Primitive Internal Coordinates (IC)

When Coord primitive, Coord dlc, or Coord tric is requested, geometry steps are taken in internal-coordinate (IC) space using:

1. Topology Detection

• Covalent-radius bond detection (38 elements, Alvarez 2008 radii)
• Bond graph traversal for angles, dihedrals, out-of-plane bending (sp² centers), and supplementary linear-angle components for near-linear triplets

2. Wilson B-Matrix

Analytic Cartesian derivatives for all primitive types:

• Bonds, angles: standard Wilson formulae
• Dihedrals / out-of-plane: analytic 4-atom gradient (cross-product formula)
• Linear-angle: stateless perpendicular-axis construction + finite-difference gradient

The B-matrix relates Cartesian displacements to IC displacements: $$\Delta \mathbf{q}=\mathbf{B}\Delta \mathbf{x}$$

3. Lindh Model Hessian

Initial diagonal estimate in IC space:

• Bonds: $0.45\times \exp (-(r-r_0{)}^2)$ per Lindh
• Angles / linear-angle: $0.15$
• Dihedrals / out-of-plane: $0.005$

4. IC Back-Transform

Iterative Newton method $({\mathbf{B}}^T{\mathbf{G}}^{-1}\Delta \mathbf{q}$ loop) with periodic-angle wrapping; returns (new_xyz, bork) convergence flag.

5. Hessian Update

MSP (mixed-mode) update blending BFGS and Murtagh–Sargent with parameter $\varphi =1-(\Delta \mathbf{y}\cdot \mathbf{\Xi }{)}^2/((\Delta \mathbf{y}\cdot \Delta \mathbf{y})(\mathbf{\Xi }\cdot \mathbf{\Xi }))$; falls back to pure BFGS.

6. Step Engine

Trust-radius method (TRM) with level-shifted Newton step; Brent bisection to satisfy $\Vert \text{step}(\lambda )\Vert =\text{trust}$.

7. Convergence

5-criterion Gaussian-style check:

• Energy change ΔE
• Gradient RMS/max
• Displacement RMS/max

Thresholds from ConvergenceCriteria (gaussian_default, gaussian_loose, gaussian_tight).

8. Trust-Radius Update

ρ-based acceptance ratio; shrink on $\rho <0.25$, grow on $\rho >0.75$.