Semi-Analytical Coulomb-Included Model

• The paper introduces a variational framework and WKB-derived corrections to model Coulomb effects, achieving quantitative agreement with particle simulations.
• Explicit analytical solutions, such as closed-form t-matrices in few-body scattering, enable efficient kernel-level substitution in complex calculations.
• The models balance analytical tractability with low-dimensional numerics to accurately capture key phenomena like charge inversion, overscreening, and spectral weight redistribution.

A semi-analytical Coulomb-included model is any theoretical or computational framework that incorporates the long-range Coulomb interaction within a system, and does so using a combination of analytical derivations and tractable, typically dimensionally reduced, numerical methods. These models are designed to preserve quantitative fidelity to physical Coulomb effects (e.g., correlations, screening, dielectric response, or singular scattering) while remaining computationally efficient. The semi-analytical character typically involves explicit, often closed-form, expressions for the dominant Coulomb contributions (sometimes via expansions, special-function solutions, or operator rearrangement), with the remaining numerical work focused on low-dimensional integrals, matrix inversion, or iterative self-consistency. Semi-analytical Coulomb-included models are prevalent across condensed matter, plasma physics, quantum chemistry, quantum transport, and strong-field atomic physics, each with problem-specific adaptation.

1. Free-Energy and Variational Principles: Strong Electrolyte and Confined Ion Systems

In the context of nanoscale electrolytes and confined ion transport, the semi-analytical Coulomb-included model developed by Ma, Xu, and Zhang introduces a variational framework by decomposing the free energy into three components: the mean-field electrostatic and entropic term, a nonlocal Coulomb correlation energy, and a hard-sphere (steric) interaction modeled by a variant of Fundamental Measure Theory (FMT). The full variational functional is

$$F[c_1,\dots,c_K] = F^{\rm mf}[c] + F^{\rm co}[c] + F^{\rm hs}[c],$$

with $F^{\rm co}[c]$ capturing correlation (beyond mean-field) via a charging process integral that involves Green's functions determined by a generalized Debye–Hückel (GDH) equation. Steric effects are included beyond lattice models through weighted density functionals.

The semi-analytical aspect is realized by solving for the GDH Green's function with a WKB approximation, yielding closed-form or semi-explicit integral expressions for the electrostatic self-energy correction, which can be efficiently evaluated in layered geometries. For systems with smoothly varying ionic strength, this delivers quantitative agreement with particle simulations, accurately capturing depletion zones, layered oscillations, and charge inversion phenomena that elude mean-field Poisson–Nernst–Planck (PNP) or classical PB models (Ma et al., 2020).

2. Explicit Analytical Solutions: Quantum Scattering and Few-Body Systems

In nuclear and atomic few-body physics, semi-analytical Coulomb-included models are constructed for the two-body Lippmann–Schwinger equation by reducing the problem to a partial-wave expansion and carrying out the remaining integrations analytically at special energies. Fock's stereographic projection maps momentum space to $S^3$, rendering angular integrals tractable. For example, Kharchenko derives closed-form analytical solutions (elementary logarithms and trigonometric functions) for the partial-wave Coulomb $t$-matrix at ground-state energies for like-charged particles,

$t_\ell(k, k'; -B_1)$

for $\ell=0,1,2$, suitable for direct implementation in kernel-based few-body codes without recourse to partial-wave summation or heavy numerics. Charge and mass enter only parametrically, and the forms remain stable in both the small and large momentum regimes (Kharchenko, 2017).

3. Semi-Analytical Techniques in Strong Correlation and Quantum Transport

In the regime of mesoscopic quantum transport—such as the double quantum dot model in the Coulomb-blockade regime—semi-analytical models exploit the equation-of-motion hierarchy and symmetries of the Hamiltonian to obtain a closed set of linear equations for all density correlators and Green's function residues. Neglecting higher-order commutators in the uncontacted or blockade regime leads to an analytic system solvable via $2\times 2$ or $7\times 7$ matrix inversion, providing explicit expressions for the many-body Green's functions

$$G^r_{i\sigma}(\omega ) = \sum_{j=1}^6 \frac{r_{i,j}}{\omega - p_{i,j} - \Delta_i(\omega)}$$

where all residues $r_{i,j}$ are explicit in the occupation numbers, themselves given by analytic solutions (Sobrino et al., 27 Jun 2024). This approach reproducibly matches advanced numerics such as non-crossing approximation (NCA) and hierarchical equations of motion (HEOM).

4. Self-Energy, Green’s Function, and Correlation Modelling

Self-energy-modified models, particularly SCF approaches for electrostatics, incorporate Coulomb correlations beyond mean-field by supplementing the classic PB or PNP framework with a self-energy correction for each ion species, defined through a generalized DH Green’s function evaluated with Born-series or WKB analysis. Ion size and space-/field-dependent dielectric response are retained, yielding a self-consistent loop: $$u_i(\mathbf r) = \frac{\widetilde u_1(\mathbf r)}{1 - \varepsilon(\mathbf r)\,\widetilde u_1(\mathbf r)\, a_i} + \frac{1}{a_i} \left[ \frac{1}{\varepsilon(\mathbf r)} - \frac{1}{\varepsilon_{\rm eff}} \right]$$ with the nonlocal term derived from the resolvent of a generalized screened Laplacian, ensuring physically finite correlation energies and inclusion of image-charge and dielectric boundary effects. These models reproduce Monte Carlo data for ion distributions, charge inversion, and dielectric response at low computational cost (Ma et al., 2014).

5. Scattering, Strong-Field, and Kinetic Models

(a) Scattering: For the description of two charged-particle scattering, especially with long-range Coulomb tails, semi-analytical treatment involves representing the interaction as a screened Yukawa potential, performing analytic azimuthal integration to reduce the Lippmann–Schwinger equation to a tractable lower-dimensional problem, and employing two-potential decomposition to handle Coulomb and short-range parts separately, with observables extrapolated as the screening parameter vanishes (Maulida et al., 2012).

(b) Strong-field physics: In atomic ionization and high-order harmonic generation (HHG) in strong fields, Coulomb-included models introduce closed-form or perturbative expressions for the Coulomb momentum transfer at tunnel exit and upon rescattering, as well as for the time shifts in ionization and return due to Coulomb symmetry constraints. Trajectories are analytically or semi-analytically propagated, and population yields or harmonic amplitudes derived with explicit scaling laws that match ab-initio TDSE results within 10–20%. For example, the amplitude scaling law $A(\Omega)\propto (\Omega/U_p)^{-3/4}$ is directly predicted by the semi-analytical model (Peng et al., 14 Dec 2025, Daněk et al., 2017).

(c) Kinetic theory: In suprathermal plasma physics and Fokker–Planck treatments with monoenergetic sources, the inclusion of the full Coulomb operator (with all velocity-space derivatives and angular diffusion) and expansion in Legendre polynomials yields semi-analytical solutions—sometimes closed-form, more often as a tridiagonal or block-diagonal system—retaining exact particle conservation, Maxwellian low-energy tails, and accurate high-energy behavior, as opposed to truncated (slowing-down) models (Goncharov, 2010).

6. Numerical Implementation and Benchmarking

Semi-analytical Coulomb-included models gain their efficacy from leveraging analytical tractability in all leading-order or symmetry-reduced channels while restricting numerics to low-dimensional grid or spectral calculations. For example, the Ma–Xu–Zhang MPNP model employs mass-conservative finite-difference time stepping with convolution-based evaluation of nonlocal chemical potentials, invoking WKB-derived formulae for the correlation self-energy at each grid point (Ma et al., 2020). These models are systematically benchmarked against fully numerical simulations (e.g., Monte Carlo, molecular dynamics, NCA/HEOM, or TDSE), and found to capture the key physical phenomena—including charge inversion, overscreening, recollision-induced amplitude suppression, and spectral weight redistribution—that are absent or incorrect in mean-field or purely perturbative schemes.

Context	Semi-Analytical Approach	Benchmark/Result
Electrolyte confinement	WKB-GDH for self-energy, FMT steric closure	Particle simulation agreement, captures surface-charge inversion
Few-body scattering	Stereographic projection, partial-wave basis	Closed s/p/d-wave $t$-matrix, kernel-level substitution in Yakubovskiĭ eqs.
Quantum dot transport	EOM hierarchy, analytic correlator system	Agrees with NCA & HEOM in blockade regime
HHG/strong-field	Tunnel-exit & recollision analytic formulae	Amplitude scaling law, attoclock calibration within 10–20% of TDSE
Kinetic plasma models	Legendre expansion, analytic Fokker–Planck	Maxwellization, particle conservation, suprathermal tails

7. Limitations and Applicability

The fidelity of semi-analytical Coulomb-included models is bounded by their foundational assumptions. Approximations such as smooth dielectric profiles, moderate coupling constants, point-charge limits, or closure at finite hierarchy order restrict their applicability at high coupling, sharp dielectric interfaces, or in systems with pronounced quantum coherence or nonlocality. Nonetheless, within their domain, such models provide essential bridges between computational feasibility and physically realistic, quantitatively predictive modeling, often serving as the basis for rapid simulation, parameter scans, and theoretical analysis in complex Coulomb-interacting systems (Ma et al., 2020, Ma et al., 2014, Maulida et al., 2012, Peng et al., 14 Dec 2025).