AxolotE: Automated code generation for real solid harmonic Gaussian basis functions in the McMurchie–Davidson framework
Description
Gaussian Type Functions (GTFs) are widely used as basis sets in electronic-structure calculations. Under many conditions, it is advantageous to express them in their Real Solid Harmonic (RSH) version, which reduce basis-set redundancy and facilitate the interpretation of results. A variation of the McMurchie–Davidson method directly utilizes RSH GTFs; however, evaluating the corresponding Hermite expansion coefficients involves intricate recurrence relations that complicate their implementation and maintenance in electronic-structure codes. To manage this, closed-source programs have used symbolic algebra tools, while public solutions are not available. We present AxolotE, an open-source framework for the automated generation of these coefficients and the corresponding C++ source code. We propose a general algorithm that focuses on the requirements imposed on indices at each recursion step rather than a fixed sequential ordering, simplifying its implementation and analysis. This tool, developed using Mathematica and Python modern symbolic algebra engines, enables reproducible derivation, validation, and code generation for arbitrary angular momentum, being tested up to ℓ=h. Benchmarks expose that the choice of symbolic algebra engine influences the compactness of the generated code and its computational performance, while preserving numerical accuracy. The resulting software will facilitate systematic comparison of Real Solid Harmonic integral implementations, providing ready-to-integrate routines for electronic-structure applications.