P-RFO (Partitioned Rational Function Optimization)

Overview

P-RFO is a second-order saddle point optimizer based on the Partitioned Rational Function Optimization algorithm. It works by partitioning the Hessian eigenspace into one mode to maximize along (the reaction coordinate) and all remaining modes to minimize along. This dual-shift approach, combined with a trust-region framework and mode-following in mass-weighted coordinates, efficiently converges to the nearest first-order saddle point from an initial guess geometry.

P-RFO is the local TS optimizer used directly with #ts(method=prfo) and as the final optimization step for NEB-TS/String-TS refinement workflows. It requires a good starting geometry that is already close to the true TS.

Parameters

Parameter	Type	Default	Description
`max_iter`	int	256	Maximum number of optimization cycles.
`trust_radius`	float	0.2	Initial trust radius in Angstrom.
`trust_min`	float	1e-3	Minimum trust radius.
`trust_max`	float	1.0	Maximum trust radius.
`eta_shrink`	float	0.75	Trust radius shrink factor when step is poor.
`eta_expand`	float	1.75	Trust radius expansion factor when step is good.
`evals_eps`	float	1e-10	Eigenvalue epsilon for numerical stability.
`mu_margin`	float	1e-8	Margin for the level-shift parameter.
`max_bisect_it`	int	60	Maximum bisection iterations for level-shift.
`f_max_th`	float	9.5e-3	Max force convergence threshold (Hartree/Angstrom).
`f_rms_th`	float	5e-3	RMS force convergence threshold.
`dp_max_th`	float	1.8e-3	Max displacement convergence threshold (Angstrom).
`dp_rms_th`	float	1.2e-3	RMS displacement convergence threshold.

Input Example

Checked-in MAPLE example: examples/ts/prfo/inp1.inp. It uses an inline TS guess geometry and writes inp1_prfo_ts.xyz plus inp1_prfo_traj.xyz.

#model=ANI-1xnr
#ts(method=prfo)
#device=gpu0

C    -0.8675   -1.1212    0.1451
C     1.2063    0.0888    0.0375
C    -0.8012    1.8039    0.7777
C     1.9270   -0.5238    1.2299
C     2.1735    0.5194   -1.0520
N     0.2436   -0.9006   -0.5570
N    -2.0080   -1.5937   -0.4191
O    -1.6073    1.4743   -0.2757
H    -1.9698    0.5875   -0.1411
H     0.0290   -0.6370   -1.5287
H    -0.8978   -0.8021    1.1842
H     0.5985    0.9809    0.3930
H    -2.6675   -1.9721    0.2647
H    -1.8829   -2.2545   -1.1927
H    -0.4288    2.8210    0.7145
H    -1.0181    1.4392    1.7832
H     2.6046    0.2089    1.6809
H     1.6503    1.0084   -1.8818
H     2.5096   -1.3982    0.9217
H     1.2101   -0.8363    1.9937
H     2.9111    1.2261   -0.6587
H     2.7050   -0.3554   -1.4408

Output Files

*_prfo_ts.xyz — Final optimized TS geometry.
*_prfo_traj.xyz — Multi-frame trajectory of the local saddle-point search.

Visualization

The P-RFO trajectory visualization uses *_prfo_traj.xyz, while the final TS panel uses *_prfo_ts.xyz. The trajectory is the local saddle-point search history, not a full reaction path.

P-RFO transition state search trajectory with energy trace — Fig. 1 — P-RFO local TS search trajectory with energy trace.

P-RFO final transition state geometry — Fig. 2 — P-RFO final optimized TS geometry.

Tips

Start from a good guess. The best starting geometry is typically the highest-energy image (HEI) from a converged NEB or String calculation. You can extract this from the *_hei.xyz output file.
Verify with a frequency calculation. After P-RFO converges, always run a frequency calculation to confirm exactly one imaginary frequency. If there are multiple imaginary frequencies, the converged structure may be a higher-order saddle point.
Use method-specific refinement keywords for automation. Set refine=nebts for NEB workflows or refine=stringts for String/GSM workflows when you want the path calculation to hand its highest-energy image to the local TS refinement stage.
Trust radius adjustment. If P-RFO oscillates or fails to converge, consider providing a geometry closer to the true TS. The trust-region mechanism will automatically adapt the step size, but a poor initial guess can cause the search to wander.