^{1}, Vanessa Audette Lynch

^{1}, Steven L. Mielke

^{2,a)}and Donald G. Truhlar

^{3,b)}

### Abstract

Practical approximation schemes for calculating partition functions of torsional modes are tested against accurate quantum mechanical results for and six isotopically substituted hydrogen peroxides. The schemes are classified on the basis of the type and amount of information that is required. First, approximate one-dimensional hindered-rotator partition functions are benchmarked against exact one-dimensional torsion results obtained by eigenvalue summation. The approximate one-dimensional methods tested in this stage include schemes that only require the equilibrium geometries and frequencies, schemes that also require the barrier heights of internal rotation, and schemes that require the whole one-dimensional torsional potential. Then, three classes of approximate full-dimensional vibrational-rotational partition functions are calculated and are compared with the accurate full-dimensional path integral partition functions. These three classes are (1) separable approximations combining harmonic oscillator-rigid rotator models with the one-dimensional torsion schemes, (2) almost-separable approximations in which the nonseparable zero-point energy is used to correct the separable approximations, and (3) improved nonseparable Pitzer–Gwinn-type methods in which approaches of type 1 are used as reference methods in the Pitzer–Gwinn approach. The effectiveness of these methods for the calculation of isotope effects is also studied. Based on the results of these studies, the best schemes of each type are recommended for further use on systems where a corresponding amount of information is available.

The authors would like to thank Hai Lin for helpful discussions. The authors would also like to thank Richard B. McClurg and H. Bernard Schlegel for providing corrections to their fifth order polynomial functions, and , Eqs. (31) and (32) of this paper. This work was supported by the National Science Foundation through Grant No. CHE03-49122, which supports the quantum mechanical path integral work, and by the U.S. Department of Energy, through Grant No. DOE-FG02-86ER13579 and by the Air Force Office of Scientific Research (AFOSR) by a Small Business Technology Transfer (STTR) grant to Scientific Applications & Research Assoc., Inc., both of which grants support the development of improved treatments of anharmonicity for reaction rate calculations.

I. INTRODUCTION

II. SYSTEM AND POTENTIAL EXPRESSIONS

III. THEORY

IV. SEPARABLE APPROXIMATIONS

A. Approximations derived from the torsional Pitzer–Gwinn-type methods

B. Methods that require geometries and frequencies

C. Methods that require geometries, frequencies, and barrier heights

1. Interpolatory treatments

2. The RPG approximation and extensions

3. Segmented methods

D. Methods that require the accurate 1D potential

1. Torsional eigenvalue summation

2. Approximations based on the Wigner–Kirkwood expression

3. Displaced-points path integrals

E. Estimates for the effective reduced moment of inertia

F. 1D Zero-point energy corrections

V. NONSEPARABLE APPROXIMATIONS

A. Full-dimensional zero-point energy corrections

B. Improved-reference Pitzer–Gwinn approximation for molecules with torsions

VI. RESULTS AND DISCUSSION

A. One-dimensional tests for

B. One-dimensional tests for isotope effects

C. Full molecular partition functions calculated by methods

D. Full molecular partition functions calculated by IRPG methods

E. Overall comparison of the best methods

VII. CONCLUDING REMARKS

### Key Topics

- Eigenvalues
- 17.0
- Zero point energy
- 12.0
- Oscillators
- 8.0
- Ground states
- 7.0
- Isotopes
- 7.0

## Figures

The potential energy along the effective torsional coordinate for as given by Eq. (1) (solid curve) and the segmented reference potential (dashed curve) are plotted as a function of . The eigenvalue spectrum of the potential of Eq. (1) is also displayed.

The potential energy along the effective torsional coordinate for as given by Eq. (1) (solid curve) and the segmented reference potential (dashed curve) are plotted as a function of . The eigenvalue spectrum of the potential of Eq. (1) is also displayed.

The C scheme moment of inertia (in ) for calculated as a function of the torsional coordinate with all other coordinates frozen at their equilibrium values (dotted line) and with all other coordinates optimized (solid line).

The C scheme moment of inertia (in ) for calculated as a function of the torsional coordinate with all other coordinates frozen at their equilibrium values (dotted line) and with all other coordinates optimized (solid line).

## Tables

Glossy of acronyms.

Glossy of acronyms.

Moments of inertia (in ) calculated by five approximate schemes for various isotopomers of .

Moments of inertia (in ) calculated by five approximate schemes for various isotopomers of .

Comparison of eigenvalues of the 1D effective torsional potential obtained with three different estimates of the reduced moment of inertia for (in and relative to the ground state) with the results of Koput *et al.* with the full-dimensional potential.

Comparison of eigenvalues of the 1D effective torsional potential obtained with three different estimates of the reduced moment of inertia for (in and relative to the ground state) with the results of Koput *et al.* with the full-dimensional potential.

Harmonic frequencies, zero-point energies (ZPE), anharmonic ZPE corrections, and tunneling splittings (all in ) for various isotopomers of .

Harmonic frequencies, zero-point energies (ZPE), anharmonic ZPE corrections, and tunneling splittings (all in ) for various isotopomers of .

Partition functions for the 1D torsional potential of calculated using various methods.

Partition functions for the 1D torsional potential of calculated using various methods.

isotope ratios for partition functions of the 1D torsional potential using various methods.

isotope ratios for partition functions of the 1D torsional potential using various methods.

Percentage errors for 1D. [Mean unsigned percentage errors (MU%E) and maximum percentage errors (Max%E) averaged over nine temperatures in the rage of for all seven isotopomers for 1D torsional partition functions with moments of inertia obtained from the C scheme at the minimum configuration. The results, for a given amount of data, are sorted in order of increasing MU%E.]

Percentage errors for 1D. [Mean unsigned percentage errors (MU%E) and maximum percentage errors (Max%E) averaged over nine temperatures in the rage of for all seven isotopomers for 1D torsional partition functions with moments of inertia obtained from the C scheme at the minimum configuration. The results, for a given amount of data, are sorted in order of increasing MU%E.]

Mean unsigned percentage errors (MU%E) (averaged over seven temperatures in the range of for all seven isotopomers) and maximum percentage errors (Max%E) full-dimensional molecular partition functions for the torsional mode treated by various methods with moments of inertia obtained from the C scheme at the minimum configuration, and the other modes treated at the quantum HO level. The results, for a given amount of data, are sorted in order of increasing sum of MU%E and Max%E.

Mean unsigned percentage errors (MU%E) (averaged over seven temperatures in the range of for all seven isotopomers) and maximum percentage errors (Max%E) full-dimensional molecular partition functions for the torsional mode treated by various methods with moments of inertia obtained from the C scheme at the minimum configuration, and the other modes treated at the quantum HO level. The results, for a given amount of data, are sorted in order of increasing sum of MU%E and Max%E.

Mean unsigned percentage errors (MU%E) (averaged over seven temperatures in the range for all seven isotopmers) and maximum percentage errors (Max%E) for full-dimensional molecular partition functions for methods that require the whole multidimensional surface.

Mean unsigned percentage errors (MU%E) (averaged over seven temperatures in the range for all seven isotopmers) and maximum percentage errors (Max%E) for full-dimensional molecular partition functions for methods that require the whole multidimensional surface.

Comparison of mean unsigned percentage errors in 1D and full-D for best methods in each class (for this purpose, “best” is defined as smallest MU%E or smallest Max%E).

Comparison of mean unsigned percentage errors in 1D and full-D for best methods in each class (for this purpose, “best” is defined as smallest MU%E or smallest Max%E).

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