Hindered-rotor harmonic oscillator

Concerning the proper treatment of torsional anharmonicity, which still represents a challenging aspect for accurate thermochemical calculations of complex molecules [257-265], a hindered-rotor anharmonic oscilattor (HRAO) model has been shown to provide accurate results[62, 72, 117, 204]. The HRAO model is based on a generalization to anharmonic force fields of the hindered-rotor harmonic oscillator (HRHO) model [257] that automatically identifies internal rotation modes and rotating groups during the normal-mode vibrational analysis. 

Note that there is nothing wrong widi Eq. (10.45). The entropy of a quantum mechanical harmonic oscillator really does go to infinity as the frequency goes to zero. What is wrong is that one usually should not apply the harmonic oscillator approximation to describe those modes exhibiting the smallest frequencies. More typically than not, such modes are torsions about single bonds characterized by very small or vanishing barriers. Such situations are known as hindered and free rotors, respectively. 

The potential functions and the resulting energy levels of a one-dimensional hindered rotor, a free rotor, and a harmonic oscillator are compared in figure 6.2. The hindered rotor energy levels can be solved numerically as solutions to the Mathieu differential equation (Abramonitz and Stegun, 1972 Wilson, 1940) and lists of values... 

The vibrational and rotational components can be calculated from the harmonic oscillator and rigid rotor models, for example, whose expressions can be found in many textbooks of statistical thermodynamics [20]. If a more sophisticated correction is needed, vibrational anharmonic corrections and the hindered rotor are also valid models to be considered. The translational component can be calculated from the respective partition function or approximated, for example, by 3I2RT, the value found for an ideal monoatomic gas. 

The effect of using different internal rotor treatments (harmonic oscillator or free rotator approximations) instead of hindered rotor treatment on the calculated reaction rate coefficient is also shown there [63]. 

In the gas phase, it is usually sufhcient to calculate the partition functions and associated thermal corrections to the enthalpy and entropy using the standard textbook formulae [31] for an ideal gas under the harmonic oscillator-rigid rotor approximation, provided one then makes explicit corrections for low-frequency torsional modes. These modes can be treated instead as one-dimensional hindered internal rotations using the torsional eigenvalue summation procedure described in Ref. [32]. Rate and equilibrium constants can then be obtained from the following standard textbook formulae [31] ... 

The BS algorithm [82] was published in 1973 and in the same year extended by Stein and Rabinovitch [83]. The latter algorithm is more flexible (e.g., in addition to harmonic oscillators it can be applied to anharmonic oscillators and hindered rotors as well) and more accurate (rounding errors in the energy discretization step are reduced), but it requires one additional array to store intermediate results. The algorithm may be described as follows ... 

The above treatment of hindered rotors assumes that a given mode can be approximated as a one-dimensional rigid rotor, and studies for small systems have shown that this is generally a reasonable assumption in those cases (82). However, for larger molecules, the various motions become increasingly coupled, and a (considerably more complex) multidimensional treatment may be needed in those cases. When coupling is significant, the use of a one-dimensional hindered rotor model may actually introduce more error than the (fully decoupled) harmonic oscillator treatment. Hence, in these cases, the one-dimensional hindered rotor model should be used cautiously. 

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