Improvement of the density fitting approach for the Gaussian electrostatic model via the use of the discrete Picard condition in Tychonov regularization

The Gaussian Electrostatic Model (GEM) is a density-based force field that employs Hermite Gaussian auxiliary basis sets (ABSs) to represent molecular electronic densities for accurate intermolecular interaction calculations. A critical step in GEM involves density fitting via Tychonov regularization, where the choice of the regularization parameter (λ) has traditionally relied on manual optimization, a tedious and non-systematic process. In this work, we present an automated approach to determine λ by implementing the Discrete Picard Condition (DPC), a method designed to stabilize solutions for ill-posed inverse problems. DPC ensures that the decay rate of the solution coefficients aligns with the singular values of the fitting matrix, enabling robust regularization without manual intervention. We validate this method by comparing intermolecular interactions for water dimers and ionic liquid systems against reference quantum mechanical and previously reported data. Results demonstrate that DPC-derived λ values yield interaction energies in close agreement with manually optimized parameters, achieving agreeable errors for most ABSs. Notably, DPC eliminates guesswork while maintaining accuracy, as evidenced by its successful integration into the GEM_fit software and the Psi4 quantum chemistry suite. This advancement streamlines the parameterization of polarizable force fields and enhances the reliability of GEM for molecular dynamics simulations.