Skip to main content
SpringerLink
Log in

The 1-D hindered rotor approximation

  • Regular Article
  • Published:
Theoretical Chemistry Accounts Aims and scope Submit manuscript

Abstract

We offer an overview of the popular one- dimensional (1-D) hindered rotor model that is often used for quantum mechanical treatment of internal rotation. This model is put in context with other methods used for treating anharmonic motions. The 1-D hindered rotor scheme is general for tops of any symmetry and has been used to provide accurate treatment of hindered rotors in a wide range of systems. One obstacle preventing wider use of the model is its lack of incorporation into common electronic structure codes. We have developed an algorithm for consistently treating all tops in a molecule, and we present simple codes which interface with electronic structure codes to provide thermochemical properties (S, C p , H) of individual species and reactions that have been corrected for internal rotations. Finally, we use this approach to give sensible advice about how the model can be used best. We show that dramatic changes in the reduced moment of inertia do not necessarily cause comparable changes in the properties of individual hindered rotors. We demonstrate that the rotational hindrance potential can be accurately determined using relatively coarse step sizes. Finally, we show that internal rotation in transition states can be treated using a “frozen transition state” approximation at a significant computational savings. We also discuss the relationship between calculated properties of hindered rotors and the choice of method and basis set used.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Schmidt M, Baldridge K, Boatz JA, Elbert S, Gordon MS, Jensen J, Koseki S, Matsunaga N, Nguyen K, Su S, Windus T, Dupuis M and Montgomery JA (1993). General atomic and molecular electronic structure system. J Comp Chem 14: 1347–1363

    Article  CAS  Google Scholar 

  2. Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, Cheeseman JR, Montgomery JA Jr, Vreven T, Kudin KN, Burant JC, Millam JM, Iyengar SS, Tomasi J, Barone V, Mennucci B, Cossi M, Scalmani G, Rega N, Petersson GA, Nakatsuji H, Hada M, Ehara M, Toyota K, Fukuda R, Hasegawa J, Ishida M, Nakajima T, Honda Y, Kitao O, Nakai H, Klene M, Li X, Knox JE, Hratchian HP, Cross JB, Bakken V, Adamo C, Jaramillo J, Gomperts R, Stratmann RE, Yazyev O, Austin AJ, Cammi R, Pomelli C, Ochterski JW, Ayala PY, Morokuma K, Voth GA, Salvador P, Dannenberg JJ, Zakrzewski VG, Dapprich S, Daniels AD, Strain MC, Farkas O, Malick DK, Rabuck AD, Raghavachari K, Foresman JB, Ortiz JV, Cui Q, Baboul AG, Clifford S, Cioslowski J, Stefanov BB, Liu G, Liashenko A, Piskorz P, Komaromi I, Martin RL, Fox DJ, Keith T, Al-Laham MA, Peng CY, Nanayakkara A, Challacombe M, Gill PMW, Johnson B, Chen W, Wong MW, Gonzalez C, Pople JA (2003) Gaussian 03, Revision C.02. Gaussian Inc., Wallingford, 2004

  3. Pitzer KS and Gwinn WD (1942). Energy levels and thermodynamic functions for molecules with internal rotation. I. Rigid frame with attached tops. J Chem Phys 10(7): 428–440

    Article  CAS  Google Scholar 

  4. Katzer G and Sax AF (2002). Beyond the harmonic approximation: Impact of anharmonic molecular vibrations on the thermochemistry of silicon hydrides. J Phys Chem A 106(31): 7204–7215

    Article  CAS  Google Scholar 

  5. Isaacson AD and Truhlar DG (1981). The accuracy of the Pitzer-Gwinn method for partition-functions of anharmonic vibrational-modes. J Chem Phys 75(8): 4090–4094

    Article  CAS  Google Scholar 

  6. Truhlar DG (1991). A simple approximation for the vibrational partition function of a hindered internal rotation. J Comp Chem 12(2): 266–270

    Article  CAS  Google Scholar 

  7. Aubanel EE, Robertson SH and Wardlaw DM (1991). Hindered rotor model for radical association reactions. J Chem Soc Faraday T 87(15): 2291–2297

    Article  CAS  Google Scholar 

  8. Gang J, Pilling MJ and Robertson SH (1996). Partition functions and densities of states for butane and pentane. J Chem Soc Faraday Trans 92(19): 3509–3518

    Article  CAS  Google Scholar 

  9. East ALL and Radom L (1997). Ab initio statistical thermodynamical models for the computation of third-law entropies. J Chem Phys 106(16): 6655–6674

    Article  CAS  Google Scholar 

  10. Gang J, Pilling MJ and Robertson SH (1997). Asymmetric internal rotation: Application to the 2-C4H9 \(\rightleftharpoons\) CH3 + C3H6 reaction. J Chem Soc Faraday Trans 93(8): 1481–1491

    Article  CAS  Google Scholar 

  11. Gang J, Pilling MJ and Robertson SH (1998). Monte Carlo calculation of partition functions for straight chain alkanes. Chem Phys 231(2–3): 183–192

    Article  CAS  Google Scholar 

  12. Ayala PY and Schlegel HB (1998). Identification and treatment of internal rotation in normal mode vibrational analysis. J Chem Phys 108(6): 2314–2325

    Article  CAS  Google Scholar 

  13. Knyazev VD (1998). Density of states of one-dimensional hindered internal rotors and separability of rotational degrees of freedom. J Phys Chem A 102(22): 3916–3922

    Article  CAS  Google Scholar 

  14. Knyazev VD and Tsang W (1998). Nonharmonic degrees of freedom: densities of states and thermodynamic functions. J Phys Chem A 102(46): 9167–9176

    Article  CAS  Google Scholar 

  15. Chen CJ and Bozzelli JW (1999). Analysis of tertiary butyl radical plus O2, isobutene plus HO2, isobutene plus OH and isobutene-OH adducts plus O2: a detailed tertiary butyl oxidation mechanism. J Phys Chem A 103(48): 9731–9769

    Article  CAS  Google Scholar 

  16. Van Speybroeck V, Van Neck D, Waroquier M, Wauters S, Saeys M and Marin GB (2000). Ab initio study of radical addition reactions: addition of a primary ethylbenzene radical to ethene. J Phys Chem A 104(46): 10939–10950

    Article  CAS  Google Scholar 

  17. Chuang YY and Truhlar DG (2000). Statistical thermodynamics of bond torsional modes. J Chem Phys 112(3): 1221–1228

    Article  CAS  Google Scholar 

  18. Van Speybroeck V, Van Neck D and Waroquier M (2002). Ab initio study of radical reactions: role of coupled internal rotations on the reaction kinetics (III). J Phys Chem A 106(38): 8945–8950

    Article  CAS  Google Scholar 

  19. Katzer G and Sax AF (2002). Numerical determination of pseudorotation constants. J Chem Phys 117(8): 8219–8228

    Article  CAS  Google Scholar 

  20. Miller TF and Clary DC (2002). Torsional path integral Monte Carlo method for the quantum simulation of large molecules. J Chem Phys 116(19): 8262–8269

    Article  CAS  Google Scholar 

  21. Miller TF and Clary DC (2003). Torsional path integral Monte Carlo method for calculating the absolute quantum free energy of large molecules. J Chem Phys 119(1): 68–76

    Article  CAS  Google Scholar 

  22. Katzer G and Sax AF (2003). A novel partition function for partially asymmetrical internal rotation. Chem Phys Lett 368(3–4): 473–479

    Article  CAS  Google Scholar 

  23. Vansteenkiste P, Van Speybroeck V, Marin GB and Waroquier M (2003). Ab initio calculation of entropy and heat capacity of gas-phase n-alkanes using internal rotations. J Phys Chem A 107(17): 3139–3145

    Article  CAS  Google Scholar 

  24. Lynch VA, Mielke SL and Truhlar DG (2004). Accurate vibrational-rotational partition functions and standard-state free energy values for H2O2 from Monte Carlo path-integral calculations. J Chem Phys 121(11): 5148–5162

    Article  CAS  Google Scholar 

  25. Barone V (2004). Vibrational zero-point energies and thermodynamic functions beyond the harmonic approximation. J Chem Phys 120(7): 3059–3065

    Article  CAS  Google Scholar 

  26. Van Speybroeck V, Vansteenkiste P, Van Neck D and Waroquier M (2005). Why does the uncoupled hindered rotor model work well for the thermodynamics of n-alkanes. Chem Phys Lett 402: 479–484

    Article  CAS  Google Scholar 

  27. Sebbarand N, Bockhorn H and Bozzelli JW (2005). Thermochemical properties, rotation barriers, and group additivity for unsaturated oxygenated hydrocarbons and radicals resulting from reaction of vinyl and phenyl radical systems with O2. J Phys Chem A 109(10): 2233–2253

    Article  CAS  Google Scholar 

  28. Katzer G and Sax AF (2005). Identification and thermodynamic treatment of several types of large-amplitude motions. J Comput Chem 26(14): 1438–1451

    Article  CAS  Google Scholar 

  29. Wong BM and Green WH Jr (2005). Effects of large-amplitude torsions on partition functions: Beyond the conventional separability assumption. Mol Phys 103(6–8): 1027–1034

    Article  CAS  Google Scholar 

  30. Van Cauter K, Van Speybroeck V, Vansteenkiste P, Reyniers MF and Waroquier M (2006). Ab initio study of free-radical polymerization: polyethylene propagation kinetics. Chem Phys Chem 7(1): 131–140

    CAS  Google Scholar 

  31. Lynch VA, Mielke SL and Truhlar DG (2006). High-precision quantum thermochemistry on nonquasiharmonic potentials: converged path-integral free energies and a systematically convergent family of generalized Pitzer-Gwinn approximations. J Phys Chem A 110(17): 5965–5965

    Article  CAS  Google Scholar 

  32. Vansteenkiste P, Van Neck D, Van Speybroeck V, Waroquier M (2006) An extended hindered-rotor model with incorporation of Coriolis and vibrational-rotational coupling for calculating partition functions and derived quantities. J Chem Phys 124(4) Art. No. 044314

  33. Ellingson BA, Lynch VA, Mielke SL and Truhlar DG (2006). Statistical thermodynamics of bond torsional modes: Tests of separable, almost-separable, and improved Pitzer-Gwinn approximations. J Chem Phys 125(8): 084305

    Article  CAS  Google Scholar 

  34. da Silva G, Kim CH and Bozzelli JW (2006). Thermodynamic properties (enthalpy, bond energy, entropy, and heat capacity) and internal rotor potentials of vinyl alcohol, methyl vinyl ether, and their corresponding radicals. J Phys Chem A 110(25): 7925–7934

    Article  CAS  Google Scholar 

  35. Wong BM, Thom RL and Field RW (2006). Accurate inertias for large-amplitude motions: improvements on prevailing approximations. J Phys Chem A 110(23): 7406–7413

    Article  CAS  Google Scholar 

  36. McClurg RB, Flagan RC and Goddard WA (1997). The hindered rotor density-of-states interpolation function. J Chem Phys 16(2): 6675–6680

    Article  Google Scholar 

  37. Bowman JM (1986). The self-consistent-field approach to polyatomic vibrations. Acc Chem Res 19(7): 202–208

    Article  CAS  Google Scholar 

  38. Handy NC (1987). The derivation of vibration-rotation kinetic-energy operators, in internal coordinates. Mol Phys 61(1): 207–223

    Article  CAS  Google Scholar 

  39. Carter S, Bowman JM and Handy NC (1998). Extensions and tests of ‘multimodes’: a code to obtain accurate vibration/rotation energies of many-mode molecules. Theor Chem Acc 100(1–4): 191–198

    CAS  Google Scholar 

  40. Handy NC and Carter S (2004). Large vibrational variational calculations using ‘multimode’ and an iterative diagonalization technique. Mol Phys 102(21–22): 2201–2205

    Article  CAS  Google Scholar 

  41. Carter S, Handy NC and Tarroni R (2005). A variational method for the calculation of spin-rovibronic energy levels of any triatomic molecule in an electronic triplet state. Mol Phys 103(6–8): 1131–1137

    Article  CAS  Google Scholar 

  42. Pitzer KS (1946). Energy levels and thermodynamic functions for molecules with internal rotation. II Unsymmetrical tops attached to a rigid frame. J Chem Phys 14(4): 239–243

    Article  CAS  Google Scholar 

  43. Kilpatrick JE and Pitzer KS (1949). Energy levels and thermodynamic functions for molecules with internal rotation. III Compound rotation. J Chem Phys 17(11): 1064–1075

    Article  CAS  Google Scholar 

  44. Sumathi R and Green WH Jr (2002). Missing thermochemical groups for large unsaturated hydrocarbons: Contrasting predictions of G2 and CBS-Q. J Phys Chem A 106: 11141–11149

    Article  CAS  Google Scholar 

  45. Gomez-Balderas R, Coote ML, Hendry D and Radom L (2004). Reliable theoretical procedures for calculating the rate of methyl radical addition to carbon-carbon double and triple bonds. J Phys Chem A 108(15): 2874–2883

    Article  CAS  Google Scholar 

  46. Coote ML, Wood GPF and Radom L (2002). Methyl radical addition to C=S double bonds: Kinetic versus thermodynamic preferences. J Phys Chem A 106(50): 12124–12138

    Article  CAS  Google Scholar 

  47. Sumathi R, Carstensen HH and Green WH Jr (2001). Reaction rate prediction via group additivity Part 1: H abstraction from alkanes by H and CH3. J Phys Chem A 105(28): 6910–6925

    Article  CAS  Google Scholar 

  48. Heuts JPA, Gilbert RG and Radom L (1996). Determination of Arrhenius parameters for propagation in free-radical polymerizations: an assessment of ab initio procedures. J Phys Chem 100(49): 18997–19006

    Article  CAS  Google Scholar 

  49. Pfaendtner J, Yu X and Broadbelt LJ (2006). Quantum chemical investigation of low-temperature intramolecular hydrogen transfer reactions of hydrocarbons. J Phys Chem A 110(37): 10863–10871

    Article  CAS  Google Scholar 

  50. McQuarrie DA and Simon JD (1999). Molecular thermodynamics. University Science Books, Sausalito

    Google Scholar 

  51. Pfaendtner J, Yu X, Broadbelt LJ (2006) Calctherm version 0.9. http://broadbelt.chem-eng.northwestern.edu

  52. Pfaendtner J, Yu X, Broadbelt LJ (2006) Calck version 0.9. http://broadbelt.chem-eng.northwestern.edu

  53. Wijaya C, Sumathi R and Green WH Jr (2003). Thermodynamic properties and kinetic parameters for cyclic ether formation from hydroperoxyalkyl radicals. J Phys Chem A 107(24): 4908–4920

    Article  CAS  Google Scholar 

  54. Izgorodina EI and Coote ML (2006). Accurate ab initio prediction of propagation rate coefficients in free-radical polymerization: acrylonitrile and vinyl chloride. J Chem Phys 324(1): 96–110

    Article  CAS  Google Scholar 

  55. Herschbach DR, Johnston HS, Pitzer KS and Powell RE (1956). Theoretical pre-exponential factors for 12 bimolecular reactions. J Chem Phys 25(4): 736–741

    Article  CAS  Google Scholar 

  56. Sumathi R and Green WH Jr (2002). A priori rate constants for kinetic modeling. Theor Chem Acc 108(4): 187–213

    CAS  Google Scholar 

  57. Carbonniere P and Barone V (2004). Coriolis couplings in variational computations of vibrational spectra beyond the harmonic approximation: implementation and validation. Chem Phys Lett 392(4–6): 365–371

    Article  CAS  Google Scholar 

  58. Nordholm S and Bacskay G (1976). Generalized finite-element method applied to bound-state calculation. Chem Phys Lett 42(2): 253–258

    Article  CAS  Google Scholar 

  59. Balint-Kurti GG, Dixon R and Marston CC (1992). Grid methods for solving the Schrödinger equation and time dependent quantum dynamics of molecular photofragmentation and reactive scattering processes. Int Rev Phys Chem 11: 317–344

    Article  CAS  Google Scholar 

  60. Balint-Kurti GG, Ward CL and Marston CC (1991). Two computer programs for solving the Schrödinger equation for bound state eigenvalues and eigenfunctions using the Fourier grid Hamiltonian method. Comput Phys Commun 67: 285–292

    Article  CAS  Google Scholar 

  61. Marston CC and Balint-Kurti GG (1989). The Fourier grid Hamiltonian method for bound state eigenvalues and eigenfunctions. J Chem Phys 91(6): 3571–3576

    Article  CAS  Google Scholar 

  62. Tafipolsky M and Schmid R (2005). Calculation of rotational partition functions by an efficient Monte Carlo importance sampling technique. J Comput Chem 26(15): 1579–1591

    Article  CAS  Google Scholar 

  63. Chempath S, Predescu C and Bell AT (2006). Quantum mechanical single molecule partition function from path integral Monte Carlo simulations. J Chem Phys 124(23): 234104

    Article  CAS  Google Scholar 

  64. Carter S, Culik SJ and Bowman JM (1997). Vibrational self-consistent field method for many-mode systems: a new approach and application to the vibrations of CO adsorbed on Cu(100). J Chem Phys 107(24): 10458–10469

    Article  CAS  Google Scholar 

  65. Carter S and Bowman JM (1998). The adiabatic rotation approximation for rovibrational energies of many-mode systems: description and tests of the method. J Chem Phys 108(11): 4397–4404

    Article  CAS  Google Scholar 

  66. Bowman JM, Carter S and Huang XC (2003). Multimode: a code to calculate rovibrational energies of polyatomic molecules. Int Rev Phys Chem 22(3): 533–549

    Article  CAS  Google Scholar 

  67. Fernandez-Ramos A, Miller JA, Klippenstein SJ and Truhlar DG (2006). Modeling the kinetics of bimolecular reactions. Chem Rev 106(11): 4518–4584

    Article  CAS  Google Scholar 

  68. Greenwald EE, North SW, Georgievskii Y and Klippenstein SJ (2005). A two transition state model for radical-molecule reactions: a case study of the addition of OH to C2H4. J Phys Chem A 109(27): 6031–6044

    Article  CAS  Google Scholar 

  69. Vansteenkiste P, VanSpeybroeck V, Verniest G, DeKimpe N and Waroquier M (2006). Applicability of the hindered rotor scheme to the puckering mode in four-membered rings. J Phys Chem A 110(10): 3838–3844

    Article  CAS  Google Scholar 

  70. Denisova TG and Denisov ET (2001). Kinetic parameters of alkyl, alkoxy and peroxy radical isomerization. Kinet Catal 42(5): 620–630

    Article  CAS  Google Scholar 

  71. Becke AD (1993). Density-functional thermochemistry 3. The role of exact exchange. J Chem Phys 98(7): 5648–5652

    Article  CAS  Google Scholar 

  72. Montgomery JA, Frisch MJ, Ochterski JW and Petersson GA (1999). A complete basis set model chemistry. VI. Use of density functional geometries and frequencies. J Chem Phys 110(6): 2822–2827

    Article  CAS  Google Scholar 

  73. Scott A and Radom L (1996). Harmonic vibrational frequencies: an evaluation of Hartree-Fock, Møller-Plesset, quadratic configuration interaction, density functional theory and semiempirical scale factors. J Phys Chem 100(41): 16502–16513

    Article  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Chemical and Biological Engineering, Northwestern University, Evanston, IL, 60208-3120, USA

    Jim Pfaendtner, Xinrui Yu & Linda J. Broadbelt

Authors
  1. Jim Pfaendtner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xinrui Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Linda J. Broadbelt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Linda J. Broadbelt.

Electronic supplementary material

Below are the electronic supplementary materials.

214_2007_376_MOESM1_ESM.pdf

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pfaendtner, J., Yu, X. & Broadbelt, L.J. The 1-D hindered rotor approximation. Theor Chem Account 118, 881–898 (2007). https://doi.org/10.1007/s00214-007-0376-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00214-007-0376-5

Keywords

Search

Navigation