Minimum Mode Following (MMF) Step

An alternative to P-RFO for saddle-point and TS searches that uses $|\lambda |$ in the denominator instead of RFO roots. This makes the step well-conditioned even when the Hessian eigenvalues have not yet converged to the correct sign pattern during the early steps of a TS search.

Formula

Given the eigendecomposition $\mathbf{H}=\mathbf{V}\mathbf{\Lambda }{\mathbf{V}}^{†}$ (eigenvalues sorted ascending), the MMF step is:

$$c_i=+\frac{{\mathbf{v}}_i^{†}\mathbf{g}}{|{\lambda }_i|+\alpha }\text{for }i<\text{order}[\text{ascent along TS modes}]$$ $$c_i=-\frac{{\mathbf{v}}_i^{†}\mathbf{g}}{|{\lambda }_i|+\alpha }\text{for }i\ge \text{order}[\text{descent along min modes}]$$

$$\mathbf{s}=\sum _i{c}_i{\mathbf{v}}_i$$

The Lagrange multiplier $\alpha \ge 0$ is determined by bisection to satisfy $\Vert \mathbf{s}(\alpha )\Vert =\text{trust}$. When the unconstrained step already satisfies the trust radius, $\alpha =0$ is used directly.

Comparison with P-RFO

Usage

Enable with Step mmf in the OPT section:

OPT
  TransitionState true
  Step     mmf
  ...
END

When the Hessian eigenvalue signs are not yet correct (e.g., very early steps on a flat surface), step=mmf is more robust than P-RFO because MMF uses $|\lambda |$ in the denominator.