  Evaluating multi-center molecular integrals with Cartesian Gaussian-type
basis sets has been a long-standing bottleneck in electronic structure theory
calculation for solids and molecules. We have developed a vector-coupling and
vector-uncoupling scheme to solve molecular Coulomb integrals with solid
harmonics basis functions(SHGO). Solid harmonics are eigenstates of angular
momentum, making it possible to factorize molecular integrals. By combining
solid harmonic addition, differential and product rules, the computationally
costly multi-center four-center integrals can be factored into an angular part
and a radial component dependent on the atomic positions. The potential
speed-up ratio in evaluating molecular nuclear Coulomb integrals in our method
can reach up to four orders of magnitude for atomic orbitals with high angular
momentum quantum numbers. The foundation underpinning the mathematical
efficiency is the quantum angular momentum theory, where both vector-coupling
and vector-uncoupling schemes correspond to unitary Clebsch-Gordan
transformations that act on quantum angular momentum states, influencing their
degree of entanglement. By incorporating quantum angular momentum through these
transformations, the entanglement of the states can be reduced, and the less
entanglement there is for a quantum system, the easier it is to simulate. The
highly efficient method unveiled here opens new avenues for accelerated
material and molecule design and discovery.
