# Barron-preconditioned Coulomb wavefunction

- ID: 52
- Canonical URL: https://synthcore.org/idea/52/barron-preconditioned-coulomb-wavefunction
- API JSON: https://synthcore.org/api/idea/52.json
- API Markdown: https://synthcore.org/api/idea/52.md
- Verification status: unverified
- Source: [arXiv:2608.22252](https://arxiv.org/abs/2608.22252)
- Category: architecture
- Solves: accuracy, sample-efficiency, stability
- ML areas: mlp, loss, training-dynamics
- Math tags: approximation-theory, harmonic-analysis, pde, functional-analysis
- Ratings: usefulness 6/10; difficulty 4/10; novelty 4/10

## Idea description

Represent a many-electron wavefunction as a fixed analytic cusp envelope multiplied by a neural residual, instead of forcing the network to learn Coulomb singularities. The paper predicts that the residual has Barron smoothness arbitrarily close to order two, which should reduce approximation difficulty and noisy local-energy derivatives near particle coalescences.

## Mathematical statement

The Coulomb Hamiltonian is H = -sum_i Delta_{x_i} + V, with V = -sum_{i,nu} Z_nu divided by |x_i - R_nu| + sum_{i<j} 1 divided by |x_i - x_j|. For an eigenfunction H psi = E psi, the paper defines phi = exp(-F_2,cut) psi and phi_3 = exp(-(F_2,cut + F_3,cut)) psi. Its main theorem states that phi and phi_3 belong to the Barron class B^s on R^(3N) for every s less than 2, while no universal factorization generally guarantees B^2. The endpoint estimate is ||u||_(B^(2-epsilon)) <= M epsilon^(-2) ||u||_(B^1), where epsilon = 2 - s, u is either phi or phi_3, and M is independent of epsilon. Here B^s is the order-s Barron or Fourier regularity norm. An implementable cusp factor is F_J(x) = sum_{i,nu} -(Z_nu/2) chi(r_i,nu) r_i,nu + sum_{i<j} (1/4) chi(r_ij) r_ij, where r_i,nu = |x_i - R_nu| and r_ij = |x_i - x_j|. The coefficients match the Kato cusp slopes for the kinetic convention -Delta.

## Key formulas

- $$\phi=e^{-F_{2,\mathrm{cut}}}\psi,\qquad \phi_3=e^{-F_{3,\mathrm{cut}}}\phi=e^{-(F_{2,\mathrm{cut}}+F_{3,\mathrm{cut}})}\psi.$$
- $$\phi,\phi_3\in\mathcal B^s(\mathbb R^{3N})\qquad\text{for every }s<2.$$
- $$\|u\|_{\mathcal B^{2-\varepsilon}}\leq\frac{M}{\varepsilon^2}\|u\|_{\mathcal B^1},\qquad \varepsilon=2-s,\qquad u\in\{\phi,\phi_3\}.$$
- $$F_J(x)=\sum_{i=1}^{N}\sum_{\nu=1}^{L}-\frac{Z_\nu}{2}\chi(r_{i\nu})r_{i\nu}+\sum_{1\leq i<j\leq N}\frac14\chi(r_{ij})r_{ij},\qquad \psi_\theta(x)=\exp\big(F_J(x)+a_\theta(x)\big).$$

## Implementation notes

Integrate the factorization at the output of a neural quantum-state MLP or permutation-equivariant electron network. The network outputs a scalar residual log-amplitude a_theta(x); for a positive ground-state experiment set u_theta(x) = exp(a_theta(x)), while excited-state experiments can use separate sign and log-amplitude heads. Compute every electron-nucleus distance r_i,nu and electron-electron distance r_ij. Use a compact smooth cutoff, for example chi(r) = (1 - r/r_c)^2 (1 + 2r/r_c) for r less than r_c and zero otherwise. Form F_J with the displayed cusp coefficients and return log psi_theta = F_J + a_theta. In variational Monte Carlo, sample configurations x, autodifferentiate log psi_theta, compute gradients and Laplacians, evaluate the local energy E_L = (H psi_theta) / psi_theta, and optimize energy or local-energy variance. Do not detach F_J: its derivatives are intended to cancel the Coulomb cusp terms. The theorem supplies the residual regularity claim and the epsilon^(-2) endpoint penalty; estimate the empirical benefit by comparing high-frequency Fourier energy or finite-difference derivative norms of residual and unfactored outputs. First test helium with two electrons and one nucleus using a three-layer MLP. Compare no Jastrow, the fixed analytic Jastrow, and a learnable Jastrow at equal parameter count and Monte Carlo budget. Measure variational energy, local-energy variance, gradient variance, and energy after a fixed number of samples. Success means lower variance and faster energy reduction at equal FLOPs, especially near coalescence points.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.