properties and energy contents of individual conformations can be studied locally within limited domains of the potential surface, a global analysis of the potential surface may reveal significantly more. In this note, some consequences of the above approach are explored, and a simple test is proposed for the detection and evaluation of the importance of multicenter interactions in conformers related to one another by bond rotations.Dedicated to Professor J. Koutecký on the occasion of his 65th birthday     

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.     

Exploitation of symmetry in periodic Self-Consistent-Field ab initio calculations: application to large three-dimensional compounds

Symmetry can dramatically reduce the computational cost (running time and memory allocation) of Self-Consistent-Field ab initio calculations for crystalline systems. Crucial for running time is use of symmetry in the evaluation of one- and two-electron integrals, diagonalization of the Fock matrix at selected points in reciprocal space, reconstruction of the density matrix. As regards memory allocation, full square matrices (overlap, Fock and density) in the Atomic Orbital (AO) basis are avoided and a direct transformation from the packed AO to the SACO (Symmetry Adapted Crystalline Orbital) basis is performed, so that the largest matrix to be handled has the size of the largest sub-block in the latter basis. We here illustrate the effectiveness of this scheme, following recent advancements in the CRYSTAL code, concerning memory allocation and direct basis set transformation. Quantitative examples are given for large unit cell systems, such as zeolites (all-silica faujasite and silicalite MFI) and garnets (pyrope). It is shown that the full SCF of 3D systems containing up to 576 atoms and 11136 Atomic Orbitals in the cell can be run with a hybrid functional on a single core PC with 500 MB RAM in about 8 h.     

M(o)bius环并苯的分子对称性

一般来说,点群理论认为M(o)bius带环分子最高的对称性只能是C2.本文讨论了由18个苯环组成的环并苯的异构体分子,包括柱面的Hückel型分子(HC-[18])和扭转180°的M(o)bius带环分子(MC-[18]).结果表明除了点对称性外,M(o)bius带环分子还存在一种可称为环面螺旋旋转(TSR)变换的对称性,为此还引用了环面正交曲线坐标系.此外,还讨论了这些分子关于TSR对称性匹配的原子集和原子轨道(AO)集.根据TSR对称性的循环群特征,可以建立此类群的不可约表示及有关特征标.这类分子的分子轨道(MO)关于TSR群的不可约表示是纯的,然而所含的相应的原子轨道对称性匹配的线性组合(SALC-AO)成分可以是多种的.     

Symmetry analysis of the potential energy surfaces for the photochemical decomposition of formaldehyde

Orbital Correspondence Analysis in Maximum Symmetry (OCAMS) is applied to the decomposition pathways of formaldehyde to H₂ + CO and to H + HCO. The symmetry adapted nuclear motions, which are preferentially incorporated in the energetically favoured fragmentation pathways on both the ground and excited state surfaces are singled out. The results of this analysis are in full agreement with those of published potential energy surfaces and consistent with the results of experimental investigations reported in the literature. The nuclear motions favouring the various processes thus appear to be deducible from considerations of orbital symmetry.This work was carried out during tenure of a Minerva Foundation grant to one of us. (E.A.H.)     

Continuous symmetry analysis of tetrahedral/planar distortions. Copper chlorides and other AB4 species

The symmetry of tetracoordinated copper complexes, especially of CuCl4(2-), is analyzed in terms of quantitative continuous symmetry. These complexes acquire structures that span from tetrahedral to planar, with all gradual variations in between, a property that is evaluated here in terms of their degrees of tetrahedricity (S(Td)) and square planarity (S(D4h)). It was found that out of the large arsenal of geometry-allowed tetrahedral structures, each of which is characterized by specific S(Td) and S(D4h) pair values, copper complexes concentrate on an extremely well-defined correlation line linking these two symmetry-content measures; that is, only very specific pairs of S(Td) and S(D4h) values that dictate each other are allowed. Furthermore, it was found that of the various routes that can lead from a tetrahedron to a planar square, the mode known as spread fits exactly in its symmetry S(Td)/S(D4h) characteristics the observed symmetry behavior of the copper complexes. Interestingly, the spread mode reflects also the (nearly) minimal possible values of S(Td)/S(D4h) pairs, namely, the minimal symmetry-distortive route. Hints that this symmetry correlation line reflects universal features that go beyond copper are provided by the symmetry-content analysis of Ni and Pt complexes, of various Zn complexes within a metalloprotein and even of CC4 fragments, all of which obey the same line. A new potential-energy surface is introduced, the axes of which are S(Td), S(D4h), and the energy. Calculating this surface for CuCl4(2-) reveals an important result: The minimal molecular symmetry distortive spread correlation line coincides with the (only) energy valley of the map. Symmetry and energy are intimately related in their drive to minimal values.