Soft Fragment-Constrained Charge Equilibration

• Soft-FQEq is a charge equilibration method that softens fragment constraints via quadratic penalties, addressing the limitations of global QEq in representing localized charge states.
• It enhances numerical robustness and enables differentiable fragment identification, which is crucial for reactive machine-learning potentials at electrochemical interfaces.
• The method bridges constrained QEq and advanced ML formulations, reducing spurious long-range charge transfer while accurately modeling donor–acceptor localization and interface gradients.

Soft Fragment-Constrained Charge Equilibration (Soft-FQEq) denotes a class of charge-equilibration methods in which fragment-resolved charge conservation is enforced by soft, typically quadratic, penalties rather than only by exact equality constraints. In the recent literature, the term is used in two closely related settings: as a penalty-based extension of constrained charge equilibration (CQEq) for constructing diabatic charge-transfer states and extracting two-state couplings, and as a differentiable fragment-constrained solver layer for reactive machine-learning interatomic potentials at electrochemical interfaces (Kundu et al., 2024, Peeketi et al., 30 Apr 2026). In both settings, Soft-FQEq addresses a structural limitation of global QEq/FQ: a single system-wide electronegativity equalization condition can admit spurious long-range charge transfer and cannot sustain fragment-specific electrochemical structure, such as donor–acceptor localization in diabatic charge transfer or the electrode–electrolyte electrochemical-potential gradient required by the double layer (Kundu et al., 2024, Peeketi et al., 30 Apr 2026).

1. Conceptual foundations

Charge equilibration models such as QEq and FQ minimize a quadratic electrostatic energy over atomic charges subject to total-charge conservation. Their computational cost is negligible relative to electronic-structure methods, which is why they underpin reactive force fields and modern charge-aware machine-learning force fields. The same efficiency, however, comes with a well-known limitation: standard QEq describes a single adiabatic ground state and enforces a global equalization condition that effectively places the entire system at one electrochemical potential, even when chemically distinct regions should remain electronically separated (Kundu et al., 2024, Peeketi et al., 30 Apr 2026).

Fragment-constrained formulations were introduced to remedy this limitation by restricting charge redistribution at the level of predefined donor, acceptor, molecular, or electrode fragments. In the diabatic-state context, CQEq imposes target fragment charges and thereby produces localized charge-transfer diabats at QEq cost, in analogy with constrained density functional theory (CDFT) but at much lower computational expense (Kundu et al., 2024). In the electrochemical-interface context, classical per-fragment charge equilibration is the established classical remedy for preventing spurious charge flow between electronically disconnected regions, but fixed molecular topology makes hard fragment definitions incompatible with reactive bond rearrangements (Peeketi et al., 30 Apr 2026).

Soft-FQEq replaces exact fragment equalities by energetic penalties. In the CQEq lineage, the penalty acts directly on deviations of fragment charges from prescribed targets; in the reactive-ML lineage, fragment membership itself becomes a differentiable function of geometry, and soft fragment constraints are enforced in a solver layer. A plausible synthesis is that Soft-FQEq is best understood as a family of fragment-resolved charge-equilibration methods whose common principle is soft enforcement of fragment charge conservation, with hard constraints recovered in appropriate limits (Kundu et al., 2024, Peeketi et al., 30 Apr 2026).

2. Variational structure and soft constraints

In the CQEq formulation, the baseline QEq energy is

$E(q) = \frac{1}{2} q^T J q + \chi^T q,$

with total-charge constraint $e^T q = Q_{\mathrm{tot}}$, where $q \in \mathbb{R}^N$ is the atomic-charge vector, $\chi$ is the electronegativity vector, $J$ is the Coulomb/Hessian matrix with diagonal hardness terms and off-diagonal interatomic Coulomb terms, and $e=(1,\ldots,1)$ (Kundu et al., 2024). The corresponding KKT system is

$$\begin{bmatrix} J & e \ e^T & 0 \end{bmatrix} \begin{bmatrix} q \ \mu \end{bmatrix} = - \begin{bmatrix} \chi \ Q_{\mathrm{tot}} \end{bmatrix}.$$

Hard fragment constraints introduce a matrix $A \in \mathbb{R}^{M \times N}$ whose rows sum atomic charges over fragments, together with targets $b \in \mathbb{R}^M$. The CQEq Lagrangian

$$\mathcal{L}(q,\lambda)=\frac{1}{2}q^T J q + \chi^T q + \lambda^T(Aq-b)$$

yields the saddle-point system

$e^T q = Q_{\mathrm{tot}}$0

Here, “fragment-constrained” means that each fragment charge $e^T q = Q_{\mathrm{tot}}$1 is fixed to a target value $e^T q = Q_{\mathrm{tot}}$2, for example a neutral donor or an oxidized donor (Kundu et al., 2024).

Soft-FQEq replaces these exact equalities by quadratic penalties,

$e^T q = Q_{\mathrm{tot}}$3

with diagonal stiffness matrix $e^T q = Q_{\mathrm{tot}}$4 and $e^T q = Q_{\mathrm{tot}}$5 (Kundu et al., 2024). Minimization gives

$e^T q = Q_{\mathrm{tot}}$6

If total charge is kept hard while fragment constraints are softened, the mixed system becomes

$e^T q = Q_{\mathrm{tot}}$7

As $e^T q = Q_{\mathrm{tot}}$8, the soft formulation recovers the hard-constraint CQEq solution; at finite $e^T q = Q_{\mathrm{tot}}$9, it improves numerical robustness and allows mixed hard/soft enforcement (Kundu et al., 2024).

The electrochemical-interface formulation uses related mathematics but different notation. There, the quadratic electrostatic term is written

$q \in \mathbb{R}^N$0

with $q \in \mathbb{R}^N$1 the Coulomb/hardness matrix from Gaussian-screened Ewald sums, and fragment constraints take the conceptual form $q \in \mathbb{R}^N$2, where $q \in \mathbb{R}^N$3 is a soft fragment-membership operator and $q \in \mathbb{R}^N$4 is obtained from learned source charges (Peeketi et al., 30 Apr 2026). This notational shift is substantial: in the CQEq literature $q \in \mathbb{R}^N$5 is the constraint incidence matrix, whereas in the ML-interface literature $q \in \mathbb{R}^N$6 is the electrostatic kernel and $q \in \mathbb{R}^N$7 or $q \in \mathbb{R}^N$8 carries fragment information.

3. Soft-FQEq for diabatic charge-transfer states

The CQEq program is motivated by electronically non-adiabatic charge transfer, where diabatic states with localized donor and acceptor charges are often more accessible than adiabatic excited states. In the two-state case, CQEq produces diabatic energies $q \in \mathbb{R}^N$9 and $\chi$0 by solving the constrained charge-equilibration problem for two fragment-charge assignments and then evaluating the original QEq energy at the resulting charge distributions. The off-diagonal diabatic coupling $\chi$1 is then extracted from the unconstrained ground-state energy $\chi$2 using

$\chi$3

and the adiabatic excited-state energy follows as

$\chi$4

Approximate adiabatic charges can also be obtained by mixing the diabatic charge vectors with populations from the two-state mixing angle $\chi$5 defined by $\chi$6 (Kundu et al., 2024).

Within this framework, Soft-FQEq is the penalty-based version of fragment-constrained diabatic construction. The crucial technical point is that physically meaningful diabatic energies must be evaluated with the unpenalized QEq functional,

$\chi$7

rather than with $\chi$8, because the penalty term is artificial and depends on $\chi$9 (Kundu et al., 2024). This distinction preserves consistency with hard CQEq in the $J$0 limit and prevents penalty-dependent biases in extracted couplings and excitation energies.

A central ingredient in the diabatic application is the use of environment-aware electronegativities,

$J$1

constructed from reference adiabatic ground-state charges $J$2. By construction, $J$3 minimizes the QEq energy, which removes the standard long-range charge-transfer pathology of QEq while leaving the same $J$4 and Slater-exponent parameters intact (Kundu et al., 2024). The paper emphasizes that this device is especially relevant when QEq-type models are embedded in reactive or machine-learning force fields.

The same study specifies the Coulomb matrix through spherically symmetric Slater-type densities,

$J$5

with three per-atom parameters: electronegativity $J$6, hardness $J$7, and Slater exponent $J$8. Typical values used in the reported calculations include, in atomic units, $J$9, $e=(1,\ldots,1)$0, $e=(1,\ldots,1)$1, $e=(1,\ldots,1)$2, $e=(1,\ldots,1)$3, together with $e=(1,\ldots,1)$4, $e=(1,\ldots,1)$5, $e=(1,\ldots,1)$6, $e=(1,\ldots,1)$7, and $e=(1,\ldots,1)$8 (Kundu et al., 2024).

4. Differentiable fragment identification in reactive machine-learning potentials

At electrochemical interfaces, the requirement is not only to constrain fragment charge, but to do so while fragments themselves may change continuously as bonds break and form. The 2026 Soft-FQEq formulation addresses this by making fragment identification differentiable. A shared HIP-NN feature network feeds four scalar readout heads: per-atom electronegativity $e=(1,\ldots,1)$9, source charge $$\begin{bmatrix} J & e \ e^T & 0 \end{bmatrix} \begin{bmatrix} q \ \mu \end{bmatrix} = - \begin{bmatrix} \chi \ Q_{\mathrm{tot}} \end{bmatrix}.$$0, short-range energy $$\begin{bmatrix} J & e \ e^T & 0 \end{bmatrix} \begin{bmatrix} q \ \mu \end{bmatrix} = - \begin{bmatrix} \chi \ Q_{\mathrm{tot}} \end{bmatrix}.$$1, and a soft bond-connectivity correction $$\begin{bmatrix} J & e \ e^T & 0 \end{bmatrix} \begin{bmatrix} q \ \mu \end{bmatrix} = - \begin{bmatrix} \chi \ Q_{\mathrm{tot}} \end{bmatrix}.$$2. The solver layer returns equilibrated charges together with per-fragment chemical potentials (Peeketi et al., 30 Apr 2026).

In that formulation, soft bond weights are first constructed as

$$\begin{bmatrix} J & e \ e^T & 0 \end{bmatrix} \begin{bmatrix} q \ \mu \end{bmatrix} = - \begin{bmatrix} \chi \ Q_{\mathrm{tot}} \end{bmatrix}.$$3

where $$\begin{bmatrix} J & e \ e^T & 0 \end{bmatrix} \begin{bmatrix} q \ \mu \end{bmatrix} = - \begin{bmatrix} \chi \ Q_{\mathrm{tot}} \end{bmatrix}.$$4 is symmetric and differentiable (Peeketi et al., 30 Apr 2026). Soft fragment membership then follows from a graph-Laplacian resolvent:

$$\begin{bmatrix} J & e \ e^T & 0 \end{bmatrix} \begin{bmatrix} q \ \mu \end{bmatrix} = - \begin{bmatrix} \chi \ Q_{\mathrm{tot}} \end{bmatrix}.$$5

$$\begin{bmatrix} J & e \ e^T & 0 \end{bmatrix} \begin{bmatrix} q \ \mu \end{bmatrix} = - \begin{bmatrix} \chi \ Q_{\mathrm{tot}} \end{bmatrix}.$$6

$$\begin{bmatrix} J & e \ e^T & 0 \end{bmatrix} \begin{bmatrix} q \ \mu \end{bmatrix} = - \begin{bmatrix} \chi \ Q_{\mathrm{tot}} \end{bmatrix}.$$7

This construction is reported to be $$\begin{bmatrix} J & e \ e^T & 0 \end{bmatrix} \begin{bmatrix} q \ \mu \end{bmatrix} = - \begin{bmatrix} \chi \ Q_{\mathrm{tot}} \end{bmatrix}.$$8 in positions and parameters, to preserve global connectivity in extended conductive regions such as electrodes, and to provide smooth fragment membership under reaction events (Peeketi et al., 30 Apr 2026).

The resulting fragment constraints are implemented through a membership operator $$\begin{bmatrix} J & e \ e^T & 0 \end{bmatrix} \begin{bmatrix} q \ \mu \end{bmatrix} = - \begin{bmatrix} \chi \ Q_{\mathrm{tot}} \end{bmatrix}.$$9, conceptually enforcing

$A \in \mathbb{R}^{M \times N}$0

together with $A \in \mathbb{R}^{M \times N}$1, where $A \in \mathbb{R}^{M \times N}$2 is derived from learned source charges. The actual implementation uses an augmented Lagrangian,

$A \in \mathbb{R}^{M \times N}$3

and an Uzawa iteration in which the inner linear solve uses

$A \in \mathbb{R}^{M \times N}$4

and the multipliers are updated as

$A \in \mathbb{R}^{M \times N}$5

until the fragment-charge residual vanishes (Peeketi et al., 30 Apr 2026).

This formulation changes the physical content of the local electrochemical potential

$A \in \mathbb{R}^{M \times N}$6

Under global QEq, stationarity enforces $A \in \mathbb{R}^{M \times N}$7 to be uniform across all atoms. Under Soft-FQEq, fragment multipliers contribute so that $A \in \mathbb{R}^{M \times N}$8 can differ across fragments while remaining equilibrated within each fragment. This is the mathematical mechanism by which the electrode–electrolyte electrochemical-potential gradient becomes representable (Peeketi et al., 30 Apr 2026).

5. Applications and empirical behavior

The diabatic-state CQEq study tests the framework on the anthracene–tetracyanoethylene charge-transfer complex and on the reductive decomposition of ethylene carbonate on a lithium metal surface. For the anthracene–TCNE system, the setup used a symmetric stack, B3LYP-D3/def2-TZVP ground-state DFT, fragment constraints on the anthracene net charge, Slater-integral $A \in \mathbb{R}^{M \times N}$9 values, the tabulated $b \in \mathbb{R}^M$0 and $b \in \mathbb{R}^M$1, and electronegativities reconstructed from reference DFT charges via $b \in \mathbb{R}^M$2. CQEq reproduces CDFT diabatic energies with discrepancies $b \in \mathbb{R}^M$3 eV over intermolecular separations $b \in \mathbb{R}^M$4. The adiabatic charge-transfer excitation energy $b \in \mathbb{R}^M$5 is $b \in \mathbb{R}^M$6 eV near the equilibrium separation $b \in \mathbb{R}^M$7 Å and increases with $b \in \mathbb{R}^M$8, while the diabatic surfaces do not cross along $b \in \mathbb{R}^M$9, consistent with small ground-state charge transfer at equilibrium (Kundu et al., 2024).

For ethylene carbonate decomposition on Li(001), a ground-state NEB on periodic PBE gives a small barrier $$\mathcal{L}(q,\lambda)=\frac{1}{2}q^T J q + \chi^T q + \lambda^T(Aq-b)$$0 eV, and cluster models with 40 Li atoms plus EC were then used for molecular CDFT/CQEq calculations at PBE with a double-$$\mathcal{L}(q,\lambda)=\frac{1}{2}q^T J q + \chi^T q + \lambda^T(Aq-b)$$1 correlation-consistent Gaussian basis. Along the entire reaction coordinate, CQEq reproduces diabatic energies and adiabatic excitation energies within $$\mathcal{L}(q,\lambda)=\frac{1}{2}q^T J q + \chi^T q + \lambda^T(Aq-b)$$2 eV. In this case the diabatic surfaces cross as the ground state changes character from neutral to ionic, indicating that the barrier arises from changing electronic character rather than purely nuclear reorganization. CQEq diabatic charges also match CDFT meta-Löwdin charges closely in both neutral and ionic diabats (Kundu et al., 2024). These results validate CQEq directly; because Soft-FQEq approaches CQEq as $$\mathcal{L}(q,\lambda)=\frac{1}{2}q^T J q + \chi^T q + \lambda^T(Aq-b)$$3, they also define the regime that the soft formulation is intended to recover.

The electrochemical-interface study applies Soft-FQEq to IrO$$\mathcal{L}(q,\lambda)=\frac{1}{2}q^T J q + \chi^T q + \lambda^T(Aq-b)$$4/H$$\mathcal{L}(q,\lambda)=\frac{1}{2}q^T J q + \chi^T q + \lambda^T(Aq-b)$$5O/Na$$\mathcal{L}(q,\lambda)=\frac{1}{2}q^T J q + \chi^T q + \lambda^T(Aq-b)$$6/ClO$$\mathcal{L}(q,\lambda)=\frac{1}{2}q^T J q + \chi^T q + \lambda^T(Aq-b)$$7 slab systems using a HIP-NN backbone implemented in the hippynn framework and trained on DFT energies, forces, and DDEC6 charges. The dataset comprises a bootstrap set of about 6800 structures spanning vacuum dimers, trimers, small molecules, IrO$$\mathcal{L}(q,\lambda)=\frac{1}{2}q^T J q + \chi^T q + \lambda^T(Aq-b)$$8 bulk structures, aqueous-electrolyte configurations, and more than 800 slab–electrolyte interfaces; the electronic-structure data were generated with VASP PBE-D3(BJ) at 450 eV cutoff, and DDEC6 charges were obtained with Chargemol (Peeketi et al., 30 Apr 2026).

In that application, the trained model attains charge accuracy of about $$\mathcal{L}(q,\lambda)=\frac{1}{2}q^T J q + \chi^T q + \lambda^T(Aq-b)$$9 e MAE overall across elements, with the largest residuals on Cl and Ir. More importantly, the electrochemical-potential profile $e^T q = Q_{\mathrm{tot}}$00 exhibits a clear electrode-to-electrolyte gradient under Soft-FQEq: the Ir sublattice has an approximately flat electrode plateau with about 30 meV standard deviation, followed by a sharp interfacial drop and a distinct bulk-water plateau. With exactly the same trained weights but the fragment-constrained solver replaced by global QEq at inference, this profile collapses to a single essentially uniform value with less than $e^T q = Q_{\mathrm{tot}}$01 eV variation. Varying Stern-layer ion composition from 4 Na$e^T q = Q_{\mathrm{tot}}$02 to 4 ClO$e^T q = Q_{\mathrm{tot}}$03 per face shifts the predicted electrode surface charge density and the electrode $e^T q = Q_{\mathrm{tot}}$04 plateau monotonically, consistent with capacitive behavior (Peeketi et al., 30 Apr 2026).

6. Relation to adjacent formalisms

Soft-FQEq belongs to a broader landscape of variable-charge models. Standard QEq/FQ minimizes a quadratic electrostatic functional under only a global total-charge constraint; it is inexpensive but can exhibit unphysical long-range charge transfer unless electronegativities are made environment-aware (Kundu et al., 2024). SQE introduces bond-charge variables to address dissociation and bond localization, but it is not directed specifically at the construction of diabatic charge-transfer states (Kundu et al., 2024). CDFT imposes electronic constraints at the ab initio level and is directly comparable in spirit to hard CQEq, but is orders of magnitude more expensive (Kundu et al., 2024).

The electrochemical-interface work contrasts Soft-FQEq with classical hard fragment-constrained QEq, where a binary membership matrix $e^T q = Q_{\mathrm{tot}}$05 imposes $e^T q = Q_{\mathrm{tot}}$06 and yields one chemical potential per fragment. That remedy is established in classical molecular dynamics but non-reactive because fragment membership changes discretely with topology (Peeketi et al., 30 Apr 2026). Soft-FQEq generalizes the same idea by allowing continuous, geometry-dependent fragment membership and by learning fragment charge targets from source charges.

A related antecedent appears in the generalized charge-transfer equilibration (QTE) framework, which formulates charge redistribution in terms of antisymmetric transfer variables rather than only atomic charges. Within that framework, a Soft-FQEq-like construction can be obtained by adding fragment penalties

$e^T q = Q_{\mathrm{tot}}$07

and, optionally, inter-fragment transfer penalties

$e^T q = Q_{\mathrm{tot}}$08

which bias the dynamics away from long-range or inter-fragment charge flow while preserving the locality encoded by overlap-weighted transfer variables (Gergs et al., 2020). In the limits $e^T q = Q_{\mathrm{tot}}$09 and $e^T q = Q_{\mathrm{tot}}$10, this construction reduces to pure QTE; in the hard limit $e^T q = Q_{\mathrm{tot}}$11 it reproduces the neutral-fragment behavior associated in that paper with QTE+ (Gergs et al., 2020). This suggests a conceptual continuity between QTE-style restricted transfer, CQEq-style fragment constraints, and recent Soft-FQEq formulations.

QTPIE and ACKS2 occupy adjacent positions in the same design space. QTPIE localizes charge transfer through overlap-weighted effective electronegativities and approximates the ideal-insulator limit, whereas QEq approximates the ideal-metal limit under global electronegativity equalization. ACKS2 uses a density-functional-inspired kernel to suppress conduction-like behavior. The QTE paper argues that a generalization based on constrained charge-transfer variables can interpolate more naturally between surface, adsorbate, and gas-phase regimes in reactive molecular dynamics (Gergs et al., 2020).

7. Numerical behavior, parameter dependence, and limitations

The numerical motivation for Soft-FQEq is as important as the physical one. Hard CQEq produces a symmetric indefinite KKT saddle-point problem, for which robust direct solvers such as sparse LDL$e^T q = Q_{\mathrm{tot}}$12 or iterative methods such as MINRES with preconditioning are recommended. Soft-FQEq converts most fragment constraints into an SPD modification of the Hessian, which is compatible with CG or L-BFGS and often better conditioned because the penalty contributes the low-rank positive-semidefinite update $e^T q = Q_{\mathrm{tot}}$13 (Kundu et al., 2024). For dense Coulomb kernels with Slater charges, the same work recommends fast multipole or Ewald-type approximations for large systems and notes that Sherman–Morrison–Woodbury updates can reuse factorizations across multiple diabatic states and penalty choices (Kundu et al., 2024).

In the ML-interface setting, the principal cost arises from dense Gaussian-Ewald electrostatics and from the graph resolvent, both of which scale as $e^T q = Q_{\mathrm{tot}}$14 in the present implementation. The reported strategy is to use float64 arithmetic, an augmented-Lagrangian Uzawa scheme with representative parameters $e^T q = Q_{\mathrm{tot}}$15 and $e^T q = Q_{\mathrm{tot}}$16, $e^T q = Q_{\mathrm{tot}}$17 unrolled iterations during training and $e^T q = Q_{\mathrm{tot}}$18–20 in inference, and Gaussian screening to keep the Coulomb matrix positive definite. The periodic electrostatics use $e^T q = Q_{\mathrm{tot}}$19 Å$e^T q = Q_{\mathrm{tot}}$20, a 7 Å real-space cutoff, and reciprocal-space accuracy $e^T q = Q_{\mathrm{tot}}$21 (Peeketi et al., 30 Apr 2026). PME-based fast Coulomb solvers and block-sparse approximations to the resolvent are identified as planned routes to scale-up.

All formulations remain parameter-dependent. In the diabatic CQEq lineage, accuracy depends on hardnesses $e^T q = Q_{\mathrm{tot}}$22, Slater exponents $e^T q = Q_{\mathrm{tot}}$23, the Coulomb kernel $e^T q = Q_{\mathrm{tot}}$24, and especially on the electronegativity construction $e^T q = Q_{\mathrm{tot}}$25. The same paper emphasizes that environment-aware $e^T q = Q_{\mathrm{tot}}$26 is essential for transferability and for eliminating long-range pathologies (Kundu et al., 2024). In the reactive ML formulation, fragment sharpness depends on the connectivity thresholds and Gaussian widths, source-charge supervision is required for identifiability of fragment targets, and auxiliary losses are needed to keep $e^T q = Q_{\mathrm{tot}}$27 physically meaningful rather than allowing the multipliers to absorb all constraint information (Peeketi et al., 30 Apr 2026).

Several physical limitations are explicit. Standard QEq Coulomb kernels do not by themselves encode metallic screening; the Li surface in the CQEq study was treated as a finite cluster, and the authors state that a periodic metallic treatment would require appropriate electrostatics such as Ewald or PME and possibly environment-dependent screening (Kundu et al., 2024). Soft fragmentation can be ambiguous in complex chemistries, especially metallic bonding regions or reconstructed oxide surfaces, and overlarge penalties can harm conditioning or create stiff fictitious dynamics (Peeketi et al., 30 Apr 2026, Gergs et al., 2020). In surface-oriented QTE simulations, a mirror boundary condition imposing $e^T q = Q_{\mathrm{tot}}$28 was proposed to accelerate thin-slab calculations by approximately a factor of 2 in COMB3, but that device addresses boundary treatment rather than the core fragment-constrained formalism itself (Gergs et al., 2020).

The most stable common lesson across these lineages is architectural rather than parametric: global QEq enforces a single electrochemical potential across the system, whereas fragment-constrained formulations introduce fragment-level chemical potentials or their soft analogues. In diabatic charge transfer, this yields localized donor–acceptor states and two-state couplings; at electrochemical interfaces, it yields the electrode–electrolyte electrochemical-potential gradient that global QEq suppresses by construction (Kundu et al., 2024, Peeketi et al., 30 Apr 2026).