Beyond DFT electronic band structures via dual-space orbital learning

Minhyuk Kang; Beom Seok Kang; Nikolai Argatoff; Miguel Nouman; Danish Khan; William A. Goddard; Anima Anandkumar

Figure 1: Dual-space orbital learning and representative applications. a, Starting from a crystal structure, a DFT calculation supplies baseline band energies and orbital matrices, including the Hamiltonian

\mathbf{H}

, overlap

\mathbf{S}

, density

\mathbf{P}

, and spin-density matrices

\mathbf{P}^{(\sigma)}_{spin}

, represented in real (

\mathbf{R}

)- and reciprocal (

\mathbf{k}

)-space. The dual-space orbital representations are processed by a periodic E(3)-equivariant graph neural network to predict state-dependent energy corrections. Target-theory (e.g., HSE06 or

G_{0}W_{0}

) band energies are reconstructed as

\hat{\varepsilon}^{\mathrm{target}}_{n\mathbf{k}\sigma}=\varepsilon^{\mathrm{base}}_{n\mathbf{k}\sigma}+\widehat{\Delta\varepsilon}_{n\mathbf{k}\sigma}

. b, Band dispersion and density of states for the displayed spin-down channel of GaO 2 , comparing PBE, HSE06, and O-kRystal. For the three-dimensional PBE-to-HSE06 task, supervision is restricted to high-symmetry points. The model generalizes to unseen path-interior points, while the density of states is reconstructed from predictions on a dense Brillouin-zone mesh. c, Spin-resolved Mn

3d

and O

2p

projected densities of states for MnO with AFM-II ordering along [111]. The O-kRystal reconstruction combines predicted HSE06 state energies with fixed PBE Mulliken projection weights, illustrating orbital-dependent changes in state-energy placement while retaining the baseline orbital character. Resolving the relative placement of transition-metal

d

and ligand

p

states is particularly important in strongly correlated oxides, where these energy scales are central to Mott and charge-transfer electronic behavior. d, Absolute band-gap errors relative to experiment for ten independently evaluated bulk materials from the mstar60 benchmark [ 47 ] , compared across PBE, HSE06, and O-kRystal

Figure 2: Large-scale prediction of HSE06 band structures with O-kRystal. a, Composition of the three-dimensional materials dataset containing 5,988 chemically diverse parent materials [ 37 ] . Element counts indicate the number of materials containing each element. Horizontal bars summarize PBE-to-HSE06 electronic-character classes and spin polarization. b, PBE and O-kRystal band energies compared with HSE06 for 575,487 matched test-state pairs within the PBE-defined 5 eV energy window for a representative held-out test split. Energies are independently referenced to the band-edge midpoint of each spectrum. The dashed line denotes parity, and the reported MAEs are state weighted. c, Entry-balanced band-energy MAEs for uncorrected PBE, scissor correction, Linear+KRR, Geometry GNN, and O-kRystal over energy windows of 2, 5, 10, and 20 eV.

\Delta

identifies the neural models trained directly on state-resolved PBE-to-HSE06 energy corrections. d, Learning curves for O-kRystal as a function of training-set size, shown separately for insulating and metallic HSE06 entries within the 5 and 20 eV windows. Values and error bars in c and d denote the mean and sample standard deviation across three independently trained models with parent-grouped splits. Band-energy evaluations in b–d use the supervised high-symmetry

\mathbf{k}

-points.

Figure 3: Generalization between and beyond high-symmetry

\mathbf{k}

-points. a, Comparison of high-symmetry-point and path-interior band-energy MAEs for 693 held-out band-structure entries within the 5 eV window. Each point represents one entry, including separate spin channels where applicable. Median MAEs are 0.122 eV and 0.121 eV, respectively, and the dashed line denotes parity. b, Distribution of normalized density-of-states (DOS) reconstruction errors for 356 insulating and 225 metallic parent materials. Spectra are reconstructed from dense off-path predictions and compared with HSE06 over

|E-E_{\mathrm{mid}}|<

5 eV using the normalized

L_{1}

metric defined in Methods 4.6 . Dashed horizontal markers indicate the PBE and O-kRystal medians. GaO 2 and WC identify the examples displayed in Figs. 1 b and 3 c, respectively. c, Band structure and density of states of WC, comparing HSE06, O-kRystal, and PBE. Marked high-symmetry points show the sampling convention used for supervision. Intermediate band-path points and the dense mesh used for DOS reconstruction are evaluated without target-theory information at inference.

Figure 4: Electronic-character recovery and orbital-dependent corrections in strongly correlated materials. a, Recovery of HSE06 electronic character for insulator-to-insulator, metal-to-insulator, and metal-to-metal systems. Bars show the fraction for which O-kRystal predicts the correct HSE06 metallic or insulating character across three independently trained models. Adjacent fractions report the number of correct predictions and total entries pooled across the corresponding test evaluations. b, Distributions of band-energy MAE for PBE, Geometry GNN, and O-kRystal in the main held-out test cohort of 693 entries and a separate benchmark of six literature-validated strongly correlated materials [ 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 ] . The correlated-material comparison is evaluated on dense Brillouin-zone meshes within the 5 eV window. Horizontal markers denote means. The annotated 9.2% and 30.8% reductions compare O-kRystal with Geometry GNN within the respective cohorts. c, Reconstruction errors relative to HSE06 for transition-metal

3d

and oxygen

2p

projected densities of states (PDOS) across six correlated materials, with spectra reconstructed using fixed PBE Mulliken projection weights to isolate differences in state-energy placement. Each row corresponds to one material, with PBE, Geometry GNN, and O-kRystal evaluated separately. The schematic summarizes the relative energy alignment of metal-centered transition-metal

3d

and ligand-derived oxygen

2p

states, which is particularly relevant to correlation-sensitive electronic structure. The MnO spectra in Figure 1 illustrate the corresponding orbital-dependent energy redistribution.

Figure 5: Band gaps and gap-informed effective-mass estimates. a, Fundamental band gaps derived from PBE and O-kRystal band energies compared with HSE06 reference gaps across the test set. The dashed line denotes parity, and the corresponding MAE and

R^{2}

values are reported for both methods. b, Material-balanced mean absolute percentage errors (MAPE) relative to experiment for 31 directional effective masses across ten bulk materials from mstar60 benchmark [ 47 ] . Results are grouped into III–V semiconductors, chalcogenides, and other semiconductors. O-kRystal* denotes a calibrated gap-informed reconstruction in which the corresponding PBE effective mass for the same carrier branch, band extremum, and crystallographic direction provides the baseline curvature information and is rescaled using the predicted-to-PBE fundamental-gap ratio. Reconstruction parameters are fitted to HSE06 masses from the other material families in a leave-one-family-out protocol.

Figure 6: Extension to

G_{0}W_{0}

quasiparticle energies. a, Five-fold cross-validated quasiparticle-state and band-gap MAEs for O-kRystal on 286 two-dimensional semiconductors from the C2DB-derived

G_{0}W_{0}

benchmark [ 38 , 39 , 41 ] . Separately trained models use either GPAW or independently calculated FHI-aims PBE eigenvalues [ 63 , 64 ] , while the

G_{0}W_{0}

labels, sampled

\mathbf{k}

points, and orbital-matrix inputs are held fixed. Each outer split uses three folds for training, one for validation and one for testing. Bars and error bars denote means and sample standard deviations across the five test folds. b, Absolute differences between FHI-aims and GPAW PBE eigenvalues for the eight highest valence (VB1–VB8) and four lowest conduction (CB1–CB4) states. Circles indicate the median discrepancy, boxes the interquartile range, and whiskers the 10th–90th percentile range. Shading highlights the frontier bands VB1 and CB1.

Stage / seed	Partition	Parents	Entries	Atoms mean [IQR]	$k_{\mathrm{HS}}$ mean [IQR]	$k_{\mathrm{full}}$ mean [IQR]	$I\!\to\!I$	$M\!\to\!I$	$M\!\to\!M$
							Parent fraction (%)
Dataset construction
Paired source	All	6576	8040	26.6 [9–32]	13.1 [12–15]	184.0 [131–220]	50.8	7.2	42.0
After magnetic-state cleaning	All	6045	6978	27.6 [9–34]	13.1 [12–15]	181.1 [130–216]	54.6	6.3	39.1
Final benchmark dataset	All	5988	6900	27.8 [10–34]	13.1 [12–15]	180.6 [129–216]	54.5	6.2	39.3
Grouped train/validation/test splits

Table S1: Dataset composition and grouped-split balance. Parent-material and band-structure-entry counts are exact. Atoms and

\mathbf{k}

-point counts are reported as entry-weighted means with interquartile ranges shown beneath in brackets.

k_{\mathrm{HS}}

and

k_{\mathrm{full}}

denote the stored high-symmetry and full-path

\mathbf{k}

-point counts, respectively. Electronic-character fractions are parent weighted. All retained datasets contain 59 chemical elements.

Category	Setting	Value
Model	Trainable parameters	8,940,709
	Branches	Real+phase affine; dual-space residual
	Interaction / decoder layers	affine 4 / 4; residual 2 / 2
Representation	Orbital operators	$H$ , $S$ , $P$ , $P_{\mathrm{spin}}$
	Reciprocal representation	Real components of orbital matrices
	AO block dimension	16

Table S2: Architecture and training settings for the three-dimensional PBE-to-HSE06 O-kRystal model.

Model	Component	Setting
Scissor	Inputs	PBE band energies and stored occupied/unoccupied assignment
	Fit	Separate global mean PBE-to-HSE06 corrections for occupied and unoccupied training states; no validation or test fitting
Linear+KRR	Inputs	Relative PBE energy, occupation indicator, band width, band standard deviation, $\log(1+\text{nearest-band spacing})$ , and $\log(N_{\mathrm{atoms}})$
	Linear stage	Training-set standardization followed by ridge regression with $\lambda_{\mathrm{ridge}}=10^{-3}N$ , where $N$ is the number of sampled training states
	Kernel stage	RBF kernel-ridge regression of the linear-model residual with $\gamma=1/6$ and regularized system $K^{\mathsf{T}}K+10^{-2}NK_{cc}+10^{-6}I$
	Sampling	Up to 768 high-symmetry states per training entry; up to 768 kernel centres sampled without replacement from the pooled training states

Table S3: Definitions and implementation details for comparison models and representation controls. For the neural learning-curve comparison, Geometry GNN, Real+Bloch phase, and O-kRystal use matched seed-specific training subsets, the same final training objective and optimization schedule, raw best_avg_win checkpoints, and the same high-symmetry-point evaluation protocol. The three neural models therefore differ primarily in their electronic representation and architecture.

Figure S1: Learning-curve comparison of geometric, real-space orbital, and dual-space orbital representations. Test-set band-energy MAE as a function of the number of training entries for states within a , 5 eV and b , 20 eV of the PBE reference energy. The Geometry GNN uses periodic crystal geometry, Bloch-phase conditioning, and the PBE band baseline and descriptors, but no orbital matrices. Real+phase additionally uses real-space orbital blocks with Bloch-phase conditioning, while O-kRystal further incorporates explicit reciprocal-space orbital matrices. All models are evaluated on the corresponding fixed test sets using matched training-data subsets. Points and error bars denote the mean and sample standard deviation across seeds 42–44. The representations perform similarly in the low-data regime, whereas the full dual-space model shows a progressively clearer advantage with increasing training-set size, particularly across the broader 20 eV energy window.

Figure S2: Dependence of unseen-point error on reciprocal-space position. Diagnostics for path-interior states within

|E-E_{\mathrm{mid}}|<5

eV using the seed-42 raw-weight checkpoint. a , Distribution of the per-entry difference between interior-point and high-symmetry-point MAE. b , Band-energy MAE as a function of reciprocal distance to the nearest high-symmetry point. c , Band-energy MAE as a function of normalized position within a path segment. d , Band-energy MAE as a function of segment length. Panels b–d pool valid states across k-point rows and are therefore state-weighted rather than material-balanced. The elevated metallic bins are concentrated in a small number of high-error materials and should not be interpreted as a universal monotonic dependence on reciprocal-space distance or segment length.

Figure S3: Representative full-path band-structure predictions. PBE, HSE06, and O-kRystal band energies are shown along identical stored

\mathbf{k}

-paths for representative materials spanning insulating, metallic, and spin-polarized electronic structures. The gallery includes GaO 2 and WC, whose dense Brillouin-zone DOS reconstructions are shown in Figs. 1 b and 3 c, respectively, together with AFM-II MnO, whose orbital-resolved spectrum is shown in Fig. 1 c. Additional examples illustrate performance across different PBE-to-HSE06 electronic-character changes and spin configurations. Values in parentheses denote the O-kRystal band-energy MAE for each displayed entry within

\lvert E-E_{\mathrm{mid}}\rvert<5

eV.

	$\leq 5$ eV			$\leq 20$ eV
Material	PBE	Geometry GNN	O-kRystal	PBE	Geometry GNN	O-kRystal
CrO 2	0.974	0.348	0.279	1.259	0.440	0.332
LaCoO 3	1.741	0.997	0.591	1.600	0.956	0.516
LaNiO 3	1.576	0.462	0.367	1.469	0.364	0.368
YTiO 3	1.036	0.198	0.102	1.201	0.177	0.121
MnO	1.678	0.692	0.605	1.537	0.500	0.489

Table S4: Correlated-material band-energy benchmark. Band-energy mean absolute errors (MAEs) are evaluated using the representative seed-42 model on dense Brillouin-zone meshes. States are selected using inclusive PBE-centered energy windows of 5 and 20 eV. For spin-polarized systems, retained states are pooled across spin channels before calculating the parent-material MAE. The final row reports the equal-material mean across the six materials. All values are in eV.

Material	HSE06	PBE	Geometry GNN	O-kRystal
CrO 2	$-0.809$	$-1.162$ (0.353)	$+1.052$ (1.861)	$+1.692$ (2.501)
LaCoO 3	$+2.669$	$-0.084$ (2.753)	$+0.735$ (1.934)	$+1.852$ (0.818)
LaNiO 3	$-2.159$	$-1.311$ (0.849)	$-0.744$ (1.416)	$-0.175$ (1.985)
YTiO 3	$+1.852$	$-0.364$ (2.216)	$+0.340$ (1.513)	$+0.536$ (1.316)
MnO	$+2.864$	$+0.803$ (2.061)	$+1.835$ (1.029)	$+2.611$ (0.253)
NiO	$+4.458$	$+0.973$ (3.485)	$+1.595$ (2.863)	$+2.109$ (2.349)

Table S5: Fundamental-gap recovery for the correlated-material benchmark. Predicted fundamental gaps are compared with the corresponding HSE06 values. Values in parentheses denote the absolute gap error relative to HSE06. Negative gaps indicate metallic electronic character and positive gaps indicate insulating character. All values are in eV.

	Transition-metal $3d$			Oxygen $2p$
Material	PBE	Geometry GNN	O-kRystal	PBE	Geometry GNN	O-kRystal
CrO 2	1.055	0.466	0.456	0.642	0.323	0.453
LaCoO 3	1.181	0.839	0.571	0.648	0.734	0.621
LaNiO 3	0.931	0.862	0.634	0.801	0.671	0.726
YTiO 3	1.477	0.780	0.450	1.309	0.396	0.255
MnO	1.938	1.163	0.715	1.378	1.109	0.676

Table S6: Orbital-resolved PDOS reconstruction errors for the correlated-material benchmark. Normalized

L_{1}

errors are evaluated relative to HSE06 within

E_{\mathrm{mid}}\pm 5

eV. All spectra use identical PBE Mulliken projection weights, such that only the state energies differ between PBE, Geometry GNN, O-kRystal, and HSE06. The final row reports the equal-material mean across the six materials. The metric is dimensionless.

Family	Materials	Materials	Observations
III–V	GaAs, GaN, InP	3	13
Chalcogenides	CdS, CdTe, 2H-MoS 2 , 2H-WS 2	4	9
Other semiconductors	Si, SiC, BN	3	9
Total	–	10	31

Table S7: Effective-mass benchmark cohort. The primary benchmark contains 31 directional effective-mass observations across ten bulk materials.

Material	Family	Mass type	$k$ point	Direction	Experiment	PBE	HSE06	O-kRystal ∗
GaAs	III–V	$m_{n}$	$\Gamma$	–	0.066	0.028	0.062	0.050
GaAs	III–V	$m_{n,\perp}$	$X_{6}$	–	0.230	0.233	0.223	0.229
GaAs	III–V	$m_{n,\perp}$	$L_{6}$	–	0.075	0.102	0.110	0.148
GaAs	III–V	$m_{n,\parallel}$	$L_{6}$	–	1.900	1.660	1.660	1.188
GaAs	III–V	$m_{p,\mathrm{hh}}$	$\Gamma$	[100]	0.395	0.324	0.309	0.257
GaAs	III–V	$m_{p,\mathrm{lh}}$	$\Gamma$	[100]	0.090	0.034	0.077	0.060

Table S8: Effective-mass observations used in the primary bulk benchmark. All masses are reported in units of the free-electron mass. O-kRystal ∗ denotes the family-held-out gap-informed reconstruction obtained from the PBE effective mass and the fundamental gap predicted by the raw seed-42 O-kRystal checkpoint. It is not a direct effective-mass output of the model.

Figure S4: State-resolved

G_{0}W_{0}

quasiparticle prediction errors. Mean absolute error for individual valence (VB8–VB1) and conduction (CB1–CB4) states across the five material-held-out folds. The two variants differ only in the PBE eigenvalue baseline used to define the state-resolved correction target. The

G_{0}W_{0}

labels,

\mathbf{k}

-point sets, and orbital inputs are otherwise identical.

Fold	Train	Validation	Test	GPAW state [ 63 ]	GPAW gap	FHI-aims state [ 64 ]	FHI-aims gap
0	171	57	58	0.1197	0.2132	0.1541	0.2374
1	172	57	57	0.1380	0.2173	0.1811	0.2229
2	172	57	57	0.1449	0.1995	0.1728	0.2018
3	172	57	57	0.1237	0.1575	0.1645	0.2021
4	171	58	57	0.1021	0.1396	0.1649	0.1654
Mean $\pm$ s.d.	–	–	–	$0.1257\pm 0.0167$	$0.1854\pm 0.0349$	$0.1675\pm 0.0101$	$0.2059\pm 0.0272$

Table S9: Five-fold

G_{0}W_{0}

quasiparticle benchmark. State and band-gap MAEs are reported in eV for independently trained models. For each outer fold, three source folds train the model, one separate fold selects the checkpoint by validation MAE, and the remaining fold is evaluated once as test data.

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