Assessing and rationalizing the performance of Hessian update schemes for reaction path Hamiltonian rate calculations

The reaction path Hamiltonian (RPH) can be used to calculate chemical reaction rate constants, going beyond transition-state theory in taking account of recrossing by providing an approximation to the dynamic transmission coefficient. However, the RPH necessitates the calculation of the Hessian matrix at a number of points along the minimum energy path; the associated computational cost stands as a bottleneck in RPH calculations, especially if one is interested in using high-accuracy electronic structure methods. In this work, four different Hessian update schemes (symmetric rank-1, Powell-symmetric Broyden, Bofill, and TS-BFGS updates) are assessed to see whether or not they reliably reproduce calculated transmission coefficients for three different chemical reactions. Based on the reactions investigated, the symmetric rank-1 Hessian update was the least appropriate for RPH construction, giving different transmission coefficients from the standard analytical Hessian approach, as well as inconsistent frequencies and coupling properties. The Bofill scheme, the Powell-symmetric Broyden scheme, and the TS-BFGS scheme were the most reliable Hessian update methods, with transmission coefficients that were in good agreement with those calculated by the standard RPH calculations. The relative accuracy of the different Hessian update schemes is further rationalized by investigating the approximated Coriolis and curvature coupling terms along the reaction-path, providing insight into when these schemes would be expected to work well. Furthermore, the associated computational cost associated with the RPH calculations was substantially reduced by the tested update schemes. Together, these results provide useful rules-of-thumb for using Hessian update schemes in RPH simulations.