On spectra of spectra

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 $${\varvec{X}}$$ .