Long-Range Inclusive Deep Potential

• Long-Range-Inclusive Deep Potential is a computational model that rigorously integrates long-range interactions, such as Coulombic tails, into atomistic simulations.
• The off-diagonal discretization method replaces divergent diagonal elements with finite off-diagonal terms, achieving exponential convergence and enhanced accuracy.
• The framework adapts to deep learning potentials across quantum, nuclear, and materials science, providing robust predictions for systems with long-range correlation effects.

A Long-Range-Inclusive Deep Potential is a class of numerical and machine-learning-based potential energy models that extends conventional deep potential (DP) frameworks by rigorously incorporating long-range interactions—especially Coulombic or other slowly decaying tails—into atomistic and quantum systems. These models overcome the limitations of strictly local representations, which, though computationally efficient, cannot capture the correct physics of systems where potential components such as 1/r or 1/rⁿ tails are non-negligible over the simulation domain. The resulting techniques provide accurate treatments of boundary conditions, collective effects, and observable properties that directly depend on long-range correlations, as required in large-scale simulations of atomic, molecular, and nuclear systems.

1. Numerical Challenges for Long-Range Potentials: Fredholm Equations and Kernel Singularities

In quantum many-body problems, the inclusion of long-range interactions such as the Coulomb potential in non-local basis expansions (e.g., with the Berggren basis) leads to Fredholm equations whose kernels are analytically integrable but possess diagonal singularities. Specifically, representing the Schrödinger equation in a basis including continua results in matrix elements of the form

$\langle u_j | V_c | u_i \rangle,$

which for the Coulomb tail ($V_c \sim 1/r$) display logarithmic divergences (e.g., $\sim \ln|k - k'|$ when expressed in a momentum basis).

Numerically, this singular behaviour precludes naive discretization: directly discretizing such kernels leads to formally infinite or undefined diagonal contributions, significantly compromising the accuracy of eigenenergies, scattering observables, and wavefunction reconstruction, especially critical in many-body nuclear physics applications involving resonant or halo states.

2. Discretization Methods for Long-Range Kernels

Three primary schemes have been devised and evaluated for the discretization of long-range kernels in the context of a Berggren basis expansion (Michel, 2010):

Method	Basic Principle	Numerical Performance/Drawbacks
Cut-off (Cut)	Truncate potential at $R$	Easiest; strongly sensitive to $R$; poor for tails
Subtraction	Subtract/add analytic divergence (in sine basis), treat residual numerically	Removes analytic singularity; residual is nonanalytic; convergence saturates with mesh
Off-diagonal	Replace divergent diagonal element by finite off-diagonal terms using a small shift in momentum	Restores smooth, analytic structure to matrix; yields exponential convergence; superior accuracy for energies and wave functions

The off-diagonal method defines, for each continuum point $k_i$, shifted states $u_i^\pm$, and replaces the problematic diagonal matrix element as

$$\langle u_i | V_c | u_i \rangle \to \langle u_i^+ | V_c | u_i^- \rangle,$$

modifying the discretized Fredholm equation for scattering states (see Eq. (8) in (Michel, 2010)). This approach is robust across numerical quadrature schemes and is highly effective for systems where a precise description of the continuum is essential.

3. Impact and Adaptation in Deep Potential and Many-Body Frameworks

Accurate representation of long-range effects is indispensable in deep potential models where the long-range tail of the interaction governs not just two-body but emergent many-body properties, such as configuration mixing, decay widths, and radial asymptotics. The "long-range-inclusive" approach, particularly the off-diagonal discretization, provides:

• Controlled, minimizable discretization error, crucial for resonant and halo systems, allowing proper configuration mixing in shell-model Hamiltonians.
• Retention of continuum completeness and proper asymptotic wavefunction behavior—essential for constructing configuration-mixed Slater determinants in Gamow shell models or related frameworks.
• Direct applicability to other kernels with similar integrable singularity structure (e.g., $1/r^n$-type tails), by constructing analogous off-diagonal replacement schemes.
• Reliable calculation of matrix elements for observables sensitive to the long-range tail, including isospin or electromagnetic observables.

This capability is critical in contexts where observables—such as cross sections, resonance widths, and occupation probabilities—are particularly dependent on the correct asymptotic behaviour of the nuclear wave function.

4. Theoretical Formulation and Key Mathematical Structures

Relevant equations structuring the long-range-inclusive approach include:

• Fredholm equation in a basis with continuum ($b$ for bound, $d$ for decaying, $k$ for continuum): $$c_n e_n + \sum_{n'\in(b,d)} c_{n'} \langle u_{n'}|V_c(\Delta Z_c, r)| u_n\rangle + \int_{l^+} c_{k'} \langle u_{k'}|V_c(\Delta Z_c, r)| u_n\rangle\, dk' = E c_n$$ (similarly for continuum states $k$).
• Coulomb potential in error function–smoothed form: $$V_c(Z_c, r) = \frac{C_c Z_c \operatorname{erf}(\alpha r)}{r}$$
• Off-diagonal replacement in momentum discretization: $k_i^\pm = k_i \pm \frac{w_i}{4\pi}$

$$\langle u_i | V_c | u_i \rangle \to \langle u_i^+ | V_c | u_i^- \rangle,$$

with the discretized equation for continuum states

$$c_i [e_i + \langle u_i^+ | V_c(\Delta Z_c, r) | u_i^- \rangle] + \sum_{j \neq i} c_j \langle u_j | V_c(\Delta Z_c, r) | u_i \rangle = E c_i$$

This mathematical infrastructure generalizes to other long-range potentials (not only Coulomb but e.g., $1/r^n$), whenever the singularity is integrable yet presents a numerical challenge in basis representation.

5. Generalization to Non-Nuclear and Deep Learning Contexts

The discretization and smoothing framework described for the Berggren basis and nuclear Hamiltonians sets the foundation for constructing deep potential models that are inclusive of long-range physics. In machine-learning-driven potentials, similar concerns arise in the transferability and accuracy of force prediction—especially when properties such as dielectric response, excitation spectrum, or long-wavelength collective modes depend on correctly incorporating tails of the potential.

Long-range-inclusive deep potentials thus employ:

• Hybrid representations where physically motivated long-range contributions are computed explicitly or via auxiliary neural-network–based surrogates, added to the traditional short-range DP terms.
• Smoothing or splitting of the potential, enabling neural network components to capture short-range correlation/interaction features, while explicit summation or analytic corrections handle asymptotic tails.
• Approaches to adaptively learn or encode kernel singularities by combining machine-learning function approximation with analytic or semianalytic corrections, avoiding double-counting and ensuring analytic continuity.

This is critical for accurate modeling in materials science, chemistry (e.g., van der Waals and electrostatic corrections), and strongly interacting quantum systems.

6. Practical Outcomes and Implications

The adoption of long-range-inclusive discretizations and modeling:

• Ensures exponential convergence for eigenenergies and wavefunction observables, outperforming traditional subtraction or cutoff methods in precision and robustness (Michel, 2010).
• Minimizes the generalization gap between training and test predictions in ML-based potentials—for instance, in surface, interface, or defect-rich environments where long-range polarization effects are pronounced.
• Enables construction of many-body expansions and configuration-mixed states that maintain correct physical asymptotics, supporting the quantitative prediction of observables in halo, resonant, or exotic (e.g., nucleonic, hadronic) states.
• Forms the mathematical and numerical underpinning for next-generation, physically sound deep learning potentials applied to systems across nuclear, atomic, and molecular scales.

7. Conclusion

Long-range-inclusive deep potential methodology, exemplified by the off-diagonal discretization method in the Berggren basis, is essential for accurate and efficient numerical treatments of systems with integrable but nonlocal potential kernels. This approach, by regularizing the singularities in an analytically controlled manner and enabling seamless extension to deep learning–based potential models, is foundational for achieving transferability, convergence, and predictive power in simulations where long-range physics is non-negligible—spanning nuclear structure, quantum chemistry, materials science, and computational physics at large (Michel, 2010).