{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T10:47:00Z","timestamp":1770288420688,"version":"3.49.0"},"reference-count":53,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,3,26]],"date-time":"2021-03-26T00:00:00Z","timestamp":1616716800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Linear molecules usually represent a special case in rotational-vibrational calculations due to a singularity of the kinetic energy operator that arises from the rotation about the a (the principal axis of least moment of inertia, becoming the molecular axis at the linear equilibrium geometry) being undefined. Assuming the standard ro-vibrational basis functions, in the 3N\u22126 approach, of the form \u2223\u03bd1,\u03bd2,\u03bd3\u21133;J,k,m\u27e9, tackling the unique difficulties of linear molecules involves constraining the vibrational and rotational functions with k=\u21133, which are the projections, in units of \u210f, of the corresponding angular momenta onto the molecular axis. These basis functions are assigned to irreducible representations (irreps) of the C2v(M) molecular symmetry group. This, in turn, necessitates purpose-built codes that specifically deal with linear molecules. In the present work, we describe an alternative scheme and introduce an (artificial) group that ensures that the condition \u21133=k is automatically applied solely through symmetry group algebra. The advantage of such an approach is that the application of symmetry group algebra in ro-vibrational calculations is ubiquitous, and so this method can be used to enable ro-vibrational calculations of linear molecules in polyatomic codes with fairly minimal modifications. To this end, we construct a\u2014formally infinite\u2014artificial molecular symmetry group D\u221eh(AEM), which consists of one-dimensional (non-degenerate) irreducible representations and use it to classify vibrational and rotational basis functions according to \u2113 and k. This extension to non-rigorous, artificial symmetry groups is based on cyclic groups of prime-order. Opposite to the usual scenario, where the form of symmetry adapted basis sets is dictated by the symmetry group the molecule belongs to, here the symmetry group D\u221eh(AEM) is built to satisfy properties for the convenience of the basis set construction and matrix elements calculations. We believe that the idea of purpose-built artificial symmetry groups can be useful in other applications.<\/jats:p>","DOI":"10.3390\/sym13040548","type":"journal-article","created":{"date-parts":[[2021,3,26]],"date-time":"2021-03-26T13:17:53Z","timestamp":1616764673000},"page":"548","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Artificial Symmetries for Calculating Vibrational Energies of Linear Molecules"],"prefix":"10.3390","volume":"13","author":[{"given":"Thomas M.","family":"Mellor","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, University College London, London WC1E 6BT, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9286-9501","authenticated-orcid":false,"given":"Sergei N.","family":"Yurchenko","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, London WC1E 6BT, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5133-7621","authenticated-orcid":false,"given":"Per","family":"Jensen","sequence":"additional","affiliation":[{"name":"Physikalische und Theoretische Chemie, Fakult\u00e4t f\u00fcr Mathematik und Naturwissenschaften, Bergische Universit\u00e4t Wuppertal, D-42097 Wuppertal, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1016\/S0009-2614(00)00407-3","article-title":"The barrier to linearity of hydrogen sulphide","volume":"322","author":"Tarczay","year":"2000","journal-title":"Chem. Phys. Lett."},{"key":"ref_2","unstructured":"Bunker, P.R., and Jensen, P. (1998). Molecular Symmetry and Spectroscopy, NRC Research Press. [2nd ed.]."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/j.jms.2017.06.012","article-title":"Bending Wavefunctions for Linear Molecules","volume":"343","author":"Hirano","year":"2018","journal-title":"J. Mol. Spectrosc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"943","DOI":"10.1080\/00268979300101741","article-title":"Vibration-rotation hamiltonians of linear-molecules","volume":"79","author":"Watson","year":"1993","journal-title":"Mol. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/0022-2852(82)90231-4","article-title":"On the calculation of vibration-rotation energy levels of quasi-linear molecules","volume":"95","author":"Carter","year":"1982","journal-title":"J. Mol. Spectrosc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"084102","DOI":"10.1063\/1.2756518","article-title":"Vibrational energy levels with arbitrary potentials using the Eckart-Watson Hamiltonians and the discrete variable representation","volume":"127","author":"Sutcliffe","year":"2007","journal-title":"J. Chem. Phys."},{"key":"ref_7","first-page":"533","article-title":"MULTIMODE: A code to calculate rovibrational energies of polyatomic molecules","volume":"22","author":"Bowman","year":"2003","journal-title":"Intern. J. Quantum Chem."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"044307","DOI":"10.1063\/1.4940780","article-title":"High-level theoretical rovibrational spectroscopy beyond fc-CCSD(T): The C3 molecule","volume":"144","author":"Schroeder","year":"2016","journal-title":"J. Chem. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1016\/0022-2852(69)90007-1","article-title":"The symmetry groups of linear molecules","volume":"32","author":"Bunker","year":"1969","journal-title":"J. Mol. Spectrosc."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Chubb, K.L., Jensen, P., and Yurchenko, S.N. (2018). Symmetry Adaptation of the Rotation-Vibration Theory for Linear Molecules. Symmetry, 10.","DOI":"10.3390\/sym10050137"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/0022-2852(70)90080-9","article-title":"Vibration-rotation problem in triatomic molecules allowing for a large-amplitude bending vibration","volume":"34","author":"Hougen","year":"1970","journal-title":"J. Mol. Spectrosc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1445","DOI":"10.1080\/00268978200101082","article-title":"A variational method for the calculation of vibrational levels of any triatomic molecule","volume":"47","author":"Carter","year":"1982","journal-title":"Mol. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0167-7977(83)90003-5","article-title":"The nonrigid bender Hamiltonian for calculating the rotation-vibration energy levels of a triatomic molecule","volume":"1","author":"Jensen","year":"1983","journal-title":"Comp. Phys. Rep."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1080\/00268978600102261","article-title":"The vibration-rotation hamiltonian\u2014A unified treatment of linear and nonlinear molecules","volume":"59","author":"Estes","year":"1986","journal-title":"Mol. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1053","DOI":"10.1080\/00268978600101801","article-title":"A generalised approach to the calculation of ro-vibrational spectra of triatomic molecules","volume":"58","author":"Sutcliffe","year":"1986","journal-title":"Mol. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1063\/1.1750455","article-title":"The kinetic energy of polyatomic molecules","volume":"7","author":"Sayvetz","year":"1939","journal-title":"J. Chem. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1063\/1.1732544","article-title":"Rotational energy levels of a linear triatomic molecule in a 2II electronic state","volume":"36","author":"Hougen","year":"1962","journal-title":"J. Chem. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1080\/00268977100100151","article-title":"The molecular hamiltonian","volume":"20","author":"Howard","year":"1971","journal-title":"Mol. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"154106","DOI":"10.1063\/5.0019546","article-title":"Treating linear molecules in calculations of rotation-vibration spectra","volume":"153","author":"Yurchenko","year":"2020","journal-title":"J. Chem. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/j.jms.2007.07.009","article-title":"Theoretical ROVibrational Energies (TROVE): A robust numerical approach to the calculation of rovibrational energies for polyatomic molecules","volume":"245","author":"Yurchenko","year":"2007","journal-title":"J. Mol. Spectrosc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"014105","DOI":"10.1063\/1.4923039","article-title":"Automatic differentiation method for numerical construction of the rotational-vibrational Hamiltonian as a power series in the curvilinear internal coordinates using the Eckart frame","volume":"143","author":"Yachmenev","year":"2015","journal-title":"J. Chem. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"11845","DOI":"10.1021\/jp9029425","article-title":"A Variationally Computed T = 300 K Line List for NH3","volume":"113","author":"Yurchenko","year":"2009","journal-title":"J. Phys. Chem. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1063\/1.3599927","article-title":"A new \u201cspectroscopic\u201d potential energy surface for formaldehyde in its ground electronic state","volume":"134","author":"Yachmenev","year":"2011","journal-title":"J. Chem. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/j.jqsrt.2014.03.012","article-title":"High temperature partition functions and thermodynamic data for ammonia and phosphine","volume":"142","author":"Hesketh","year":"2014","journal-title":"J. Quant. Spectrosc. Radiat. Transf."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2337","DOI":"10.1093\/mnras\/stu2246","article-title":"ExoMol line lists\u2014VII. The rotation-vibration spectrum of phosphine up to 1500 K","volume":"446","author":"Tennyson","year":"2015","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Underwood, D.S., Yurchenko, S.N., Tennyson, J., and Jensen, P. (2014). Rotational spectrum of SO3 and theoretical evidence for the formation of sixfold rotational energy-level clusters in its vibrational ground state. J. Chem. Phys., 140.","DOI":"10.1063\/1.4882865"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1704","DOI":"10.1093\/mnras\/stv091","article-title":"ExoMol line lists\u2014VIII. A variationally computed line list for hot formaldehyde","volume":"448","author":"Yachmenev","year":"2015","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1649","DOI":"10.1093\/mnras\/stu326","article-title":"ExoMol line lists\u2014IV. The rotation-vibration spectrum of methane up to 1500 K","volume":"440","author":"Yurchenko","year":"2014","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/j.jms.2015.10.004","article-title":"A variationally calculated room temperature line-list for H2O2","volume":"318","author":"Ovsyannikov","year":"2015","journal-title":"J. Mol. Spectrosc."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Owens, A., Yurchenko, S.N., Yachmenev, A., Tennyson, J., and Thiel, W. (2015). Accurate ab initio vibrational energies of methyl chloride. J. Chem. Phys., 142.","DOI":"10.1063\/1.4922890"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"244317","DOI":"10.1063\/1.4938563","article-title":"A global potential energy surface and dipole moment surface for silane","volume":"143","author":"Owens","year":"2015","journal-title":"J. Chem. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"244306","DOI":"10.1063\/1.4938253","article-title":"Ro-vibrational averaging of the isotropic hyperfine coupling constant for the methyl radical","volume":"143","author":"Adam","year":"2015","journal-title":"J. Chem. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3191","DOI":"10.1093\/mnras\/stv869","article-title":"Accurate prediction of the ammonia probes of a variable proton-to-electron mass ratio","volume":"450","author":"Owens","year":"2015","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1012","DOI":"10.1093\/mnras\/stw1295","article-title":"ExoMol line lists\u2014XV. A new hot line list for hydrogen peroxide","volume":"461","author":"Polyansky","year":"2016","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3890","DOI":"10.1093\/mnras\/stw849","article-title":"ExoMol molecular line lists\u2014XIV. The rotation-vibration spectrum of hot SO2","volume":"459","author":"Underwood","year":"2016","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1016\/j.jqsrt.2016.06.037","article-title":"A global ab initio dipole moment surface for methyl chloride","volume":"184","author":"Owens","year":"2016","journal-title":"J. Quant. Spectrosc. Radiat. Transf."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Owens, A., Yurchenko, S.N., Yachmenev, A., Tennyson, J., and Thiel, W. (2016). A highly accurate ab initio potential energy surface for methane. J. Chem. Phys., 145.","DOI":"10.1063\/1.4962261"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"3002","DOI":"10.1093\/mnras\/sty1542","article-title":"ExoMol line lists\u2014XXIX. The rotation-vibration spectrum of methyl chloride up to 1200 K","volume":"479","author":"Owens","year":"2018","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"3220","DOI":"10.1093\/mnras\/sty1239","article-title":"ExoMol molecular line lists\u2014XXVII. Spectra of C2H4","volume":"478","author":"Mant","year":"2018","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"3264","DOI":"10.1039\/C8CP07110A","article-title":"A variationally computed room temperature line list for AsH3","volume":"21","author":"Coles","year":"2019","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"e1581951","DOI":"10.1080\/00268976.2019.1581951","article-title":"The infrared spectrum of PF3 and analysis of rotational energy clustering effect","volume":"118","author":"Mant","year":"2019","journal-title":"Mol. Phys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1111\/j.1365-2966.2012.21440.x","article-title":"ExoMol: Molecular line lists for exoplanet and other atmospheres","volume":"425","author":"Tennyson","year":"2012","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1002\/qua.25190","article-title":"The ExoMol project: Software for computing large molecular line lists","volume":"117","author":"Tennyson","year":"2017","journal-title":"Int. J. Quantum Chem."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1016\/j.cpc.2017.01.013","article-title":"GPU Accelerated INtensities MPI GAIN-MPI: A new method of computing Einstein-A coefficients","volume":"214","author":"Yurchenko","year":"2017","journal-title":"Comput. Phys. Commun."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1927","DOI":"10.1093\/mnras\/staa1287","article-title":"ExoMol line lists\u2014XXXVIII. High-temperature molecular line list of silicon dioxide (SiO2)","volume":"495","author":"Owens","year":"2020","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"5282","DOI":"10.1093\/mnras\/staa1874","article-title":"ExoMol line lists\u2014XXXIX. Ro-vibrational molecular line list for CO2","volume":"496","author":"Yurchenko","year":"2020","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"12476","DOI":"10.1039\/D0CP01667E","article-title":"Detection of electric-quadrupole transitions in water vapour near 5.4 and 2.5 \u03bcm","volume":"22","author":"Campargue","year":"2020","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"4368","DOI":"10.1021\/acs.jctc.7b00506","article-title":"Symmetry adapted ro-vibrational basis functions for variational nuclear motion: TROVE approach","volume":"13","author":"Yurchenko","year":"2017","journal-title":"J. Chem. Theory Comput."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"592","DOI":"10.1093\/mnras\/84.8.592","article-title":"A method of extrapolation of perturbations","volume":"84","author":"Noumerov","year":"1924","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_50","first-page":"188","article-title":"M\u00e9thode nouvelle de la d\u00e9termination des orbites et le calcul des \u00e9ph\u00e9m\u00e9rides en tenant compte des perturbations","volume":"Volume 2","author":"Noumeroff","year":"1923","journal-title":"Trudy Glavnoi Rossiiskoi Astrofizicheskoj Observatorii"},{"key":"ref_51","first-page":"363","article-title":"An Improved eigenvalue corrector formula for solving the Schr\u00f6dinger equation for central fields","volume":"15","author":"Cooley","year":"1961","journal-title":"Math. Comp."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Kurzweil, H., and Stellmacher, B. (2004). The Theory of Finite Group, Springer.","DOI":"10.1007\/b97433"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Mellor, T.M., Yurchenko, S.N., Mant, B.P., and Jensen, P. (2019). Transformation Properties under the Operations of the Molecular Symmetry Groups G36 and G36(EM) of Ethane H3CCH3. Symmetry, 11.","DOI":"10.3390\/sym11070862"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/4\/548\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:32:48Z","timestamp":1760362368000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/4\/548"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,26]]},"references-count":53,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,4]]}},"alternative-id":["sym13040548"],"URL":"https:\/\/doi.org\/10.3390\/sym13040548","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,26]]}}}