Where the Error Lives: Spectral Representations of Learned Kohn-Sham Hamiltonians

Fang Sun; Hekai Cheng; Ruoyan Li; Yuanzhou Chen; Wei Wang; Jason Cong; Yizhou Sun

physics.chem-phOct 27, 2026

Where the Error Lives: Spectral Representations of Learned Kohn-Sham Hamiltonians

Authors: Fang Sun, Hekai Cheng, Ruoyan Li, Yuanzhou Chen, Wei Wang, Jason Cong, Yizhou Sun

Organizations: Computer Science Department, University of California, Los Angeles

Abstract

Kohn-Sham density functional theory (DFT) is the standard method for computing the electronic structure of molecules and materials, but its self-consistent-field (SCF) iterations make it expensive for large systems and high-throughput screening. Machine-learned Hamiltonian models offer a shortcut: an equivariant network predicts the Kohn-Sham Hamiltonian matrix in one forward pass, and diagonalizing the prediction yields orbital energies, the HOMO-LUMO gap, molecular orbitals and the electron density. These models are trained to match the reference matrix entry by entry, yet a small entry-wise error does not guarantee accurate eigenvalues or eigenvectors. Two physics-informed alternatives go partway: a recently proposed wavefunction-alignment loss adds a spectral penalty without diagonalizing the prediction, and matching the electron density of the diagonalized prediction supervises it directly. Neither is complete: by representing the prediction error in the eigenbasis of the reference Hamiltonian, we show that each sees only part of the error, and that shifting the predicted orbital energies leaves the electron density exactly unchanged. The same analysis identifies a complete training signal: the cumulative local density of states (cumulative LDOS), the number of electrons at each point in space with energy below a given threshold. The cumulative LDOS is a single physical observable that contains the electron density as a special case, responds to every component of the error, determines the Hamiltonian uniquely, and yields gradients that do not vanish for large orbitalenergy errors. Fine-tuning three Hamiltonian networks on QH9 in this way, which we call FermiScope, reduces the HOMO-LUMO gap error without increasing the element-wise error. The gains carry over to out-of-distribution and molecular-dynamics splits, where the wavefunction-alignment loss helps less.

Figures & tables

Figure 1: Overview. (a–c) A Hamiltonian network maps a molecule, QH9 $\#23470$ (C 8 H 5 N), to its Kohn–Sham Hamiltonian $\hat{\mathbf{H}}$ , whose eigendecomposition gives the orbital energies and HOMO–LUMO gap, the electron density and the molecular orbitals. All visualizations in panels (a–c) use reference DFT data for this molecule. (d) In the reference orbital basis, the prediction error splits into three sectors: orbital-energy shifts ( A , yellow-orange diagonal), occupied–virtual mixing ( B , blue off-diagonal block) and rotations within the occupied or the virtual space ( C , purple off-diagonal blocks). The tags in panel (c) mark the sectors each downstream property depends on. (e) The sectors each training signal sees. WALoss is from Li et al. (2025) ; density matching, Equation 4 , is the $E=\mu$ slice of our loss. (f) At a point $\mathbf{r}$ , the cumulative LDOS $N(\mathbf{r},E)$ is a staircase with one step per orbital. An energy error shifts a step ( A ), and a shape error changes step heights ( B , C ). The shape error shown changes the LUMO step but not the slice at $\mu$ , which is the density, so it is of type C and invisible to density matching; Figure 2 shows all three types.

Figure 2: How each kind of error shows up in the cumulative LDOS: the first-order change of

N(\mathbf{r},E)

at one point

\mathbf{r}

as a function of energy (schematic levels with the median QH9 gap,

\sigma=0.02

Ha; sign and scale arbitrary). An orbital-energy error ( A , left) adds a bump at its level. An occupied–virtual coupling ( B , middle) adds a plateau between the two levels, which crosses the Fermi level

\mu

. A coupling of two occupied orbitals ( C , right) adds a plateau that ends below

\mu

. Density matching reads only the slice

E=\mu

(dashed line), where only the B plateau is nonzero; the cumulative LDOS loss reads the whole energy window and sees all three (Theorem 3.5 (ii)).

backbone	role	method	$H$	diag	non-diag	HOMO $\downarrow$	LUMO $\downarrow$	gap $\downarrow$	$\bm{\varepsilon}_{\mathcal{O}}$ $\downarrow$	$\psi$ $\uparrow$
		pretrained	77.56	129.25	73.68	1041	9445	9632	1206	96.28
	baseline	+ WALoss	78.04	119.03	74.95	847	5458	5835	570	96.79
	ablation	+ density only ( $w{=}100$ )	76.82	118.70	73.65	719	1575	2081	356	97.87
QHNet	ours	+ cumulative LDOS (ours)	74.81	109.54	72.19	678	1482	2027	341	97.91
		pretrained	36.60	44.79	35.38	345	1309	1554	380	99.06
	baseline	+ WALoss	37.28	46.03	35.98	341	957	1259	288	99.17

Table 1: Results on the QH9-stable-id test set (13,084 molecules) after 30k fine-tuning steps. Energies in

\mu

Ha,

\psi

in %. For each backbone, the gray rows are the two baselines, the pretrained model and fine-tuning with WALoss ( Li et al., 2025 ) . The violet row, density only , is the ablation at weight

w=100

that keeps only the

E=\mu

slice of our loss, which is density matching (Proposition B.4 ). The blue row is our cumulative LDOS loss, which adds the energy resolution. All fine-tuning runs share every other setting, and the peak-shaped variant is in Appendix G.4 . Bold marks the best downstream metric per backbone.

Figure 3: Reduction of the HOMO–LUMO gap error relative to each pretrained model on the three shifted QH9 splits (higher is better; negative values mean that the error grew). The cumulative LDOS loss (ours, trained as

\mathcal{L}_{\mathrm{elem}}+w\,\mathcal{L}_{\mathrm{cum}}

) lowers the gap error on every backbone and split, and it also improves every other downstream metric; Table 6 in Appendix G.1 lists all metrics.

Appendix figures & tables23 assets

Supplementary material from the paper’s appendix.

Appendix

term	meaning
atomic number $Z$	Number of protons of a nucleus; it identifies the element (H 1, C 6, N 7, O 8, F 9).
basis set, basis function, atomic orbital	A fixed set of functions $\phi_{\mu}$ centred on the atoms, in which the molecular orbitals are expanded; each function is called an atomic orbital.
def2-SVP	A standard Gaussian basis set of split-valence quality with polarization functions; it gives 5 functions per H atom and 14 per C, N, O or F atom.
molecular orbital $\psi_{i}$	A one-electron wavefunction of the molecule, a linear combination of basis functions; $\|\psi_{i}(\mathbf{r})\|^{2}$ is the probability density of an electron in that orbital.
Kohn–Sham DFT	Density functional theory in which the interacting electrons are replaced by non-interacting electrons in an effective potential that depends on the electron density.
exchange–correlation functional, B3LYP	The approximate part of the effective potential that accounts for many-electron effects; B3LYP is a widely used hybrid functional, which mixes in exact exchange.

Appendix

Table 2: Quantum-chemistry terms used in this paper.

Figure 4: (a) The self-consistent-field loop of Kohn–Sham DFT: the Kohn–Sham matrix is built from the current density matrix and diagonalized, and the occupied orbitals give the next density matrix, until the two agree. (b) A Hamiltonian network predicts the converged matrix in one forward pass; one diagonalization gives the orbital energies, orbitals and density, and the prediction can also initialize SCF. Iteration counts are medians over 50 QH9-stable-ood test molecules (PySCF, B3LYP/def2-SVP).

loss	energies $d_{i}$	$\bm{\Delta}_{\mathcal{O}\mathcal{V}}$ (density)	$\bm{\Delta}_{\mathcal{O}\mathcal{O}},\bm{\Delta}_{\mathcal{V}\mathcal{V}}$	weighting	eigenpairs of $\hat{\mathbf{H}}$
element-wise $\mathcal{L}_{\mathrm{elem}}$	$\checkmark$	$\checkmark$	$\checkmark$	uniform	–
WALoss, full ( Li et al., 2025 )	$\checkmark$	$\checkmark$	$\checkmark$	uniform	–
WALoss, diagonal-only ( Li et al., 2025 )	first order	–	–	occupied orbitals and LUMO	–
density matching $\mathcal{L}_{\rho}$	–	$\checkmark$	–	frontier, $1/(\varepsilon_{a}-\varepsilon_{i})$	$\checkmark$
cumulative LDOS $\mathcal{L}_{\mathrm{cum}}$ (ours)	$\checkmark$	$\checkmark$	$\checkmark$	frontier and energy window	$\checkmark$

Appendix

Table 3: Which sectors of the error each loss responds to (first order), how it weights them, and whether it uses the eigenpairs of the prediction. The element-wise loss is uniform up to the overlap metric (Lemma B.1 ).

split	selection rule	train	val	test	tests
stable-id	random $80/10/10$	104,664	13,083	13,084	interpolation
stable-ood	$\leq 20$ / $21$ – $22$ / $\geq 23$ atoms	104,001	17,495	9,335	molecule size
dynamic-geo	geometries within a molecule	79,920	9,990	9,990	unseen geometry
dynamic-mol	disjoint molecules	79,900	9,900	10,100	unseen molecule

Appendix

Table 4: The four QH9 splits. Sizes are the exact record counts read from the released split masks. The size split thresholds are on the total atom count, hydrogens included; the random split uses a fixed permutation with seed 43.

quantity	median	range	min–max
atoms per molecule	18	–	5–29
heavy atoms (C, N, O, F)	9	–	1–9
basis functions $n$	171	141–196	34–226
occupied orbitals $n_{\mathrm{occ}}$	33	29–35	5–37
occupied fraction $f=n_{\mathrm{occ}}/n$	0.193	0.178–0.219	0.147–0.256
HOMO–LUMO gap (Ha)	0.248	0.173–0.305	0.079–0.463

Appendix

Table 5: QH9-stable composition and reference spectra, measured on a uniform 4,000-molecule sample. Ranges are

5

–

95\%

intervals unless a min–max is stated. The last row is the exact fraction of the parameters of a symmetric residual that lie in the two sectors the density loss cannot resolve.

Figure 5: QH9-stable statistics from a uniform 4,000-molecule sample. (a) Basis size. (b) Number of occupied orbitals. (c) Reference HOMO–LUMO gap. (d) Exact partition of the

n(n+1)/2

parameters of a symmetric residual

\bm{\Delta}

into the three sectors of Proposition 3.1 , at fixed

n=171

, as a function of the occupied fraction

f

. Only the occupied–virtual slice changes the density; at the QH9 median occupied fraction it accounts for

31\%

of the parameters, and the remaining

69\%

is invisible to

\mathcal{L}_{\rho}

Figure 6: Reference electron density for twelve QH9-stable molecules, on the best-fit plane through the nuclei, logarithmic colour scale, all panels on the same length scale and the same colour scale. Bonds are drawn from covalent-radius criteria; nuclei more than $0.7\,a_{0}$ off the plane are drawn open, so a panel with many open markers is a non-planar molecule seen in projection. Molecular formulae are in Hill notation.

Figure 7: Anatomy of a reference Kohn–Sham Hamiltonian, molecule $\#12345$ (C 7 H 10 O, $n=162$ , $n_{\mathrm{occ}}=30$ ). (a) $|\mathbf{H}^{\star}_{\mu\nu}|$ and (b) $|\mathbf{S}_{\mu\nu}|$ on a logarithmic colour scale; thin white rules mark heavy-atom block boundaries. Entries below $10^{-5}$ are shown at the floor of the scale. (c) Magnitude of the Hamiltonian elements against the distance between the atoms carrying the two basis functions, with same-atom pairs placed at zero and jittered. The orange curve is the binned median and the orange marker the same-atom median. The decay over six orders of magnitude is what local message-passing architectures exploit.

Figure 8: Real-space reference quantities for molecule $\#23470$ , 2,5-diethynyl-1H-pyrrole (C 8 H 5 N, $n=151$ , $n_{\mathrm{occ}}=30$ ), which is exactly planar. (a) Electron density in the nuclear plane, logarithmic scale, with the molecular skeleton overlaid and the nitrogen drawn open. (b,c) HOMO and LUMO amplitudes sampled $1\,a_{0}$ above the nuclear plane and normalized by their own maxima; both vanish identically in the plane itself. (d) Upper envelope ( $99.5$ th percentile per radial bin) of $\rho^{\star}$ and of the core-suppressed target $\rho^{\star}/\mathrm{div}$ against distance to the nearest nucleus; the two coincide beyond $2\,a_{0}$ , where $\mathrm{div}=1$ . (e) The 256 sampled points reproduce the radial distribution of the full 14,112-point quadrature grid. (f) Relative error of the atomic-superposition density against distance to the nearest nucleus.

Figure 9: Spectral reference quantities for molecule

\#23470

. (a) Generalized eigenvalues with the loss window shaded; the energy axis is broken so that the nine core levels are visible. (b) Cumulative LDOS

N(\mathbf{r},E)

of Equation 7 at five of the 256 sample points, spanning three orders of magnitude in local density. (c) Proposition B.4 tested pointwise:

N(\mathbf{r},\mu)

against

\rho^{\star}(\mathbf{r})/2

at all 256 sample points. (d) The two energy profiles at one sample point, with the constant contribution of the levels below the window removed and each curve normalized by its own maximum. Ticks mark the orbital energies.

Figure 10: Sector content of a physically realized Hamiltonian change, and two numerical properties of the estimator. (a) $|\bm{\Delta}|$ for one pair of QH9-dynamic frames of the same molecule separated by $0.30$ Å RMSD, in the eigenbasis of the reference frame; red rules mark the occupied/virtual boundary at $n_{\mathrm{occ}}=34$ , and values are clipped to the colour range at both ends. (b) Median share of $\|\bm{\Delta}\|_{F}^{2}$ per sector against displacement, over 640 frame pairs from 214 trajectories. The virtual–virtual share, which contains the LUMO energy error and is invisible to $\mathcal{L}_{\rho}$ , doubles from $15\%$ to $30\%$ as the displacement grows. (c) Relative standard error of the per-molecule loss estimate when $|\mathcal{R}|$ of the 256 cached sample points are used, over 400 draws; solid lines are a random $75$ $\mu$ Ha element-wise residual, dashed lines a $10$ mHa LUMO shift, and the grey guide is the $|\mathcal{R}|^{-1/2}$ rate. (d) Observed amplification $\|\bm{\Delta}\|_{F}/\|\delta\mathbf{H}\|_{F}$ against the bound $\|\mathbf{S}^{-1}\|_{2}$ of Lemma B.1 , same frame pairs.

backbone	split	method	$H$	diag	non-diag	HOMO $\downarrow$	LUMO $\downarrow$	gap $\downarrow$	$\bm{\varepsilon}_{\mathcal{O}}$ $\downarrow$	$\psi$ $\uparrow$
QHNet	stable-ood	pretrained	73.53	122.32	70.78	561	4632	4522	740	95.54
		+ WALoss	72.70	115.00	70.30	475	5511	5671	393	95.04
		+ cumulative LDOS (ours)	71.10	110.81	68.87	442	1420	1506	290	96.17
	dynamic-geo	pretrained	98.4	161.4	93.6	960	5561	6090	840	96.66
		+ WALoss	100.0	162.2	95.3	976	6950	7441	597	96.06
		+ cumulative LDOS (ours)	95.6	148.5	91.6	771	1517	2132	400	97.85

Appendix

Table 6: Results on the shifted QH9 splits (test sets; energies in

\mu

Ha,

\psi

in %). Same protocol as Table 1 . Bold marks the best downstream metric per block.

backbone	method	$H$	diag	non-diag	HOMO $\downarrow$	LUMO $\downarrow$	gap $\downarrow$	$\bm{\varepsilon}_{\mathcal{O}}$ $\downarrow$	$\psi$ $\uparrow$
QHNet	pretrained	77.56	129.25	73.68	1041	9445	9632	1206	96.28
	+ element-wise loss only (control)	74.44	108.11	71.90	1280	14946	14769	868	95.57
	+ WALoss	78.04	119.03	74.95	847	5458	5835	570	96.79
	+ cumulative LDOS (ours)	74.81	109.54	72.19	678	1482	2027	341	97.91
SPHNet	pretrained	36.60	44.79	35.38	345	1309	1554	380	99.06
	+ element-wise loss only (control)	36.55	44.67	35.35	349	1311	1555	381	99.05

Appendix

Table 7: QH9-stable-id, all three backbones (Table 1 with the element-wise-only control added; test set,

\mu

Ha,

\psi

in %). Bold marks the best downstream metric per backbone.

split	method	$H$	diag	non-diag	HOMO $\downarrow$	LUMO $\downarrow$	gap $\downarrow$	$\bm{\varepsilon}_{\mathcal{O}}$ $\downarrow$	$\psi$ $\uparrow$
stable-id	pretrained	77.56	129.25	73.68	1041	9445	9632	1206	96.28
	+ element-wise loss only (control)	74.44	108.11	71.90	1280	14946	14769	868	95.57
	+ WALoss	78.04	119.03	74.95	847	5458	5835	570	96.79
	+ cumulative LDOS (ours)	74.81	109.54	72.19	678	1482	2027	341	97.91
stable-ood	pretrained	73.53	122.32	70.78	561	4632	4522	740	95.54
	+ element-wise loss only (control)	70.86	109.25	68.70	783	15801	15638	582	94.35

Appendix

Table 8: QHNet on all four QH9 splits (the QHNet rows of Tables 1 and 6 with the element-wise-only control added; test sets,

\mu

Ha,

\psi

in %). Bold marks the best downstream metric per split.

method	$H$	diag	non-diag	HOMO	LUMO	gap	$\bm{\varepsilon}_{\mathcal{O}}$	$\psi$
pretrained	77.56	129.25	73.68	1041	9445	9632	1206	96.28
element-wise loss only (control)	74.44	108.11	71.90	1280	14946	14769	868	95.57
WALoss $w{=}0.03$	76.01	112.51	73.26	823	5728	6072	593	96.81
WALoss $w{=}0.1$	78.04	119.03	74.95	847	5458	5835	570	96.79
density only $w{=}1$	77.72	126.93	74.02	780	1923	2376	571	97.44
density only $w{=}10$	79.01	138.44	74.55	802	2500	2896	373	97.43

Appendix

Table 9: QHNet on QH9-stable-id: all metrics against the auxiliary weight

w

(test set,

\mu

Ha). WALoss is the full form used throughout the paper; the weight of the cumulative loss,

w=30

, is selected on the validation split.

method	$H$	diag	non-diag	HOMO	LUMO	gap	$\bm{\varepsilon}_{\mathcal{O}}$	$\psi$
pretrained	77.56	129.25	73.68	1041	9445	9632	1206	96.28
diagonal-only WALoss $w{=}1$	74.48	108.29	71.93	1130	12504	12373	797	95.88
diagonal-only WALoss $w{=}10$	75.20	110.03	72.58	1090	14208	13701	977	95.59
diagonal-only WALoss $w{=}100$	77.84	117.44	74.86	1862	28282	26813	1573	93.66

Appendix

Table 10: QHNet on QH9-stable-id with the diagonal-only form of WALoss at three weights (test set,

\mu

Ha).

Figure 11: Weight sensitivity on QHNet (entries of Tables 9 and 10 ). (a) Gap error and (b) occupied-orbital energy error against the auxiliary weight

w

, with the pretrained checkpoint, the element-wise-only control and WALoss (

w=0.1

) as horizontal references. The diagonal-only WALoss gets worse as its weight grows; the density-only and cumulative losses improve or stay flat, and the cumulative loss is the lowest at every weight.

loss	LUMO $+2$	LUMO $+1$	LUMO	HOMO	HOMO $-5$	HOMO $-19$	HOMO $-21$	HOMO $-23$
pretrained	3455	3838	9593	1005	403	635	1029	1049
peak LDOS	2287	1698	7685	1053	593	5955	11364	20979
cumulative LDOS	1032	994	1567	690	235	195	251	367

Appendix

Table 11: Per-orbital energy error on QHNet (1,000 test molecules,

\mu

Ha) for the peak-shaped and the cumulative loss at the same weight (

w=30

), window and

\sigma

. The peak-shaped loss corrects the virtual orbitals inside the window but lets the deep occupied orbitals drift; the cumulative loss improves every orbital.

method	SCF iterations	SCF time	total time
pretrained	0.6732	0.8115	0.8619
element-wise loss only (control)	0.6735	0.8139	0.8655
density only ( $w=100$ )	0.6710	0.8171	0.8682
cumulative LDOS ( $w=100$ )	0.6713	0.8028	0.8505

Appendix

Table 12: SCF acceleration on 50 QH9-stable-id test molecules: ratio of SCF iterations and of wall time relative to the minao initial guess (lower is better; total time includes model inference).

auxiliary loss	seconds/step	overhead	peak GPU memory (GB)
none (element-wise only)	1.47	–	31.2
density only	1.92	$+0.45$ s ( $+30\%$ )	31.2
peak LDOS	2.28	$+0.81$ s ( $+55\%$ )	31.2
cumulative LDOS	2.19	$+0.72$ s ( $+49\%$ )	31.2

Appendix

Table 13: Training cost of the auxiliary losses on QHNet (QH9-stable-id), measured on one NVIDIA L40S with the same code, data order and batch size (32). Seconds per step are averaged over steps 50–300 without validation; memory is the peak reserved CUDA memory. The eigendecompositions run on the CPU in double precision.

Figure 12: (a) Response of each loss to controlled perturbations of the reference Hamiltonian of one QH9 molecule ( $\#12345$ ): shifting the LUMO or HOMO energy by 2 mHa with all orbitals fixed, or rotating the HOMO and LUMO into each other with all energies fixed. Density matching does not respond to energy shifts at all (values at the level of floating-point round-off); both LDOS losses respond to both kinds of error. (b) Mismatch between a displaced and a reference orbital as a function of the energy error $\delta$ , for a Gaussian peak and for a cumulative step (closed forms of Theorem 3.5 (iv)): the peak mismatch saturates, the step mismatch keeps growing. (c) The corresponding gradient with respect to $\delta$ , normalized by its maximum: the peak gradient vanishes beyond a few $\sigma$ , whereas the step gradient approaches a constant.

statement	what is checked	result	reference
generalized eigenbasis	$\\|\hat{\mathbf{C}}^{\top}\mathbf{S}\hat{\mathbf{C}}-\mathbf{I}\\|_{\infty}$	$4.8\times 10^{-14}$	–
cache consistency	recomputed vs. released $\mathbf{S}$	$2.2\times 10^{-7}$	App. C
cache consistency	$\int\rho^{\star}\,dV$ vs. $N_{e}=60$	$59.978$	App. C
Prop. B.4	$N(\mathbf{r},\mu)$ vs. $\rho^{\star}(\mathbf{r})/2$ , all 256 points	$6.4\times 10^{-8}$ rel. (float32 cache)	Fig. 9 c
Prop. B.6	closed forms $g(\delta)$ , $c(\delta)$ vs. quadrature	$9.3\times 10^{-16}$ rel.	Fig. 12 b
energy-only perturbation	induced $\Delta\varepsilon_{i}$	$2.000$ mHa (exact)	Fig. 12 a

Appendix

Table 14: Numerical checks. Molecule

\#12345

and

\#23470

are the two QH9-stable geometries used in the appendix figures; the lower block also uses molecule

\#18

. “Machine zero” means the response is at or below the double-precision noise floor of the quantity being differenced.

	QHNet	SPHNet / WANet
optimizer	AdamW, $\beta=(0.99,0.999)$	native
learning rate	$1\times 10^{-5}$	$1\times 10^{-5}$
warm-up steps	1,000	native
fine-tuning steps	30,000	30,000
batch size	32	64
gradient clipping	norm 5.0	norm 5.0

Appendix

Table 15: Fine-tuning settings. Every method of a given backbone shares all settings except the auxiliary loss and its weight; unlisted settings follow each backbone’s released training recipe (“native”). The cumulative LDOS weight is selected on the validation split; Table 9 reports its test-set sensitivity on QHNet.

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Abstract

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