Delocalized & TRIC Coordinates

Delocalized Internal Coordinates (DLC)

DLC forms a non-redundant basis from the redundant primitive ICs:

1. SVD rank detection on the primitive B-matrix builds a non-redundant delocalized basis
2. Projection and back-transform operate in the reduced space

The DLC basis matrix $\mathbf{U}$ (rank × m) is obtained via SVD of the B-matrix: $$\mathbf{B}=\mathbf{U}\mathbf{\Sigma }{\mathbf{V}}^{†}$$

Gradients and Hessians are transformed: $${\mathbf{g}}_{\text{DLC}}=\mathbf{U}{\mathbf{g}}_{\text{prim}}$$ $${\mathbf{H}}_{\text{DLC}}=\mathbf{U}{\mathbf{H}}_{\text{prim}}{\mathbf{U}}^{†}$$

The primitive Hessian ${\mathbf{H}}_{\text{IC}}$ is retained in full redundant IC space and updated via MSP as usual, providing stable curvature propagation across steps.

Translation-Rotation Internal Coordinates (TRIC)

TRIC adds rigid-body translational and rotational modes via weighted internal coordinates:

1. Modified Gram-Schmidt removes translational/rotational modes before and after DLC projection
2. Rigid-body components suppressed to machine precision

TRIC is required for:

• Rigid true option
• RemoveTr true option (also works with DLC/HDLC)

Native TRIC Coordinate System

The GeomeTRIC optimizer backend builds an explicit translation-rotation internal-coordinate system rather than removing TR modes from a DLC basis. For each disconnected fragment it appends six rigid-body coordinates in the order $[T_x,T_y,T_z,R_x,R_y,R_z]$:

• Translation coordinates are the fragment centroid components.
• Rotation coordinates are an exponential-map representation of the optimal rotation relative to the geometry captured when the system is built. The exponential map avoids gimbal-lock singularities, and its analytic derivatives supply the rotational rows of the Wilson $\mathbf{B}$-matrix.

Dihedral and rotation coordinates are periodic: internal-coordinate differences for these wrap into $(-\pi ,\pi ]$ during gradient projection and the internal-to-Cartesian back-transform. See the GeomeTRIC Optimizer chapter for the full step engine, initial-Hessian strategy, and trust-radius control.

HDLC (Hybrid Delocalized)

HDLC adds Cartesian components to the DLC basis, providing a complete spanning set while retaining the benefits of delocalization for the internal degrees of freedom.

Usage

OPT
  Coord  dlc      # or tric, hdlc, tric-p
  ...
END

TRIC is recommended for systems with floppy modes or when exact translational/ rotational separation is needed.