A minimal auxiliary basis fills in the Hartree–Fock CBS limit: one minimal-basis calculation plus kernel ridge regression cuts error to 7.03 mHa
Synopsis
Approaching the complete basis set (CBS) limit in quantum chemistry usually means several mid-quality calculations plus extrapolation. Here a single Hartree–Fock calculation in the minimal pcseg-0 basis is augmented with an optimized, minimal complementary auxiliary basis set (CABS) singles correction, and kernel ridge regression learns the residual. On GDB-BSIE, CABSmin/pcseg-0 with atomic detrending gives an RMSE of 7.03 mHa against RHF/cc-pV5Z, at about 16% of the cost of a cc-pVDZ single point.
Interpretation
Estimating the complete-basis-set limit of Hartree–Fock energies no longer has to start from several mid-quality basis-set calculations and an extrapolation; one minimal-basis calculation plus a physical correction and a machine-learning regression can supply it. The usual route extrapolates from two or more cardinal numbers, and its reliability degrades when the chosen basis pair is not fully inside the asymptotic convergence regime; extrapolating from a minimal basis is not suitable, since SZ/DZ extrapolation leaves a 4.95 mHa residual, close to the 5.47 mHa of cc-pVDZ alone. On GDB-BSIE with RHF/cc-pV5Z as the target, CABSmin/pcseg-0 with atomic detrending reaches an RMSE of 7.03 mHa and CABSsmall/pcseg-0 reaches 5.99 mHa, against 19.62 mHa for uncorrected pcseg-0.
The construction treats the minimal-basis calculation as a physical baseline and lets the model learn only what remains. The complementary auxiliary basis set (CABS) is a set of functions that never enters the Hartree–Fock calculation but is added in postprocessing to cover degrees of freedom the original basis misses; here it is shrunk to a minimum, and atomic or pair detrending removes known systematic trends first. CABS exponents were optimized on 200 molecules stratified from GDB-BSIE to span C, H, N, O, F, S and Cl; CABSmin, using only D functions, adds about 5% to the pcseg-0 cost, while CABSsmall costs about twice a pcseg-0 calculation.
Kernel ridge regression training is itself accelerated, making broad hyperparameter cross-validation feasible on a single core. Canonical kernel ridge regression refactorizes the kernel matrix for every hyperparameter combination, and for local representations each kernel matrix element costs quadratically in the number of atoms. Combining kernel centering, the Schur complement and eigendecomposition, and approximating the exponential in local kernels with Chebyshev polynomials, trains models 10× to 50× faster than canonical kernel ridge regression; at 4096 training points the global representation drops to 17.5% of the original cost and the local one to 2.4%.
Perspective
The demonstrated scope is Hartree–Fock energy and gradient estimation on the GDB-BSIE dataset with RHF/cc-pV5Z as the target; spatial gradients are evaluated at xTB-GFN2 minimum geometries and alchemical gradients directly on GDB-BSIE geometries, over more than 500 molecules.
The authors note that the speedups assume the decompositions and the Chebyshev approximation apply to every hyperparameter pair, which is not formally guaranteed for every dataset, so performance curves may vary with dataset and hyperparameter grid; for gradients, CABS exponents are not interpolated between elements, effectively assuming the gradients are already at CBS quality.