Computationally Efficient Molecular Integrals of Solid Harmonic Gaussian Orbitals Using Quantum Entanglement of Angular Momentum

Abstract: 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.