Accurate and scalable exchange-correlation with deep learning

Motivation

• Current exchange-correlation approximations remain insufficient for reliably predicting experimental outcomes, with chemical accuracy commonly requiring errors below ∼1 kcal/mol.These limitations cause screening pipelines to pass too many candidates to experimental validation and constrain downstream methods trained on DFT data.Source: §1 · pp. 1–2

Background

• Traditional Jacob’s ladder approximations trade accuracy for cost by adding increasingly expensive non-local, wavefunction-like ingredients to semi-local information.Lower-rung methods retain O(N 3) scaling but have limited accuracy, whereas higher-rung methods can scale as O(N 4), O(N 5) or higher.Source: §1 · p. 2

Related work

• Earlier machine-learning functionals did not shift this trade-off because high-accuracy training data were scarce and most models relied on handcrafted features.Efforts beyond handcrafted descriptors remained largely confined to model systems or narrowly defined problems.Source: §1 · p. 3

Contribution

• Skala achieves higher accuracy than leading hybrid functionals across broad main-group chemistry while retaining the asymptotic complexity of semi-local DFT.Its scalable neural-network formulation also naturally enables GPU acceleration and matches or exceeds hybrid functionals for equilibrium structures and dipole moments.Source: §1 · p. 3

Method

• Skala combines approximately 400,000 high-accuracy energy differences spanning diverse chemistry with a neural architecture that learns non-local electronic representations from simple semi-local inputs.The training data were generated using an efficient wavefunction-based protocol and augmented with comparable public datasets.Source: §1 · p. 3

Novelty

• Skala’s accuracy improves systematically as increasingly diverse high-accuracy data are added, without increasing inference-time computational complexity.This scaling behavior motivates a versioned, continuously evolving framework in which later releases supersede earlier models through demonstrable gains in accuracy and robustness.Source: §1 · p. 3

2 Deep learning the XC functional: basic challenges and solutions

• The XC functional maps the three-dimensional electron density to an energy, requiring a well-defined infinitely dense-grid limit and good convergence under grid refinement.Semi-local DFT evaluates the XC energy on integration grids typically containing approximately 10^4–10^6 points for molecules with up to several hundred atoms.Source: §2 · p. 3

2.1 The two-stage training protocol

• Skala uses pre-training on fixed B3LYP densities followed by SCF fine-tuning, avoiding backpropagation through SCF while improving self-consistent energies and densities simultaneously.Pre-training uses wavefunction-level reaction-energy references and scalable approximate densities; fine-tuning uses model-generated SCF densities and monitors energy gaps and dipole moments.Source: §2.1 · p. 4

2.2 The XC functional architecture of Skala

• Skala’s neural functional learns non-local representations through coarse atom-centered points while preserving semi-local DFT’s asymptotic computational complexity and grid-convergence behavior.The architecture incorporates constraints including uniform coordinate scaling, size consistency, and the Lieb–Oxford lower bound, and uses a fixed D3 correction rather than learning dispersion explicitly.Source: §2.2 · pp. 4–5
• Learned non-locality is intended to disrupt Jacob’s ladder’s accuracy–cost trade-off, enabling high main-group accuracy without increasing computational cost and allowing systematic improvement with more data.This approach replaces increasingly costly hand-engineered ingredients with deep representation learning while retaining scalable semi-local DFT complexity.Source: §2.2 · pp. 4–5

2.3 The training data

• MSR-ACC provides Skala-1.1 with about 400,000 chemically diverse energy differences labeled using high-accuracy wavefunction methods, including approximately 120,000 total atomization energies.The collection combines W1-F12 and W1w data with additional accurate public datasets spanning affinities, ionization potentials, reaction pathways, non-covalent chemistry, structures, conformers, and reaction barriers.Source: §2.1 · p. 3, §2.3 · p. 5

3 Accuracy of Skala

• Skala achieves the lowest overall WTMAD-2 error on GMTKN55, surpassing all compared hybrid functionals at semi-local cost and performing best on 32 of 55 subsets.It outperforms the best hybrid across nearly all thermochemistry and kinetics benchmarks without Hartree–Fock exchange.Source: §3 · pp. 6–7, Table 1 · p. 6
• Skala reaches an MAE of 0.92 kcal/mol on the single-reference W4-17 subset and 1.23 kcal/mol overall, including multi-reference cases absent from training.The W4-17 test set contains 200 diverse atomization energies, whereas the single-reference subset contains 183 reactions.Source: §3 · pp. 6–7

3.1 The importance of learning nonlocal interactions

• The non-local branch reduces average WTMAD-2 error on Diet GMTKN55 by about 50% across five seeds compared with local models.This supports learning essential non-local effects directly from data without Hartree–Fock exchange.Source: §3.1 · pp. 7–8

3.2 Skala’s accuracy improves systematically with training data

• Increasing training-data size and chemical diversity systematically lowers Diet GMTKN55 WTMAD-2 error and inter-seed variance, with non-covalent interactions and conformers providing further improvement.Set A gives GGA-level accuracy, set B adds a modest improvement, and set C further reduces the error.Source: §3.2 · p. 8

3.3 The emergence of learned physical constraints with training data

• Training on the full dataset makes Skala consistently satisfy the positivity of the kinetic correlation component Tc, unlike training on MSR-ACC/TAE25 alone.The resulting Tc values are close to reverse-engineered CCSD-density values and remain close to pure Kohn–Sham values despite evaluation in the GKS framework.Source: §3.3 · p. 8

4.1 Densities

• Dipole errors fall below B3LYP and the best range-separated hybrids after self-consistent fine-tuning, while energies and densities improve simultaneously.Fine-tuning is initially erratic below approximately 2,000 steps; afterward, all systems converge and both properties improve.Source: Figure 3 · p. 9, §4.1 · p. 9

4.2 Equilibrium geometries

• Approximately 20,000 W1w-labeled distorted structures improve Skala’s equilibrium geometries beyond the GGA-level accuracy obtained from equilibrium and reaction-path training alone.The distortion labels encode minima information at CCSD(T)/CBS accuracy because the original equilibrium structures came from approximate B3LYP geometries.Source: §4.2 · p. 9
• Skala’s optimized geometries achieve accuracy on par with or exceeding the best hybrid functionals across light- and heavy-main-group and small-molecule benchmarks.The benchmarks include bond lengths and, for selected datasets, bond angles, compared against functionals across Jacob’s ladder and GFN2-xTB.Source: §4.2 · pp. 9–10, Figure 3 · p. 9
• Skala’s average GEO error is approximately 0.02 kcal/mol on W4-11-GEOM, indicating little added energy error from using its own optimized geometries.The GEO metric compares energies on a functional’s optimized geometry against a reference geometry.Source: §4.2 · p. 10
• The MSR-ACC/Distortion dataset also enables prediction of highly strained conformations in larger molecules, including adenosine, benzylpenicillin, and efavirenz.These applications demonstrate generalization beyond equilibrium-geometry benchmarks.Source: §4.2 · p. 10

• Skala has the same asymptotic scaling as meta-GGA semi-local DFT, but prefactors, non-compute bottlenecks, and hardware optimizations require empirical cost verification.Methods with identical asymptotic scaling can differ by orders of magnitude in practical cost, as illustrated by more expensive local hybrids.Source: §5 · p. 10
• Within 30% of r2SCAN on GPUs, Skala’s XC evaluation cost is substantially lower than hybrid functionals, which are roughly 3–6× more expensive.On CPUs, Skala costs approximately 2× r2SCAN for smaller systems, with the prefactor decreasing toward unity as atomic-orbital evaluation dominates for larger systems.Source: §5 · p. 11
• Skala’s grid-level non-locality is incompatible with grid-point-independent XC libraries such as libXC, creating a production-code adoption barrier.The paper argues that substantially higher DFT accuracy without increased cost requires production-code support for deep-learning functionals that learn non-locality.Source: §5 · p. 11
• GauXC provides a scalable Gaussian-basis integration pathway already used by Psi4 and CP2K, while direct checkpoint integration supports PySCF and other codes through Python, C/C++, and Fortran.GauXC supports CPU and GPU architectures, and direct integration works with any basis set.Source: §5 · p. 11

• Skala outperforms state-of-the-art range-separated hybrids across broad main-group chemistry while retaining the computational cost of semi-local DFT.Its large-scale training set, scalable protocol, and low-cost non-local architecture break the traditional accuracy–cost trade-off.Source: §6 · p. 11
• Expanding training data is expected to improve Skala’s accuracy and generality, while minimal transition-metal training already yields performance on par with established functionals.The preview covers selected 3d and 4d transition-metal benchmarks, but further gains require substantially broader and more diverse high-accuracy data.Source: §6 · p. 11
• Extending Skala to multi-reference and strongly correlated systems remains a key challenge because accurate reference data at scale is difficult to generate.Learned functionals could transfer wavefunction-method accuracy to medium-large systems and generate data for force fields and other large-scale models.Source: §6 · p. 11
• Skala’s physically constrained exchange-correlation design generalizes accurately across most thermochemistry benchmarks and defaults to standard semi-local DFT in the worst case.The Kohn–Sham framework captures dominant energy contributions, while the learned functional supplies a smaller correction transferable to unseen elements and larger systems.Source: §6 · p. 11

7.1 Skala: A model for scalable non-local representation learning

• Skala uses a coarse-grid intermediary to learn non-local interactions from meta-GGA features while avoiding the prohibitive cost of all-to-all communication on fine DFT grids.The architecture combines spin symmetrization, equivariant coarse-grid processing, and atom-centered grids to preserve computational scalability; its bounded enhancement factor enforces the Lieb–Oxford lower bound.Source: §7.1 · p. 17

7.2 Training data

• The training set contains ∼400k CCSD(T)/CBS-level or higher reaction energies spanning atomization, conformational, non-covalent, reaction, and atomic data.Publicly available datasets supplement MSR-ACC, while molecular-graph-based subtraction reduces overlap with GMTKN55 and W4-17, and a reserved holdout supports model selection.Source: §7.2 · pp. 17–18
• Pre-training uses fixed B3LYP densities and multiple numerical integration grids, followed by fine-tuning with denser grids and varied partitioning weights to regularize grid-dependent behavior.The fixed-density procedure replaces B3LYP exchange-correlation energies with Skala predictions when computing reaction energies.Source: §7.2 · p. 18

7.3 Data availability

• MSR-ACC subsets have documented release status, while many supplemental datasets, benchmark sets, geometry collections, and evaluation resources are publicly available.The released resources include MSR-ACC/TAE25 and MSR-ACC/Water, alongside W4-CC, NCI, reaction, atomic, benchmark, dipole, and geometry datasets.Source: §7.3 · p. 18

7.4 Code availability

• The Skala model and inference code are released under an MIT license with integrations for PySCF, GPU4PySCF, ASE, Azure AI Foundry, Accelerated DFT, and GauXC.The repository provides the PyTorch implementation and quantum-chemistry package hookups.Source: §7.4 · p. 19

Architecture

• Skala uses spin-invariant meta-GGA features, three equivariant non-local interaction layers, and coarse-point spherical-harmonic message passing to generate the exchange-correlation enhancement factor.The architecture projects grid features through spherical harmonics and radial bases, mixes higher-body-order correlations at nuclear coarse points, and reconstructs non-local grid functions with density scaling and skip connections.Source: Figure 5 · p. 22, Method References · p. 22

Training datasets

• The training resources span atomization, conformational, proton-affinity, ionization-potential, electron-affinity, reaction, water, distortion, and non-covalent-interaction datasets, with overlap removed from benchmark tests.Extended Data Figure 6 defines dataset categories for the data-ablation study and reports effective counts after removing benchmark overlap and validation splits.Source: Figure 6 · p. 23

Generalization

• Holdout evaluations show that Skala generalizes beyond its training data across diverse chemical properties.The holdout analysis reports absolute-error distributions and MAEs for publicly available sets, including MSR-ACC/TAE25 and Water2510-related data.Source: Figure 7 · p. 24

Atomization and distortion energies

• Skala accurately predicts held-out atomization energies and benefits substantially from distortion training data when modeling relative energies along vibrational modes.The evaluation covers W4-17 atomization energies, multi-reference and single-reference subsets, and Wiggle150 strained-conformer energies, including an ablation over independent seeds.Source: Method References · p. 24, Figure 8 · p. 25

Specialized benchmarks

• Skala’s learned non-local effects support accurate treatment of non-local exchange, SN2 barriers, and transition-metal reactions despite minimal 3d and 4d training coverage.The cited benchmarks include the molecular thermometer, W1-SN2-BH, and transition-metal reaction sets, where Skala is reported to perform on par with some leading hybrid and meta-GGA functionals.Source: Figure 9 · p. 26, Method References · p. 26, Figure 10 · p. 27

A.1 In theory

• The Kohn–Sham energy contains an unknown exchange-correlation functional Exc[ρ], making its accurate approximation the central modeling challenge.Exc maps the electron density to a scalar exchange-correlation energy.Source: §A.1 · p. 29
• Skala parameterizes the exchange-correlation energy with a deep neural-network enhancement factor multiplying the LDA exchange energy density.This form facilitates enforcing high-density uniform coordinate scaling, size consistency, and the Lieb–Oxford lower bound.Source: §A.1 · p. 29

A.2 In practice

• In practice, Skala evaluates density features on a classical integration grid of points and weights to numerically approximate exchange-correlation energies and potentials.The point-cloud representation is independent of the basis set.Source: §A.2 · p. 29
• The neural network receives seven semi-local features per grid point, including Kohn–Sham kinetic-energy density and the two spin channels.The resulting input is a tensor in R^G×7, with G typically depending on system size.Source: §A.2 · p. 29
• Semi-local features alone cannot represent the exact functional, which requires non-local dependence on the density.The two strategies are adding hand-designed cross-point features or allowing the model to mix information across grid points.Source: §A.2 · pp. 29–30
• Skala chooses to learn longer-range dependencies by mixing information across grid points, moving beyond traditional Jacob’s-ladder approaches and hand-designed features.This approach infers non-locality through data and the model.Source: §A.2 · p. 30
• Because dispersion is very long-range and typically post-corrected, training on B3LYP densities includes its D3 correction and targets other shorter-range non-local effects.The D3 correction is included as part of the total energy during training.Source: §A.2 · p. 30

A.3 Neural network architecture

• The architecture extracts spin-order-invariant features, applies three non-local layers, and outputs an enhancement factor from atom-centered integration grids.The model uses Dhid = 256, Dnonl = 16, and L = 3; its scaled-sigmoid output is centered near the LDA exchange limit.Source: §A.3 · pp. 30–31

A.4 Non-local interaction through coarse points

• Coarse points make non-local message passing tractable by aggregating information from each atom’s grid points onto an atomic-center coarse point, processing it, and sending features back.The model uses ℓmax = 3, and exp(−ρi) suppresses non-local effects on higher-density grid points, such as regions near nuclei.Source: §A.4 · pp. 31–32

A.5 Atomic partition of the energy density

• Per-atom processing produces atom-dependent energy densities that are combined using merged grid weights, with partition-scheme augmentation preventing dependence on a particular partition function.Each grid point communicates only with its associated atom’s coarse point, while the density features carry cross-atom information.Source: §A.4 · p. 31, §A.5 · pp. 32–33

A.6 Intuition on the structure of the non-local layer

• A downsampling–upsampling pass can approximate two-body interactions, while symmetric contraction extends the layer to higher-body interactions and enables up to four-body interactions when ν = 3.The theoretical motivation connects the construction to rotationally invariant basis expansions and identifies atomic centers as natural expansion points.Source: §A.6 · pp. 33–34

A.7 Related work

• The design builds on ACE-style systematic expressivity and resembles fast multipole methods, while a related nuclei-centric message-passing approach overfit preliminary Diet GMTKN55 evaluations.Expressivity can increase through larger basis sets and higher interaction correlation order; coarse points additionally provide computational aggregation.Source: §A.7 · p. 35
• Neural operators map input functions to output functions at arbitrary discretizations, motivating a grid-adapted non-local layer that shares similarities with low-rank kernel neural operators.The layer is tailored to irregular atomic grids, where standard neural-operator techniques have limited applicability.Source: §A.7 · p. 35

B.1 Training objective

• The molecular total-energy construction uses a precomputed B3LYP exchange-correlation-subtracted quantity that includes D3 dispersion energy.This quantity enters the reaction-energy calculation alongside the stoichiometric coefficients of the reaction molecules.Source: §B.1 · p. 35
• Training minimizes a weighted mean-squared regression loss on reference reaction energies, with reactions sampled through the hierarchical procedure described in Sec. B.2.Reaction energies are weighted sums of stoichiometric coefficients applied to molecular total energies.Source: §B.1 · p. 35

B.2 Dataset sampling and model selection

• Adaptive sampling prioritizes underrepresented and weakly learned datasets, updating probabilities every 25,000 steps with η = 0.025 using relative excess loss.Two-level sampling first selects a dataset and then samples a reaction within it; datasets already matching the best baseline are downweighted to limit overfitting reference noise.Source: §B.2 · p. 36
• Relative excess loss evaluates each holdout every 25,000 steps against baseline-functionals’ minimum and median MAEs, diagnosing progress across datasets.Its aggregate value softly counts datasets where the model has not yet reached best-baseline accuracy.Source: §B.2 · p. 36

B.3 Parameter initialization

• Local network weights use Xavier initialization with zero biases, while non-local parameters use a modified Xavier scheme accounting for shared tensor-feature weights.Source: §B.3 · p. 36

B.4 Optimization

• Muon optimizes hidden weight matrices, whereas Adam optimizes biases and the final output layer, with independent cosine schedules and linear warmup.Muon applies momentum SGD followed by orthogonalized updates; Adam uses betas (0.9, 0.95) and ϵ = 10−10.Source: §B.4 · pp. 36–37
• Ablation models train for 1,000,000 steps on 8 A100 GPUs with minibatch size 8 and use exponential moving averages with decay 0.9999.Source: §B.4 · p. 37

B.5 Self-consistent fine-tuning

• Self-consistent fine-tuning transfers predictive performance by training on densities produced by the model itself after initial training on B3LYP features.It uses a learning rate of 10−5, RMS gradient clipping at 10−4, EMA decay 0.999, minibatch size 1, one A100 GPU, and 20,000 steps.Source: §B.5 · p. 37
• Self-consistent densities are generated with PySCF and DIIS using Treutler-Ahlrichs grids, grid level 3, def2-QZVP densities, and a density threshold of 10−8.The SCF procedure starts from a B3LYP guess, uses density fitting, and terminates at specified energy, gradient, or 40-step limits.Source: §B.5 · pp. 37–38

C Training data: computational details

• The training-data section defines the computational protocols used to produce the reference training data.Source: C Training data: computational details · p. 38

C.1 MSR-ACC/TAE dataset

• MSR-ACC/TAE contains W1-F12-labeled atomization energies for filtered, lowest-energy equilibrium structures generated from covalent graphs, combining single-fragment and multi-fragment molecules.Structures with triplet ground states or %TAE[(T)] above 6% are excluded; 90.7% are single covalently bound fragments.Source: §C.1 · p. 38
• Approximately 41k larger GDB9-derived structures extend MSR-ACC/TAE through element substitutions, geometry relaxation, total atomization reactions, and W1w labeling.The substitutions replace second-row elements with third-row elements and H with Li, Na, F, or Cl.Source: §C.1 · p. 38

C.2 MSR-ACC/IP, /EA, /PA, /Conf, /NCI, /Water, /Distortion and /Reactions datasets

• MSR-ACC/IP, /EA, /PA, /Conf, /NCI, /Water, /Distortion, and /Reactions cover ionization, electron attachment, protonation, conformers, noncovalent clusters, water, vibrational distortions, and reaction barriers.These datasets are derived from MSR-ACC/TAE, sampled molecular fragments, published water structures, and reaction datasets, with specialized structure-generation protocols.Source: §C.2 · p. 38

C.3 Atomic datasets

• Atomic datasets provide CCSD(T)/CBS total energies, ionization potentials up to triple ionization, and electron affinities for atoms up to argon, excluding Li and Be.The HF and correlation components are extrapolated from aug-cc-pCVQZ and aug-cc-pCV5Z basis sets.Source: §C.3 · p. 39

C.4 Density features on the grid

• Density-feature augmentation evaluates densities on four distinct integration grids and four radial schemes to regularize the model against numerical grid variations.The radial schemes are Treutler, Mura–Knowles, Gauss-Chebyshev, and Delley.Source: §C.4 · p. 39

C.5 Augmentation of the space-partitioning schemes during training

• Training randomizes among Becke, SSF, and LKO space-partitioning schemes, consistently applying each sampled scheme across reactions to reduce overfitting to partitioning artifacts.The LKO scheme caps interatomic distances at a cutoff radius, enabling linear-scaling cost for large molecules.Source: §C.5 · p. 39
• Reaction calculations use shared basis sets and integration grids for error cancellation, fixed B3LYP densities, and anionic-reaction-specific ma-def2-QZVP basis sets.Relative reaction energies are computed from B3LYP total energies without the exchange-correlation energy.Source: §C.5 · p. 39

D Evaluation protocols

• The evaluation protocols cover four benchmark categories: reaction energies, dipole moments, geometry optimization, and computational cost.These categories define the paper’s evaluation settings.Source: D Evaluation protocols · p. 39

D.1 Evaluation sets

• Skala is evaluated on GMTKN55 and additional atomization, strained-conformation, reaction-barrier, transition-metal, holdout, and specialized datasets.A Diet GMTKN55 subset is used for architecture and dataset-composition ablations, while dipole, geometry, and computational-cost benchmarks are also included.Source: §D.1 · p. 39

D.7 Computational cost: GPU and CPU implementation

• The Skala functional is integrated with GauXC for TorchScript-based exchange-correlation energy and potential evaluation on both GPU and CPU implementations.Cost timings use 38 molecules, def2-TZVP, and specified grid and integration settings on Azure A100 and E32ADS V5 systems.Source: §D.1 · p. 39, Figure 11 · p. 40, §D.7 · pp. 42–43

D.2 Baselines and dispersion correction settings

• Baseline comparisons include semi-local and hybrid functionals with explicitly specified D3(BJ), D3(0), or VV10 dispersion corrections.The compared methods include revPBE, B97M-V, r2SCAN, B3LYP, M06-2X, ωB97X-V, and ωB97M-V.Source: §D.2 · p. 40

D.3 Basis sets

• Reaction energies use def2-QZVP or ma-def2-QZVP for anions, dipoles use aug-pc-4, geometry optimization uses def2-TZVP, and all evaluations use density fitting.The auxiliary basis is def2-universal-jkfit.Source: §D.3 · p. 40

D.4 SCF retry protocol

• For reaction-energy benchmarks and dipole computations, Skala starts SCF from MINAO, uses a Treutler level-3 grid, applies tolerances of 5 · 10−6 Eh and 10−3 Eh, and retries after 60 steps with DIIS, damping, and conditional level shifts.Source: §D.4 · pp. 40–41

D.5 SCF with orbital gradient descent fallback

• A gradient-descent fallback searches orbital coefficients with line search when DIIS fails to lower the energy, using double precision and, in some attempts, projection from def2-TZVP to def2-QZVP.The line search uses the Armijo rule and bounded trial steps.Source: §D.5 · pp. 41–42

D.6 Geometry optimization: implementation

• Geometry optimization includes Hellmann–Feynman, Pulay, grid-position, and coarse-point exchange-correlation contributions, with tightened energy and gradient tolerances for geomeTRIC optimization.The successful initial SCF method is retained during subsequent optimization steps.Source: §D.6 · p. 42

E.1 Spin-symmetry broken solutions for multi-reference systems; E.2 GMTKN55 benchmark; E.3 Convergence with respect to the grid size

• Allowing spin-symmetry breaking improves atomization energies for 1BN, B2, and C2, while Skala-1.1 shows grid-convergence behavior similar to traditional functionals on W4-17 and Diet GMTKN55.The spin-broken calculations use 50 random seeds per molecule with tighter convergence criteria, and remaining deviations are attributed to tighter convergence and increased global-minimum sampling.Source: §E.1 · pp. 43, 45, Figure 12 · p. 43, §E.3 · p. 45

Emergent constraints

• Neural-network functionals can learn exact constraints from sufficient data, whereas directly imposing them may restrict model expressivity.The paper frames emergent constraint satisfaction as an alternative to traditionally building constraints into approximate functional forms.Source: §E.4 · pp. 45, 47

Scaling analysis

• Uniform coordinate-scaling constraints require special treatment because Skala’s meta-GGA feature τ is evaluated within generalized rather than pure Kohn–Sham calculations.In generalized Kohn–Sham theory, only Ts + Exc is well defined, and the scaled meta-Kohn–Sham orbitals do not have a well-defined scaling.Source: §E.4 · p. 47

Emergent constraints

• Skala learns a positive GKS correlation kinetic energy close to the pure-KS value, consistently across five random seeds.This behavior is remarkable because training only on MSR-ACC/TAE25 can otherwise allow the GKS correlation kinetic energy to become negative.Source: §E.4 · pp. 47–48

Scaling analysis

• The physical constraint T_c^GKS[ρ] ≤ T_c[ρ] is expected only for identical densities, which are not obtained here.This qualification limits direct interpretation of the observed closeness between GKS and pure-KS correlation kinetic energies.Source: §E.4 · p. 48

E.5 Ablation of the performance on geometry optimization

• The GEO metric is defined as the energy difference between a functional evaluated at the reference geometry and at its own optimized geometry.It directly measures the extra error introduced when the functional is also used to optimize molecular geometries.Source: §E.5 · p. 48
• Training without TAEs for vibrationally distorted geometries yields bond-length, bond-angle, and GEO performance around revPBE, whereas including them boosts performance to Skala-1.1’s current level.The ablation compares checkpoints trained with and without MSR-ACC/Distortion data.Source: §E.5 · p. 48, Figure 15 · p. 49
• The ablation supports the conclusion that training on distorted molecular geometries is crucial for accurate geometry optimization.This conclusion follows from the degradation toward revPBE-level performance when MSR-ACC/Distortion data are removed.Source: §E.5 · p. 48
• Overall, Skala-1.1 is expected to perform well when reference molecular geometries are unavailable and geometries must be optimized with the functional.This conclusion summarizes the geometry-optimization results presented in the section.Source: §E.5 · p. 48
• Skala-1.1 matches state-of-the-art hybrids such as ωB97M-V on the GEO metric for W4-11-GEOM geometry optimization.The GEO metric measures the additional energy error from evaluating a functional at its own optimized geometry instead of the reference geometry.Source: §E.5 · p. 48, Figure 15 · p. 49
• Geometry-optimization accuracy is evaluated across W4-11-GEOM and, for checkpoint comparisons, LMGB35 and HMGB1 using bond-length and bond-angle errors.The reported errors are average absolute deviations from ground-truth bond lengths in Ångstrom and bond angles in degrees.Source: Figure 15 · p. 49, §E.5 · p. 49