Identification and treatment of internal rotation in normal mode vibrational analysis

TLDR

The authors present an automated procedure that identifies internal rotation modes and rotating groups during normal mode vibrational analysis and introduces an improved approximation for thermodynamic corrections. The method requires no user intervention, uses redundant internal coordinates to fix stretching, bending, and out‑of‑plane motions, solves the constrained vibrational problem to obtain rigid‑rotor internal rotation modes, identifies normal modes by comparison, automatically determines rotating group composition, computes the kinetic energy matrix via Wilson‑G or Kilpatrick‑Pitzer, and derives periodicity, symmetry numbers, and well‑multiplicity using simple rules that can be user‑adjusted. The resulting analytical approximation to the one‑dimensional hindered rotor partition function reproduces Pitzer and Gwinn values within ±0.4% (max error 2.1%) and outperforms previous approximations over a broader range, while its extension to multidimensional rotors provides useful starting approximations for further studies.

Abstract

A procedure that automatically identifies internal rotation modes and rotating groups during the normal mode vibrational analysis is outlined, and an improved approximation to the corrections for the thermodynamic functions is proposed. The identification and the characterization of the internal rotation modes require no user intervention and make extensive use of the information imbedded in the redundant internal coordinates. Rigid-rotor internal rotation modes are obtained by fixing stretching, bending, and out-of-plane bending motions and solving the vibrational problem for the constrained system. Normal vibrational modes corresponding to internal rotations are identified by comparing them with the constrained modes. The atomic composition of the rotating groups is determined automatically and the kinetic energy matrix for internal rotation is given by either the constrained Wilson-G matrix or the Kilpatrick and Pitzer protocol. The potential periodicity, the rotating tops’ symmetry numbers, and the well-multiplicity are obtained using simple rules. These parameters can be altered by user input. An improved analytical approximation to the partition function for a one-dimensional hindered internal rotation has been developed that reproduces the accurate values tabulated by Pitzer and Gwinn to ±0.4% with a maximum error of 2.1%. This approximation is shown to behave better than previously available approximations over a wider range of regimes. The one-dimensional rotor treatment is generalized to give useful approximations to the multidimensional rotor thermodynamic functions that can be a good start for more thorough studies.