XIAM uses the extended Internal Axis Method proposed by Woods to treat internal rotation in an asymmetric top molecule and the principal features are:

• up to three symmetric internal rotors
• up to one quadrupolar nucleus with weakly interacting nuclear quadrupole coupling
• centrifugal distortion up to sixth order for the pure rotational part
• centrifugal distortion up to fourth order between internal and overall rotation
• some top-top coupling terms for analysis of excited states of internal rotation
• high speed of operation due to suitable basis transformations and matrix factorisation

        The recommended reference for citing the use of XIAM is:

        H.Hartwig and H.Dreizler, Z. Naturforsch 51a, 923-932 (1996).

        Definition of the empirical internal rotation-overall rotation distortion operator programmed into XIAM as terms Dpi2J, Dpi2K and Dpi2-. is in Eq.(6) of

        N.Hansen, H.Mader and T.Bruhn, Molec. Phys. 97, 587-595 (1999).

        m-methylanisole: S.Herbers and V.L.Nguyen, J.Mol.Spectrosc. 370, 111289 (2020).
        4-methylacetophenone: S.Herbers et al., J.Chem.Phys. 152, 074301 (2020).

        ERHAM sets up and solves the "Effective rotational Hamiltonian for molecules with two periodic large-amplitude motions". It allows to fit spectroscopic constants to observed transition frequencies (usually to experimental precision) and to predict the spectrum.

        The reference for citing the use of ERHAM is: P. Groner, J. Chem. Phys. 107, 4483-4498 (1997).
        A review of the theory and the performance of the effective rotational Hamiltonian is also available: P. Groner, J. Mol.Spectrosc. 278, 52-67 (2012).
           

        Principal features:

• One or two internal rotors, not restricted to threefold rotors
• Models and symmetry groups:
  1. Equivalent rotors: C_2v, C₂, C_s
  2. Non-equivalent rotors: C_s, C₁
  3. Single rotor: C_s, C₁
• max(J) = 120
• Number of transitions in fit < 8191
• Modular input for “tunneling parameters”
• Tunneling energy parameters e_qq
• Tunneling contributions to rotational and distortion constants
• Quartic and sextic centrifugal distortion constants (A-reduction); higher order CD terms may be defined using the “tunneling parameter input” which can also be used to define terms for the S-reduction
• Global fit of several non-interacting vibrational states to the same r-vector parameters
• high speed of operation due to suitable basis transformations and matrix factorisation

        ERHAM has been used in numerous investigations, which can be treated as worked examples for the various areas of its applicability.  Published applications involving its author (GS = ground state, ETS = torsional excited state):

• Dimethyl ether (GS): P. Groner et al., Astrophys. J. 500, 1059-1063 (1998)
• 3-Methyl-1,2-butadiene (global fit of GS and 1^st ETS): S. Bell et al., J. Phys. Chem. A 104, 514-520 (2000)
• Acetone (GS): P. Groner et al., Astrophys. J. Suppl. Ser. 142, 145-151 (2002)
• Ethyl methyl ether (GS, nonequivalent): U. Fuchs et al., Astrophys. J. Suppl. Ser. 144, 277-286 (2003)
• Dimethyl diselenide (GS, isotopomers with C₂ or C₁ symmetry): P. Groner et al., J. Mol. Spectrosc. 226, 169-181 (2004)
• Acetone-¹³C (equivalent, non-equivalent): F. J. Lovas & P. Groner, J. Mol. Spectrosc. 236, 173-177 (2006)
• Acetone (1^st ETS): P. Groner et al., J. Mol. Struct. 795, 173-178 (2006)
• Methyl carbamate (1 rotor, GS) P. Groner et al., Astrophys. J. Suppl. Ser. 169, 28-36 (2007)
• Methyl formate-1-¹³C (1 rotor, GS) A. Maeda et al., Astrophys. J. Suppl. Ser. 175, 138-146 (2008)
• Acetone (2^nd ETS): P. Groner et al., J. Mol. Spectrosc. 251, 180-184 (2008)
• CHClF₂-H₂O Chlorodifluoromethane-water (1 top - two-fold, GS): B.J. Bills et al., J. Mol. Spectrosc. 268, 7-15 (2011)
• 1,1-difluoroacetone (1 top, GS): G.S. Grubbs, II et al.,  J. Mol. Spectrosc. 280, 21-26 (2012)
• Dimethyl ether-d₁ (1 top, 2 conformers, GS): C. Richard et al., A & A 552, A117 (2013)

Other authors:

• Propane (GS & 2 ETS) Drouin et al. J. Mol. Spectrosc. 240, 227-237 (2006)
• Pyruvic acid (1 rotor, GS & several non-interacting excited states) Z. Kisiel et al., J. Mol. Spectrosc. 241, 220-229 (2007)
• Methyl formate-¹²C & -1-¹³C  (1 rotor, ETS) A. Maeda et al. J. Mol. Spectrosc. 251, 293-300 (2008)
• Pyruvic acid (1 rotor, GS & several non-interacting excited states) Z. Kisiel et al., J. Mol. Spectrosc. 241, 220-229 (2007)
• Dimethyl ether (GS) Endres et al. A&A 504, 635-640 (2009)
• Pyruvonitrile (1 rotor, GS & several non-interacting excited states) Krasnicki et al., J. Mol. Spectrosc. 260, 57-65 (2010)
• Dimethyl carbonate (GS): F.J. Lovas, et al., J. Mol. Spectrosc. 264, 10-18 (2010)
• Isopropenyl acetate (GS): H.L.V. Nguyen et al., J. Mol. Spectrosc. 264, 120-124 (2010)
• Dimethyl sulfate (GS): L.B. Favero et al., Chem. Phys. Lett. 517, 139-145 (2010)
• CF₃(CF₂)₃O-CH₃ & (CF₃)₂CFCF₂OCH₃ (1 top each, GS): G. S. Grubbs, II, et al., J. Phys. Chem. A, 115, 1086–1091 (2011)

        BARRIER has been created primarily to derive barriers to internal rotation for a single rotor from the tunneling  splittings determined by ERHAM.

• BELGI-Cs - program for molecules containing an internal rotor (of C_3v symmetry) which can turn relative to the rest of the molecule (of C_s symmetry)
• BELGI-C1 - program for molecules containing an internal rotor (of C_3v symmetry) which can turn relative to the rest of the molecule (with no symmetry)
• several utility programs for both versions of BELGI

This program has a long history, detailed in the readme, and the authors (in chronological order) are: 

• I. Kleiner from Laboratoire Interuniversitaire des Systèmes Atmosphériques, LISA, (Université paris 7 et Paris 12 et CNRS, Créteil, France)
• M. Godefroid from the "Laboratoire de Chimie Quantique et Photophysique" , Free University of Brussels (Belgium),
• J. T. Hougen from the National Institute for Standards and Technology (NIST, Gaithersburg, USA),
• L-H. Xu from Department of Physical Sciences, University of New Brunswick,
• J. Ortigoso from Instituto de Estructura de la Materia, CSIC (Madrid, Spain),
• V. Ilyushin from the Radio Astronomy Institute of NASU, Kharkov (Ukraine)
• M. Carvajal-Zaera from the Departamento de Fisica Aplicada, University of Huelva (Spain)

       The reference for citing the use of BELGI-Cs is:

• J. T. Hougen, I. Kleiner and M. Godefroid, J. Mol. Spectrosc., 163, 559-586 (1994). 
  

       Extensive listing of previous applications of  BELGI-Cs is available and those papers contain many different examples of the use of this program.

        Principal characteristics of BELGI-Cs:

• Fit one internal rotor of C_3v symmetry (like a CH₃ group), while the rest of the molecule possesses a plane of symmetry (C_s).
• J_max = 30
• Up to 80000 lines to fit or to calculate
• Up to 80 parameters of fit in each vibrational state
• Up to 2 vibrational states

• A two-step diagonalisation with:
  1. the diagonalisation of a 21x21 torsional matrix for each K and s value (K is the projection of J on the symmetry axis of the molecule and s is the symmetry with s = 0 for the A states and s = 1 for the E states), and
  2. the diagonalisation of the rotation, centrifugal distortion and rotation-torsion coupling terms of the Hamiltonian (dimension (9)*(2J+1) x (9)*(2J+1))

• A global fit of the A and E species corresponding to ALL the torsional levels (up to the 9^th torsional state v_t 0, 1…8)

       The references for citing BELGI-C1 are:

1. I. Kleiner and J. T. Hougen, J. Chem. Phys. 119, 5505 (2003) 
2. R. J. Lavrich, A. R. Hight Walker, D. F. Plusquellic, I. Kleiner, R. D. Suenram, , J. T. Hougen and G. T. Fraser, J. Chem. Phys. , 119, 5497-5504 (2003).

        You can also check the listing and a listing of previous applications of  BELGI-Cs is given here.       
Principal characteristics of BELGI-C1:

• can fit one internal rotor of C_3v symmetry (like a CH₃ group), the rest of the molecule may not possess a plan of symmetry (C₁). Complex algebra used.
• J_max = 30
• Max 20000 lines to fit or to calculate
• Max 80 parameters to fit in each vibrational states
• A two-step diagonalisation with:
  1. the diagonalisation of a 21x21 torsional matrix for each K and s value (K is the projection of J on the symmetry axis of the molecule and s is the symmetry with s = 0 for the A states and s =1 for the E states) and
  2. the diagonalisation of the rotation, centrifugal distortion and rotation-torsion coupling terms of the Hamiltonian (dimension (9)*(2J+1) x (9)*(2J+1))

• A Global fit of the A and E species corresponding to ALL the torsional levels (up to the 9^th torsional state v_t 0, 1…8)

• Sixfold rotation case with a C_3v internal rotor and a C_2v frame (such as toluene and nitromethane)
  
• Threefold rotation case with a C_3v internal rotor and a C_s frame (such as acetic acid, acetamide or methyl formate)

        The feature distinguishing the two programs is that RAM36 deals with the most common case of hyperfine-free transitions, while RAM36hf allows also inclusion of hyperfine structure from the presence of a single quadrupolar atom.

        The reference for citing the use of RAM36 is:

• V.V. Ilyushin, Z. Kisiel, L. Pszczółkowski, H. Mäder, J.T. Hougen,  “A New Torsion-Rotation Fitting Program for Molecules with a Six-Fold Barrier: Application to the Microwave Spectrum of Toluene”, Journal of Molecular Spectroscopy 259, 26-38 (2010).
  

        This paper describes both the program and its application to the analysis of the lowest m-states in the rotational spectrum of toluene, which is a rather demanding low-barrier case. Description of the steps taken to increase the speed of RAM36 operation relative to its predecessors is given in:

• V. V. Ilyushin, C. P. Endres, F. Lewen, S. Schlemmer, B. J. Drouin "Submillimeter wave spectrum of acetic acid",  Journal of Molecular Spectroscopy 290, 31 - 41 (2013).

        Another example of application of the sixfold barrier mode of RAM36 can be found in:    

• V.V. Ilyushin, L.B. Favero, W.Caminati, J-U. Grabow “Intertorsional Interactions Revealing Absolute Configurations: The V6 Internal Rotation Heavy-Top Case of Benzotrifluoride”, ChemPhysChem 11, 2589 – 2593.(2010). 

        An example of the use of RAM36 for a problem with a threefold barrier is described in:

• V.Ilyushin, R.Rizzato, L.Evangelisti, G.Feng, A.Maris, S.Melandri, W.Caminati', “Almost free methyl top internal rotation: rotational spectrum of 2-butynoic acid”, Journal of Molecular Spectroscopy  267, 186 - 190 (2011).

        The references for citing the use of RAM3hf are:

• V.V. Ilyushin, “Millimeter wave spectrum of nitromethane”, Journal of Molecular Spectroscopy 345, 64-69 (2018).

• A.Belloche, A.A.Mescheheryakov, R.T.Garrod, V.V.Ilyushin, E.A.Alekseev, R.A.Motiyenko, I.Margules, H.S.P.Muller, K.M.Menten, "Rotational spectroscopy, tentative interstellar detection, and chemical modelling of N-methylformamide", Astronomy & Astrophysics, 601, A49:41pp (2017).

        The first paper describes a sixfold barrier case application, and the second describes a threefold barrier case application.  Data files for both cases are provided below.

• METHOD 1: Fourier series expansion based on the Mathieu equation description of the Internal Axis System (IAS)  Hamiltonian for internal rotation [1].  The .PAR parameter file for SPFIT is set up by means of two preprocessor programs, first MOIAM (input file .INP),  then IAMCALC (input file .IAM). This results in a .PAR file with an extensive set of linked parameters.  This file is completely unreadable and its size can run to many Megabytes. Fortunately, in the final SPFIT output these expansions are brought together into the original leading parameters as specified in the .IAM file.  Direct program documentation  seems to be limited to that in moiam.c and iamcalc.c source codes, but there are quite a number of published applications to serve as worked examples.  These include studies of HNO3 [2], methyl formate [3], propane [4], acetaldehyde [5], hydroxyacetone [6], and methyl carbamate [7].  Note that in [4] there is also a comparison of this way of using SPFIT with ERHAM.
  

1. H.M.Pickett,  J,Chem.Phys.107, 6732 (1997)
2. D.T.Petkie, T.M.Goyette, P.Helminger, H.M.Pickett, F.C.De Lucia, J.Mol.Spectrosc. 208, 121 (2001); this appears to be the first explicit mention of MOIAM and IAMCALC in a publication.
3. Methyl Formate = species  c060003 in jpl spectral line catalog: PDF entry (includes a short description of IAMCALC),  IAM file, PAR file (warning: 18MB) , LIN file
4. B.J.Drouin, J.C.Pearson, A.Walters, V.Lattanzi, J.Mol.Spectrosc. 240, 227 (2006) = propane, species c044013 in jpl spectral line catalog: PDF entry, PAR file, LIN file
5. Acetaldehyde = species c044003 in jpl spectral line catalog: PDF entry (includes a short description of IAMCALC),  IAM file, PAR file (warning: 8.8MB) , LIN file
6. A.J.Apponi, J.J.Hoy, D.T.Halfen, L.M.Ziurys, Astrophy. J. 652, 1787 (2006) = Hydrohxyacetone, species c074003 in jpl spectral line catalog: PDF entry,  PAR file, LIN file
7. Methyl Carbamate = species c075004 in jpl spectral line catalog: PDF entry (includes a short description of IAMCALC),  PAR file (warning: 4MB), LIN file

• METHOD 2: Effective single state fits based on perturbation approximations.  These will only work sensibly in specific cases but may be all that is required for the relatively small, low-J data sets obtained in supersonic expansion measurements.  The approach is based on the Principal Axis Method approach (PAM) and depends on the fact that for a sizable threefold barrier the A states are sufficiently well treatable by the standard asymmetric rotor Hamiltonian.  For the E states the torsion-rotation interaction may be sufficiently well described by terms of the type D_aP_a and their centrifugal distortion expansion, with the choice of terms depending on the orientation of the internal rotation axis relative to the inertial axes.  These terms have direct SPFIT indices.  The approach has been well described and used to treat supersonic expansion data for o-chlorotoluene [1], and later for much higher-J, mmw data for pyruvic acid [2] and pyruvonitrile [3].  The discussion in Ref.[1] constitutes a nice tutorial in the use of the method, including how to derive the barrier height V3 from such fits by using tabulated perturbation coefficients in Appendix C of  [4].  In [2-3] this method is compared to the results from XIAM and ERHAM and the advantages and disadvantages of this simple approach are discussed.  The supplementary material for [2-3] also contains the input data files for SPFIT.
  

1. D.Gerhard, A.Hellweg, I.Merke, W.Stahl, M.Baudelet, D.Petitprez, G.Wlodarczak, J.Mol.Spectrosc. 220, 234  (2003).
2. Z.Kisiel, L.Pszczolkowski, E.Bialkowska-Jaworka, S.B.Charnley, J.Mol.Spectrosc. 241, 220 (2007).
3. A.Krasnicki, Z.Kisiel, L.Pszczolkowski,  J.Mol.Spectrosc. 260, 57 (2010).
4. D.R.Herschbach, J.Chem.Phys. 31, 91 (1959).