A version-one arXiv preprint reports that projected q-shells (PQS), a way of organizing nested gausslet basis functions, showed smaller numerical errors than the standard White-Lindsey (WL) nesting in tests of the hydrogen molecular ion, H2 and restricted C2. The clearest gains appeared in compact basis settings, while the difference narrowed at the highest orders tested.
The study asks whether PQS can retain full local order without increasing the nominal shell dimension. In standard nesting, a joining shell is divided into disjoint faces, edges and corners, reducing polynomial completeness from order q to q minus 2. Here, q means the parent or local polynomial order. PQS is intended to restore the two lost orders.
An ideal match, then a small gap
For an ideal, undistorted parent, the mathematical construction has an exact property: the PQS span is equivalent to the span of an overcomplete basis of overlapping patches. The span is the range of functions the construction can represent, so under those conditions the two descriptions cover the same space.
The numerical check of that ideal case found exactly 98 directions per added shell. The final space was 419-dimensional, and the projectors used to compare it with the overlapping construction agreed to 1.4 × 10⁻¹⁵, a near-machine-precision match.
The mapped construction was close but not exact. In the reported 419-function PQS space, the normalized residual from the union of overlapping patches was 2.353 × 10⁻³, and the maximum principal angle from the leading union subspace was 0.334 degrees. In other words, the mapped case retained a close but not exact match to the overlapping space.
Projection also left small, measurable moment defects. In these diagnostics, a moment mismatch tracks how the constructed function's center and shape differ from the target. The outward-growing shell-center mismatch reached 0.0061 function widths in the undistorted test. With standard coordinate distortion, the mean mismatch ranged from 0.050 to 0.079 widths; t2 stayed below 0.067 and t11 below 2.6 × 10⁻⁴.
A smoother test
A separate smooth-function test grouped results by degree and active Cartesian directions, creating 25 comparison groups. PQS had lower mean loss in every group, and the median loss reduction was more than fourfold.
The molecular comparisons
The molecular benchmark moved from a one-electron representation test to a correlated two-electron problem and then to a multi-orbital Hartree-Fock state. It used the hydrogen molecular ion, H2 and restricted C2. That progression checked the basis construction first on a simple electronic state and then on more involved molecular calculations.
In the one-electron test, PQS was more accurate for both states throughout the reported comparison. At the compact order labeled ns = 4, it reduced the g-state error by a factor of 6.9 and the u-state error by 5.8 relative to WL. For the odd-parity state, the PQS error fell from 2.17 mHa at ns = 5 to 0.244 mHa at ns = 6. The corresponding WL error fell from 8.02 to 1.505 mHa.
H2 restricted Hartree-Fock results were less straightforward. Across 39 geometry and order points, PQS had smaller errors under the integral diagonal approximation, or IDA, and lower orbital energies when evaluated with the parent Hamiltonian. Direct RHF energies were lower for WL in 29 cases, a pattern associated with negative WL IDA errors. Against the extrapolated reference, however, PQS was closer in both panels at all 39 points.
In a finite-basis full-configuration-interaction comparison for H2, projected FCI errors were smaller for PQS at every plotted geometry and order. The advantage was largest at ns = 4, remained substantial at ns = 5 and became small at ns = 6. The tested difference was therefore more modest once the basis order was high.
Restricted C2 showed a large compact-order separation. At ns = 5, the WL error magnitude was about 7.5 times the PQS value at R = 2.35 bohr and about six times as large at R = 3.00 bohr. PQS remained better at ns = 6, while both constructions entered the sub-mHa regime by ns = 7. The signed errors were nonmonotonic, which may reflect cancellation between finite-basis and diagonal-interaction errors.
A result with limits
The exact span statement has a narrow scope. It applies to an ideal, undistorted parent; the mapped 419-function comparison was close but not exact, and the moment checks showed projection-related mismatches. The numerical construction therefore retained the reported span advantage without preserving every ideal property exactly.
The energy results should be read in the same way. The molecular evidence covers the hydrogen molecular ion, H2 and restricted C2, with the strongest differences at compact orders and smaller gaps at higher order. It is evidence from those reported benchmarks rather than a universal performance claim.
The document is an arXiv version-one preprint dated 28 Aug 2026. It says the work was supported by the U.S. NSF under Grant DMR-2412638, reports no conflicts of interest, and makes the study data available from the author on reasonable request. The open-source PQS implementation is available in GaussletBases.jl.
Paper data and sources
Original title: Projected q-Shells for Nested Gausslet Bases
Authors: Steven R. White
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text