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.     

Full spin and spatial symmetry adapted technique for correlated electronic hamiltonians: Application to an icosahedral cluster

One of the long standing problems in quantum chemistry had been the inability to exploit full spatial and spin symmetry of an electronic Hamiltonian belonging to a non‐Abelian point group. Here, we present a general technique which can utilize all the symmetries of an electronic (magnetic) Hamiltonian to obtain its full eigenvalue spectrum. This is a hybrid method based on Valence Bond basis and the basis of constant z‐component of the total spin. This technique is applicable to systems with any point group symmetry and is easy to implement on a computer. We illustrate the power of the method by applying it to a model icosahedral half‐filled electronic system. This model spans a huge Hilbert space (dimension 1,778,966) and in the largest non‐Abelian point group. The C₆₀ molecule has this symmetry and hence our calculation throw light on the higher energy excited states of the bucky ball. This method can also be utilized to study finite temperature properties of strongly correlated systems within an exact diagonalization approach. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Determining symmetry changes during a chemical reaction: the case of diazene isomerization

Analysis of reaction paths in terms of the Continuous Symmetry Measure (CSM) provides an alternative way to analyze the geometrical changes that take place during a reaction. Unique symmetry-profiles, describing the symmetry changes along the internal reaction coordinate were calculated for the cis-trans isomerization reaction of N₂H₂ and for its halogeno derivatives. A “symmetry transition point” is identified at the extremum point along the symmetry-profiles. At this point, the deviation of the molecule from the rotational symmetry of the reactant is the same as its deviation from the rotational symmetry of the product. In a second application we show that the CSM can be used as an alternative reaction coordinate. Calculations at the MP2 and DFT levels result in similar symmetry profiles.     

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.     

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.     

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     

New implementation of molecular double point‐group symmetry in four‐component relativistic Gaussian‐type spinors

A new, practical implementation of double‐group symmetry to relativistic Gaussian spinors is presented for four‐component relativistic molecular calculations. We show that the systematic adaptability to irreducible representations under arbitrary point‐group symmetry, as well as Kramers (time‐reversal) symmetry, is inherent in the present basis spinors, which possess the analytic structure of Dirac atomic spinors. The implementation of double‐group symmetry entails significant computational efficiencies in the relativistic second‐order Møller–Plesset perturbation calculation on Au₂ and the density functional theory (DFT) calculation with the B3LYP functional on octahedral UF₆, in which the highest symmetries used are, respectively, C and D. The four‐component B3LYP equilibrium geometry of UF₆ is reported. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

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.     

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.     

Evaluating molecular asymmetry

A method for calculating the asymmetry parameters of molecules based on Avnir’s CSM approach combined with the “dissymmetry function” method is suggested. The performance of the approach is demonstrated on various geometrical models — high-symmetry antiprisms of S₁₀ and D₅ symmetry groups, helices, and molecular objects. It is shown that the MCSM method unambiguously determines the symmetry element or estimates the degree of asymmetry for molecules from different structural classes. A. V. Bogatskii Physiocochemical Institute, Ukrainian Academy of Sciences. Odessa State University. Translated fromZhurnal Strukturnoi Khimii, Vol. 39, No. 3, pp. 547–552, May–June, 1998.     

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.     

Understanding topological symmetry: a heuristic approach to its determination

An algorithm based on heuristic rules for topological symmetry perception of organic structures having heteroatoms, multiple bonds, and any kind of cycle, and configuration, is presented. This algorithm identifies topological symmetry planes and sets of equivalent atoms in the structure, named symmetry atom groups (SAGs). This approach avoids both the need to explore the entire graph automorphism groups, and to encompass cycle determination, resulting in a very effective computer processing. Applications to several structures, some of them highly symmetrical such as dendrimers, are presented.     

Utilization of ‘pseudo-lattice symmetry’ in cluster calculations

Cluster calculations which model chemisorption on a surface are often composed of substrate atoms arranged in a periodic manner. This pseudo-lattice symmetry of a cluster is used to reduce the number of 2-electron integrals computed in a SCF calculation by evaluating only unique integrals identified by lattice displacement vectors. The method, without using any explicit symmetry, is shown to be competitive with calculations which utilize point group symmetry. It is also demonstrated that the pseudo-lattice method markedly reduces the number of 2-electron integrals in multi-layer clusters which have little or no symmetry.     

Approximate symmetry characteristics using fuzzy-subset theory study for chiral transitions of allene-1,3,-dihalides

Based on our study of the application of fuzzy-subset theory to the characterization of imperfect symmetry in some stable molecular systems and simple dynamic molecular systems, we analyze the internal rotation process of allene-1,3- dihalides. Allene-1,3-dihalides (CHX=C=CHY, where X and Y may be the same or different halogen atoms) are optically chiral nonplanar molecules. The two end-groups may internally rotate about the near straight linear C=C=C axis, and the molecule may change its chirality. The internal rotation process may pass through two different planar transition state (TS): cis-TS and trans-TS, which belong to C_2v and C_2h point groups (as X and Y to be same), respectively. The intrinsic reaction coordinate (IRC) corresponding to the two TS processes is denoted as cis-IRC and trans-IRC. However, for the whole IRC reaction process, only their subgroup C₂ well-defined symmetry remains. Other symmetry transformations in C_2v and C_2h point groups can only be examined in terms of imperfect symmetry, although there appear certain reaction reversal joint point group G(R_cC_2v) and G(R_tC_2h) well-defined symmetry in the dynamics through the IRC processes. If X and Y are different, the stable molecule has no conventional nontrivial point group symmetry. The internal rotation processes may pass through two different planar TS’s (cis-and trans-TS). The TS will still be a planar molecule belonging to C_S point group with the molecule plane as its symmetry plane. Other states in the IRC may belong to certain reaction reversal joint point groups, G(RM)_C and G(RM)_T. We have thus examined the approximate symmetry of MO’s related to C₂ point group. Moreover, we have also analyzed the membership functions, representation components, and their relationships shown in the MO fuzzy main representation correlation diagrams.     

The combinatorics of symmetry adaptation

A method is developed for obtaining the generating functions for the equivalence classes of orbitals wherein only orbitals within an equivalence class participate in symmetry adaptation. It is shown that using Williamson's combinatorial theorem the generating functions for the symmetry species contained in each equivalence class can be obtained. The method is illustrated with Porphindianion.     

Exploiting space‐group symmetry in fragment‐based molecular crystal calculations

Recent developments in fragment‐based methods make it increasingly feasible to use high‐level ab initio electronic structure techniques to molecular crystals. Such studies remain computationally demanding, however. Here, we describe a straightforward algorithm for exploiting space‐group symmetry in fragment‐based methods which often provides computational speed‐ups of several fold or more. This algorithm does not require a priori specification of the space group or symmetry operators. Rather, the symmetrically equivalent fragments are identified automatically by aligning the individual fragments along their principle axes of inertia and testing for equivalence with other fragments. The symmetry operators relating equivalent fragments can then be worked out easily. Implementation of this algorithm for computing energies, nuclear gradients with respect to both atomic coordinates and lattice parameters, and the nuclear hessian is described. © 2014 Wiley Periodicals, Inc.     

The fuzzy D<Subscript>6h</Subscript>-symmetries of azines molecules and their molecular orbitals

Based on our previous study on the elementary characterization of fuzzy symmetry, we inquire into the fuzzy symmetries of some simple linear and plane molecules. These systems belong to point groups that include the identity and twofold symmetry elements, but not include higher multi-fold symmetry ones, and their molecular orbitals (MOs) only belong to one-dimension irreducible representations. In this paper, we take the azines as a typical model to examine the fuzzy symmetry in relation to the D_6h point group. As this group includes multi-fold symmetry elements such as a sixfold rotation axis, some of the MOs may belong to two-dimensional irreducible representations. We inquire into the fuzzy symmetry of these molecules and their MOs in terms of membership functions, representation components and correlation diagrams. In addition to these neutral closed shell molecules, pyridine hydride radical, anion, and cation are also analyzed.     

Isomerism and symmetry of bridged polymeric phthalocyanines

Bridged polymeric phthalocyanines exist in different isomeric forms of varying symmetry. The number of isomers and the symmetry depends on the structure, i. e. the size and shape, of the polymeric phthalocyanine. Equations are derived for calculating the number of isomers for bridged polymeric phthalocyanines, and symmetry point groups for various shapes (linear, square, rectangular) are discussed.