Adaptive-Partitioning Multilayer Dynamics Simulations: 2. Implementations of the Permuted and Interpolated Adaptive-Partitioning Gradients

Recently, an adaptive-partitioning multilayer Q1/Q2/MM method was proposed, where Q1 and Q2 denote, respectively, two distinct quantum-mechanical levels of theory and MM, the molecular-mechanical force fields. Such a multilayer model resembles the ONIOM (our own N-layered integrated molecular orbital and molecular mechanics) model by Morokuma and co-workers, but it is distinguished by on-the-fly reclassifying atoms to be Q1, Q2, or MM in dynamics simulations. To smoothly blend the levels of descriptions of the atoms, buffer zones are introduced between adjacent layers, and the energy is smoothly interpolated. In particular, the Q1/Q2 interaction energy was expressed in two different formalisms: permuted and interpolated adaptive-partitioning (PAP and IAP), respectively. While the PAP energy is based on a weighted many-body expansion, the IAP energy is derived via alchemical quantum calculations with interpolated Fock and overlap matrices. In this article, we examine in-depth the irregularities in the IAP energy near the boundary between the buffer and Q2 zones, which were found prominent in some calculations. These irregularities are due to basis-set linear dependencies, which can be effectively suppressed using a cutoff for the weighted atomic orbital coefficients. Furthermore, we derived and implemented the gradients for both PAP and IAP. Test calculations on a series of water cluster models show perfectly smooth gradients in PAP, while a minor discontinuity occurs in IAP gradients at the buffer/Q2 boundary. The energy and gradient discontinuities in IAP become smaller when moving the buffer/Q2 boundary further away from the Q1 center and when increasing the size of the basis sets used. Overall, those discontinuities are controllable, and possible ways to further diminish them are discussed