Economized path integrals.

The Hessian of the ring polymer spring potential in the standard Trotter path integral is a P × P symmetric circulant matrix with a centroid eigenvalue of zero. All such matrices commute and are diagonalized by the same bead-to-normal mode transformation matrix, and their eigenvalues contain ⌈P/2⌉ - 1 degenerate pairs by symmetry. However, this still leaves some freedom to improve on the Trotter approximation: one can optimize the remaining ⌊P/2⌋ independent non-zero normal mode frequencies to fit the exact quantum mechanical radii of gyration of harmonic ring polymers with frequencies in the range 0 ≤ ω ≤ ωmax, where ωmax is the maximum physical frequency in the problem of interest. The optimization involves solving a simple least squares problem for the optimum (economized or "Eco") internal mode frequencies. The remainder of the calculation then proceeds in the same way as a Trotter path integral calculation. An example application to hexagonal ice shows that the convergence of the Eco path integral is comparable to that of the fourth order Suzuki-Chin path integral, but with purely second order Trotter effort. There is no need to calculate the projected Hessians that arise in the Suzuki-Chin method by finite differences; there is no need to develop any new estimators for observables, and once the Eco frequencies have been calculated, the implementation of the Eco path integral involves changing just a few lines of a Trotter path integral code. To provide a more impressive example, we have implemented the Eco method in GPUMD and used it to converge the (negative) thermal expansion coefficient and the constant pressure heat capacity of MOF-5 with a machine-learned neuroevolution potential.