Skip to main content
SpringerLink
Log in

Classification of Shapes and Deformations of Large Systems by Invariant Coordinates

  • Conference paper
  • First Online:
Computational Science and Its Applications – ICCSA 2020 (ICCSA 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12255))

Included in the following conference series:

  • 2031 Accesses

Abstract

The use of hyperspherical coordinates is widespread in reactive scattering studies, allowing for a symmetric representation of the quantum dynamics of reactive processes. Indeed, among the variants of hyperspherical coordinates, the so called “symmetric” ones are “democratic” with respect to the asymptotic channels and so are the corresponding basis sets, since basis functions can be symmetrized with respect to particle exchange, acting on just a reduced subset of coordinates. Applications to scattering problems are limited to few-atom systems, due to computational cost. An extension of the representation to many-body classical dynamics is possible and has been proposed in a series of papers, where different aspects have been investigated. Here we recall the possibility of defining shape coordinates invariant with respect to the remaining degrees of freedom, which are suitable for systematic classification of structures of clusters and large biomolecules. The definition of shape parameters and to provide examples of their application are the purposes of the present paper.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Zhao, B., Guo, H.: State-to-state quantum reactive scattering in four-atom systems. WIREs Comput. Mol. Sci. 7, e1301 (2017)

    Google Scholar 

  2. Skouteris, D., Castillo, J., Manolopoulos, D.E.: Abc: a quantum reactive scattering program. Comput. Phys. Comm. 133, 128–135 (2000)

    MATH  Google Scholar 

  3. Lepetit, B., Launay, J.M.: Quantum-mechanical study of the reaction He+H\(_{2}^+ \rightarrow \) HeH\(^+\) + H with hyperspherical coordinates. J. Chem. Phys. 95, 5159–5168 (1991)

    Google Scholar 

  4. Aquilanti, V., Beddoni, A., Cavalli, S., Lombardi, A., Littlejohn, R.: Collective hyperspherical coordinates for polyatomic molecules and clusters. Mol. Phys. 98(21), 1763–1770 (2000)

    Google Scholar 

  5. Aquilanti, V., Beddoni, A., Lombardi, A., Littlejohn, R.: Hyperspherical harmonics for polyatomic systems: basis set for kinematic rotations. Int. J. Quantum Chem. 89(4), 277–291 (2002)

    Google Scholar 

  6. Aquilanti, V., Lombardi, A., Littlejohn, R.: Hyperspherical harmonics for polyatomic systems: basis set for collective motions. Theor. Chem. Acc. 111(2–6), 400–406 (2004)

    Google Scholar 

  7. Kuppermann, A.: Quantum reaction dynamics and hyperspherical harmonics. Isr. J. Chem. 43, 229 (2003)

    Google Scholar 

  8. De Fazio, D., Cavalli, S., Aquilanti, V.: Benchmark quantum mechanical calculations of vibrationally resolved cross sections and rate constants on ab initio potential energy surfaces for the F + HD reaction: comparisons with experiments. J. Phys. Chem. A 120, 5288–5299 (2016)

    Google Scholar 

  9. Aquilanti, V., Cavalli, S.: The quantum-mechanical hamiltonian for tetraatomic systems insymmetric hyperspherical coordinates. J. Chem. Soc., Faraday Trans. 93, 801–809 (1997)

    Google Scholar 

  10. Lombardi, A., Laganà, A., Pirani, F., Palazzetti, F., Faginas-Lago, N.: Carbon oxides in gas flows and earth and planetary atmospheres: state-to-state simulations of energy transfer and dissociation reactions. In: Murgante, B., Misra, S., Carlini, M., Torre, C., Nguyen, H.Q., Taniar, D., Apduhan, B., Gervasi, O. (eds.) Computational Science and Its Applications - ICCSA 2013. Lecture Notes in Computer Science, vol. 7972, pp. 17–31. Springer, Berlin Heidelberg (2013). https://doi.org/10.1007/978-3-642-39643-4_2

    Chapter  Google Scholar 

  11. Lago, N.F., Albertí, M., Laganà, A., Lombardi, A.: Water (H\(_2\)O)\(_m\) or benzene (C\(_6\)H\(_6\))\(_n\) aggregates to solvate the K\(^+\). In: Murgante, B., et al. (eds.) ICCSA 2013. LNCS, vol. 7971, pp. 1–15. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39637-3_1

    Chapter  Google Scholar 

  12. Faginas-Lago, N., Albertí, M., Costantini, A., Laganá, A., Lombardi, A., Pacifici, L.: An innovative synergistic grid approach to the computational study of protein aggregation mechanisms. J. Mol. Model. 20(7), 2226 (2014)

    Google Scholar 

  13. Faginas-Lago, N., Yeni, D., Huarte, F., Alcamì, M., Martin, F.: Adsorption of hydrogen molecules on carbon nanotubes using quantum chemistry and molecular dynamics. J. Phys. Chem. A 120, 6451–6458 (2016)

    Google Scholar 

  14. Faginas-Lago, N., Lombardi, A., Albertí, M., Grossi, G.: Accurate analytic intermolecular potential for the simulation of Na\(^+\) and K\(^+\) ion hydration in liquid water. J. Mol. Liq. 204, 192–197 (2015)

    Google Scholar 

  15. Albertí, M., Faginas Lago, N.: Competitive solvation of K\(^{+}\) by C\(_6\)H\(_6\) and H\(_2\)O in the K\(^{+}\)-(C\(_6\)h\(_6\))\(_n\)-(H\(_2\)O)\(_m\) (n = 1–4; m = 1–6) aggregates. Eur. Phys. J. D 67, 73 (2013)

    Google Scholar 

  16. Albertí, M., Faginas Lago, N.: Ion size influence on the ar solvation shells of M\(^+\)-C\(_6\)F\(_6\) clusters (m = na, k, rb, cs). J. Phys. Chem. A 116, 3094–3102 (2012)

    Google Scholar 

  17. Albertí, M., Faginas Lago, N., Pirani, F.: Ar solvation shells in K\(^+\)-HFBz: From cluster rearrangement to solvation dynamics. J. Phys. Chem. A 115, 10871–10879 (2011)

    Google Scholar 

  18. Lago, N.F., Albertí, M., Laganà, A., Lombardi, A., Pacifici, L., Costantini, A.: The molecular stirrer catalytic effect in methane ice formation. In: Murgante, B., et al. (eds.) ICCSA 2014. LNCS, vol. 8579, pp. 585–600. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-09144-0_40

    Chapter  Google Scholar 

  19. Faginas-Lago, N., Huarte Larrañaga, F., Albertí, M.: On the suitability of the ilj function to match different formulations of the electrostatic potential for water-water interactions. Eur. Phys. J. D 55(1), 75 (2009)

    Google Scholar 

  20. Bartolomei, M., Pirani, F., Laganà, A., Lombardi, A.: A full dimensional grid empowered simulation of the CO\(_2\)+ CO\(_2\) processes. J. Comp. Chem. 33, 1806 (2012)

    Google Scholar 

  21. Lombardi, A., Faginas-Lago, N., Laganà, A., Pirani, F., Falcinelli, S.: A bond-bond portable approach to intermolecular interactions: Simulations for n-methylacetamide and carbon dioxide dimers. In: Murgante, B., et al. (eds.) Computational Science and Its Applications - ICCSA 2012. Lecture Notes in Computer Science, vol. 7333, pp. 387–400. Springer, Berlin Heidelberg (2012). https://doi.org/10.1007/978-3-642-31125-3_30

    Chapter  Google Scholar 

  22. Albertí, M., Faginas-Lago, N., Laganà, A., Pirani, F.: A portable intermolecular potential for molecular dynamics studies of nma-nma and nma-H\(_2\)O aggregates. Phys. Chem. Chem. Phys. 13(18), 8422–8432 (2011)

    Google Scholar 

  23. Albertí, M., Faginas-Lago, N., Pirani, F.: Benzene water interaction: from gaseous dimers to solvated aggregates. Chem. Phys. 399, 232 (2012)

    Google Scholar 

  24. Falcinelli, S., Rosi, M., Candori, P., Vecchiocattivi, F., Bartocci, A., Lombardi, A., Faginas-Lago, N., Pirani, F.: Modeling the intermolecular interactions and characterization of the dynamics of collisional autoionization processes. In: Murgante, B., et al. (eds.) Computational Science and Its Applications - ICCSA 2013. Lecture Notes in Computer Science, vol. 7971, pp. 69–83. Springer, Berlin Heidelberg (2013). https://doi.org/10.1007/978-3-642-39637-3_6

    Chapter  Google Scholar 

  25. Lombardi, A., Faginas-Lago, N., Pacifici, L., Costantini, A.: Modeling of energy transfer from vibrationally excited CO\(_2\) molecules: cross sections and probabilities for kinetic modeling of atmospheres, flows, and plasmas. J. Phys. Chem. A 117(45), 11430–11440 (2013)

    Google Scholar 

  26. Lombardi, A., Pirani, F., Laganà, A., Bartolomei, M.: Energy transfer dynamics and kinetics of elementary processes (promoted) by gas-phase CO\(_2\)-N\(_2\) collisions: selectivity control by the anisotropy of the interaction. J. Comp. Chem. 37, 1463–1475 (2016)

    Google Scholar 

  27. Pacifici, L., Verdicchio, M., Faginas-Lago, N., Lombardi, A., Costantini, A.: A high-level ab initio study of the n2 + n2 reaction channel. J. Comput. Chem. 34(31), 2668–2676 (2013)

    Google Scholar 

  28. Lombardi, A., Faginas-Lago, N., Pacifici, L., Grossi, G.: Energy transfer upon collision of selectively excited CO\(_2\) molecules: state-to-state cross sections and probabilities for modeling of atmospheres and gaseous flows. J. Chem. Phys. 143, 034307 (2015)

    Google Scholar 

  29. Celiberto, R., et al.: Atomic and molecular data for spacecraft re-entry plasmas. Plasma Sources Sci. Technol. 25(3), 033004 (2016)

    Google Scholar 

  30. Faginas-Lago, N., Lombardi, A., Albertí, M.: Aqueous n-methylacetamide: new analytic potentials and a molecular dynamics study. J. Mol. Liq. 224, 792–800 (2016)

    Google Scholar 

  31. Palazzetti, F., Munusamy, E., Lombardi, A., Grossi, G., Aquilanti, V.: Spherical and hyperspherical representation of potential energy surfaces for intermolecular interactions. Int. J. Quantum Chem. 111(2), 318–332 (2011)

    Google Scholar 

  32. Lombardi, A., Palazzetti, F.: A comparison of interatomic potentials for rare gas nanoaggregates. J. Mol. Struc-THEOCHEM 852(1–3), 22–29 (2008)

    Google Scholar 

  33. Barreto, P.R., 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(2), 028111 (2011)

    Google Scholar 

  34. Barreto, P.R., et al.: Potential energy surfaces for interactions of H\(^2\)O with H\(_2\), N\(_2\) and O\(_2\): a hyperspherical harmonics representation, and a minimal model for the H\(_2\)O-rare-gas-atom systems. Comput. Theor. Chem. 990, 53–61 (2012)

    Google Scholar 

  35. Lombardi, A., Pirani, F., Bartolomei, M., Coletti, C., Laganà, A.: A full dimensional potential energy function and the calculation of the state-specific properties of the CO+ N\(_2\) inelastic processes within an Open Molecular Science Cloud perspective. Front. Chem. 7, 309 (2019)

    Google Scholar 

  36. Faginas Lago, N., Lombardi, A., Vekeman, J., Rosi, M., et al.: Molecular dynamics of CH\(_4\)/N\(_2\) mixtures on a flexible graphene layer: adsorption and selectivity case study. Front. Chem. 7, 386 (2019)

    Google Scholar 

  37. Nakamura, M., et al.: Dynamical, spectroscopic and computational imaging of bond breaking in photodissociation: roaming and role of conical intersections. Faraday Discuss. 177, 77–98 (2015)

    Google Scholar 

  38. Aquilanti, V., Lombardi, A., Yurtsever, E.: Global view of classical clusters: the hyperspherical approach to structure and dynamics. Phys. Chem. Chem. Phys. 4(20), 5040–5051 (2002)

    Google Scholar 

  39. Sevryuk, M.B., Lombardi, A., Aquilanti, V.: Hyperangular momenta and energy partitions in multidimensional many-particle classical mechanics: the invariance approach to cluster dynamics. Phys. Rev. A 72(3), 033201 (2005)

    MathSciNet  Google Scholar 

  40. Castro Palacio, J., Velazquez Abad, L., Lombardi, A., Aquilanti, V., Rubayo Soneira, J.: Normal and hyperspherical mode analysis of no-doped kr crystals upon rydberg excitation of the impurity. J. Chem. Phys. 126(17), 174701 (2007)

    Google Scholar 

  41. Lombardi, A., Palazzetti, F., Aquilanti, V.: Molecular dynamics of chiral molecules in hyperspherical coordinates. In: Misra, S., et al. (eds.) ICCSA 2019. LNCS, vol. 11624, pp. 413–427. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-24311-1_30

    Chapter  Google Scholar 

  42. Lombardi, A., Palazzetti, F., Sevryuk, M.B.: Hyperspherical coordinates and energy partitions for reactive processes and clusters. In: AIP Conference Proceedings. Volume 2186, p. 030014. AIP Publishing LLC (2019)

    Google Scholar 

  43. Lombardi, A., Palazzetti, F.: Chirality in molecular collision dynamics. J. Condens. Matter Phys. 30(6), 063003 (2018)

    Google Scholar 

  44. Lombardi, A., Palazzetti, F., Peroncelli, L., Grossi, G., Aquilanti, V., Sevryuk, M.: Few-body quantum and many-body classical hyperspherical approaches to reactions and to cluster dynamics. Theor. Chem. Acc. 117(5–6), 709–721 (2007)

    Google Scholar 

  45. Aquilanti, V., Grossi, G., Lombardi, A., Maciel, G.S., Palazzetti, F.: Aligned molecular collisions and a stereodynamical mechanism for selective chirality. Rend. Fis. Acc. Lincei 22, 125–135 (2011)

    Google Scholar 

  46. Lombardi, A., Faginas-Lago, N., Aquilanti, V.: The invariance approach to structure and dynamics: classical hyperspherical coordinates. In: Misra, S., et al. (eds.) ICCSA 2019. LNCS, vol. 11624, pp. 428–438. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-24311-1_31

    Chapter  Google Scholar 

  47. Caglioti, C., Dos Santos, R.F., Lombardi, A., Palazzetti, F., Aquilanti, V.: Screens displaying structural properties of aminoacids in polypeptide chains: alanine as a case study. In: Misra, S., et al. (eds.) ICCSA 2019. LNCS, vol. 11624, pp. 439–449. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-24311-1_32

    Chapter  Google Scholar 

  48. Caglioti, C., Ferreira, R.d.S., Palazzetti, F., Lombardi, A., Aquilanti, V.: Screen representation of structural properties of alanine in polypeptide chains. In: AIP Conference Proceedings. Volume 2186, p. 030015, AIP Publishing LLC (2019)

    Google Scholar 

  49. Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. University Press, Cambridge (1990)

    MATH  Google Scholar 

  50. Gatti, F., Lung, C.: Vector parametrization of the \(n\)-atom problem in quantum mechanics. i. jacobi vectors. J. Chem. Phys. 108(21), 8804–8820 (1998)

    Google Scholar 

  51. Aquilanti, V., Lombardi, A., Yurtsever, E.: Global view of classical clusters: the hyperspherical approach to structure and dynamics. Phys. Chem. Chem. Phys. 4, 5040–5051 (2002)

    Google Scholar 

  52. Aquilanti, V., Lombardi, A., Sevryuk, M.B.: Phase-space invariants for aggregates of particles: hyperangular momenta and partitions of the classical kinetic energy. J. Chem. Phys. 121, 5579 (2004)

    Google Scholar 

  53. Aquilanti, V., Carmona Novillo, E., Garcia, E., Lombardi, A., Sevryuk, M.B., Yurtsever, E.: Invariant energy partitions in chemical reactions and cluster dynamics simulations. Comput. Mat. Sci. 35, 187–191 (2006)

    Google Scholar 

  54. Aquilanti, V., Lombardi, A., Sevryuk, M.B., Yurtsever, E.: Phase-space invariants as indicators of the critical behavior of nanoaggregates. Phys. Rev. Lett. 93, 113402 (2004)

    Google Scholar 

  55. Calvo, F., Gadea, X., Lombardi, A., Aquilanti, V.: Isomerization dynamics and thermodynamics of ionic argon clusters. J. Chem. Phys. 125, 114307 (2006)

    Google Scholar 

  56. Lombardi, A., Aquilanti, V., Yurtsever, E., Sevryuk, M.B.: Specific heats of clusters near a phase transition: energy partitions among internal modes. Chem. Phys. Lett. 30, 424–428 (2006)

    Google Scholar 

  57. Lombardi, A., Maciel, G.S., Palazzetti, F., Grossi, G., Aquilanti, V.: Alignment and chirality in gaseous flows. J. Vacuum Soc. Japan 53(11), 645–653 (2010)

    Google Scholar 

  58. Palazzetti, F., et al.: Aligned molecules: chirality discrimination in photodissociation and in molecular dynamics. Rendiconti Lincei 24(3), 299–308 (2013)

    Google Scholar 

  59. Littlejohn, R.G., Mitchell, A., Aquilanti, V.: Quantum dynamics of kinematic invariants in tetra-and polyatomic systems. Phys. Chem. Chem. Phys. 1, 1259–1264 (1999)

    Google Scholar 

  60. Wales, D.G., et al.: The Cambridge Cluster Database (2001)

    Google Scholar 

Download references

Acknowledgments

The authors acknowledge financial support from MIUR PRIN 2010–2011 (contract 2010ERFKXL\(\_\)002) and from “Fondazione Cassa di Risparmio di Perugia (Codice Progetto: 2015.0331.021 Ricerca Scientifica e Tecnologica)”. Thanks are due to the Dipartimento di Chimica, Biologia e Biotecnologie dell’Università di Perugia (FRB, Fondo per la Ricerca di Base 2017) and to the MIUR and the University of Perugia for the financial support of the AMIS project through the program “Dipartimenti di Eccellenza”. A. L. acknowledges financial support from MIUR PRIN 2015 (contract 2015F59J3R\(\_\)002). A.L. thanks the OU Supercomputing Center for Education & Research (OSCER) at the University of Oklahoma, for allocated computing time.

Author information

Authors and Affiliations

  1. Dipartimento di Chimica, Biologia e Biotecnologie, Università di Perugia, Perugia, Italy

    Lombardi Andrea & Noelia Faginas-Lago

Authors
  1. Lombardi Andrea
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Noelia Faginas-Lago
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lombardi Andrea .

Editor information

Editors and Affiliations

  1. University of Perugia, Perugia, Italy

    Osvaldo Gervasi

  2. University of Basilicata, Potenza, Potenza, Italy

    Beniamino Murgante

  3. Chair- Center of ICT/ICE, Covenant University, Ota, Nigeria

    Sanjay Misra

  4. University of Cagliari, Cagliari, Italy

    Chiara Garau

  5. University of Cagliari, Cagliari, Italy

    Ivan Blečić

  6. Clayton School of Information Technology, Monash University, Clayton, VIC, Australia

    David Taniar

  7. Department of Information Science, Kyushu Sangyo University, Fukuoka, Japan

    Bernady O. Apduhan

  8. University of Minho, Braga, Portugal

    Ana Maria A. C. Rocha

  9. Polytechnic University of Bari, Bari, Italy

    Eufemia Tarantino

  10. Polytechnic University of Bari, Bari, Italy

    Carmelo Maria Torre

  11. Department of Neurology, University of Massachusetts Medical School, Worcester, MA, USA

    Yeliz Karaca

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Andrea, L., Faginas-Lago, N. (2020). Classification of Shapes and Deformations of Large Systems by Invariant Coordinates. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2020. ICCSA 2020. Lecture Notes in Computer Science(), vol 12255. Springer, Cham. https://doi.org/10.1007/978-3-030-58820-5_40

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-58820-5_40

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-58819-9

  • Online ISBN: 978-3-030-58820-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation