**Machine Learning in Electronic-Structure Theory**

## Adaptive Diagonal Basis Sets for Electronic Structure

**
Michael Lindsey, University of California, Berkeley (UC Berkeley)
**

**Monday, March 25, 2024**

**Abstract**:

Electronic structure calculations usually begin with the choice of a truncated basis set for functions of a single space variable, followed by a Galerkin projection onto the associated antisymmetrized tensor product basis. This projection determines a second-quantized "quantum chemistry Hamiltonian" in which the problem details are encoded into one matrix and one four-index tensor, which respectively encode the single-electron and two-electron contributions to the energy. In particular, the four-index tensor of electron repulsion integrals (ERI) can be difficult to manage computationally. Diagonal basis sets---smooth, orthogonal basis sets that behave like delta functions---can be constructed to yield ERI that are highly structured in a way that simplifies many downstream calculations. However, these constructions are grid-based and can require a large basis set to yield acceptable accuracy. We describe two new approaches for constructing adaptive diagonal basis sets that conform to individual problem geometries. The first of these computes a highly accurate solution of a mass transport problem to define a grid deformation and then constructs a quantum chemistry Hamiltonian using pseudospectral insights that sidestep expensive numerical integrations. The second defines a nested generalization of the recently introduced gausslet basis set. This latter construction relies on new mathematical insights into diagonal 1D basis sets. (The approaches are based on collaborations with Sandeep Sharma and Steve White, respectively.)