Design: self-consistent long-range gamma for GFN2-xTB-SECCM

Status (former-tracker issue 444): non-molecular 3-D ewald_gamma=True with the default gamma_form="klopman_ohno" fails closed before supercell matrix assembly. Exact zero-image molecular delegation is unchanged. The implemented opt-in gamma_form="elstner" permits 1-D, 2-D and 3-D calculations with an exponential remainder. It changes the finite-distance pair functional; it is not the published GFN2 kernel or a validated full-model GFN2 replacement. Finite WS truncation does not cure the Klopman-Ohno thermodynamic-limit defect.

The SECCM gamma code, tests and changelog call this defect #444. That number belongs to the issue tracker used before the repository split and was not carried over. The current tracker’s #444 is an unrelated SCC-DFTB-SECCM fixture issue, so this note is the live description of the defect.

Literature and the long-range/short-range partition

GFN2-xTB, Bannwarth, Ehlert and Grimme (2019) defines a shell-resolved charge interaction with Klopman-Ohno damping. For shell hardnesses (h_a,h_b), the implemented pair kernel is

eta_ab = 2 / (h_a + h_b)
gamma_ab(R) = 1 / sqrt(R^2 + eta_ab^2)
gamma_ab(R) - 1/R = -eta_ab^2 / (2 R^3) + O(R^-5).

Here eta_ab is a damping length, not the hardness itself. A consistent point-Coulomb partition at a finite WS boundary can be written

Gamma_ab = Ewald_Coulomb_ab
         + sum_WS w [gamma_ab(R_image) - 1/R_image]
         + on_site_ab.
E_charge = (1/2) dq^T Gamma dq.

Nonzero-distance records enter the bracket; the same-atom on-site block is handled separately, with the Ewald self convention retained. The one-half in the energy avoids pair double counting. This identity separates point electrostatics from the damped remainder; it does not establish convergence of the latter as the cluster grows.

The corresponding CCM embedding partition is the exact Ewald potential minus the bare point-Coulomb contribution already owned by the WS cell. Janetzko, Bredow and Jug, JCP 116, 8994-9004 (2002), especially equations 19-20, supplies that long-range CCM construction. Bredow, Geudtner and Jug (2001) uses finite Evjen-weighted point-charge shells and defers exact Ewald embedding on page 93. It is not the source of the later exact operator. The citation database retains both works under their distinct roles; janetzko_ccm_long_range_2002 is the existing long-range entry.

Why the 3-D WS remainder has no thermodynamic limit

The number of sites in a three-dimensional radial shell grows as R^2 dR, so the leading R^-3 remainder accumulates as dR/R. For a repeated charge pattern its leading charge-weighted coefficient is proportional to sum_ab dq_a dq_b eta_ab^2. Neutrality sum_a dq_a = 0 does not generally cancel a pair-dependent coefficient. Shell-dependent hardness also means a single chemical element is not a general escape from the problem.

For the neutral first-shell Mg4O4 patterns below the coefficient is 16 [1/h_Mg^2 + 1/h_O^2 - 8/(h_Mg+h_O)^2], strictly positive for unequal positive hardnesses. The regression checks this analytic non-cancellation independently of the two finite-size energies; the latter are not used to fit an asymptotic rate.

WS ownership makes every fixed-cluster matrix finite. Increasing the cluster grows the owned remainder, however, and can accumulate the same logarithmic defect. Neither Ewald-alpha independence, finite-cutoff parity with another implementation, small charges, nor SCC convergence proves the existence of the thermodynamic limit. The earlier version of this design incorrectly called the WS convention convergent in 3-D; that claim is withdrawn.

In one and two dimensions the corresponding far-field absolute remainder scales as integral R^(d-4) dR and converges. This dimensional argument does not certify every other GFN2 channel or solve an SCC instability. The shipped 1-D wire and 2-D Parry/Heyes paths, including the exact pairwise 2-D K=0 term, remain unchanged. The separate SCC-DFTB contracts tracked by former-tracker issues 211 and 425 are not modified by this GFN2 containment.

An exponential remainder such as the density-derived SCC-DFTB gamma in Elstner et al. (1998), equations 17-18 has a convergent tail. The existing opt-in Elstner implementation uses tau = 16/5 U, retains the on-site parameter block, and replaces the nonzero-distance pair kernel. Convergence of that remainder does not validate the modified GFN2 functional, its parameters, or all other cyclic channels. Klopman-Ohno remains the SECCM default; D4 dispersion and SECCM nuclear/cell derivatives remain absent. Molecular GFN2 acceptance with coupled D4 cannot qualify this dispersion-free cyclic model.

Fixed-charge cluster discriminator

The native diagnostic seam _gfn2_seccm_ewald_shell_gamma_from_records is tested without an SCC solve. Both probes use a cubic side of 4.212 angstrom, the site order

(0,0,0), (.5,.5,0), (.5,0,.5), (0,.5,.5),
(.5,.5,.5), (0,0,.5), (0,.5,0), (.5,0,0).

The historical synthetic Mg4O4 probe assigns [Mg,O,O,O,Mg,Mg,Mg,O]. It is not B1 rocksalt: sites 0 and 5 are nearest neighbors of the same species. The physical B1 control assigns [Mg,Mg,Mg,Mg,O,O,O,O] to the same sites. Each conventional cell is replicated n x n x n. Only each atom’s first enumerated shell carries dq = -1 for Mg or +1 for O; every other shell carries zero. This is a prescribed ionic diagnostic, not self-consistent occupations.

Probe	n=1, 8 atoms	n=2, 64 atoms	Difference
Synthetic cubic Mg4O4	+0.361531884674	+0.464251887966	+0.102720003292
B1 rocksalt control	+0.144635757778	+0.255217985354	+0.110582227576

Values are 0.5 dq^T Gamma dq / n^3 in Ha per conventional cell, with alpha = sqrt(pi) / volume^(1/3). These are total fixed-charge kernel energies, not the remainder alone. Independent contraction of WS records isolates the size step entirely to KO-1/R; the on-site and point-Coulomb contributions per cell are invariant. Two points discriminate cluster drift but are not a fitted asymptotic logarithmic rate or a prediction of its sign at all sizes. No n=3 value or extrapolated limit is claimed.

Production boundary and regression scope

The public Python facade and raw native calculation reject non-molecular 3-D ewald_gamma with Klopman-Ohno, including a group-order-one topology with nonzero image records. Explicit Elstner selects the alternative kernel and is not subject to this particular refusal. The guard follows option/restart validation and exact molecular delegation, and precedes matrix assembly. Existing validation precedence, #410 failed-state classification, #421 diagnostic-only symmetry reporting, and molecular Newton refusal are preserved. include_aes=False does not create a bypass: it selects the cyclic engine even for zero-image records.

The internal kernel still pins finite-cluster bookkeeping, the eta-to-zero point-Coulomb anchor, and skew-cell enumeration. Those tests do not authorize production 3-D Klopman-Ohno use. Named former-tracker issue 130 Cu recipes now assert the explicit rejection; their old energy, size and state assertions remain historical evidence, not reachable production expectations. Earlier MgO/corundum bulk convergence and lattice claims likewise do not establish thermodynamic validity. Molecular, 1-D/2-D and non-ewald_gamma calculations are not reparameterized.