Ewald summation in periodic systems¶

In 3D periodic solids the Coulomb series \(\sum_{g,A,B} Z_A Z_B / |R_{AB} + g|\) is conditionally convergent: different truncation shapes give different answers. The standard fix is Ewald summation, which splits each \(1/r\) term into a short-range part (evaluated in real space, where it converges exponentially) plus a long-range part (evaluated in reciprocal space, where the Gaussian damping makes the sum converge fast too). vibe-qc’s full 3D Ewald infrastructure — nuclear repulsion, nuclear attraction \(V(g)\), and the electron-repulsion Coulomb \(J(\mu\nu)\) — is shipped through a single Coulomb-method dispatcher; this page tells you how to use it.

The user-facing recommendation¶

For any 3D-periodic SCF, set the Coulomb method to EWALD_3D and call the dispatcher:

import vibeqc as vq

opts = vq.PeriodicSCFOptions()
opts.lattice_opts.coulomb_method = vq.CoulombMethod.EWALD_3D

# Multi-k:
result = vq.run_rhf_periodic_scf(system, basis, kmesh, opts,
                                 omega=0.5, spacing_bohr=0.3)
# Γ-only:
result = vq.run_rhf_periodic_gamma_scf(system, basis, opts,
                                       omega=0.5, spacing_bohr=0.3)

The dispatcher routes on opts.lattice_opts.coulomb_method to the appropriate backend:

`CoulombMethod`	Routed backend (multi-k)	Routed backend (Γ-only)	Status
`DIRECT_TRUNCATED`	`run_rhf_periodic`	`run_rhf_periodic_gamma`	Default. Half-summed direct truncation — exact in the molecular-limit regime (large unit cell, cutoff excludes all non-zero cells). Conditionally convergent in 3D bulk.
`EWALD_3D`	`run_rhf_periodic_multi_k_ewald3d`	`run_rhf_periodic_gamma_ewald3d`	Full Gaussian-charge Ewald — unconditionally convergent. The native FFT long-range solver now supports skew cells via a full reciprocal-lattice metric; the legacy EWALD_3D SCF route remains a parity/debug path while FFTDF/GDF production work is in progress.
`SLAB_EWALD_2D`	—	—	Not yet wired into SCF dispatch (raises `NotImplementedError`).
`NEUTRALIZED_1D`	—	—	Not yet wired into SCF dispatch.

Nuclear repulsion routes through the same enum: when EWALD_3D is selected, nuclear_repulsion_per_cell automatically uses Ewald summation, so the electronic and nuclear contributions stay self-consistent.

Same dispatcher for KS and UKS¶

The pattern above is the HF dispatcher. The KS (Kohn-Sham DFT) and UKS (open-shell DFT) families ship the same dispatcher shape and the same CoulombMethod enum:

Driver family	Dispatcher (multi-k)	Dispatcher (Γ-only)	Phase
Closed-shell HF	`run_rhf_periodic_scf`	`run_rhf_periodic_gamma_scf`	12e-c-4
Closed-shell KS	`run_rks_periodic_scf`	`run_rks_periodic_gamma_scf`	15c-1, 15c-2
Open-shell UKS	`run_uks_periodic_scf`	`run_uks_periodic_gamma_scf`	15c-3

Each accepts the same keyword arguments (omega, grid_shape, origin, spacing_bohr, linear_dep_threshold) and routes on options.lattice_opts.coulomb_method identically. Pick the options class that matches your method (PeriodicSCFOptions / PeriodicRHFOptions for HF, PeriodicKSOptions for RKS/UKS), set the Coulomb method, and call the dispatcher.

The Ewald-specific keyword arguments to the dispatcher (omega, grid_shape, origin, spacing_bohr, linear_dep_threshold) are forwarded to the EWALD_3D backend and ignored on the DIRECT_TRUNCATED path. Defaults: omega=0.5 bohr⁻¹ (the short/long-range splitting parameter), spacing_bohr=0.3 (reciprocal-space FFT grid spacing). Most users never need to touch these — the only knob that matters in practice is the CoulombMethod choice itself.

A complete worked example¶

import vibeqc as vq

# Conventional cubic LiH rocksalt — 8 atoms per cell, lattice 4.084 Å.
a_ang = 4.084
a = a_ang / 0.529177210903
import numpy as np
lat = a * np.eye(3)

# Off-FFT-grid shift: atoms exactly at the corner of the box inflate
# the long-range residual ~100x; a small (0.05 bohr) shift fixes it.
shift = 0.05
unit_cell = []
for z in (3, 1):                                       # 3 = Li, 1 = H
    base = 0.5 if z == 1 else 0.0
    for fx, fy, fz in [(0, 0, 0), (0, 0.5, 0.5),
                        (0.5, 0, 0.5), (0.5, 0.5, 0)]:
        x = (fx + base) * a + shift
        y = (fy + base) * a + shift
        z_ = (fz + base) * a + shift
        unit_cell.append(vq.Atom(z, [x, y, z_]))

system = vq.PeriodicSystem(dim=3, lattice=lat, unit_cell=unit_cell)
basis = vq.BasisSet(system.unit_cell_molecule(), "sto-3g")

# Γ-only SCF with Ewald
opts = vq.PeriodicSCFOptions()
opts.lattice_opts.coulomb_method = vq.CoulombMethod.EWALD_3D
opts.lattice_opts.cutoff_bohr = 12.0          # short-range AO cutoff
opts.lattice_opts.nuclear_cutoff_bohr = 25.0   # nuclear-Ewald real cutoff

result = vq.run_rhf_periodic_gamma_scf(
    system, basis, opts,
    omega=0.5,
    spacing_bohr=0.4,
)

print(f"Converged:    {result.converged}")
print(f"Total energy: {result.energy:.8f} Ha per cell")
print(f"SCF iters:    {result.n_iter}")

For multi-k, build a :class:vibeqc.KPoints (or, for back-compat, a raw BlochKMesh) and call run_rhf_periodic_scf(system, basis, kpoints, opts). The result mirrors the Γ-only structure plus per-k orbital coefficients. KPoints exposes Monkhorst–Pack, Γ-centered, IBZ-reduced (via spglib), HPKOT band paths (via seekpath), explicit user lists, and density-based auto-meshes (KPPRA / kspacing / VASP Auto) — see user_guide/k_points.md for the full reference.

What “ω-invariance” means and why we test it¶

Ewald has one free parameter — the splitting \(\omega\) that decides how much of \(1/r\) is sent to real-space vs reciprocal-space. The total energy must be independent of \(\omega\) to whatever the real- and reciprocal-space cutoffs allow. This is the primary correctness test of any Ewald implementation. vibe-qc’s bulk-crystal benchmark suite verifies it explicitly across H₂ / LiH / MgO / Ne fixtures: scanning \(\omega \in \{0.3, 0.4, 0.5, 0.6, 0.7\}\) recovers the same total energy to within \(10^{-8}\) Ha when the cutoffs are adequate. If your run is \(\omega\)-sensitive at the \(10^{-4}\) level, the cutoffs are too tight — increase cutoff_bohr and nuclear_cutoff_bohr and re-test.

Choosing cutoffs and the FT mesh¶

The EWALD_3D path has three convergence knobs:

opts.lattice_opts.cutoff_bohr — real-space cutoff for AO pair / ERI integrals and the AO-pair-FT Bloch cell list. Default 15 bohr; can usually drop to 10–12 for valence-only bases like sto-3g, pob-DZVP.
opts.lattice_opts.nuclear_cutoff_bohr — real-space cutoff for the nuclear-Ewald real-space term. Default 25 bohr; auto-derived from omega if you pass omega explicitly.
VIBEQC_J_EWALD3D_KE — kinetic-energy cutoff (Hartree) for the analytical AO-pair-FT Hartree J mesh. Default 200 Ha. Increase for benchmark-quality all-electron tight-core cells.

The historical FFT-Poisson collocation backend is still available for diagnostics with VIBEQC_J_EWALD3D_BACKEND=grid. In that mode only, the dispatcher spacing_bohr= kwarg controls the real-space FFT grid.

The auto-derived \(\alpha\) (Ewald splitting parameter) is chosen so that erfc\((\alpha \cdot R_{\text{cut}}) \approx 10^{-12}\). That makes both the real- and reciprocal-space sums converge fast on the supplied cutoffs without you having to tune \(\alpha\) by hand.

Warning

The legacy multi-k RKS EWALD_3D path stores periodic AO values for every real-space density cell, and GGA functionals store three AO gradient arrays as well. Conventional-cell, pob-TZVP-scale ionic crystals can therefore hit a tens-of-GiB memory cliff before the first SCF iteration. The driver now raises MemoryError with an allocation estimate instead of letting the operating system kill the process. Use the GDF periodic route (run_krhf_periodic_gdf / run_krks_periodic_gdf) where available, or coarsen the DFT grid only for small debug runs. The current multi-k GDF spike uses PySCF’s k-point-symmetry classes for weighted IBZ meshes when available, with a full-mesh fallback for older PySCF / shifted meshes. A fully native GDF loop remains the long-term performance follow-up.

Low-level Ewald primitives¶

The dispatcher is the right entry point for SCF work. For quasi-classical lattice-electrostatics calculations (Madelung constants, point-charge potentials, charge-only models), use the primitives directly:

from vibeqc import EwaldOptions, ewald_point_charge_energy
import numpy as np

# NaCl-like rocksalt with ±1 charges
a = 5.6 / 0.529177210903                       # 5.6 Å in bohr
lattice = 0.5 * a * np.array([[0, 1, 1],
                              [1, 0, 1],
                              [1, 1, 0]], dtype=float).T
positions = np.column_stack([[0, 0, 0],
                             [0.5 * a, 0, 0]])
charges = np.array([+1.0, -1.0])

opts = EwaldOptions()
opts.real_cutoff_bohr = 30.0          # auto-derive α and recip cutoff
energy = ewald_point_charge_energy(lattice, positions, charges, opts)
print(f"NaCl Madelung / pair: {energy:.8f} Ha   (expected -0.330275485)")

Other low-level entry points:

vq.ewald_point_charge_potential — Madelung potential at given probe points.
vq.ewald_nuclear_potential / vq.ewald_nuclear_repulsion — nuclei-only versions.
vq.compute_nuclear_lattice_ewald — nuclear-attraction matrix elements via Ewald.
vq.build_j_ewald_3d — direct J(μν) Coulomb build at a single density (used inside the SCF dispatcher).

These all operate on the same EwaldOptions / LatticeSumOptions machinery; the dispatcher just wires them together at SCF time.

Madelung constant for arbitrary Bravais cells¶

vibeqc.madelung.madelung_constant_for_cell(system, *, precision=1e-12, eta=None) returns the Ewald-Madelung constant ξ for any Bravais lattice via a proper real-space + reciprocal-space Ewald self-energy sum with a neutralising background. It is η-invariant at convergence and bit-exact to PySCF’s pbc.tools.madelung to 6+ significant figures on cubic, primitive FCC, and intermediate cells.

import vibeqc as vq
from vibeqc.madelung import madelung_constant_for_cell

xi = madelung_constant_for_cell(system)

Prior to commit d7f4b3bd, this routine used the cubic Wigner shortcut α_M ≈ 2.837297 / V^(1/3), which is correct for conventional cubic cells but off by ~1.77 % for primitive FCC cells. The 1.77 % gap propagated through the exxdiv='ewald' K-shift to ~16 mHa of SCF bias on LiH primitive FCC. The proper Ewald sum closes that gauge defect and was a key contributor to the v0.9 PBC GDF chemical-accuracy milestone on LiH (see periodic_methods.md § 3.2.1).

Theory¶

Splitting the Coulomb sum¶

For a 3D-periodic distribution of point charges \(\{Z_A\}\) at positions \(\{R_A\}\) in a unit cell with translations \(\{g\}\), the energy

\[ E_{\text{nn}} = \tfrac{1}{2} \sum_{A, B} {\sum_{g}}' \frac{Z_A Z_B}{|R_A - R_B + g|} \]

(primed sum excludes the \(A = B, g = 0\) term) is conditionally convergent. Ewald’s 1921 trick is to split each \(1/r\) as

\[ \frac{1}{r} = \frac{\operatorname{erfc}(\alpha r)}{r} + \frac{\operatorname{erf}(\alpha r)}{r} \]

The first term decays exponentially in real space; the second is smooth enough to Fourier-transform and sum in reciprocal space. The total energy under the tin-foil + jellium convention becomes

\[ E_{\text{nn}} = E_{\text{real}} + E_{\text{recip}} + E_{\text{self}} + E_{\text{bg}} \]

with

\(E_{\text{real}} = \tfrac{1}{2} \sum_{A,B} {\sum_{g}}' Z_A Z_B \cdot \operatorname{erfc}(\alpha \, r) / r\) — exponentially convergent real-space sum.
\(E_{\text{recip}} = \tfrac{2\pi}{V} \sum_{G \neq 0} |S(G)|^2 \, e^{-|G|^2/(4\alpha^2)} / |G|^2\) — k-space sum with Gaussian damping. \(S(G) = \sum_A Z_A \, e^{iG \cdot R_A}\) is the nuclear structure factor.
\(E_{\text{self}} = -(\alpha/\sqrt{\pi}) \sum_A Z_A^2\) — removes the spurious self-interaction from each nucleus’s Gaussian-smeared image.
\(E_{\text{bg}} = -(\pi/(2 \alpha^2 V)) (\sum_A Z_A)^2\) — background correction for a non-neutral cell (the “jellium” prescription).

From point charges to AO densities¶

For an AO basis the analogue construction extends the same idea to Gaussian charge distributions. Each AO pair \(\chi_\mu \chi_\nu^g\) is itself a sum of Gaussians; their \(1/r\)-mediated interaction with the rest of the lattice splits the same way:

Near field (AO pairs within a short-range cutoff): exact analytic Gaussian-Gaussian Coulomb integrals, no multipole expansion.
Far field: the AO pair is approximated by its multipole expansion (monopole + dipole + quadrupole) and the multipoles are Ewald-summed in reciprocal space via FFT.

This is the Saunders–Dovesi multipolar splitting used in CRYSTAL. The FFT grid is built in fractional coordinates, and the reciprocal kernel uses the full lattice metric, so primitive skew cells can run through the native long-range solver. The legacy EWALD_3D SCF code is still not the production recommendation for tight ionic crystals; the active parity track is native FFTDF/GDF with benchmarks against CRYSTAL and PySCF.

Nuclear-attraction \(V(g)\)¶

The same erfc / erf splitting applies to the nuclei-electron attraction matrix elements

\[ V_{\mu\nu}(g) = \sum_{A, h} -Z_A \, \langle \chi_\mu | 1 / r_{A,h} | \chi_\nu^g \rangle \]

The short-range piece is computed by compute_nuclear_erfc_lattice (real-space, exponentially convergent in \(h\)). The long-range piece adds the smooth erf\((\omega r)/r\) kernel, which integrates exactly over Gaussian AO pairs in reciprocal space. The dispatcher wires both together inside the SCF Fock build.

References¶

Ewald, P. P. “Die Berechnung optischer und elektrostatischer Gitterpotentiale,” Ann. Phys. 64, 253 (1921). The original Ewald paper.
Saunders, V. R.; Freyria-Fava, C.; Dovesi, R.; Salasco, L.; Roetti, C. “On the electrostatic potential in crystalline systems where the charge density is expanded in Gaussian functions,” Mol. Phys. 77, 629 (1992). Multipolar splitting used in CRYSTAL — the basis of vibe-qc’s far-field treatment.
Toukmaji, A. Y.; Board Jr., J. A. “Ewald summation techniques in perspective: a survey,” Comput. Phys. Commun. 95, 73 (1996). Modern Ewald review covering the tin-foil convention, \(\alpha\) optimization, and the various flavors used in MD codes.
Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids, 2nd ed., Oxford University Press (2017). Clear textbook treatment with the \(\alpha\) auto-tuning formula vibe-qc uses.
Pisani, C.; Dovesi, R.; Roetti, C. Hartree-Fock Ab Initio Treatment of Crystalline Systems. Lecture Notes in Chemistry vol. 48, Springer (1988). Foundational text for periodic LCAO HF with the Ewald/multipolar treatment of the lattice Coulomb sums CRYSTAL is built on.