Skip to main content
SpringerLink
Log in

The Invariance Approach to Structure and Dynamics: Classical Hyperspherical Coordinates

  • Conference paper
  • First Online:
Computational Science and Its Applications – ICCSA 2019 (ICCSA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11624))

Included in the following conference series:

Abstract

The hyperspherical coordinate systems have been extensively adopted in the study of few-body scattering problems of nuclear and molecular physics, for example in the practical implementation of demanding quantum calculations typical of chemical reactions. Hyperangular momenta are the dynamical quantities used in this representation and the hyperspherical harmonics are the corresponding basis functions. The use of such formalism is limited to the treatment of three- or four-center problems, due to the exceedingly high computational cost of quantum dynamics calculations. To circumvent this restriction, an hyperspherical formulation has been developed in a series of works during the last decades, suitable for the simulation of cluster and large molecular system dynamics. Such a hyperspherical formulation is based on classical definitions of the hyperangular momenta and on the search of invariant dynamical quantities. Exploiting invariance of hyperspherical shape coordinates with respect to rotations and kinematic rotations, we consider the use of the classical hyperspherical representation as a tool for the static analysis of geometry and minimum energy structures of atomic and molecular structures, showing some applications to relatively simple systems.

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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. Comp. Phys. Comm. 133, 128–135 (2000)

    Google Scholar 

  3. Lepetit, B., Launay, J.M.: Quantum mechanical study of the reaction He+\({\rm {H}}^{+}\)\(_{2}\rightarrow \)HeH\(^{+}\)+H with hyperspherical coordinates. J. Chem. Phys. 97, 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. Theoret. 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. Laganà, A., Crocchianti, S., Faginas Lago, N., Pacifici, L., Ferraro, G.: A nonorthogonal coordinate approach to atom-diatom parallel reactive scattering calculations. Collect. Czech. Chem. Commun. 68, 307–330 (2003)

    Google Scholar 

  11. Lago, N.F., Laganà, A., Garcia, E., Gimenez, X.: Thermal rate coefficients for the N + N2 reaction: quasiclassical, semiclassical and quantum calculations. In: Gervasi, O., et al. (eds.) ICCSA 2005. LNCS, vol. 3480, pp. 1083–1092. Springer, Heidelberg (2005). https://doi.org/10.1007/11424758_113

    Chapter  Google Scholar 

  12. Faginas-Lago, N., Laganá, A.: A comparison of semiclassical IVR and exact quantum collinear atom diatom transition probabilities for mixed reactive and non reactive regimes. In: AIP Conference Proceedings, vol. 762, p. 920 (2005)

    Google Scholar 

  13. Faginas-Lago, N., Laganà, A.: On the semiclassical initial value calculation of thermal rate coefficients for the N + N\(_2\) reaction. J. Chem. Phys. 125, 114311 (2006)

    Google Scholar 

  14. Faginas-Lago, N., Costantini, A., Huarte-Larrañaga, F.: Direct calculation of the rate coefficients on the grid: exact quantum versus semiclassical results for N+ N\(_2\). Int. J. Quantum Chem. 110(2), 422–431 (2010)

    Google Scholar 

  15. Faginas, N., Huarte-Larranaga, F., Laganà, A.: Full dimensional quantum versus semiclassical reactivity for the bent transition state reaction N + N\(_2\). Chem. Phys. Lett. 464, 249–255 (2008)

    Google Scholar 

  16. Rampino, S., Faginas-Lago, N., Laganà, A., Huarte-Larrañaga, F.: An extension of the grid empowered molecular simulator to quantum reactive scattering. J. Comput. Chem. 33, 708–714 (2012)

    Google Scholar 

  17. Laganà, A., Faginas-Lago, N., Rampino, S., Huarte-Larrañaga, F., García, E.: Thermal rate coefficients in collinear versus bent transition state reactions: the N + N\(_2\) case study. Phys. Scr. 78(5), 058116 (2008)

    Google Scholar 

  18. Lombardi, A., Lago, N.F., 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.) ICCSA 2012. LNCS, vol. 7333, pp. 387–400. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31125-3_30

    Chapter  Google Scholar 

  19. Lombardi, A., Laganà, A., Pirani, F., Palazzetti, F., Lago, N.F.: Carbon oxides in gas flows and earth and planetary atmospheres: state-to-state simulations of energy transfer and dissociation reactions. In: Murgante, B., et al. (eds.) ICCSA 2013. LNCS, vol. 7972, pp. 17–31. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39643-4_2

    Chapter  Google Scholar 

  20. 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 

  21. 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 

  22. 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 

  23. 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 

  24. 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 

  25. 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 

  26. 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 

  27. 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 

  28. 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 

  29. Bartolomei, M., Pirani, F., Laganà, A., Lombardi, A.: J. Comput. Chem. 33, 1806 (2012)

    Google Scholar 

  30. 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 

  31. Albertí, M., Faginas-Lago, N., Pirani, F.: J. Phys. Chem. A 115(40), 10871–10879 (2011)

    Google Scholar 

  32. Albertí, M., Faginas-Lago, N.: Eur. Phys. J. D 67, 73 (2013)

    Google Scholar 

  33. Albertí, M., Faginas-Lago, N., Pirani, F.: Chem. Phys. 399, 232 (2012)

    Google Scholar 

  34. Falcinelli, S., et al.: Modeling the intermolecular interactions and characterization of the dynamics of collisional autoionization processes. In: Murgante, B., et al. (eds.) ICCSA 2013. LNCS, vol. 7971, pp. 69–83. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39637-3_6

    Chapter  Google Scholar 

  35. 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 

  36. 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. Comput. Chem. 37, 1463–1475 (2016)

    Google Scholar 

  37. Pacifici, L., Verdicchio, M., Faginas-Lago, N., Lombardi, A., Costantini, A.: A high-level ab initio study of the N\(_2\) + N\(_2\) reaction channel. J. Comput. Chem. 34(31), 2668–2676 (2013)

    Google Scholar 

  38. 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 

  39. 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, 11430–11440 (2013)

    Google Scholar 

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

    Google Scholar 

  41. 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 

  42. 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 

  43. Lombardi, A., Palazzetti, F.: A comparison of interatomic potentials for rare gas nanoaggregates. J. Mol. Struct. (Thoechem) 852(1–3), 22–29 (2008)

    Google Scholar 

  44. 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 

  45. 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. Theoret. Chem. 990, 53–61 (2012)

    Google Scholar 

  46. 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 

  47. 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 

  48. 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 

  49. 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 

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

    Google Scholar 

  51. 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. Theoret. Chem. Acc. 117(5–6), 709–721 (2007)

    Google Scholar 

  52. 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 

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

    Google Scholar 

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

    Google Scholar 

  55. 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 

  56. 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 

  57. 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. Mater. Sci. 35, 187–191 (2006)

    Google Scholar 

  58. 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 

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

    Google Scholar 

  60. 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 

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

    Google Scholar 

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

    Google Scholar 

  63. 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 

  64. Wales, D.J., Doye, J.P.K., Dullweber, A., Hodges, M.P., Naumkin, F.Y., Calvo, F., Hernández-Rojas, J., Middleton, T.F.: The Cambridge Cluster Database. http://www-wales.ch.cam.ac.uk/CCD.html

  65. Doye, J.P.K., Wales, D.J.: Global minima for transition metal clusters described by Sutton-Chen potentials. New J. Chem. 22, 733–744 (1998)

    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)”. They also acknowledge the Italian Ministry for Education, University and Research, MIUR, for financial supporting through SIR 2014 “Scientific Independence for young Researchers” (RBSI14U3VF). 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

    Andrea Lombardi, Noelia Faginas-Lago & Vincenzo Aquilanti

  2. Consortium for Computational Molecular and Materials Sciences (CMS)², Via Elce di Sotto, 8, 06123, Perugia, Italy

    Andrea Lombardi & Noelia Faginas-Lago

Authors
  1. Andrea Lombardi
    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

  3. Vincenzo Aquilanti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrea Lombardi .

Editor information

Editors and Affiliations

  1. Covenant University, Ota, Nigeria

    Sanjay Misra

  2. University of Perugia, Perugia, Italy

    Osvaldo Gervasi

  3. University of Basilicata, Potenza, Italy

    Beniamino Murgante

  4. Saint Petersburg State University, Saint Petersburg, Russia

    Elena Stankova

  5. Saint Petersburg State University, Saint Petersburg, Russia

    Vladimir Korkhov

  6. Polytechnic University of Bari, Bari, Italy

    Carmelo Torre

  7. University of Minho, Braga, Portugal

    Ana Maria A.C. Rocha

  8. Monash University, Clayton, VIC, Australia

    David Taniar

  9. Kyushu Sangyo University, Fukuoka, Japan

    Bernady O. Apduhan

  10. Polytechnic University of Bari, Bari, Italy

    Eufemia Tarantino

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Lombardi, A., Faginas-Lago, N., Aquilanti, V. (2019). The Invariance Approach to Structure and Dynamics: Classical Hyperspherical Coordinates. In: Misra, S., et al. Computational Science and Its Applications – ICCSA 2019. ICCSA 2019. Lecture Notes in Computer Science(), vol 11624. Springer, Cham. https://doi.org/10.1007/978-3-030-24311-1_31

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-24311-1_31

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-24310-4

  • Online ISBN: 978-3-030-24311-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation