Notes: Atomic Charge Methods and MLFF Long-Range Prior Art#

Date:

2026-06-26

Status:

Reference material supporting Plan: Long-Range Electrostatics for Short-Range MLFFs. Background, not a work plan.

The one distinction that matters: what the scheme consumes#

Atomic charges are not quantum observables – there is no unique way to partition the electron density among atoms, hence the zoo of schemes. For this campaign the reliability question and the molecular-vs-periodic question collapse onto a single axis: what input does the scheme consume?

  • Basis-/wavefunction-based schemes are tied to a molecular code’s internal data and (mostly) to localized basis sets -> molecular codes only.

  • Real-space density-grid schemes consume a charge density on a grid (or a cube/wfx), which every code can emit and which is periodicity-agnostic -> they work identically for VASP and for molecular codes.

The grid-based methods are therefore both the more transferable/reliable choice and the ones that work for periodic systems – which is why DDEC6 and Bader anchor Subcampaign 1 of the plan.

Method landscape#

Method

Input

Periodic?

Reliability / use

Code

Mulliken / Löwdin

basis

molecular

Basis-set unstable; do not trust quantitatively

native everywhere

NPA (NBO)

basis

molecular

Stable, good chemical charges; not ESP-faithful

NBO (Gaussian/Orca/Psi4 iface)

CHELPG / Merz-Kollman / RESP

ESP on grid (vacuum)

molecular only

ESP-faithful -> force-field charges; buried-atom & conformer issues

native (Gaussian/Orca/Psi4)

Hirshfeld

density grid

yes

Small / underestimated charges

Multiwfn, Critic2, native

Hirshfeld-I (iterative)

density grid

yes

Much better than Hirshfeld; captures charge transfer

Multiwfn, HORTON, Critic2

CM5

density (Hirshfeld + params)

yes (mol. mainly)

Cheap, good dipoles, general-purpose molecular

native (Gaussian/Psi4), Multiwfn

MBIS

density grid

yes

Modern stockholder; favored for FF/ML (OpenFF)

Psi4 (native), HORTON, GPU4PySCF

Bader / QTAIM

density grid

yes

Rigorous, basis-independent; charges exaggerated but consistent

Henkelman bader (CHGCAR), Critic2, Multiwfn

DDEC6

density grid (+ core)

yes, incl. metals/MOFs

ESP-faithful and chemically meaningful and transferable

Chargemol (VASP, Gaussian, Orca, …)

REPEAT

periodic ESP

yes

Periodic analog of CHELPG

a few codes / scripts

Recommendations#

  • One method to span everything: DDEC6 (Chargemol). The only widely-used scheme that simultaneously reproduces the ESP, gives transferable chemical charges, and is designed for molecular and periodic systems including metals. It also solves the “ESP charges for VASP” problem that CHELPG/MK cannot (no vacuum region) – so REPEAT is not separately needed. This is the training target for the charge model.

  • Bader (Henkelman) as the robust, well-understood grid companion, especially for solid-state users who expect it. Charges run large/ionic and are not ESP-faithful, so it is a companion, not the MLFF training target.

  • RESP / CHELPG remain the molecular force-field path; MBIS is the rising FF/ML choice and Psi4 does it natively.

  • Treat Mulliken / Löwdin as diagnostics only.

Bader vs DDEC6 (why both, and how they differ)#

Both consume the same density grid, both are periodic- and molecular-capable, and both need the same density export from the QM step – which is why they share Subcampaign 1 of the plan.

  • Bader / QTAIM partitions real space into basins separated by zero-flux surfaces (where \(\nabla\rho\cdot\mathbf{n}=0\)); each basin holds one nucleus, and integrating \(\rho\) over it gives the atom’s electron count, so \(q_A = Z_A - N_{\text{basin}}\). No basis functions are involved, so it is basis-robust and code-agnostic, and it is the solid-state standard. But the charges are not fit to the electrostatic potential and come out large / very ionic (e.g. water O near -1.1 to -1.2). Good for charge-transfer / oxidation-state-style analysis, less ideal as force-field point charges.

  • DDEC6 is a stockholder-type partition whose weights are optimized so the charges simultaneously approximate the ESP and stay chemically transferable – the “best of both” you want for FF/ML charges.

Density export caveat (both methods). For VASP they need the all-electron density, not just the pseudized valence CHGCAR: set LAECHG=.TRUE. to also write AECCAR0 (core) and AECCAR2 (valence). Without the core density the basins / weights are misassigned around nuclei. Molecular codes export .wfx / cube / molden.

MLFF long-range prior art#

The range-separation / delta-learning scheme in the plan is well-precedented:

  • DPLR (Deep Potential Long Range) – predicts charges (via Deep Wannier centroids), computes the long-range Ewald energy from spread (Gaussian) charges so the subtracted term is smooth, subtracts it from the DFT data, fits the short-range model to the remainder, and adds it back. Implemented in DeePMD-kit + LAMMPS. This is the closest precedent to the plan’s scheme.

  • 4G-HDNNP (Ko, Finkler, Goedecker, Behler) – a fourth-generation HDNNP using charge equilibration for non-local charge transfer; the reference point for “do the charges right, then the electrostatics.”

  • Latent Ewald Summation (LES) / CACE-LR (Cheng group) – learns latent charges and adds an Ewald term, without needing reference charges; already coupled to MACE. An alternative worth knowing, though this campaign deliberately uses physical DDEC6 charges for interpretability and reusable, inspectable charge sets.

Two design points that fall out of the prior art and are load-bearing for the plan (stated in full there): range-separate the subtracted Coulomb so only the smooth long-range part is removed (DPLR uses Gaussian-spread charges for exactly this), and keep train-subtract identical to inference-add (same charge model, same Ewald routine) – trivially satisfied by doing the electrostatics in the MDI engine with a single shared torch-pme routine.

References#

  • Plan: Long-Range Electrostatics for Short-Range MLFFs

  • DDEC6: Manz & Limas, “Introducing DDEC6 atomic population analysis: part 1. Charge partitioning theory and methodology,” RSC Adv. 6, 47771 (2016), doi:10.1039/C6RA04656H. (Part 2, doi:10.1039/C6RA05507A, covers periodic and nonperiodic materials.) DDEC6 is the method; Chargemol is the program implementing it.

  • Bader analysis on grids: Henkelman group bader code; Critic2; Multiwfn (tools, no single canonical paper – cite the specific code’s own reference when used)

  • DPLR: Zhang, Wang, Muniz, Panagiotopoulos, Car & E, “A deep potential model with long-range electrostatic interactions,” J. Chem. Phys. 156, 124107 (2022); arXiv:2112.13327

  • 4G-HDNNP: Ko, Finkler, Goedecker & Behler, “A fourth-generation high-dimensional neural network potential with accurate electrostatics including non-local charge transfer,” Nat. Commun. 12, 398 (2021), doi:10.1038/s41467-020-20427-2

  • LES: B. Cheng, “Latent Ewald summation for machine learning of long-range interactions,” npj Comput. Mater. (2025), doi:10.1038/s41524-025-01577-7; arXiv:2408.15165. CACE (Cartesian Atomic Cluster Expansion), the MLIP it is coupled with, is at https://github.com/BingqingCheng/cace

  • torch-pme – differentiable PME/Ewald in PyTorch (lab-cosmo / COSMO @ EPFL; metatensor ecosystem): lab-cosmo/torch-pme