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Scaling Density Functional Theory with Gaussian Splatting

Andrés Guzmán-Cordero, Cindy Zhang, Majdi Hassan, Marta Skreta, Kirill Neklyudov, Matija Medvidović

arXiv:2609.31483v1physics.chem-phcond-mat.mtrl-scics.LGphysics.comp-ph

TL;DR

Fixed atom-centered bases limit how DFT accuracy and cost scale, and can constrain representation of physical regimes such as anions and stretched bonds. GS-DFT optimizes floating Gaussian orbitals with direct energy minimization and adaptive density fitting; the paper reports improved accuracy scaling, diffuse-physics results, and memory efficiency, with force-reference comparisons carrying a stated caveat.

  • Problem

    Fixed-basis parameterization constrains DFT scalability and physical fidelity, while basis errors can also affect densities and forces.

    Source: §1 · pp. 1–2

  • Method

    GS-DFT optimizes floating anisotropic Gaussian orbitals by direct variational energy minimization and uses screened adaptive density fitting for two-electron terms.

    Source: §1 · p. 2, §4 · p. 6

  • Results

    GS-DFT reports 2.3× better accuracy scaling across systems, reproduces or surpasses augmented-baseline results for anions and stretched bonds, and reduces density-fitting memory to O(M^2).

    Source: §1 · p. 2

  • Takeaways & Limitations

    Adaptive density fitting brings a roughly 10,000-electron system within single-node optimization, while the conclusion identifies longer convergence time as a constraint.

    Source: §1 · p. 2, §6 · p. 9

    Forces may have different absolute error ordering against CBS and ISC references; an independently discretized, converged accurate method would resolve absolute force accuracy.

    Source: §5.1 · p. 7

Abstract

from arXiv · show

Density functional theory (DFT) strikes a practical balance between accuracy and computational cost in many problems of computational chemistry and materials science. However, many DFT calculations are limited by fixed atom-centered basis sets, which dictate how accuracy and cost scale with system size. We propose Gaussian Splatting for Density Functional Theory (GS-DFT), which represents molecular orbitals as a cloud of Gaussians whose positions, shapes, and mixing coefficients are optimized jointly by gradient descent to minimize the energy without training data. Conceptually, GS-DFT is 3D Gaussian splatting with the renderer replaced by quantum mechanics. We introduce two key solver components: adaptive density fitting with screening for efficient evaluation of two-electron integrals, and a regularized differentiable orthogonalization of the molecular orbitals. Empirically, the optimized basis reaches the accuracy of the largest conventional basis sets with a fraction of the parameters, converging systematically in energy, density, and nuclear forces. At equal parameter count, it captures the stretched-bond and anion physics that fixed bases only recover with specialized basis augmentation. The resulting solver exhibits quadratic peak memory scaling in the cloud size, allowing us to simulate systems of up to 2,742 atoms (10,406 electrons) without any modifications at triple-zeta scale using a single four-GPU node.

1 INTRODUCTION

Fixed atom-centered bases constrain DFT’s scalability and physical fidelity, while errors in orbitals can also affect densities and forces. GS-DFT instead optimizes floating Gaussian orbitals directly and reports improved accuracy scaling, diffuse-physics performance, and memory efficiency.

1 INTRODUCTION

  • Fixed basis functions constrain scalability and physical fidelity, and orbital errors can be amplified in densities and forces.Forces converge more slowly with basis size than energies and require additional correction terms for atom-centered bases.Source: §1 · pp. 1–2
  • GS-DFT represents orbitals with floating anisotropic Gaussians and minimizes total DFT energy directly using automatic differentiation.This replaces the conventional self-consistency cycle with a parallelizable direct optimization procedure.Source: §1 · p. 2
  • 2.3× better accuracy scaling is reported for GS-DFT across all systems.The reported scaling also holds across common exchange-correlation functional types.Source: §1 · p. 2
  • At augmented-basis budgets, GS-DFT reproduces or surpasses augmented-baseline results for anions and stretched bonds.Fixed bases require expert-triggered augmentation for these regimes.Source: §1 · p. 2
  • Adaptive density fitting reduces memory cost to O(M^2), enabling optimization of a roughly 10,000-electron system on one node.The reported memory footprint is reduced by approximately 20×.Source: §1 · p. 2

2 BACKGROUND

The background frames Kohn–Sham DFT as an energy minimization over orbitals, commonly represented in fixed atom-centered Gaussian bases. It describes the computational costs of those bases, density fitting, and the conceptual parallel with 3D Gaussian splatting.

2 BACKGROUND

  • Kohn–Sham DFT models a surrogate non-interacting system whose density matches the full theory, and finds orbitals by minimizing an energy functional.The functional includes kinetic, external, Hartree, exchange-correlation, and background contributions.Source: §2 · p. 3
  • Traditional orbitals are linear combinations of fixed Gaussian-type atomic orbitals, with coefficients optimized while basis functions remain atom-centered.The self-consistent-field iteration repeatedly diagonalizes a B × B effective Hamiltonian at O(B^3) cost.Source: §2 · p. 3
  • Naive evaluation of the electron-repulsion tensor has O(B^4) entries, creating O(B^4) computation and memory costs.The integrals over Gaussian products are analytically tractable, but the full tensor is expensive to evaluate.Source: §2 · p. 3
  • Density fitting approximates the quadratic density expansion with a linear auxiliary-basis expansion fitted by least squares.Its approximation error is second order in the Coulomb-metric projection residual, while choosing an auxiliary basis introduces a separate error source.Source: §2 · p. 4
  • GS-DFT shares 3D Gaussian splatting’s Gaussian-mixture optimization structure, replacing photometric loss with the DFT total-energy functional.The paper contrasts projections onto image planes with projections onto the occupied electronic-state subspace.Source: §2 · p. 4

3 RELATED WORK

Prior work has explored floating Gaussian orbitals, adaptive spatial discretizations, and learned DFT components. GS-DFT differs by directly targeting the orbital basis rather than relying on fixed or lightly relaxed atom-centered bases.

3 RELATED WORK

  • Floating-Gaussian DFT approaches, including Pederson et al. (1988), remained inaccurate and unscalable without a suitable density-fitting scheme.Adaptive finite-element, wavelet, multiwavelet, and gausslet methods refine space but multiply degrees of freedom rather than moving them.Source: §3 · p. 4
  • Differentiable direct-optimization and learned-orbital methods generally retain atom-centered bases; GS-DFT targets the basis itself and can compose with them.The surveyed learned methods also predict interatomic potentials, exchange-correlation functionals, densities, Hamiltonians, or properties.Source: §3 · p. 4

4 METHOD

GS-DFT uses shared floating Gaussian primitives to represent occupied orbitals, with differentiable orthogonalization and adaptive density fitting supporting direct energy optimization. The method refreshes a screened auxiliary basis during optimization and obtains forces by automatic differentiation.

4 METHOD

  • GS-DFT represents occupied Kohn–Sham orbitals with a learnable cloud of Gaussian primitives.Each anisotropic primitive has a center, log-eigenvalues, and orientation, and the orbitals are formed by linearly mixing shared primitives.Source: §4 · pp. 4–5
  • Löwdin orthonormalization enforces C^T S(Θ) C = I while allowing gradients to propagate through the constraint.The backward pass regularizes near-degenerate eigenvalue differences; away from degeneracies, the regularization has no effect at convergence.Source: §4 · p. 5
  • Adaptive density fitting selects splat pair-products as auxiliary Gaussians, reducing an exact O(M^4) Hartree evaluation to a screened set.Pairs are retained when |S_μν| > τ; practical overlap neighborhoods yield N_aux = O(M).Source: §4 · p. 6
  • The auxiliary basis is frozen for K optimization steps, then refreshed after those gradient updates to the Gaussian primitives.The fit coefficients are obtained analytically by least-squares fitting.Source: §4 · p. 6
  • The differentiable energy calculation enables direct gradient-based optimization instead of the self-consistent-field algorithm.The auxiliary metric is factorized once per refresh and reused through K triangular solves, amortizing its dominant cubic operation.Source: §4 · pp. 5–6
  • Because no basis function is anchored to an atom, forces can be computed by automatic differentiation without the Pulay correction.The force evaluation is performed after the GS-DFT run converges.Source: §4 · p. 6
  • The method’s reported observables include energy residual per atom, density variation per electron, and maximum nuclear-force deviation against CBS and ISC references.These quantities are plotted against the number of free parameters.Source: Figure 3 · p. 7

5 EXPERIMENTS

The experiments assess GS-DFT’s accuracy across observables and functionals, its representation of diffuse physics, and its accuracy and memory scaling. They compare splats with conventional Gaussian bases and test large systems against reference calculations.

5.1 ACCURACY AND FUNCTIONAL COVERAGE

  • Across water, ethanol, and alanine dipeptide, the experiments track energy, density, and forces; density predictions converge reliably and are similar to or better than GTO references.Absolute force-error ordering may depend on the chosen CBS or ISC baseline; a third independently discretized and converged method would resolve absolute force accuracy.Source: §5.1 · p. 7
  • For ethanol at matched function count M = 174, GS-DFT energies are closer to the CBS limit than cc-pVTZ for all four tested functionals.For PBE and r2SCAN, optimized energies are in the range of cc-pV5Z using 29% of its functions; Table 1 covers ethanol’s ∆CBS and %M.Source: §5.1 · p. 7, Table 1 · p. 8
  • At matched function count, GS-DFT reproduces GTO references for GW100 ionization potentials, while both methods retain errors against CCSD(T) and experiment attributed to orbital relaxation and missing electronic correlation.Table 2 reports −εHOMO in eV under Hartree–Fock and identifies CCSD(T) and experimental GW100 references.Source: §5.1 · p. 8, Table 2 · p. 8

5.2 SPLATS AS A PHYSICAL REPRESENTATION

  • At matched unaugmented function count, splats achieve lower energies than cc-pVTZ and aug-cc-pVTZ for stretched LiF, while the Figure 4 caption reports a growing margin over the unaugmented basis.For fluoride, Figure 4(a) reports basis-set convergence using the energy residual from the CBS limit in mHa.Source: §5.2 · p. 8, Figure 4 · p. 8

5.3 ACCURACY AND MEMORY SCALING

  • 2.3× sharper energy-error scaling exponents on average than GTOs, with similar scaling improvements across the studied systems.Figure 5 compares GTO rungs against the CBS energy and splat clouds against the ISC energy, fitting limits over four runs with different parameter counts.Source: §5.3 · p. 9, Figure 5 · p. 9
  • GS-DFT tackles the same large FMO systems as the references using a fraction of the memory estimated for density-fitted cc-pVTZ and avoiding inter-node communication.Table 3 reports energy in Ha and memory in GB; one 4×H200 GPU node has 564 GB in total.Source: §5.3 · p. 9, Table 3 · p. 9

6 CONCLUSION

GS-DFT replaces fixed atom-centered orbital bases with floating Gaussians optimized by gradient descent on variational energy. It achieves lower energies, correct observables, and adaptive representation of diffuse physics, but requires more optimization steps and time to converge.

6 CONCLUSION

  • GS-DFT replaces fixed atom-centered bases with floating Gaussian orbitals whose parameters are optimized by gradient descent on variational energy.The conclusion describes the resulting representation as adaptive.Source: §6 · p. 9
  • The adaptive representation achieves lower energies, delivers correct observables, and automatically captures diffuse physics.Source: §6 · p. 9
  • GS-DFT requires more optimization steps and therefore more time to converge.Physics-informed natural-gradient optimization is identified as a future research direction.Source: §6 · p. 9

(Supplementary Material)

The supplementary theoretical-properties section frames questions about which guarantees floating Gaussians retain or change. It examines molecular symmetries, forces, adaptive density fitting, and regularized eigensolving.

A THEORETICAL PROPERTIES OF GS-DFT

A.1 SYMMETRY OF THE REPRESENTATION

The splat representation preserves energy under rigid motions through an explicit transformation of its parameters, making the symmetry a property of the architecture rather than something learned by optimization. This holds for the full orthogonal group, although Adam’s optimization trajectories need not be rotation equivariant.

A.1 SYMMETRY OF THE REPRESENTATION

  • An explicit transformation of splats and nuclei preserves the orbitals and density under rigid motion while keeping the coefficients fixed.The transformation applies to the splat parameterization and preserves every term of the energy, including density-fitted Hartree energy.Source: §A.1 · p. 18
  • Rigid motions preserve the overlap matrix, occupied Gram matrix, Löwdin coefficients, and density matrix.The change of variables preserves integrals, leaving the overlap and subsequent orbital orthogonalization quantities unchanged.Source: §A.1 · p. 19
  • The energy terms remain invariant through preservation of norms, distances, density-based scalars, and Coulomb integrals.The Hartree and background terms depend on preserved distances, while semilocal exchange-correlation terms depend on invariant density quantities.Source: §A.1 · p. 19
  • The adaptive-fit auxiliary functions transform with the splats, leaving the fit matrices unchanged.Each auxiliary function is a pair product of splats.Source: §A.1 · p. 19
  • The symmetry holds for the full orthogonal group O(3), including reflections, which require no special quaternion treatment.Precision matrices are quadratic in the frame, so Y and −Y act identically; centers and orientations transform, while log-eigenvalue scales remain invariant.Source: §A.1 · p. 19
  • Energy invariance does not make Adam’s per-coordinate preconditioning rotation equivariant, so rotated copies optimized from scratch may follow different trajectories.A converged cloud can instead be mapped to a rigidly moved geometry with the same energy and no re-optimization.Source: §A.1 · p. 19

A.2 NUCLEAR FORCES

Floating splats make nuclear forces the explicit derivative of the energy with respect to nuclear coordinates, avoiding Pulay corrections. With finite optimization residuals, force accuracy converges more slowly than energy accuracy, and the summed center gradient supplies a reference-free convergence check.

A.2 NUCLEAR FORCES

  • Floating splats remove the Pulay force, leaving the electrostatic force from the explicit nuclear-coordinate dependence of the energy.Unlike atom-centered bases, floating basis motion is part of the inner optimization, so the envelope theorem removes its contribution.Source: §A.2 · p. 20
  • The force result assumes a smooth manifold of minimizers and a Hessian positive definite on directions normal to that manifold.The minimizer need not be isolated because the energy has continuous and discrete symmetries and may have other flat directions.Source: §A.2 · p. 20
  • At an exact minimizer, the ground-state energy derivative is the explicit partial with respect to nuclear coordinates.The response of optimized parameters does not contribute because the energy is stationary in them.Source: §A.2 · p. 20
  • The computed net force equals the summed splat-center gradient for any parameter values, making nonzero net force a reference-free measure of incomplete optimization.An exact force field has zero net force, and this check measures incomplete optimization rather than another error source.Source: §A.2 · pp. 21–22
  • Between auxiliary-basis refreshes, the optimizer holds that basis fixed, omitting its dependence on auxiliary functions from the gradient.The omitted term is proportional to the Coulomb-norm fit residual controlled by Section A.3.Source: §A.2 · p. 21
  • Energy error is quadratic in the optimization residual, whereas force error is linear, so force labels need a tighter stopping criterion.The optimizer’s gradient norm bounds the force error.Source: §A.2 · p. 22

A.3 ADAPTIVE DENSITY FITTING

The appendix formalizes adaptive density fitting and regularized orthogonalization, then relates floating splats to fixed Gaussian bases and clarifies parameter-count comparisons.

A.3 ADAPTIVE DENSITY FITTING

  • Density fitting replaces the quartic six-dimensional Hartree integral with a quadratic auxiliary-basis fit, whose optimized energy error is second-order in coefficient error.The error is also half the squared Coulomb distance from the density to the auxiliary span.Source: §A.3 · pp. 22–23
  • At refreshes, screening confines fit error to discarded density pairs; longer refresh intervals amortize cubic factorization but permit greater density drift.For semilocal functionals the fitted energy is below the exact-integral energy; for hybrid functionals and Hartree–Fock it is not a bound in either direction.Source: §A.3 · pp. 23–24

A.4 REGULARIZED EIGENSOLVE

  • Regularized Löwdin retraction floors small Gram eigenvalues and modifies near-degenerate gradients, dropping directions coupling floored and unfloored modes.When the converged Gram matrix is sufficiently well-conditioned and has no near-degenerate eigenvalues, the regularization has no effect at convergence.Source: §A.4 · pp. 24, 26

B FROM GAUSSIAN-TYPE ORBITALS TO FLOATING SPLATS

  • The appendix relates floating splats to fixed Gaussian bases, earlier floating-orbital models, and 3D Gaussian splatting, while identifying where the correspondences stop.It also frames basis comparisons through parameter counting.Source: B FROM GAUSSIAN-TYPE ORBITALS TO… · p. 26

B.1 CONTRACTED GAUSSIAN BASES

  • For water cc-pVDZ, matched function count gives a 1.21 cloud-to-basis parameter ratio; counting tabulated basis values avoids a hidden-parameter advantage.The cloud overhead is 21% for water, while counting only Gaussian-side coefficients would give a ratio of 2.8.Source: §B.1 · p. 28, Table 4 · p. 28

B.2 GAUSSIAN-TYPE ORBITALS AS LIMITS OF SPLATS

This appendix relates Gaussian splats to Gaussian-type orbitals and 3D Gaussian splatting, and describes GS-DFT’s implementation, numerical procedures, integral evaluation, and scaling mechanisms.

B.2 GAUSSIAN-TYPE ORBITALS AS LIMITS OF SPLATS

  • Signed combinations of displaced splats approximate Gaussian shells in H1, but exact angular factors require increasingly ill-conditioned cancellation as displacements shrink.The construction uses 4 splats per p primitive and 10 per d primitive; floating-point accuracy limits angular momentum l to displacements h with u h^-l ≪ 1.Source: §B.2 · pp. 29–30

B.3 FLOATING-ORBITAL MODELS

  • GS-DFT generalizes floating-Gaussian models beyond M = Nocc by addressing integral cost, gradient evaluation, optimization scaling, and linear dependence.Adaptive density fitting, reverse-mode automatic differentiation, Adam, and a regularized eigensolve address these obstacles; the earlier models remained minimal because they made larger clouds too expensive.Source: §B.3 · pp. 30–31

B.4 CORRESPONDENCE WITH 3D GAUSSIAN SPLATTING

  • GS-DFT shares 3DGS’s Gaussian parameterization chart, but its orbitals are linear in signed coefficients and its density is quadratic, unlike alpha-composited rendering.Coulomb interactions are long-range and orthonormality is global, while the 3DGS rendering projection is only first-order; GS-DFT instead uses exact three-dimensional overlap, kinetic, and Coulomb integrals, with quadrature for exchange-correlation.Source: §B.4 · pp. 32–33

C IMPLEMENTATION

  • The implementation appendix covers the software stack, integral derivations, operation costs and memory, multi-device execution, training settings, evaluation, and hardware and code layout.Source: C IMPLEMENTATION · p. 33

C.1 SOFTWARE AND NUMERICS

  • GS-DFT uses dftax for native JAX DFT functions and puts splat-specific operations in gs-dft; double precision and reverse-mode differentiation provide gradients for all parameters.Hand-written reverse passes cover the regularized eigensolve, streamed Coulomb contractions, and Coulomb quadrature.Source: §C.1 · p. 34

C.2 INTEGRALS

  • Most energy terms reduce to Gaussian integrals over splat products, while Coulomb quantities use a one-dimensional quadrature and exchange-correlation uses grid quadrature.The same Coulomb formulation covers the auxiliary metric and fit vector as well as exact Hartree evaluation.Source: §C.2 · pp. 34–35
  • Range-separated hybrids use a finite-interval quadrature for the long-range kernel, with a 12-node Gauss–Legendre rule and an unrolled fused kernel whose reverse pass is analytic.The node loop stores no per-node intermediate.Source: §C.2 · p. 35
  • Exchange–correlation is the only grid-evaluated energy term; VV10 correlation is a grid double sum streamed in chunks with an analytic reverse pass.GGA functionals use density and its gradient, while meta-GGAs such as r2SCAN also use kinetic-energy density.Source: §C.2 · pp. 35–36

C.3 COMPLEXITY AND MEMORY

  • For semilocal functionals, auxiliary-metric factorization is the method’s cubic operation and is amortized over K steps; hybrids add an exchange metric solve at every step.The passage says this is why hybrid calculations are limited to smaller systems.Source: §C.3 · p. 36
  • On insulin, reducing chunk size c from 2048 to 512 halves peak memory per step from 35.4 to 17.6 GB, costs 2.8% more time, and leaves energy unchanged to twelve digits.At fixed neighbor degree k, screened rows grow linearly with cloud size M; protein-scale runs use c = 512.Source: §C.3 · p. 37

C.4 MULTI-DEVICE SHARDING

  • Sharded energy and gradients equal their single-device counterparts up to floating-point summation order.Work is split over splat rows, retained pairs, and grid points, while parameters are replicated across devices.Source: §C.4 · p. 37
  • Adding devices shortens each device’s share of work but does not reduce peak memory; the sharded path lacks kinetic-energy density evaluation, so meta-GGAs run on one device.The auxiliary-metric factor and fit-vector chunk remain replicated or per-device constant in memory.Source: §C.4 · p. 38

C.5 TRAINING RECIPE

The section specifies GS-DFT training choices and evaluation procedures, then describes the differentiable DFT frameworks used for comparison.

C.5 TRAINING RECIPE

  • Runs use Adam with warmup and cosine decay, reference-free geometry-and-element initializations, level-3 XC grids (level 2 for proteins), and screening for Figure 3 or M ≥ 300.Observables in Figure 3 are evaluated on a common level-5 grid; screening is applied to every Figure 3 run.Source: §C.5 · pp. 38–39

C.6 EVALUATION PROTOCOL

  • Molecular energies use exact post-training Hartree evaluation, whereas protein-scale energies use the fitted training objective; force and ionization-potential evaluations have separate procedures.Protein-scale forces use unpolished checkpoints, while other force checkpoints are polished; ionization potentials come from one Fock build and an occupied-subspace eigensolve without another self-consistent iteration.Source: §C.6 · pp. 39–40

D COMPARISON WITH DIFFERENTIABLE DFT FRAMEWORKS

D.1 CAPABILITIES

The comparison tests correctness on water and scaling on a cc-pVTZ size ladder. GS-DFT reaches insulin while differentiable baselines stop at ethanol or earlier, though it is slower than a converged SCF on these systems.

D.1 CAPABILITIES

  • GS-DFT is the only compared differentiable code that distributes over multiple GPUs; none of the others offers both density fitting and GPU forces.At cc-pVTZ on one node, the largest completed system among the other differentiable codes is ethanol in DQC, with 9 atoms.Source: §D.1 · p. 41, Table 10 · p. 41

D.2 PROTOCOL

  • The protocol checks water correctness, then tests a cc-pVTZ ladder from water through alanine chains and insulin, recording energy, peak GPU memory, iterations, and time.Runs use one node and continue until the first out-of-memory error or crash.Source: §D.2 · p. 41

D.3 AGREEMENT ON A SMALL MOLECULE

  • At matched count M = nao = 24, GS-DFT ends 66.6 mHa above the cc-pVDZ water reference, below the crossing point of the two scaling laws.D4FT in double precision agrees with PySCF to 0.04 mHa at cc-pVDZ and 0.05 mHa at cc-pVTZ; MESS fails once polarization functions are present.Source: §D.3 · p. 42, Table 11 · p. 42

D.4 SCALING WITH SYSTEM SIZE

  • 17,448 functions fit in 18.4 GB for GS-DFT at insulin; every differentiable baseline stops at ethanol or earlier.Across the ladder, peak memory grows by a factor of 102 while the basis grows by a factor of 300.Source: §D.4 · p. 42

E ABLATIONS

The ablations vary one setting at a time from the production configuration. They show that regularized orthogonalization prevents stalling on some symmetric molecules, initialization improves energies and seed consistency, and refresh and screening settings strongly affect cost without ordering the resulting energies.

E.1 REGULARIZED EIGENSOLVE

  • More than 40 mHa above regularized results, methane and benzene stall without regularization; CO2 and N2 differ by less than 2 mHa.At matched wall-clock time, the regularized and unregularized results also agree on a 45-residue alanine chain; which symmetries trigger stalling remains open.Source: §E.1 · pp. 42–43, Table 13 · p. 43

E.2 OPTIMIZER

  • Adamax finishes 13.7 mHa below Adam; Adam, five variants, and LAMB finish within 14 mHa of Adam, while Yogi, RMSProp, and AMSGrad finish 75 to 94 mHa higher.Other optimizers finish 0.2 to 1.0 Ha higher, and Lion and Adan diverge.Source: §E.2 · p. 43, Table 14 · p. 43

E.3 INITIALIZATION

  • 27 mHa lower on water and 35 mHa lower on ethanol than random placement, the full chemically informed initialization also reduces seed spread.Seed spread falls from 11.7 to 2.1 mHa on water and from 4.6 to 0.1 mHa on ethanol; the initialization adds exponent ranges, bond-centered splats, bond-aligned anisotropy, and nuclear-charge-proportional allocation.Source: §E.3 · pp. 43–44, Table 15 · p. 44

E.4 REFRESH PERIOD

  • 76% more costly than rebuilding every 200 steps, rebuilding every step offers little advantage: K = 10 is already within 8% of the cheapest setting.Energies span 86 mHa without ordering by refresh period; the authors use K = 10.Source: §E.4 · p. 44, Table 16 · p. 44

E.5 SCREENING THRESHOLD

  • At τ = 10−8, screening removes half the pairs and 46% of wall-clock time; four further decades remove only 12% more.Energies span 63 mHa without ordering by threshold; the authors use τ = 10−7, near the knee of the pair count.Source: §E.5 · p. 44, Table 17 · p. 44

F ADDITIONAL RESULTS

On the ionic dissociation ladders, continued splat clouds track below the matched-count basis ladder. For bare anions, splats with plain-basis function counts outperform same-cardinality augmented bases from TZ onward.

F.1 BARE ANIONS

  • From TZ onward, splats with plain-basis function counts lie below same-cardinality augmented bases for both anions; at QZ, they lie below aug-cc-pV5Z.The clouds receive no diffuse functions or anion-specific information beyond nuclei and electron count; the optimizer moves splats outward from the nuclei.Source: §F.1 · p. 45, Table 18 · p. 45

F.2 DISSOCIATION ENERGY LADDERS

  • Below the entire matched-count ladder, continued splat curves track the ionic diabat for LiH and LiF.At LiH, R = 6 Å, the augmented QZ energy is 335 mHa above the augmented DZ energy; this baseline convergence failure is excluded from matched-count margins.Source: §F.2 · p. 45, Figure 6 · p. 46

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