Improved Methods for Feynman Path Integral Calculations of Vibrational−Rotational Free Energies and Application to Isotopic Fractionation of Hydrated Chloride Ions

We present two enhancements to our methods for calculating vibrational−rotational free energies by Feynman path integrals, namely, a sequential sectioning scheme for efficiently generating random free-particle paths and a stratified sampling scheme that uses the energy of the path centroids. These improved methods are used with three interaction potentials to calculate equilibrium constants for the fractionation behavior of Cl⁻ hydration in the presence of a gas-phase mixture of H₂O, D₂O, and HDO. Ion cyclotron resonance experiments indicate that the equilibrium constant, K_eq, for the reaction Cl(H₂O)⁻ + D₂O ⇌ Cl(D₂O)⁻ + H₂O is 0.76, whereas the three theoretical predictions are 0.946, 0.979, and 1.20. Similarly, the experimental K_eq for the Cl(H₂O)⁻ + HDO ⇌ Cl(HDO)⁻ + H₂O reaction is 0.64 as compared to theoretical values of 0.972, 0.998, and 1.10. Although Cl(H₂O)⁻ has a large degree of anharmonicity, K_eq values calculated with the harmonic oscillator rigid rotator (HORR) approximation agree with the accurate treatment to within better than 2% in all cases. Results of a variety of electronic structure calculations, including coupled cluster and multireference configuration interaction calculations, with either the HORR approximation or with anharmonicity estimated via second-order vibrational perturbation theory, all agree well with the equilibrium constants obtained from the analytical surfaces.