Abstract
A great variety of data on molecular structure and changes, accumulated both experimentally and theoretically, need be compacted and classified to extract the information arguably relevant to understand the basic mechanisms of chemical transformations. Here a screen for displaying four-center processes is developed and as an illustration applied to conformations involving torsions around O – O and S – S bonds, extending the structural properties previously calculated in this laboratory. The construction of the screen follows from connections recently established between the classical kinematic mechanism – the four-bar linkage – and the basic ingredient of quantum angular momentum theory – the 6j symbol.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Maciel, G.S., Bitencourt, A.C.P., Ragni, M., Aquilanti, V.: Alkyl peroxides: effect of substituent groups on the torsional mode around the O – O bond. Int. J. Quant. Chem. 107, 2697–2707 (2007)
Maciel, G.S., Bitencourt, A.C.P., Ragni, M., Aquilanti, V.: Quantum study of peroxidic bonds and torsional levels for ROOR’ molecules (R, R’ = H, F, Cl, NO, CN). J. Phys. Chem. A 111, 12604–12610 (2007)
Aquilanti, V., Ragni, M., Bitencourt, A.C.P., Maciel, G.S., Prudente, F.V.: Intramolecular dynamics of RS – SR’ systems (R, R’ = H, F, Cl, CH3, C2H5): torsional potentials, energy levels, partition functions. J. Phys. Chem. A 113, 3804–3813 (2009)
Aquilanti, V., Grossi, G., Lombardi, A., Maciel, G.S., Palazzetti, F.: The origin of chiral discrimination: supersonic molecular beam experiments and molecular dynamics simulations of collisional mechanisms. Phys. Scr. 78, 058119 (2008)
Barreto, P.R.P., Vilela, A.F.A., Lombardi, A., Maciel, G.S., Palazzetti, F., Aquilanti, V.: The hydrogen peroxide-rare gas systems: Quantum chemical calculations and hyperspherical harmonic representation of the potential energy surface for atom-floppy molecule interactions. J. Phys. Chem. A 111, 12754–17762 (2008)
Barreto, P.R.P., Palazzetti, F., Grossi, G., Lombardi, A., Maciel, G.S., Vilela, A.F.A.: Range and strength of intermolecular forces for van der waals complexes of the type H2Xn-Rg, with X = O, S and n = 1, 2. Int. J. Quant. Chem. 110, 777–786 (2010)
Barreto, P.R.P., Albernaz, A.F., Palazzetti, F., Lombardi, A., Grossi, G., Aquilanti, V.: Hyperspherical representation of potential energy surfaces: intermolecular interactions in tetra-atomic and penta-atomic systems. Phys. Scr. 84, 028111 (2011)
Maciel, G.S., Barreto, P.R.P., Palazzetti, F., Lombardi, A., Aquilanti, V.: A quantum chemical study of H2S2: Intramolecular torsional mode and intermolecular interactions with rare gases. J. Chem. Phys. 129, 164302 (2008)
Barreto, P.R.P., Albernaz, A.F., Palazzetti, F.: Potential energy surfaces for van der Waals complexes of rare gases with H2S and H2S2: Extension to xenon interactions and hyperspherical harmonics representation. Int. J. Quant. Chem. 112, 834–847 (2012)
Lombardi, A., Palazzetti, F., MacIel, G.S., Aquilanti, V., Sevryuk, M.B.: Simulation of oriented collision dynamics of simple chiral molecules. Int. J. Quant. Chem. 111, 1651–1658 (2011)
Maciel, G.S., Bitencourt, A.C.P., Ragni, M., Aquilanti, V.: Studies of the dynamics around the O – O bond: orthogonal mods of hydrogen peroxide. Chem. Phys. Lett. 432, 383–390 (2006)
Palazzetti, F., Munusamy, E., Lombardi, A., Grossi, G., Aquilanti, V.: Spherical and hyperspherical representation of potential energy surfaces for intermolecular interactions. Int. J. Quant. Chem. 111, 318–332 (2011)
Moss, G. P.: Pure Appl. Chem. 68, 2193–2222 (1996)
Bitencourt, A.C.P., Ragni, M., Littlejohn, R.G., Anderson, R., Aquilanti, V.: The Screen Representation of Vector Coupling Coefficients or Wigner 3j Symbols: Exact Computation and Illustration of the Asymptotic Behavior. In: Murgante, B., Misra, S., Rocha, A.M.A.C., Torre, C., Rocha, J.G., Falcão, Maria Irene, Taniar, D., Apduhan, B.O., Gervasi, Osvaldo (eds.) ICCSA 2014. LNCS, vol. 8579, pp. 468–481. Springer, Cham (2014). doi:10.1007/978-3-319-09144-0_32
Aquilanti, V., Bitencourt, A., da S. Ferreira, C., Marzuoli, A., Ragni, M.: Quantum and semiclassical spin networks: from atomic and molecular physics to quantum computing and gravity. PhysicaScripta 78, 058103 (2008)
Aquilanti, V., Bitencourt, A., da S. Ferreira, C., Marzuoli, A., Ragni, M.: Combinatorics of angular momentum recoupling theory: spin networks, their asymptotics and applications. Theor. Chem. Acc. 123, 237 (2009)
Ponzano, G., Regge, T.: Semiclassical limit of Racah coefficients. In: Bloch, F. et al (eds.) Spectroscopic and Group Theoretical Methods in Physics, pp. 1–58. North Holland, Amsterdam (1968)
Neville, D.: A technique for solving recurrence relations approximately and its application to the 3 j and 6 j symbols. J. Math. Phys. 12, 2438 (1971)
Schulten, K., Gordon, R.: Semiclassical approximations to 3j- and 6j-coefficients for quantum-mechanical coupling of angular momenta. J. Math. Phys. 16, 1971–1988 (1975)
Schulten, K., Gordon, R.: Exact recursive evaluation of 3j- and 6j-coecients for quantum mechanical coupling of angular momenta. J. Math. Phys. 16, 1961–1970 (1975)
Ragni, M., Bitencourt, A.C., Aquilanti, V., Anderson, R.W., Littlejohn, R.G.: Exact computation and asymptotic approximations of 6j symbols: Illustration of their semiclassical limits. Int. J. Quantum Chem. 110(3), 731–742 (2010)
Aquilanti, V., Cavalli, S., Coletti, C.: Angular and hyperangular momentum recoupling, harmonic superposition and Racah polynomials. A recursive algorithm. Chem. Phys. Lett. 344, 587–600 (2001)
Littlejohn, R.G., Yu, L.: Uniform semiclassical approximation for the Wigner 6j-symbol in terms of rotation matrices. J. Phys. Chem. A 113, 14904–14922 (2009)
Aquilanti, V., Haggard, H.M., Hedeman, A., Jeevangee, N., Littlejohn, R., Yu, L.: Semiclassical mechanics of the Wigner 6j-symbol. J. Phys. A 45, 065209 (2012). arXiv:1009.2811v2
Aquilanti, V., Capecchi, G.: Harmonic analysis and discrete polynomials. From semiclassical angular momentum theory to the hyperquantization algorithm. Theor. Chem. Accounts 104, 183–188 (2000)
De Fazio, D., Cavalli, S., Aquilanti, V.: Orthogonal polynomials of a discrete variable as expansion basis sets in quantum mechanics. the hyperquantization algorithm. Int. J. Quantum Chem. 93, 91–111 (2003)
Aquilanti, V., Cavalli, S., De Fazio, D.: Angular and hyperangular momentum coupling coecients as Hahn polynomials. J. Phys. Chem. 99(42), 15694–15698 (1995)
Koekoek, R., Lesky, P., Swarttouw, R.: Hypergeometric Orthogonal Polynomials and Their q-Analogues. Springer, Heidelberg (2010)
Bitencourt, A.C.P., Marzuoli, A., Ragni, M., Anderson, R.W., Aquilanti, V.: Exact and asymptotic computations of elementary spin networks: classification of the quantum–classical boundaries. In: Murgante, B., Gervasi, O., Misra, S., Nedjah, N., Rocha, A.M.A.C., Taniar, D., Apduhan, B.O. (eds.) ICCSA 2012. LNCS, vol. 7333, pp. 723–737. Springer, Heidelberg (2012). doi:10.1007/978-3-642-31125-3_54
Varshalovich, D., Moskalev, A., Khersonskii, V.: Quantum Theory of Angular Momentum. World Scientific, Singapore (1988)
Aquilanti, V., Haggard, H.M., Littlejohn, R.G., Yu, L.: Semiclassical analysis of Wigner 3 j -symbol. J. Phys. A 40(21), 5637–5674 (2007)
Anderson, R.W., Aquilanti, V.: The discrete representation correspondence between quantum and classical spatial distributions of angular momentum vectors. J. Chem. Phys. 124, 214104 (2006). (9 pages)
Anderson, R.W., Aquilanti, V., da Silva Ferreira, C.: Exact computation and large angular momentum asymptotics of 3nj symbols: semiclassical disentangling of spin networks. J. Chem. Phys. 129, 161101–161105 (2008)
Anderson, R.W., Aquilanti, V., Bitencourt, A.C.P., Marinelli, D., Ragni, M.: The Screen Representation of Spin Networks: 2D Recurrence, Eigenvalue Equation for 6j Symbols, Geometric Interpretation and Hamiltonian Dynamics. In: Murgante, B., Misra, S., Carlini, M., Torre, C.M., Nguyen, H.-Q., Taniar, D., Apduhan, B.O., Gervasi, O. (eds.) ICCSA 2013. LNCS, vol. 7972, pp. 46–59. Springer, Heidelberg (2013). doi:10.1007/978-3-642-39643-4_4
Aquilanti, V., Marinelli, D., Marzuoli, A.: Hamiltonian dynamics of a quantum of space: hidden symmetries and spectrum of the volume operator, and discrete orthogonal polynomials. J. Phys. A: Math. Theor. 46, 175303 (2013). arXiv:1301.1949v2
Arruda, M.S., Santos, R.F., Marinelli, D., Aquilanti, V.: Spin-coupling diagrams and incidence geometry: a note on combinatorial and quantum-computational aspects. In: Gervasi, O., Murgante, B., Misra, S., Rocha, A.M.A.C., Torre, C., Taniar, D., Apduhan, B.O., Stankova, E., Wang, S. (eds.) ICCSA 2016. LNCS, vol. 9786, pp. 431–442. Springer, Cham (2016). doi:10.1007/978-3-319-42085-1_33
Ragni, M., Littlejohn, R.G., Bitencourt, A.C.P., Aquilanti, V., Anderson, R.W.: The screen representation of spin networks: images of 6j symbols and semiclassical features. In: Murgante, B., Misra, S., Carlini, M., Torre, C.M., Nguyen, H.-Q., Taniar, D., Apduhan, B.O., Gervasi, O. (eds.) ICCSA 2013. LNCS, vol. 7972, pp. 60–72. Springer, Heidelberg (2013). doi:10.1007/978-3-642-39643-4_5
Lombardi, A., Palazzetti, F., Aquilanti, V., Pirani, F., Casavecchia, P.: The astrochemical observatory: experimental and computational focus on the chiral molecule propylene oxide as a case study. In: ICCSA 2017, Part V, LNCS, vol. 10408, pp. 1-14. doi:10.1007/978-3-319-62404-4_20
Aquilanti, V., Anderson, R. W.: Spherical and hyperbolic spin network: the q-extensions of wigner racah 6j coefficients and general orthogonal discrete basis sets in applied quantum mechanics. In: ICCSA 2017, Part V, LNCS, vol. 10408, pp. 1-16. doi:10.1007/978-3-319-62404-4_25
Acknowledgements
The authors acknowledge the Italian Ministry for Education, University and Research, MIUR for financial support through SIR 2014 Scientific Independence for Young Researchers (RBSI14U3VF).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Appendix. Quadrilaterals, Quadrangles, Tetrahedra: The 6-Distance Representation
Appendix. Quadrilaterals, Quadrangles, Tetrahedra: The 6-Distance Representation
We extend here geometrical considerations alluded to in Sect. 2.1. The six distances that define the stereogenic unit of peroxides and persulfides, i.e. the sequence of bonds R1 – O1 – O2 – R2 (for the sake of simplicity we refer only to the peroxide case, but it can be simply applied to persulfides), individuate a tetrahedron, whose edges are the distances O1R1, O1R2, O2R2, O2R1, O1O2, and R1R2. The planar projection of the tetrahedron permits to define a quadrilateral, whose diagonals can be chosen among the pairs of opposite distances R1O1 and R2O2; R1O2 and R2O1; O1O2 and R1R2. The tetrahedron and quadrilateral coincide when the stereogenic unit has a planar geometry, e.g. for the cis and trans configurations. The length of the diagonals vary within a range, keeping the four sides of the quadrilateral fixed. In these terms, we represent the variation of the diagonals in a two-dimensional diagram, the screen. We are here inspired by the Wigner-Racah-Regge approach to theory of the most basic ingredients of quantum angular momentum and of spin networks, the 6j symbol. In a similar fashion, let’s arrange the six distances as follows,
the 6d symbols. In the application to peroxides, in the text the diagonals correspond to the first column, since in the specific case of peroxides and persulfides, the variation of R1 – R2 distance is the most suitable to monitor the chirality change transition. Two other choices of diagonals is possible, since by taking into account that the symbol is invariant under permutation of the three columns:
(there are 3! = 6 ways identical symbols).
The first row represents a triad (a triangular face of the tetrahedron into a vertex), while the second row exhibits the convergence of three edges of the tetrahedron. One can have four triads, \( \left\{ {{\text{O}}_{1} {\text{R}}_{1} , {\text{O}}_{1} {\text{R}}_{2} , {\text{O}}_{1} {\text{O}}_{2} } \right\} \), \( \left\{ {{\text{O}}_{2} {\text{R}}_{2} , {\text{O}}_{2} {\text{R}}_{1} , {\text{O}}_{1} {\text{O}}_{2} } \right\}, \left\{ {{\text{O}}_{1} {\text{R}}_{1} , {\text{O}}_{2} {\text{R}}_{1} , {\text{R}}_{1} {\text{R}}_{1} } \right\}, \left\{ {{\text{O}}_{2} {\text{R}}_{2} , {\text{O}}_{1} {\text{R}}_{2} , {\text{R}}_{1} {\text{R}}_{2} } \right\} \). The invariance with respect to permutations of the four triads and the related triangles generate the invariance of the 6j under the interchange of upper and lower arguments of any two columns, for example
(four ways). A total of 24 symmetries can be enumerated in this way. Each symbol has in addition the six Regge symmetries replicas, for a total of 144 symmetries. The important relationship of these quantum mechanically discovered (1959) symmetries were later found. Surprisingly they apply to properties of Euclidean (Ponzano and Regge, 1968) and non-Euclidean tetrahedra and to the operating rules of the most venerable of the kinematic mechanisms. For further discussions, see companion papers in this volume [38, 39].
This “distance only” formulation may be suitable to describe peroxides and similar classes of molecules for which often mapping based on pairs of dihedral angle (e.g. Ramachandran plot) is not applicable. In turn, as a viewpoint in the same spirit of the displays on screen examined in the mean text, it can be arguably extendible to other systems, such as molecules characterized by the asymmetric carbon covalently bound to four different ligands, or to describe folding involving the peptidic bonds.
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Aquilanti, V., Caglioti, C., Lombardi, A., Maciel, G.S., Palazzetti, F. (2017). Screens for Displaying Chirality Changing Mechanisms of a Series of Peroxides and Persulfides from Conformational Structures Computed by Quantum Chemistry. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2017. ICCSA 2017. Lecture Notes in Computer Science(), vol 10408. Springer, Cham. https://doi.org/10.1007/978-3-319-62404-4_26
Download citation
DOI: https://doi.org/10.1007/978-3-319-62404-4_26
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-62403-7
Online ISBN: 978-3-319-62404-4
eBook Packages: Computer ScienceComputer Science (R0)