Journal of Mathematical Chemistry

, Volume 54, Issue 3, pp 806–822 | Cite as

On spectra of spectra

  • Péter Árendás
  • Tibor Furtenbacher
  • Attila G. Császár
Original Paper

Abstract

Spectroscopic networks (SNs), where the vertices are discrete, rovibronic energy levels and the edges are transitions among the levels allowed by quantum mechanics, serve as useful models helping to understand high-resolution spectra of molecular systems. The experimental SNs of the \(^{12}\hbox {C}_{2}, \hbox {H}_{3}^{+}\), \(\hbox {H}_{2}\hbox {D}^{+}\), \(\hbox {HD}_{2}^{+}\), \(^{14}\hbox {NH}_{3}\), and \(\hbox {H}_{2}{}^{16}\hbox {O}\) molecules, containing a single copy of the known measured and assigned transitions, are investigated via the corresponding network representation matrices, including the Ritz matrix \({\varvec{X}}\), the adjacency matrix \({\varvec{A}}\), the combinatorial Laplacian matrix \({\varvec{L}}^{\mathrm{C}}\), and the normalized Laplacian matrix \({\varvec{L}}^{\mathrm{N}}\). Using elements of graph (network) theory and the eigenvalue spectra of the matrices mentioned, several interesting results relevant for high-resolution molecular spectroscopy are revealed about the structure of the investigated SNs. For example, as long as the parity selection rule of molecular rovibrational transitions is not violated, the experimental SNs investigated not only contain, with the exception of \(^{12}\hbox {C}_{2}\), two principal components but they are all bipartite networks, as proven by the symmetry of the eigenvalue spectrum of \({\varvec{A}}\) about the origin. Furthermore, the PageRank ordering system is introduced to molecular spectroscopy to identify the most important vertices of SNs. The rankings provided by the degree of the levels and by PageRank may differ significantly; it appears that PageRank provides the more useful ranking. The connectors of relatively dense clusters of the SNs are identified and analysed via spectral clustering techniques based on \({\varvec{L}}^{\mathrm{C}}\) and \({\varvec{L}}^{\mathrm{N}}\). The identification of connectors becomes especially important when judging the true accuracy of the experimental rovibrational energy levels obtained through the Measured Active Rotational-Vibrational Energy Levels (MARVEL) approach, built with the help of the Ritz matrix \({\varvec{X}}\).

Keywords

Spectroscopic network Matrix representations Eigenvalues Bipartiteness PageRank Spectral clustering 

References

  1. 1.
    A.G. Császár, G. Czakó, T. Furtenbacher, E. Mátyus, Annu. Rep. Comp. Chem. 3, 155 (2007)CrossRefGoogle Scholar
  2. 2.
    T. Furtenbacher, A.G. Császár, J. Tennyson, J. Mol. Spectrosc. 245, 115 (2007)CrossRefGoogle Scholar
  3. 3.
    T. Furtenbacher, A.G. Császár, J. Quant. Spectrosc. Rad. Transfer 113, 929 (2012)CrossRefGoogle Scholar
  4. 4.
    A.G. Császár, T. Furtenbacher, J. Mol. Spectrosc. 266, 99 (2011)CrossRefGoogle Scholar
  5. 5.
    T. Furtenbacher, A.G. Császár, J. Mol. Struct. 1009, 123 (2012)CrossRefGoogle Scholar
  6. 6.
    T. Furtenbacher, P. Árendás, G. Mellau, A.G. Császár, Sci. Rep. 4, 4654 (2014)CrossRefGoogle Scholar
  7. 7.
    J. Tennyson, P.F. Bernath, L.R. Brown, A. Campargue, M.R. Carleer, A.G. Császár, L. Daumont, R.R. Gamache, J.T. Hodges, A. Jenouvrier, O.V. Naumenko, O.L. Polyansky, L.S. Rothman, R.A. Toth, A.C. Vandaele, N.F. Zobov, A.Z. Fazliev, T. Furtenbacher, I.F. Gordon, S.-M. Hu, S.N. Mikhailenko, B. Voronin, J. Quant. Spectrosc. Rad. Transf. 111, 2160 (2010)CrossRefGoogle Scholar
  8. 8.
    J. Tennyson, P.F. Bernath, L.R. Brown, A. Campargue, M.R. Carleer, A.G. Császár, R.R. Gamache, J.T. Hodges, A. Jenouvrier, O.V. Naumenko, O.L. Polyansky, L.S. Rothman, R.A. Toth, A.C. Vandaele, N.F. Zobov, L. Daumont, A.Z. Fazliev, T. Furtenbacher, I.F. Gordon, S.N. Mikhailenko, S.V. Shirin, J. Quant. Spectrosc. Rad. Transf. 110, 573 (2009)CrossRefGoogle Scholar
  9. 9.
    J. Tennyson, P.F. Bernath, L.R. Brown, A. Campargue, A.G. Császár, L. Daumont, R.R. Gamache, J.T. Hodges, O.V. Naumenko, O.L. Polyansky, L.S. Rothman, A.C. Vandaele, N.F. Zobov, A.R. Al Derzi, C. Fábri, A.Z. Fazliev, T. Furtenbacher, I.E. Gordon, L. Lodi, I.I. Mizus, J. Quant. Spectrosc. Rad. Transf. 117, 29 (2013)Google Scholar
  10. 10.
    J. Tennyson, P.F. Bernath, L.R. Brown, A. Campargue, A.G. Császár, L. Daumont, R.R. Gamache, J.T. Hodges, O.V. Naumenko, O.L. Polyansky, L.S. Rothman, A.C. Vandaele, N.F. Zobov, N. Dénes, A.Z. Fazliev, T. Furtenbacher, I.E. Gordon, S.-M. Hu, T. Szidarovszky, I.A. Vasilenko, J. Quant. Spectrosc. Rad. Transf. 142, 93 (2014)CrossRefGoogle Scholar
  11. 11.
    J. Tennyson, P.F. Bernath, L.R. Brown, A. Campargue, A.G. Császár, L. Daumont, R.R. Gamache, J.T. Hodges, O.V. Naumenko, O.L. Polyansky, L.S. Rothman, A.C. Vandaele, N.F. Zobov, Pure Appl. Chem. 86, 71 (2014)Google Scholar
  12. 12.
    L.S. Rothman, J. Quant. Spectrosc. Rad. Transf. 111, 1565 (2010)CrossRefGoogle Scholar
  13. 13.
    L.S. Rothman, I.E. Gordon, A. Barbe, D. C. Benner, P.F. Bernath, M. Birk, V. Boudon, L.R. Brown, A. Campargue, J.-P. Champion, K. Chance, L.H. Coudert, V. Dana, V.M. Devi, S. Fally, J.-M. Flaud, R.R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W.J. Lafferty, J.-Y. Mandin, S.T. Massie, S.N. Mikhailenko, C.E. Miller, N. Moazzen-Ahmadi, O.V. Naumenko, A.V. Nikitin, J. Orphal, V.I. Perevalov, A. Perrin, A. Predoi-Cross, C.P. Rinsland, M. Rotger, M. Simeckova, M.A.H. Smith, K. Sung, S.A. Tashkun, J. Tennyson, R.A. Toth, A.C. Vandaele, J. Vander Auwera, J. Quant. Spectrosc. Rad. Transf. 110, 533 (2009)Google Scholar
  14. 14.
    L.S. Rothman, I.E. Gordon, R.J. Barber, H. Dothe, R.R. Gamache, A. Goldman, V.I. Perevalov, S.A. Tashkun, J. Tennyson, J. Quant. Spectrosc. Rad. Transf. 111, 2139 (2010)Google Scholar
  15. 15.
    N. Jacquinet-Husson, N.A. Scott, A. Chédin, K. Garceran, R. Armante, A.A. Chursin, A. Barbe, M. Birk, L.R. Brown, C. Camy-Peyret, C. Claveau, C. Clerbaux, P.F. Coheur, V. Dana, L. Daumont, M.R. Debacker-Barilly, J.M. Flaud, A. Goldman, A. Hamdouni, M. Hess, A. Nikitin, D. Newnham, A. Perrin, V.I. Perevalov, L. Régalia-Jarlot, A. Rublev, F. Schreier, I. Schult, K.M. Smith, S.A. Tashkun, J.L. Teffo, R.A. Toth, Vl.G. Tyuterev, J. Vander Auwera, P. Varanasi, G. Wagner, J. Quant. Spectrosc. Rad. Transf. 95, 429 (2005)Google Scholar
  16. 16.
    K.P. Dere, E. Landi, P.R. Young, G. Del Zanna, M. Landini, H.E. Mason, Astron. Astrophys. 498, 915 (2009)CrossRefGoogle Scholar
  17. 17.
    H.S.P. Müller, F. Schlöder, J. Stutzki, G. Winnewisser, J. Mol. Struct. 742, 215 (2005)CrossRefGoogle Scholar
  18. 18.
    H.S.P. Müller, S. Thorwirth, D.A. Roth, G. Winnewisser, Astron. Astrophys. 370, L49 (2001)CrossRefGoogle Scholar
  19. 19.
    H.M. Pickett, R.L. Poynter, E.A. Cohen, M.L. Delitsky, J.C. Pearson, H.S.P. Müller, J. Quant. Spectrosc. Rad. Transf. 60, 883 (1998)CrossRefGoogle Scholar
  20. 20.
    N. Jacquinet-Husson, L. Crepeau, R. Armante, C. Boutammine, A. Chédin, N.A. Scott, C. Crevoisier, V. Capelle, C. Boone, N. Poulet-Crovisier, A. Barbe, A. Campargue, D. Chris Benner, Y. Benilan, B. Bézard, V. Boudon, L.R. Brown, L.H. Coudert, A. Coustenis, V. Dana, V.M. Devi, S. Fally, A. Fayt, J.-M. Flaud, A. Goldman, M. Herman, G.J. Harris, D. Jacquemart, A. Jolly, I. Kleiner, A. Kleinböhl, F. Kwabia-Tchana, N. Lavrentieva, N. Lacome, L.-H. Xu, O.M. Lyulin, J.-Y. Mandin, A. Maki, S. Mikhailenko, C.E. Miller, T. Mishina, N. Moazzen-Ahmadi, H.S.P. Müller, A. Nikitin, J. Orphal, V. Perevalov, A. Perrin, D.T. Petkie, A. Predoi-Cross, C.P. Rinsland, J.J. Remedios, M. Rotger, M.A.H. Smith, K. Sung, S. Tashkun, J. Tennyson, R.A. Toth, A.-C. Vandaele, J. Vander Auwera, J. Quant. Spectrosc. Rad. Transf. 112, 2395 (2011)Google Scholar
  21. 21.
    M.L. Dubernet, V. Boudon, J.L. Culhane, M.S. Dimitrijevic, A.Z. Fazliev, C. Joblin, F. Kupka, G. Leto, P. Le Sidaner, P.A. Loboda, H.E. Mason, N.J. Mason, C. Mendoza, G. Mulas, T.J. Millar, L.A. Nuñez, V.I. Perevalov, N. Piskunov, Y. Ralchenko, L.S. Rothman, E. Roueff, T.A. Ryabchikova, A. Ryabtsev, S. Sahal-Bréchot, B. Schmitt, S. Schlemmer, J. Tennyson, V.G. Tyuterev, N.A. Walton, V. Wakelam, C. Zeippen, J. Quant. Spectr. Rad. Transf. 111, 2265 (2010)CrossRefGoogle Scholar
  22. 22.
    S.A. Tashkun, V.I. Perevalov, J.-L. Teffo, A.D. Bykov, N.N. Lavrentieva, J. Quant. Spectrosc. Rad. Transf. 82, 165 (2003)CrossRefGoogle Scholar
  23. 23.
    O.L. Polyansky, A.G. Császár, S.V. Shirin, N.F. Zobov, P. Barletta, J. Tennyson, D.W. Schwenke, P.J. Knowles, Science 299, 539 (2003)CrossRefGoogle Scholar
  24. 24.
    R.J. Barber, J. Tennyson, G.J. Harris, R.N. Tolchenov, Mon. Not. R. Astron. Soc. 368, 1087 (2006)CrossRefGoogle Scholar
  25. 25.
    T. Sochi, J. Tennyson, Mon. Not. R. Astron. Soc. 405, 2345 (2010)Google Scholar
  26. 26.
    S.V. Shirin, N.F. Zobov, R.I. Ovsyannikov, O.L. Polyansky, J. Tennyson, J. Chem. Phys. 128, 224306 (2008)CrossRefGoogle Scholar
  27. 27.
    B.A. Voronin, J. Tennyson, R.N. Tolchenov, A.A. Lugovskoy, S.N. Yurchenko, Mon. Not. R. Astron. Soc. 402, 492 (2010)CrossRefGoogle Scholar
  28. 28.
    S.N. Yurchenko, R.J. Barber, A. Yachmenev, W. Thiel, P. Jensen, J. Tennyson, J. Phys. Chem. A 113, 11845 (2009)CrossRefGoogle Scholar
  29. 29.
    A.-L. Barabási, Linked (Penguin, 2003)Google Scholar
  30. 30.
    M.E.J. Newman, Networks (Oxford University Press, Oxford, 2010)CrossRefGoogle Scholar
  31. 31.
    R. Diestel, Graph Theory, 3rd edn. (Springer, Berlin, 2005)Google Scholar
  32. 32.
    F. Harary, Graph Theory (Addison-Wesley, Reading, 1969)Google Scholar
  33. 33.
    R.J. Wilson, Introduction to Graph Theory, 3rd edn. (Longman, Harlow, 1985)Google Scholar
  34. 34.
    T. Furtenbacher, I. Szabó, A.G. Császár, P.F. Bernath, S.N. Yurchenko, J. Tennyson, (manuscript in preparation)Google Scholar
  35. 35.
    T. Furtenbacher, T. Szidarovszky, C. Fábri, A.G. Császár, Phys. Chem. Chem. Phys. 15, 10181 (2013)CrossRefGoogle Scholar
  36. 36.
    T. Furtenbacher, T. Szidarovszky, E. Mátyus, C. Fábri, A.G. Császár, J. Chem. Theory Comput. 9, 5471 (2013)CrossRefGoogle Scholar
  37. 37.
    C. Fábri, E. Mátyus, T. Furtenbacher, B. Mihály, T. Zoltáni, L. Nemes, A.G. Császár, J. Chem. Phys. 135, 094307 (2011)CrossRefGoogle Scholar
  38. 38.
    A.R. Al Derzi, T. Furtenbacher, J. Tennyson, S.N. Yurchenko, A.G. Császár, J. Quant. Spectrosc. Rad. Transf. 116, 117 (2015)CrossRefGoogle Scholar
  39. 39.
    A.G. Császár, T. Furtenbacher, Phys. Chem. Chem. Phys. 18, 1092 (2016)Google Scholar
  40. 40.
    S.N. Yurchenko, R.J. Barber, J. Tennyson, Mon. Not. R. Astron. Soc. 413, 1828 (2011)CrossRefGoogle Scholar
  41. 41.
    J.-M. Flaud, C. Camy-Peyret, J.P. Maillard, Mol. Phys. 32, 499 (1976)CrossRefGoogle Scholar
  42. 42.
    S.A. Tashkun, V.I. Perevalov, J.L. Teffo, A.D. Bykov, N.N. Lavrentieva, J. Quant. Spectrosc. Rad. Transf. 82, 165 (2003)CrossRefGoogle Scholar
  43. 43.
    F. Chung, Spectral Graph Theory (CBMS Lecture Notes, AMS, Providence, RI, 1997)Google Scholar
  44. 44.
    M. Bolla, Spectral Clustering and Biclustering: Learning Large Graphs and Contingency Tables (Wiley, New York, 2013)CrossRefGoogle Scholar
  45. 45.
    C. Lanczos, J. Res. Natl. Bur. Stand. 45, 255 (1950)CrossRefGoogle Scholar
  46. 46.
    J.K. Cullum, R.A. Willoughby, Lanczos Algorithms for Large Symmetric Eigenvalue Computations (Birkhauser, Boston, 1985)Google Scholar
  47. 47.
    F. Chung, L. Lu, V. Vu, Ann. Comb. 7, 21 (2003)CrossRefGoogle Scholar
  48. 48.
    M. Bolla, Lin. Alg. Appl. 402, 228 (2005)CrossRefGoogle Scholar
  49. 49.
    T.F. Móri, Combin. Probab. Comput. 14, 339 (2005)CrossRefGoogle Scholar
  50. 50.
    R.A. Brualdi, The mutually beneficial relationship of graphs and matrices, volume 115 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI (2011)Google Scholar
  51. 51.
    G. Kirchhoff, Ann. Phys. Chem. 72, 497 (1847)CrossRefGoogle Scholar
  52. 52.
    S. Chaiken, J. SIAM, Alg. Disc. Methods 3, 319 (1982)CrossRefGoogle Scholar
  53. 53.
    L. Page, S. Brin, R. Motwani, T. Winograd, The PageRank Citation Ranking: Bringing Order to the Web. Technical Report, Stanford InfoLab (1999)Google Scholar
  54. 54.
    A. Clauset, M.E.J. Newman, C. Moore, Phys. Rev. E 70, 066111 (2004)CrossRefGoogle Scholar
  55. 55.
    S. Fortunato, Phys. Rep. 486, 75 (2010)CrossRefGoogle Scholar
  56. 56.
    M. Bolla, Phys. Rev. E 84, 016108 (2011)CrossRefGoogle Scholar
  57. 57.
    http://snap.stanford.edu/snap/, Last accessed on October 1, (2015)

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Péter Árendás
    • 1
    • 2
  • Tibor Furtenbacher
    • 1
    • 3
  • Attila G. Császár
    • 1
    • 3
  1. 1.MTA-ELTE Complex Chemical Systems Research GroupBudapestHungary
  2. 2.Department of Algebra and Number Theory, Institute of MathematicsEötvös UniversityBudapest 112Hungary
  3. 3.Laboratory of Molecular Structure and Dynamics, Institute of ChemistryEötvös UniversityBudapest 112Hungary

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