An arXiv preprint reports a computational test of NEO-sDFT, a subsystem method designed to treat selected protons quantum mechanically inside larger molecular calculations. In its biggest NEO-FDE demonstration, the method reached a system of up to 909 atoms, with 303 protons treated quantum mechanically. The calculations produced a geometry-dependent accuracy picture: differences from NEO-DFT were small at one tested displacement and much larger at the shortest one.
NEO-sDFT partitions a molecular calculation into electron- and proton-density subsystems and uses embedding potentials to represent interactions between them. That gives the method a way to assign quantum treatment to selected protons while describing the system through linked subsystems. The central question was whether this approach could provide an accurate and efficient treatment as molecular systems grow.
Water dimers exposed the geometry effect
To test accuracy, the authors used four water-dimer proton-assignment scenarios: no quantum protons; one non-coordinating quantum proton; one coordinating quantum proton; and two quantum protons. The comparisons examined interaction-energy differences between NEO-sDFT and NEO-DFT across the tested intermolecular displacements.
In the electronic comparison, sDFT curves nearly coincided with DFT for displacements larger than 0.2 Å. At 0.0 Å, the difference was about 6 meV, but at −0.4 Å it was about 62 meV.
The quantum-proton scenarios did not produce one uniform error level. In the second scenario, NEO-sDFT differed from NEO-DFT by 5 meV at 0.0 Å and 59 meV at −0.4 Å. In the third and fourth scenarios, the corresponding differences at 0.0 Å were about 2 and 1 meV, while at −0.4 Å they were 63 and 59 meV.
Taken together, the pointwise results show a sharp change with displacement: the reported deviations were around 1–6 meV at 0.0 Å in the tested scenarios, but about 59–63 meV at −0.4 Å. The preprint reports these as approximate deviations and gives no uncertainty intervals, so the figures describe the tested calculations rather than a statistical estimate of error.
Energy was not the only check. In the dimer, NEO-sDFT and NEO-DFT produced very similar proton-density distributions. Compared with the isolated monomer, the dimer densities were slightly more delocalized and shifted toward oxygen. The authors concluded that the effects of neglected proton non-additive kinetic and electron–proton correlation contributions were a few meV or less.
The larger test followed computing time
For the larger-system exercise, the authors used NEO-FDE on water clusters containing M = 15, 30, 70, 145, 237 and 303 water molecules, with M denoting the cluster size. The cluster calculations used three freeze-and-thaw cycles.
Wall time—the elapsed time required for the calculation—followed a power-law fit of O(M^1.68) for the smaller subset of 15, 30 and 70 molecules. When all six cluster sizes were included, the fitted cost reached cubic scaling. The reported fits had R² values of at least 0.9994, indicating that the measured times tracked the fitted curves closely; R² here describes fit quality, not uncertainty in the exponent.
The scaling test was paired with a particle-count demonstration. The largest computation treated 303 protons and 3,030 electrons quantum mechanically. The largest reported molecular system contained 909 atoms and 303 quantum protons. These figures describe the scale reached in the reported calculations.
Where the evidence stops
The main technical boundary is interaction strength. The authors identify the approximation of the electronic non-additive kinetic energy as the likely main source of NEO-sDFT error and say that it limits the method’s applicability to weakly interacting subsystems. The water-dimer tests therefore show how the method behaved in the configurations examined, while leaving its accuracy for strongly interacting subsystems unresolved.
The cluster results also represent a particular implementation test. The reported scaling used the six stated water-cluster sizes and the three-cycle freeze-and-thaw setup. The preprint’s evidence remains centered on water dimers and water clusters.
The document is an arXiv preprint, version 1, dated 20 August 2026. The authors say the program implementation will soon be available as open source, but no current release details are reported. The work was supported by the German Research Foundation under project 545861628, with computing resources from the HPC Service of FUB-IT at Freie Universität Berlin.
Taken as a methods report, the study demonstrates a 909-atom water-cluster calculation and close proton-density agreement with NEO-DFT in the dimer tests. It also shows that energy deviations grew at the shortest tested displacement, while the authors identify weakly interacting subsystems as the method’s current applicability range.
Paper data and sources
Original title: Nuclear-Electronic Orbital Subsystem Density Functional Theory
Authors: Denis G. Artiukhin
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text