Dimer Method

Why saddle points?

A first-order saddle point on the PES is a transition state: the geometry where the system passes from one local minimum to another. The energy at the saddle determines the reaction barrier, which controls the rate. Finding transition states is therefore central to understanding reaction mechanisms and computing rate constants.

Unlike minima (where all eigenvalues of the Hessian are positive), a saddle has exactly one negative eigenvalue. The corresponding eigenvector points along the reaction coordinate. The dimer method exploits this by following the lowest curvature mode uphill while relaxing in all other directions.

How the dimer works

A “dimer” is a pair of points separated by a small distance (dimer_sep) along an orientation vector. The dimer does two things at each step:

Rotation (finding the lowest mode)

The curvature along the dimer axis is estimated from the gradient difference between the two images:

\[C \approx \frac{(\mathbf{G}_{1} - \mathbf{G}_{0}) \cdot \hat{\mathbf{n}}}{\Delta R}\]

where \(\mathbf{G}_{0}\) and \(\mathbf{G}_{1}\) are the gradients at the two dimer images, \(\hat{\mathbf{n}}\) is the dimer orientation, and \(\Delta R\) is the dimer separation. Rotation adjusts the orientation to minimize this curvature, aligning the dimer with the softest mode. This is equivalent to finding the lowest eigenvector of the Hessian, but without computing the full Hessian.

Translation (walking to the saddle)

The translational force drives the midpoint toward the saddle. The force construction depends on the curvature sign:

• Negative curvature (already straddling the saddle): the force along the dimer orientation is reversed. The midpoint walks uphill along the reaction coordinate while minimizing perpendicular to it.

• Positive curvature (not yet at the saddle): the midpoint steps along the orientation toward the region of negative curvature.

For small molecular systems, both forces and orientations are projected onto the non-translational subspace (removing 6 COM + rotation degrees of freedom in 3D).

The dimer method builds on the GP machinery introduced in earlier tutorials: the kernel (T3), hyperparameter training (T4), and FPS/RFF scalability (T5). The GP-Dimer and OTGPD variants use the same PredModel dispatch and trust region clipping as GP minimization.

Three variants

Standard Dimer

Evaluates the oracle at every rotation and translation step. Reliable but expensive: ~45 oracle calls on LEPS.

GP-Dimer

The outer loop follows the same rotation + translation, but oracle calls happen only at outer iterations:

1. Train GP on all oracle data (FPS subset for hyperparameters, PredModel for predictions on full data).

2. Run rotation and translation on the GP surrogate.

3. Oracle call at the proposed midpoint position.

4. Add new data, retrain, repeat.

This replaces dozens of per-step oracle calls with GP predictions. On LEPS: ~9 oracle calls.

OTGPD

Adds an adaptive threshold: the oracle is called only when the GP-predicted force exceeds a threshold that tightens as the optimizer approaches convergence. See OTGPD for details. On LEPS: ~13 oracle calls.

GP rotation considerations

With small dimer_sep (e.g., 0.005), the kernel correlation between the midpoint and the dimer image approaches 1.0 (~0.99985 for MolInvDistSE). The GP predicts nearly identical gradients at both points, making the finite-difference curvature estimate unreliable (the sign can flip from numerical noise).

When to skip rotation: if the initial orientation comes from a known source (NEB climbing image tangent, Hessian eigenmode analysis), rotation is unnecessary. Set max_rot_iter = 0. This saves oracle calls and avoids the correlation issue entirely.

When rotation is needed: if the initial orientation is a random guess, rotation is essential. Use larger dimer_sep (0.01-0.1) where the GP can resolve the gradient difference, or use the oracle directly for rotation steps.

LEPS example

cargo run --release --example leps_dimer

Starting from 0.05 A displaced from the known LEPS saddle along the negative eigenmode (from Hessian), GP-Dimer converges in ~9 oracle calls and OTGPD in ~13. The standard dimer needs ~45 calls.

Translation force vs oracle calls for standard dimer, GP-Dimer, and OTGPD on the LEPS surface. GP-Dimer converges fastest because the GP learns the curvature near the saddle quickly. OTGPD is slightly slower due to the adaptive threshold mechanism but more robust for difficult surfaces.

The starting configuration avoids the rotation issue by providing the correct dimer orientation from the outset (max_rot_iter = 0).

Configuration Reference

Parameter	Default	Description
conv_tol	0.1	Force norm convergence threshold
dimer_sep	0.005	Half-length of the dimer
max_iter	200	Maximum outer iterations
max_rot_iter	0	Rotation steps per outer iter (0 = skip)
max_oracle_calls	0 (unlimited)	Oracle call budget
rff_features	0 (exact GP)	RFF feature count for PredModel
fps_history	0 (use all)	FPS subset size
trust_radius	0.1	Trust region radius