Computation of rate coefficients in solutions based on transition state theory combined with a heuristically corrected polarizable continuum model: intermolecular Diels–Alder reactions as case studies

Abstract

Transition state theory (TST) based on activation parameters computed using quantum mechanics calculations combined with the polarizable continuum model (QM/PCM) is a fundamental tool for investigating reaction rates in the liquid phase. In conventional QM/PCM methods, thermodynamic data and partition functions for a solute are often derived from a quasi-ideal gas treatment (IGT) widely implemented in commercially available computation packages. This approach tends to overestimate entropy because calculations of thermodynamic parameters in the liquid phase ignore hindered translational and rotational modes in real solutions. The present work formulated partition functions for more realistic solutes hindered by surrounding solvent molecules in conjunction with the basic QM/PCM concept. In addition, a configuration partition function for solute molecules at a standard concentration of 1 mol dm⁻³ was incorporated using a simple lattice model. The canonical partition function and thermodynamic functions were derived based on statistical thermodynamics for localized systems. Expressions for rate coefficients within TST were also derived with a consistent formalism based on the standard state selected in partition function calculations. The performance of the proposed method was assessed by predicting rate coefficients for three different Diels–Alder reactions and comparing these with experimental results. QM/PCM calculations at the G4//ωB97X-D/6-311++G(d,p)/IEF-PCM level of theory with corrections for the dispersion and repulsion energies were performed to obtain the electronic structures of stationary points on potential energy surfaces as a means of finding activation enthalpy, entropy and Gibbs energy values based on revised partition functions as well as predicting rate coefficients. The activation Gibbs energies obtained from our proposed method were lower than those obtained from the IGT method due to reasonable entropy computations. The proposed method overestimated the rate coefficients by one to two orders of magnitude compared to the experimental values, whereas the IGT method underestimated them by the same amount. This discrepancy arises because the proposed method calculates the partition function from the viewpoint of a localized system, whereas the IGT method calculates it from the viewpoint of a non-localized system. Given that actual liquids exist in a state between non-localized and localized systems, it is essential to formulate the partition function in a way that more accurately represents the liquid state.