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This paper introduces direct real and complex solid-harmonic integral engines that generate Coulomb tensors more efficiently than traditional Cartesian Gaussian methods by working directly in the spherical function space. This approach reduces the number of functions needed for calculations in higher angular momentum shells while maintaining the essential angular structure and properties of the tensors. The authors demonstrate that this method not only streamlines integral evaluation but also enhances applications in low-rank factorization and quantum simulation, providing a more efficient framework for many-electron computations.
Solid-harmonic integral engines cut the size of Coulomb tensors while preserving crucial angular properties, revolutionizing many-electron computation efficiency.
Electron-repulsion integrals describe the Coulomb interaction between charge distributions built from orbital basis functions. Most integral algorithms generate these quantities through Cartesian Gaussian functions, whose angular shapes are written as powers of $x$, $y$, and $z$, and then transform the result to spherical functions. This route is effective, but from $d$ shells onward the Cartesian representation contains more functions than the spherical space required by the calculation. Direct real or complex solid-harmonic engines work in that target space from the beginning. They therefore produce a smaller final Coulomb tensor while preserving the ordering, phase, and magnetic-quantum-number labels that describe its angular structure. Following this structure beyond integral evaluation reveals direct connections to the algorithms that use the tensor. Simple analytical counts quantify tensor size, angular blocks, radial Slater--Condon parameters, and pair-space work. These quantities guide low-rank factorization, local Hamiltonian construction, quantum simulation, and transformations to spinor or effective-model bases. In this way, solid-harmonic integral engines provide a direct bridge between efficient integral generation and structured many-electron computation.
