Anharmonic Vibrational Properties from Intrinsic n‐Mode State Densities

A method for calculating fully anharmonic vibrational state counts, state densities, and partition functions for molecules is presented. The method makes use of a new quantity, the intrinsic density of states, which is associated with the states that uniquely arise from a given mode, mode pairing, or higher-order mode coupling. By using only low-order intrinsic densities, the fully coupled anharmonic vibrational result can be constructed, as shown by our application of the method to methane, CH4, and cyclopropene, C3H4. Truncation of the intrinsic expansion at the coupling of pairs of modes yields greatly improved scaling over direct evaluation of the full-dimensional result and recovers a large fraction of the total anharmonicity. We also discuss the relation of the new quantities to the structure of the potential energy surface. SECTION: Kinetics and Dynamics A predictions of kinetics and thermodynamics depend critically on accurate evaluation of the rovibrational density of states, ρ(E), and its related quantities. In the classic Rice, Ramsperger, Kassel, and Marcus (RRKM) theory of unimolecular reactions, the rate constant is proportional to the number of states at the transition state divided by the density of states of the reactant. Similarly, in bimolecular canonical transition state theory (TST) the rate constant is dependent on the density of states through the appearance of partition functions corresponding to the reactants and to the transition state. Unfortunately, these quantities are difficult to evaluate accurately for several reasons, including the need for proper quantization, couplings between rotations and vibrations, and anharmonic couplings among the vibrations. In this letter we present a general and efficient scheme for including the important anharmonic coupling. A common approximation, which makes the computation of ρ(E) quite amenable, is to use a separable model in which every vibrational degree of freedom is assumed to be totally uncoupled from any of the others. Most often each mode is further assumed to be harmonic, although sometimes modes are given special treatment as Morse oscillators, hindered rotors, etc. Algorithms and computer implementations of these methods, e.g., the semiclassical Whitten-Rabinovitch (WR) approximation, the Beyer−Swinehart (BS) and Stein−Rabinovitch (SR) state counting algorithms, or the steepest-descent method, are readily available. As might be expected, at higher energies these approximations can significantly underestimate the anharmonicity because of the neglect of the coupling between vibrational modes. This is true even when the underlying independent vibrations are treated as anharmonic oscillators. Many attempts have been made to improve on the separable approximation by looking at the specific coupling between different modes. For instance, the coupling of bends to stretches has been studied and empirical models describing this coupling have been constructed, the role of stretch− stretch coupling has been investigated in triatomic systems, numerous methods for treating torsional motions have been developed, and Monte Carlo integration has been applied to calculation of quantum vibrational states using a spectroscopic (e.g., Dunham) expansion, which includes some anharmonic terms. It is also worth noting the recent semiempirical work of Schmatz as well as the thermodynamic method, which relies on experimental data, the density correlation function method of Jeffreys et al., and the use of path-integral methods to calculate the quantum partition function directly. The accurate inclusion of the coupling terms, however, remains an open issue. Despite its associated problems, separability has certain nice features that we wish to retain. It allows a complex problem, the calculation of the full-dimensional coupled density of states, ρ(E), to be broken down into a set of small, readily computable quantities that can be then reassembled to yield the full result. As such, it renders large, potentially intractable problems amenable to computation. In this Letter, we Received: June 8, 2013 Accepted: July 10, 2013 Published: July 10, 2013 Letter