Stationarity Constraints Improve Both Energies and Densities of Machine-Learned Exchange-Correlation Functionals

Yuhao Chen; Donald G. Truhlar; Xiao He

physics.chem-phOct 21, 2026

Stationarity Constraints Improve Both Energies and Densities of Machine-Learned Exchange-Correlation Functionals

Authors: Yuhao Chen, Donald G. Truhlar, Xiao He

Organizations: Shanghai Engineering Research Center of Molecular Therapeutics and New Drug Development, Shanghai Frontiers Science Center of Molecule Intelligent Syntheses, School of Chemistry and Molecular Engineering, East China Normal University, Shanghai 200062, China · Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, USA · Anhui Provincial Key Laboratory of Advanced Catalysis and Energy Materials, Anhui Ultra High Molecular Weight Polyethylene Fiber Engineering Research Center, School of Chemistry and Chemical Engineering, Anqing Normal University, Anqing 261433, China · New York University–East China Normal University Center for Computational Chemistry, New York University Shanghai, Shanghai 200062, China

Abstract

Empirical exchange-correlation functionals are fitted on fixed orbitals but used self-consistently, so better energies often come with worse densities. We show that the Brillouin residual of a trained functional on the Kohn-Sham orbitals of an accurate density controls, to first order, both the self-consistent density shift and the gap between fitted and self-consistent energies. Penalizing it requires no self-consistent-field solution during training. Trained on 6 energies and 11 CCSD densities, a neural meta-GGA surpasses SCAN in both atomization energies and densities of unseen molecules.

Figures & tables

Figure 1: Training XC functionals on fixed orbitals. (a) Wu-Yang inversion of the CCSD density gives fixed KS orbitals, on which the occupied-virtual Fock block $F_{ai}$ of the trained functional yields the predicted SCF rotation $\kappa_{ai}$ (stationarity loss $\mathcal{L}_{G}$ ) and the first-order density change $\delta n$ (density loss $\mathcal{L}_{D}$ ); the energy loss $\mathcal{L}_{E}$ uses the same orbitals. One Fock build per system gives the parameter gradient, with no SCF. (b) Energy along one occupied-virtual rotation. Energy-only training (E) matches the reference energy at $\kappa=0$ , but the orbital gradient $2F_{ai}$ drives the SCF solution to $\kappa^{*}=-F_{ai}/\Delta_{ai}$ , causing an energy error $\Delta_{ai}\kappa^{*2}$ and a density shift $\delta n$ . EGD makes the reference orbitals stationary.

Figure 2: Absolute errors of self-consistent atomization and ionization energies on the test sets. Boxes: interquartile range; bars: medians; diamonds: MAEs. “Energy loss only” denotes E training; the other NN functionals use the full loss, Eq. ( 9 ) (EGD). The stationarity and density terms lower the errors of both NN functionals on all atomization-energy sets, most strongly for new bond pairs. CCSD uses the aug-cc-pVTZ basis.

Figure 3: (a) GGA and (b) meta-GGA functionals in the plane of AE103 atomization-energy MAE and DP99 density error. Open circles: conventional functionals; filled blue circles: Pareto-optimal conventional functionals; blue line: Pareto front; red diamonds: NN functionals trained with $L=E$ or $L=EGD$ . The two constraints move both NN functionals from the interior to the low-density-error end of the front. Values for all functionals are in the SM [ 38 ] .

	Training	Validation
Atomization	H 2 , LiH, O 2 , CO	N 2 , NO, F 2 , HF
Density	H, Li, Ne, H 2 , Li 2 , LiH, B 2 , O 2 , CO, O + , Ne +	He, Be, N, N 2 , F 2 , HF
Ionization	O, Ne	N, F

Table 1: Training and validation systems. Density systems use CCSD

\Lambda

densities and their Wu-Yang orbitals, entering

\mathcal{L}_{G}

and

\mathcal{L}_{D}

(training) or the density error (validation). Atomic energies of C and O needed for atomization energies are evaluated on fixed orbitals but do not enter

\mathcal{L}_{G}

\mathcal{L}_{D}

. The partition follows Ref. [ 32 ] , with O + and Ne + added to the density set.

Density Functional Theory

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Machine-learned (ML) exchange-correlation (XC) functionals aim to replace human-designed density functional approximations by learning directly from reference data, but they still do not consistently outperform traditional

\mathcal{O}(N^4)

-scaling hybrid functionals. We study a hybrid-distillation setting in which

\mathcal{O}(N^3)

-scaling ML-XC functionals are trained to reproduce B3LYP/def2-SVP targets. We introduce Derivative Informed XC-Loss (DI-Loss), a loss that incorporates additional information from the reference hybrid functional by supervising first and second derivatives of the energy on the Grassmannian of admissible density matrices. Rather than only matching the self-consistent fixed point, DI-Loss aligns the local first- and second-order response of the learned functional with that of the target functional. Across four evaluated architectures, DI-Loss consistently improves the main energy metrics. Averaged uniformly across architectures, the total-energy MAE decreases by 66% relative to energy and density supervision alone. The density-sensitive mean-field energy metric

E_ρ

improves from

1.2

0.8

mEh on average, while dipole and

\mathcal{L}_2

density errors do not improve uniformly. We further show that densities from the distilled functionals reduce hybrid-functional SCF iterations by up to 50%. In downstream TDDFT calculations, Hessian supervision improves excited-state predictions, with XCdiff reducing the mean excitation-energy MAE by 19 - 35%.

Abstract

Density Functional Theory

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