Quantum Chemical Free Energies: Structure Optimization and Vibrational Frequencies in Normal Modes

A computational protocol is presented that uses normal mode coordinates for structure optimization and for obtaining harmonic frequencies by numerical differentiation. It reduces numerical accuracy problems encountered when density functional theory with plane wave basis sets is applied to systems with flat potential energy surfaces. The approach is applied to calculate Gibbs free energies for adsorption of methane, ethane, and propane on the Brønsted acidic sites of zeolite H-CHA. The values obtained (273.15 K, 0.1 MPa,), -0.25, -5.95, and -16.76 kJ/mol, respectively, follow the trend of the experimental values, which is not the case for results obtained with the standard approach (Cartesian optimization, frequencies from Cartesian distortions). Anharmonicity effects have been approximately taken into account by solving one-dimensional Schrödinger equations along each normal mode. This leads to a systematic increase of the Gibbs free energy of adsorption of 4.5, 5.0, and 3.1 kJ/mol for methane, ethane, and propane, respectively, making adsorption at a given pressure and temperature less likely. This is due to an increase of low vibrational frequencies associated with hindered translations and rotations of the adsorbed molecules and the floppy modes of the zeolite framework.