A Pruned Collocation-Based Multiconfiguration Time-Dependent Hartree Approach Using a Smolyak Grid for Solving the Schrödinger Equation with a General Potential Energy Surface

Abstract

Standard multiconfiguration time-dependent Hartree (MCTDH) calculations use a direct product basis and rely on the potential being a sum of products (SOPs). The size of the direct product MCTDH basis scales exponentially with the number of atoms. Accurate potentials may not be SOPs. We introduce an MCTDH approach that uses a pruned basis and a collocation grid. Pruning the basis significantly reduces its size. Collocation makes it possible to do calculations using a potential that is not a SOP. The collocation point set is a Smolyak grid. Strategies using pruned MCTDH bases already exist, but they work only if the potential is a SOP. Strategies for using MCTDH with collocation also exist, but they work only if the MCTDH basis is a direct product. In this paper, we combine a pruned basis with collocation. This makes it possible to mitigate the direct-product basis size problem and do calculations when the potential is not a SOP. Because collocation is used, there are no integrals and no need for quadrature. All required matrix-vector products can be evaluated sequentially. We use nested sets of collocation points and hierarchical basis functions. They permit efficient inversion of the (large) matrix whose elements are basis functions evaluated at points, which is necessary to transform values of functions at points to basis coefficients. The inversion technique could be used outside of chemical physics. We confirm the validity of this new pruned, collocation-based (PC-)MCTDH approach by calculating the first 50 vibrational eigenenergies of CH2NH.