cond-mat.mtrl-sciAug 14, 2026
SaveWhen do machine-learned exchange-correlation improvements inherit into density-functional tight binding?
Organizations: Department of Computer and Electrical Engineering, Texas A&M University, College Station, TX, USA · Department of Electrical and Computer Engineering, Texas A&M University at Qatar, Doha, Qatar · Department of Prosthetics and Orthotics, Ankara University, Ankara, Turkey · College of Science and Engineering, Hamad Bin Khalifa University, Doha, Qatar
Abstract
Machine-learned exchange-correlation functionals correct band gaps at near-semilocal cost, while density-functional tight binding reaches the --atom regime; combining them assumes that a better parent yields a better parameterization, but we show it does not. Current-generation functionals are orbital-dependent generalized Kohn-Sham operators, whereas the parameterization channel is built on a multiplicative potential, preventing exact representation. Using the transfer ratio, the surviving fraction of a parent-level change, we find anti-transfer: coherently negative ratios across four covalent semiconductors move the gap in the wrong direction, consistent with a molecular proxy and an rSCAN control. The minimal-basis overgap is dominated by the on-site convention rather than basis incompleteness; correcting the on-site block removes most of it, while one -polarization shell closes a further -, depending on the placement of the empty level, which no free-atom eigenvalue uniquely fixes. Occupied-manifold enhancements, ionic and closed-shell repulsive potentials, and rocksalt-oxide gaps inherit, whereas elemental and III-V covalent networks inherit neither gaps nor repulsive potentials and oxide networks inherit only the latter. We screen 23 elements and release the parameter sets, showing that the transfer ratio provides a cheap pre-test before any parameterization campaign.
Figures & tables
Figure 1: The functional-agnostic parameterization bridge, and the three checks that certify it. a , The pipeline, with what each stage produces. The functional enters at exactly one point, the Kohn–Sham potential inversion, which is also where the compression becomes lossy: the two-center integrator consumes a multiplicative and nothing else, and an orbital-dependent functional admits no such . b , Fidelity of the inversion against a stored reference potential. The route the bridge uses and a full potential reconstruction both recover it to within a few meV; the density-gradient-only approximation fails by two orders of magnitude and was rejected. c , The confinement sweep for the carbon reference. The maximum overlap deviation has a sharp minimum at the certified radius, which is what fixes the convention; the Hamiltonian root-mean-square difference does not, falling monotonically through it. The convention is certified against and against alone. d , Coverage. Elements passing the atomic feasibility gates across three element sets; the periodic machine-learned self-consistent field is the binding constraint.
| Parent DFT | DFTB (free-atom) | Gap transfer ratio | ||
| System | (eV) | (eV) 1 | Free-atom | Confined 2 |
| Covalent semiconductors | ||||
| Diamond | ||||
| Silicon | ||||
| 3C-SiC | ||||
| BP | ||||
Table 1: The band-gap correction does not merely fail to survive, it moves the tight-binding gap the wrong way. All free-atom entries are evaluated on the fundamental band-path gap at both levels. The magnesium-oxide gap change is of order eV, so the ionic case is functional-insensitive rather than positively transferring. A ratio of one is full inheritance and the useful-transfer threshold; the confined ratios, which the on-site convention error drove toward zero, are given for comparison in the last column.
Figure 2: The band-gap correction reverses under the compression, and what survives instead. a , Input against output: the parent gap opens (grey; dashed outlines mark full inheritance) while the compressed DFTB gap closes (red). The upper segment carries the scale this is read against, the residual overgap of the same tables under the same convention: , and eV for diamond, silicon and 3C-SiC. b , The same result as a ratio, against the useful-transfer threshold. Gap ratios for the covalent solids, the near-zero ionic case (MgO), and the bandwidth ratios (a width quantity, convention-independent) reported for the three solids that carry them; none clears the gate. c , The molecular proxy on N 2 , CO and CO 2 , with a parent correction five times larger. Parent and tight-binding responses are drawn as ranges. d , What survives. The free-atom on-site – opening persists as an occupied-width enhancement into the dimer and solid while the solid gap ratio barely moves, the partition the theory predicts.
Figure 3: Why the parameterization channel cannot carry an orbital-dependent functional, and the two measurements that decide whether it does. is the dimensionless number the functional fixes at the nucleus. a , The -resolved local representative with the cusp condition imposed, at . Each angular channel sees a different potential and the difference diverges as , so no multiplicative lies in the shaded band; only the divergence is parameter-free. b , The cusp coefficients against . Both are linear and vanish at different ( and ), so they never vanish together; the ratio at only. c , Where the discarded gap-shift term acts. Sketched for a homonuclear pair; the argument does not run for ionic solids. d , Two measurements on a common scale. At the production inversion’s matching radius the channel splitting is eV, three orders above that inversion’s meV noise floor; under PBE the theory predicts exactly zero and the negative control confirms it at the floor.
Figure 4: The overgap of the unfitted tables, what a single shell removes of it, and the two things that would undo the claim. a , -point gaps; red rules mark the parent (identical settings). The minimal tables overgap the parent by eV; adding a -polarized set (confinement radii and on-site eigenvalues held fixed) reduces the overgap several-fold (per-solid values in Table 2 ). The published-parameter baseline is given in Fig. 6 ; under the free-atom convention the excess is shared with fitted sets, not a property of our convention alone. b , The closure and its residual, drawn at the confined on-site convention, which isolates the basis-completeness effect; under the free-atom convention adopted in this work the closure is smaller and depends on the placement of the empty level, as Fig. 5 shows. The residual grows with polarity and is intrinsic to the two-center on-site approximation, not removed by further basis enlargement (per-solid closures and residuals in Table 2 ). c , The on-site convention, tested and corrected. The standard scheme puts free -atom eigenvalues on the diagonal ; the confined convention places confined eigenvalues there instead (every on-site ). Rebuilding with free-atom eigenvalues collapses the overgap, confirming the convention was the dominant contributor to panel a .
Figure 5: Under the convention the overgap closure tracks the placement of the empty level, which no free-atom eigenvalue fixes. a , Mean and per-solid closure as the carbon is swept from the confined value ( eV) toward the valence shell, with silicon shifted in proportion. The dashed rule marks the closure obtained when the confined convention is used on both the and sides. b , The Mulliken population across the same sweep, against the eight valence electrons of the cell; placements further down, at the free-atom level, take the manifold to a valence-shell population and are excluded on that basis. All gaps are fundamental band-path gaps; solids, confinement radii and free-atom and on-site energies held fixed throughout.
| Parent | Minimal (eV) | -polarized (eV) | Closure | ||||
| System | DFT (eV) | Gap | Overgap 1 | Gap | Residual 2 | par. | (%) 3 |
| Diamond | |||||||
| Silicon | |||||||
| 3C-SiC | |||||||
| Mean | – | – | – | – | – | – | |
Table 2: One shell removes most of the overgap at fixed on-site energies, and what it leaves behind grows with polarity. Gaps are reported at the confined-atom on-site convention, held fixed between the and sets so that this table isolates the basis-completeness effect alone. Both columns are -point gaps for parent and model alike, so this is an overgap decomposition and not a transfer ratio. The free-atom closure and its dependence on the -channel placement are given in Fig. 5 .
Figure 6: Band structures, densities of states, and the fitted-table baseline, in the free-atom on-site convention. a–c , Band structures along L– –X–W–K– with the total density of states, for diamond, silicon, and 3C-SiC under the semilocal reference (PBE, and ) and the machine-learned functional (CIDER23X, ); energies referenced to the valence-band maximum. d–f , Fundamental gaps of each functional-consistent table and of the fitted published sets, run in the same DFTB+ binary, against the experimental gap (dashed).
Figure 7: The bonding-class structure of the transfer failure. a , Gap inheritance per compound, against the eV gate. Only the rocksalt oxide and chloride approach it; every covalent network diverges, and two ionic compounds do so as well, by distinct minimal-basis mechanisms. b , Held-out repulsive-potential transfer, scored as the mean-centered correlation between a potential fitted on one configuration set and tested on another, separating cleanly by bonding class. Mg–O is drawn in its own class colour as the one ionic pair that fails; the text diagnoses it as a near-flat residual rather than a fitting instability. c , The class separation and the swept criterion, which are different things. The shaded bands are the two class ranges; the gap between them is where a class-separating bar must fall. The swept criterion lies inside the covalent band and does not separate the classes, moving only the sulfur dimer. d , The per-battery means against the gate.
Figure 8: The application boundary: what the compressed model can and cannot be asked for. a , Three regimes on a common scale of deviation from the parent: same-manifold spin flips track it closely, gap-crossing promotions carry the ground-state overgap with them, and multi-determinant multiplets return an identically zero splitting under single-reference SCF, which is a method boundary rather than a parameterization one. b , The silicon divacancy gap against supercell size, the inter-functional difference changing sign as the cell grows.
| Quantity (Ha) | Carbon | Silicon | Permitted by the residual identity 1 |
|---|---|---|---|
| -resolved , median | – | ||
| at bond, machine-learned 2 | to | ||
| at bond, semilocal | – |
Table S1 : -resolved wall signature for carbon against silicon. Median -resolved on-site potential difference from separate - and -channel inversions of the machine-learned confined atom, and the two-center integral at the bond length under the inverted machine-learned and the semilocal potentials.
| System | (eV) | (eV) | |
|---|---|---|---|
| Diamond | |||
| Silicon | |||
| 3C-SiC |
Table S2 : Occupied-bandwidth changes under the change of functional (PBE CIDER23X), and the transfer ratio they form. Parent-DFT values are GPAW occupied bandwidths; DFTB values are the change in the tight-binding occupied bandwidth at the same geometry.
| CIDER23X | CIDER24Xe | PBE control | |||||||
|---|---|---|---|---|---|---|---|---|---|
| Atom | |||||||||
| C | |||||||||
| N | |||||||||
| O | |||||||||
| Si | |||||||||
Table S3 : Percentiles of the node-masked channel-potential asymmetry (Ha), for two machine-learned functionals and the PBE control.
| Prediction | What a failure would mean | Outcome | |
|---|---|---|---|
| P1 | exactly under PBE, where | the inversion is ill-conditioned, not the channel | holds: median and meV, at or below the meV noise floor |
| P2 | for every ; and linear in with the functional’s own | the near-nucleus law is not the mechanism | holds: – , far from the degenerate ; naive excluded |
Table S4 : The two predictions the theory makes that this study tests, and what a failure of each would mean. Both are computable from quantities the parameterization already produces, namely the functional evaluated on its own converged density and the confined-atom radial solutions, so neither requires new data. P1 is parameter-free: it decides whether the inversion is orbital-faithful at all, and everything downstream rests on it passing.
| Parent gap (eV) | Parent | Transfer ratio | |||
|---|---|---|---|---|---|
| System | PBE | r 2 SCAN | (eV) | Free-atom | Confined |
| Diamond | |||||
| Silicon | |||||
| 3C-SiC | |||||
Table S5 : Independent meta-GGA control. r 2 SCAN, carried through the same pipeline, anti-transfers like the machine-learned functionals: the parent gap opens while the compressed DFTB gap closes, giving a negative transfer ratio for every solid under the physical free-atom convention. Parent gaps are GPAW band-path fundamental gaps; the transfer ratio is the DFTB gap change divided by the parent gap change, .
| Set | Elements | Files | Expected 1 | Parent functional |
|---|---|---|---|---|
| PBE | C, N, O, Si | PBE (semilocal reference) | ||
| CIDER23X | C, Si | nonlocal-feature meta-GGA, energy-trained | ||
| CIDER24Xe | C, N, O | density-matrix features, gap-trained (molecular) | ||
| Total | 25 | 29 |
Table S6 : Released Slater–Koster parameter sets. Files counts the extended-format .skf files (homonuclear plus ordered heteronuclear pairs). Provenance is the parent functional bridged through the confined-atom inversion.