^{1,a)}and Ahren W. Jasper

^{1,b)}

### Abstract

An algorithm is presented for calculating fully anharmonic vibrational state counts, state densities, and partition functions for molecules using Monte Carlo integration of classical phase space. The algorithm includes numerical evaluations of the elements of the Jacobian and is general enough to allow for sampling in arbitrary curvilinear or rectilinear coordinate systems. Invariance to the choice of coordinate system is demonstrated for vibrational state densities of methane, where we find comparable sampling efficiency when using curvilinear z-matrix and rectilinear Cartesian normal mode coordinates. In agreement with past work, we find that anharmonicity increases the vibrational state density of methane by a factor of ∼2 at its dissociation threshold. For the vinyl radical, we find a significant (∼10×) improvement in sampling efficiency when using curvilinear z-matrix coordinates relative to Cartesian normal mode coordinates. We attribute this improved efficiency, in part, to a more natural curvilinear coordinate description of the double well associated with the H2C–C–H wagging motion. The anharmonicity correction for the vinyl radical state density is ∼1.4 at its dissociation threshold. Finally, we demonstrate that with trivial parallelizations of the Monte Carlo step, tractable calculations can be made for the vinyl radical using direct ab initio potential energy surface evaluations and a composite QCISD(T)/MP2 method.

Financial support from the Division of Chemical Sciences, Geosciences, and Biosciences, Office of Basic Energy Sciences, U.S. Department of Energy (Grant No. DE-AC04-94-AL85000) is gratefully acknowledged.

I. INTRODUCTION

II. THEORY

III. RESULTS AND DISCUSSION

A. Methane

B. Vinyl radical

IV. CONCLUSIONS

### Key Topics

- Normal modes
- 16.0
- Electron densities of states
- 11.0
- Potential energy surfaces
- 9.0
- Dissociation
- 7.0
- Methane
- 6.0

## Figures

Comparison of the anharmonicity correction to ρ for CH4 computed using Cartesian normal-mode and z-matrix coordinates.

Comparison of the anharmonicity correction to ρ for CH4 computed using Cartesian normal-mode and z-matrix coordinates.

(a) Relative standard deviation and (b) relative maximum error for W at three energies: 10 000 cm−1, 15 000 cm−1, and 20 000 cm−1 using both coordinate systems. Each point on the plot corresponds to the convergence metrics evaluated for one of the three energies using the specified coordinate system and for some value of M. The solid curves are the best fits of the data points to the form a · M b .

(a) Relative standard deviation and (b) relative maximum error for W at three energies: 10 000 cm−1, 15 000 cm−1, and 20 000 cm−1 using both coordinate systems. Each point on the plot corresponds to the convergence metrics evaluated for one of the three energies using the specified coordinate system and for some value of M. The solid curves are the best fits of the data points to the form a · M b .

(a) One-dimensional state density for the H2C–C–H vibration for two choices of the bending coordinate and the harmonic approximation to it. The energy of the barrier between the two equivalent minima is shown as the pink vertical line. (b) The potential energy curves along two coordinates.

(a) One-dimensional state density for the H2C–C–H vibration for two choices of the bending coordinate and the harmonic approximation to it. The energy of the barrier between the two equivalent minima is shown as the pink vertical line. (b) The potential energy curves along two coordinates.

(a) Relative standard deviation and (b) relative maximum error for W at two energies, 5000 cm−1 and 10 000 cm−1, using both coordinate systems. The solid curves are the best fits of the data points to the form a · M −0.5.

(a) Relative standard deviation and (b) relative maximum error for W at two energies, 5000 cm−1 and 10 000 cm−1, using both coordinate systems. The solid curves are the best fits of the data points to the form a · M −0.5.

Anharmonicity corrections to the (a) density of states and (b) partition function for the vinyl radical.

Anharmonicity corrections to the (a) density of states and (b) partition function for the vinyl radical.

## Tables

Comparison of vibrational partition functions obtained using the harmonic approximation, the Pitzer-Gwinn8 corrected harmonic partition function, and VCI. Results above 1500 K are not shown for VCI because it became difficult to converge sufficient vibrational states at high enough energies to converge Q.

Comparison of vibrational partition functions obtained using the harmonic approximation, the Pitzer-Gwinn8 corrected harmonic partition function, and VCI. Results above 1500 K are not shown for VCI because it became difficult to converge sufficient vibrational states at high enough energies to converge Q.

Comparison of partition functions calculated with and without the inclusion of coupling between vibrations and rotations. All partition functions are calculated classically and numbers in square brackets correspond to powers of ten. The columns, respectively, are: the purely vibrational harmonic oscillator partition function, Q vib,har; the purely vibrational anharmonic sampled partition function, Q vib,anh; the rovibrational partition function using the harmonic vibrations and the rigid rotor approximation, Q rovib,anh–rr; the rovibrational partition function using the anharmonic sampled vibrations and the rigid rotor approximation, Q rovib,anh–rr; and the rovibrational partition function computed fully using the sampling algorithm, Q rovib,anh. The anharmonicity corrections are defined as f vib = Q vib,anh/Q vib,har, f rot = Q rovib,anh/Q rovib,anh–rr, and f rovib = Q rovib,anh/Q rovib,har–rr.

Comparison of partition functions calculated with and without the inclusion of coupling between vibrations and rotations. All partition functions are calculated classically and numbers in square brackets correspond to powers of ten. The columns, respectively, are: the purely vibrational harmonic oscillator partition function, Q vib,har; the purely vibrational anharmonic sampled partition function, Q vib,anh; the rovibrational partition function using the harmonic vibrations and the rigid rotor approximation, Q rovib,anh–rr; the rovibrational partition function using the anharmonic sampled vibrations and the rigid rotor approximation, Q rovib,anh–rr; and the rovibrational partition function computed fully using the sampling algorithm, Q rovib,anh. The anharmonicity corrections are defined as f vib = Q vib,anh/Q vib,har, f rot = Q rovib,anh/Q rovib,anh–rr, and f rovib = Q rovib,anh/Q rovib,har–rr.

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