Hindered rotation and bending anharmonicity in aluminum alkyls: implications for methylaluminoxane thermodynamics

Accurate thermodynamic calculations for aluminum alkyls require proper treatment of low-frequency vibrations poorly described by the harmonic approximation (HA). Here, we present a systematic investigation of hindered rotation and out-of-plane bending in aluminum trichloride (ATC) and its methyl derivatives, employing advanced computational methods to perform anharmonic entropy corrections, such as torsional eigenvalue summation (TES), the extended two-dimensional torsion method (E2DT), the multi-structural approximation with torsional anharmonicity (MS-T), and Fourier grid Hamiltonian (FGH). Our results reveal distinct structure-dependent behaviors: monomers exhibit near-free methyl rotations where the HA overestimates entropy by 20-30 J K^-1 mol^-1, while dimers show more hindered rotations adequately described by the HA around room temperature. Out-of-plane bending modes in dimers display increasing energy level spacing that reduces their thermodynamic contribution compared to HA predictions. Simple quasi-harmonic approaches reduce HA inaccuracy for monomers but systematically underestimate the entropy of low-frequency modes in dimers. By applying appropriate methods to ATC and trimethylaluminum (TMA), we achieve excellent agreement with experimental entropy values (within 3.5 J K^-1 mol^-1). This validation supports extending our approach to methylaluminoxane (MAO), critical for understanding polyolefin catalysis. The dimer species studied here directly relate to MAO edge sites involved in metallocene catalyst activation. Our findings suggest that mode-specific anharmonic corrections are essential for accurate thermodynamic modeling of MAO and similar aluminum-containing systems.