The partition-of-unity quantum (PUQ) method for wavepacket dynamics

We introduce the Partition-of-Unity Quantum (PUQ) method for wavepacket dynamics, in which a smoothly merged, piecewise representation is constructed from local, overlapping nodes. The partition-of-unity (PU) comprises compactly supported weight functions centered on each node, enriched by a small set of basis functions attached to the node. Both the placement and the enrichment functions can be chosen independently for each node, facilitating ad hoc construction of a representation tailored to the local physics. Equations of motion follow from the Dirac-Frenkel variational principle, with the norm and energy conserved by a Crank-Nicolson propagator. On a two-dimensional coupled multi-well potential with complex dynamics, even the bare PU representation (no enrichment) achieves high fidelity, demonstrating that the PU framework itself is an effective basis for quantum dynamics. Adding local enrichment functions dramatically reduces the required node density, but the choice of enrichment becomes critical. In a variant based on normal modes (NM-PUQ), enrichment functions are chosen according to the Hessian at each node center: harmonic-oscillator eigenstates along confining directions and standing waves along soft directions. Both unenriched PUQ and NM-PUQ achieve near-reference fidelity with roughly 4× fewer degrees of freedom than a global plane wave representation and about half those of a low-rank SVD contraction. Because PUQ basis functions have compact support, all operator matrices are sparse and scale linearly with the number of nodes.