^{1,a)}, Mirco Ragni

^{1}, Glauciete S. Maciel

^{1}, Vincenzo Aquilanti

^{1}and Frederico V. Prudente

^{2}

### Abstract

In view of the particular attention recently devoted to hindered rotations, we have tested reduced kinetic energy operators to study the torsional mode around the O–O bond for and for a series of its derivatives (HOOCl, HOOCN, HOOF, HOONO, HOOMe, HOOEt, MeOOMe, ClOOCl, FOOCl, FOOF, and FOONO), for which we had previously determined potential energy profiles along the dihedral angle [ , Et ]. We have calculated level distributions as a function of temperature and partition functions for all systems. Specifically, for the system we have used two procedures for the reduction in the kinetic energy operator to that of a rigid-rotor-like one and the calculated partition functions are compared with previous work. Quantum partition functions are evaluated both by quantum level state sums and by simple classical approximations. A semiclassical approach, using a linear approximation of the classical path and a quadratic Feynman–Hibbs approximation of Feynman path integral, introduced in previous work and here applied to the torsional mode, is shown to greatly improve the classical approximations. Further improvement is obtained by the explicit introduction of the dependence of the moment of inertia from the torsional angle. These results permit one to discuss the characteristic time for chirality changes for the investigated molecules either by quantum mechanical tunneling (dominating at low temperatures) or by transition state theory (expected to provide an estimate of racemization rates in the high energy limit).

We thank the Agenzia Spaziale Italiana (ASI) for support and the Italian Ministero per l ’Universitá e la Ricerca (MIUR) for a PRIN grant. A.C.P.B. acknowledges a CAPES Brazilian fellowship. We thank D. G. Truhlar and S. L. Mielke for insightful correspondence and an anonymous referee for constructive criticism. F.V.P. thanks CNPq (Brazil) for financial support.

I. INTRODUCTION

II. TORSIONAL LEVELS

A. Valence-type coordinates

B. Orthogonal coordinates

C. Symmetry classes

III. TORSIONAL PARTITION FUNCTION

A. Quantum formulas

B. Classical formulas

C. Semiclassical formulas

IV. RESULTS AND DISCUSSION

A. Hydrogen peroxide

B. Other systems

C. Intramolecular chirality changing rates

V. FURTHER REMARKS AND CONCLUSIONS

### Key Topics

- Chiral symmetries
- 6.0
- Tunneling
- 6.0
- Boltzmann equations
- 4.0
- Angular momentum
- 3.0
- Level splitting
- 3.0

## Figures

Illustration of the HOOF molecule in the (a) *cis* and , (b) *trans* , and both chiral equilibrium configurations (c) and (d) . The figure also shows the torsional potential and the levels for the HOOF molecule [designated as according the nomenclature of the previous papers (Refs. 2 and 3)].

Illustration of the HOOF molecule in the (a) *cis* and , (b) *trans* , and both chiral equilibrium configurations (c) and (d) . The figure also shows the torsional potential and the levels for the HOOF molecule [designated as according the nomenclature of the previous papers (Refs. 2 and 3)].

Illustration of the representations of the structure of the molecules in terms of the usual valence-type coordinates, where , , and are the interatomic distances, and are the bond angles, and is the dihedral angle.

Illustration of the representations of the structure of the molecules in terms of the usual valence-type coordinates, where , , and are the interatomic distances, and are the bond angles, and is the dihedral angle.

Illustration of the representations of the structure of the molecule in terms of the orthogonal local coordinates or diatom-diatom vectors. and coincide with the OH bonds but joins the centers of mass of the two OH groups.

Illustration of the representations of the structure of the molecule in terms of the orthogonal local coordinates or diatom-diatom vectors. and coincide with the OH bonds but joins the centers of mass of the two OH groups.

Torsional energy profile as a function of the dihedral angle and energy levels , for HOOCN.

Torsional energy profile as a function of the dihedral angle and energy levels , for HOOCN.

Same as Fig. 4 for MeOOMe (labels , omitted).

Same as Fig. 4 for MeOOMe (labels , omitted).

## Tables

Equilibrium geometries and cis and trans barriers calculated with the method (Refs. 2 and 3). and are calculated using Eqs. (4) and (5). The bond lengths are expressed in angstrom, the angles in degree, and the barriers and in . In the case of the HOOH we have also used the diatom-diatom approach (Fig. 3) and , , and .

Equilibrium geometries and cis and trans barriers calculated with the method (Refs. 2 and 3). and are calculated using Eqs. (4) and (5). The bond lengths are expressed in angstrom, the angles in degree, and the barriers and in . In the case of the HOOH we have also used the diatom-diatom approach (Fig. 3) and , , and .

Coefficients of (Fig. 2) of the expansion [Eq. (1)] in .

Coefficients of (Fig. 2) of the expansion [Eq. (1)] in .

Distribution of the levels for the system, evaluated with the orthogonal method (diatom-diatom approach).

Distribution of the levels for the system, evaluated with the orthogonal method (diatom-diatom approach).

Torsional partition functions at different temperature for the system. and are the quantum and classical partition functions using the diatom-diatom approach. and are calculated using Eq. (12) directly from the levels of the symmetries and . is calculated using Eqs. (10) and (13). uses Eq. (15) and uses Eq. (16). is calculated using the LCP/QFH approach.

Torsional partition functions at different temperature for the system. and are the quantum and classical partition functions using the diatom-diatom approach. and are calculated using Eq. (12) directly from the levels of the symmetries and . is calculated using Eqs. (10) and (13). uses Eq. (15) and uses Eq. (16). is calculated using the LCP/QFH approach.

Partition functions for system at low temperatures evaluated using valence coordinates.

Partition functions for system at low temperatures evaluated using valence coordinates.

Torsional partition functions at different temperatures for the HOOCl, HOOCN, HOOF, and HOONO. are calculated using Eq. (12) directly from the levels of the symmetries and . uses Eq. (15) and uses Eq. (16). is calculated using the LCP/QFH approach.

Torsional partition functions at different temperatures for the HOOCl, HOOCN, HOOF, and HOONO. are calculated using Eq. (12) directly from the levels of the symmetries and . uses Eq. (15) and uses Eq. (16). is calculated using the LCP/QFH approach.

Torsional partition functions at different temperatures for the ClOOCl, FOOCl, FOOF, and FOONO. are calculated using Eq. (12) directly from the levels of the symmetries and . uses Eq. (15) and uses Eq. (16). is calculated using the LCP/QFH approach.

Torsional partition functions at different temperatures for the ClOOCl, FOOCl, FOOF, and FOONO. are calculated using Eq. (12) directly from the levels of the symmetries and . uses Eq. (15) and uses Eq. (16). is calculated using the LCP/QFH approach.

Torsional partition functions at different temperatures for the HOOMe, MeOOMe, and HOOEt. are calculated using Eq. (12) directly from the levels of the symmetries and . uses Eq. (15) and uses Eq. (16). is calculated using the LCP/QFH approach.

Torsional partition functions at different temperatures for the HOOMe, MeOOMe, and HOOEt. are calculated using Eq. (12) directly from the levels of the symmetries and . uses Eq. (15) and uses Eq. (16). is calculated using the LCP/QFH approach.

Tunneling splittings and racemization times for some systems. and are in .

Tunneling splittings and racemization times for some systems. and are in .

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