[HaoZeke]

A converged saddle does not say which two minima it joins.
Process search checks it with two minimizations displaced along the
mode.
These can slide into a minimum the saddle does not connect.
irc_after = true in [Saddle Search] follows the intrinsic reaction
coordinate (IRC) with rgsaddle.
The path goes downhill from the saddle in both directions, and the eOn
minimizer relaxes both ends.

Six commits on develop 630ccad2 add the path, the relaxed ends, a
dimer fix and the config.yaml keys, 29 files, +1762/-10.
A diff against main also shows develop's own commits that are not on
main yet.

2f464bd1 follows the intrinsic reaction coordinate, in 8 files with
+430.
cf269a18 describes the saddle irc keys in the schema, in 3 files with
+67.
417c404f covers the irc_after ends and default bytes, in 3 files with
+235.
bf0f5ae4 drops the torque stop from the improved rgsaddle dimer, in 6
files with +35/-10.
f36b286d adds a how-to for the reaction path after a saddle, in 11
files with +970.
dff34824 adds the saddle irc keys to config.yaml, in 1 file with +25.

Rebase onto develop

The series now sits on the develop commit that refuses prfo_refine
before the first force call.
The rebase had conflicts in two files.
SaddleSearchJob.cpp includes both PrfoRefine.h and XtsciIrc.h.
Its run still calls requirePrfoRefine for a min-mode search.
runPrepared keeps the IRC check as it was.
In client/meson.build, test_prfo_refine stays outside the rgsaddle
block, where develop moved it.
test_rgsaddle_irc stays inside that block.
No commit became empty.

The sixth commit adds the four irc keys to eon/config.yaml.
The schema drift test compares that file with the schema, and it failed
from the second commit on.

Keys and outputs

The keys are
irc_after, irc_step_size, irc_converged_force and irc_max_steps.
The first is off by default, and the last defaults to 1000.
The step defaults to 0.1 amu^1/2 Angstroms, where amu is the atomic mass
unit.
The force threshold defaults to 0.05
electronvolts/Angstrom (eV/Angstrom).
Sella's IRC stops its path there, and so does the rgsaddle session.
A branch ends at a minimum when rgsaddle reports one.
rgsaddle does so only after the first step and where the largest atomic
force is at or below the threshold.
The job writes irc.con, irc.dat, irc_min1.con and irc_min2.con,
plus seven results.dat entries.
With the option off, a saddle search writes the same files as before.

The IRC needs rgsaddle C application binary interface (ABI) minor 14 or
newer.
Both the build and the library loaded at run time must meet it.
The rgsaddle crate stayed at 0.2.2 from minor 5 to 18.
So eOn checks RGSADDLE_ABI_MINOR and the ABI version the library
reports, not the pkg-config version.
Other builds refuse irc_after before the saddle search starts.

Dimer rotation

The fourth commit fixes the dimer search itself, and it would apply
without the IRC.
With rgsaddle, eOn hands both of its dimers to XtsciMinMode.
That class passed
[Dimer] torque_min, 0.1 by default, to rgsaddle as rotation_tol.
rgsaddle 0.3.0 and newer read a rotation_tol of 1e-4 or more as the
classic torque window.
Once the rotational force is below 1, that window stops a rotation after
its first step.
So a request for the improved dimer ran a classic one.
On the Pt island of examples/akmc-pt, develop against rgsaddle main
fe6bed89 ends at another saddle.
It stops at -1774.804074 eV after 449 force calls, against -1775.190088
eV on the pin.

The commit passes rotation_tol = 0 when [Dimer] improved is true.
Lanczos keeps torque_min, since it reads rotation_tol as its
residual stop.
The improved dimer then rotates until
converged_angle or rotations_max, as eOn's own ImprovedDimer does.
It needs no ABI change, so it applies to the pin 06c55c5 and to the
releases 0.3.0 to 0.3.2.
With it, the pin and main find the same saddle on all five inputs
below, the island included.

Four of the five saddles that the pinned rgsaddle finds move.
LJ7 and LJ13 are the 7- and 13-atom Lennard-Jones clusters.
The numbers below compare develop and this series, both against the pin.

For the Morse Pt adatom the saddle energy is -1462.008706 eV, the force
calls go from 38 to 46, and the eigenvalue goes from -1.013461 to
-1.010819.
For LJ7 the saddle energy is -15.283414 eV, the force calls go from 748
to 750, and the eigenvalue goes from -9.880619 to -9.880954.
For the Pt island the saddle energy is -1775.190088 eV, the force calls
go from 471 to 486, and the eigenvalue goes from -0.612079 to -0.613662.
For the how-to input the saddle energy is -1462.008706 eV, the force
calls go from 32 to 39, and the eigenvalue goes from -1.010681 to
-1.010010.

The energies agree to six decimals, and each moved saddle lies within
1e-4 Angstroms of develop's.
LJ13 keeps its saddle and its 8 force calls.
Two job tests bound the force calls of the Morse Pt search.
Under WITH_RGSADDLE the bounds rise from 39 to 47 and from 69 to 75,
and the expected eigenvalue becomes -1.010819.
The bounds without rgsaddle stay as they were.
Once rgsaddle releases C ABI minor 26, a separate follow-up will set the
torque window through rgsaddle_minmode_set_torque_window.

The pinned rgsaddle

The rgsaddle that continuous integration (CI) pins, 06c55c5, can
report a minimum beside the saddle.
On the Morse Pt saddle at the defaults it stops the branch along the
mode 0.0152 Angstroms from the saddle.
The largest force there is 0.017 eV/Angstrom, already below 0.05.
It can also run a branch to irc_max_steps with its force below the
threshold.
LJ13 at step 0.5 shows this.
The LJ13 step-0.5 numbers come from runs on an earlier revision of this
series.
Its eOn code differs from this tip only in rebases, one local variable
name, and comment, docs and config.yaml text.
The how-to shows the first case.
It uses step 0.5 and threshold 0.01, the cheapest of its runs that trace
both branches with the pin.
The schema note for irc_converged_force says so too.

The fix is one rgsaddle commit on main fe6bed89, not yet opened as a
pull request.
It stops a branch on the force and on the curvature along the launch
mode.
eOn needs no change for it, because eOn reads only the converged status.
With it, the LJ13 step-0.5 case reaches the minimum on both branches at
both thresholds.
On the Pt island the pin stops both branches about one step from the
saddle.
The relaxed ends still reach the right minima there.
The island path with the fix runs slowly, a separate known issue in the
rgsaddle IRC stop.
The LJ13 and island sentences on the fix also come from runs on that
earlier revision.

Tests

On the tip, meson test --suite eon passes 77 tests against the pin and
74 without rgsaddle.
test_prfo_refine now runs in both builds, as on develop.
Only the pinned build has test_rgsaddle_band, test_rgsaddle_irc and
test_rgsaddle_neb.
The Python tests give 789 passed and 17 skipped, with 206 subtests
passed.
The eon-schema tests give 56 passed and 1 failed,
test_published_tree_matches_generator.
Develop 630ccad2 fails it the same way.
Its published index.json, instanton.json and
nudged-elastic-band.json lack the develop keys tunnel_wall_points
and symmetrize.
This series does not touch those keys, and its own irc keys match the
generator.

One nudged elastic band (NEB) test compares a band with Subversion (SVN)
reference data.
Against rgsaddle main that test, NEB LJ matches SVN reference, fails
its status == 0 check.
Develop fails it the same way against main, and passes it against the
pin.
It is the only failure against rgsaddle 0.3.2, on the tip and on develop
alike.
The failure comes from rgsaddle e253dab, which changed the
climbing-image rule of the band.
A pending rgsaddle change, not yet on main, keeps the climb off an
endpoint that slopes into the band.
This series does not touch the band.

test_rgsaddle_irc follows the IRC from the LJ13 saddle that the NEB
test refines.
It checks that each relaxed end matches one of the two NEB minima.
The [irc] job cases check three things.
With irc_after off, setting the irc keys leaves every returned file
unchanged.
A build without the IRC refuses the option.
On the Morse Pt saddle the IRC leaves saddle.con, mode.dat and the
saddle force calls unchanged.
The two job cases with raised bounds pass with and without rgsaddle.

Docs

docs/source/tutorials/systems/irc_after_saddle.md is a how-to on the
Morse Pt saddle.
Its log line, results.dat lines and figure come from the run in
data/irc_pt/.
A table of six runs shows how step size and threshold trade path detail
for force calls.
A cost section compares the IRC with the displaced minimizations on four
systems.
These are the Morse Pt adatom saddle, LJ7, LJ13 and the Pt island of
examples/akmc-pt.
On all four the IRC reached the same minima and used more force calls.
On the island, with 175 free atoms, the path took about 120 s for 16
force calls.
The user guide gains a short IRC section, and a towncrier fragment
records the keys.
docs/source/user_guide/optimizer.md and a second fragment record the
dimer change.

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Codecov Report

❌ Patch coverage is 22.22222% with 84 lines in your changes missing coverage. Please review.

Files with missing lines	Patch %	Lines
client/SaddleSearchJob.cpp	8.33%	77 Missing ⚠️
client/XtsciIrc.cpp	22.22%	7 Missing ⚠️

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