GeomeTRIC Optimizer

The GeomeTRIC backend is a quasi-Newton geometry optimizer that works natively in translation-rotation internal coordinates (TRIC). It targets both energy minima and first-order saddle points, drives an adaptive trust-radius loop, and back-transforms each internal-coordinate step to Cartesian space iteratively.

Select it from the OPT section:

OPT
  Algorithm  geometric      # aliases: geomet, tric
  Coord      tric | tric-p  # tric-p uses primitives only
  ...
END

Coordinate System

The optimizer builds a [TricSystem] from molecular topology:

1. Topology primitives — bonds, angles, dihedrals, and out-of-plane terms inferred from the connectivity graph. A linear/chain fallback is used when the topology yields no primitives.
2. Per-fragment external coordinates — for the tric model, each disconnected fragment contributes six rigid-body coordinates ordered as $[T_x,T_y,T_z,R_x,R_y,R_z]$:
   • Translation is the fragment centroid.
   • Rotation is encoded as an exponential map relative to the geometry captured when the system was built.

The tric-p model uses topology primitives only (no rigid-body coordinates), matching the Primitive coordinate kind.

Rotation Coordinates

Rotational internal coordinates use a quaternion/exponential-map algebra:

• The optimal rotation quaternion between the current fragment geometry and its reference is obtained from the eigenvector of a $4\times 4$ correlation matrix.
• The quaternion is mapped to a three-component exponential map that is free of gimbal singularities over the working range.
• Analytic derivatives of the exponential map with respect to Cartesian coordinates supply the rotational rows of the Wilson $\mathbf{B}$-matrix.

B-Matrix and Gradient Projection

The Wilson $\mathbf{B}$-matrix collects $\partial q_i/\partial x$ for every internal coordinate. Primitive rows come from the topology set; the centroid translation rows are constant ($1/N_{\text{frag}}$), and the rotation rows come from the analytic exponential-map derivatives.

Cartesian gradients are projected into internals through the Wilson $\mathbf{G}$-matrix:

$${\mathbf{g}}_q={\mathbf{G}}^{-1}\mathbf{B}{\mathbf{g}}_x,\mathbf{G}=\mathbf{B}{\mathbf{B}}^{†}$$

with ${\mathbf{G}}^{-1}$ formed as an SVD pseudo-inverse to handle the redundancy of the internal set.

Initial Hessian

initial_ic_hessian seeds the internal-coordinate Hessian:

• From a Cartesian Hessian (recommended for TS): transformed into internals via ${\mathbf{H}}_q={\mathbf{G}}^{-1}\mathbf{B}{\mathbf{H}}_x{\mathbf{B}}^{†}{\mathbf{G}}^{-1}$. For minimization it is forced positive-definite; for TS the single negative eigenvalue (the reaction coordinate) is preserved.
• Model guess (fallback): the Lindh diagonal for primitives plus weak positive curvature ($0.05$ a.u.) on each translation/rotation coordinate. This fallback also covers degenerate B-matrices from near-linear fragments.

Step Engine

Each cycle proposes a step with the trust-radius engine:

• Minimization uses the trust-radius Newton-Raphson (TRM) step $\Delta y=-(\mathbf{H}+v\mathbf{I}{)}^{-1}\mathbf{g}$ with BFGS Hessian updates by default.
• Transition states use restricted-step partitioned RFO (RS-P-RFO), which maximizes along the lowest Hessian eigenmode (transition vector) and minimizes along the rest, with Bofill Hessian updates by default.

The trust_step driver scales the proposed step to the current trust radius via a damped Hebden iteration. The iteration is bounded (oscillation detection plus a hard MAX_ITER cap) and returns the best step found. The RS-P-RFO metric uses a strictly positive scaling parameter; non-finite or non-positive values are clamped to a small positive floor so the symmetric eigensolver always converges.

Internal-to-Cartesian Back-Transform

ic_to_cartesian recovers Cartesian coordinates that reproduce a target internal displacement through the iteration

$$\Delta x={\mathbf{B}}^{†}{\mathbf{G}}^{-1}\Delta q$$

with periodic-aware residuals (dihedral and rotation coordinates wrap into $(-\pi ,\pi ]$), per-atom step capping, and adaptive damping. A bork flag is raised when the residual stays large, which the optimizer treats as a rejected step.

Trust-Radius Control

The step quality factor $\rho =\Delta E_{\text{actual}}/\Delta E_{\text{predicted}}$ drives an adaptive trust radius:

• $\rho >0.75$ at the trust boundary → grow trust (up to TMax).
• $\rho <0.25$ → shrink trust (down to the floor).
• Bad steps are rejected and retried with a halved trust radius; once the floor is reached the step is accepted to keep making progress.

Minimization rejects energy-raising steps against a confident prediction; TS searches legitimately raise the energy along the reaction coordinate and only reject pathological quality factors or failed back-transforms.

Convergence

Uses the same 5-criterion Gaussian-style test as the other backends (energy change, gradient RMS/max, displacement RMS/max). See Geometry Optimization Algorithms for the shared convergence presets.